**SUMMARY**

**INTRODUCTION. 5D Theoretical minimum to understand ¬Algebra as a mirror of the Universe.**

__Non-Euclidean geometry: __The fractal point. The lineal network Vital Topology. Organic Spacetime planes.

∆ST. The fractal Universe: Its 5D metric, S=T relativity.

Minds=Monads: Its Supœrganisms, actions and territories of order.

__¬Ælgebra____ of exist¡ence__: Its worldcycle & fractal generator. ¡ts entropic limits.

**BOOK II. THE SCALAR VIEW OF MATHEMATICS. ¬ALGEBRA OF NUMBERS.**

**I . Philosophy of ¬Æ Mathematics. **

Its subdisciplines: Non-Euclidean Geometry and Non-Aristotelian ¬Algebra

A measure of ¬∆@ST, stœps, Dimotions & Symmetries.

¬Algebra as a mirror language within Mathematics within the fractal Universe.

** **

**3 Ages of ¬Algebra**

1st age: Arithmetic, numbers.

2nd Age: operands and functions. Analysis.

3rd Age: Groups, sets and functionals

±¡ Ages: Existential ¬Ælgebra and Boolean ¬Algebra.

** **

**III. 1 ^{st} young ∆-1 age:**

**Number theory, Operands, the complex plane. Polynomials.**

Numbers as regular forms. Prime, even and odd numbers. Constants.

The five families of numbers.

The Symmetric inverted Operands=Dimotions of space-time translated into ¬Algebra.

Dimensional growth. Equations. Polynomial Functions & its 3 operands: ±; x÷; xª

Sinusoidal functions.

The new operands of Calculus: Derivatives and integrals. Its functions.

Finitesimals & Power Series: from Archimedes to Newton.

Finitesimals in time, space & scale. Its wholes.

The fundamental theorem of ¬Algebra. Complex numbers.

**2**^{nd}classic ∆º**age of equations. The function of existience.**

Operands of equivalence and relations of order.

The function of existience. Its curvature, probabilities & statistics.

Pentalogic on the function of existence its 5 Dimotions.

Its Stœps and a(nti)symmetries: S=T

** **

**Equations of simplex Dimotions. Polynomials. Statistics=Probabilities.**

Polynomials approximating calculus of Dimotions and ∆±¡ Planes.

Probabilities: discrete finitesimals in time.

Statistics: Discrete wholes in space.

** **

** **

**3**^{rd}informative**age. ∆+1 Wholes: §œt. Groups, functionals.**

Group theory. Present, entangled ∆@st-symmetries. Groups in space, time and scale.

Sets. From Cantor’s Sets to existential §œ.Ts. Its proper use and hyperbolic, ∞ egocy.

Functionals: Probing new planes of existence. Hilbert Spaces & quantum physics.

Philosophy of ¬Algebra in its 3^{rd} informative, old age: Hilbert’s axiomatic method. Its incompleteness.

** **

**VII. The 2 futures. -¡: Digital Thought: AI-lgebra. Transhumanism.**

Humind’s entropic death. The metalmind

Its 1^{st} age: Binary, Boolean ¬Algebra. Reticules of thought.

Its 2^{nd} age: AI: Algorithms of Information. Its 2 gender brain.

Its 3^{rd} age: A conscious metal mind. ¡ts function of existence.

2036: chip Homoctonos, 2^{nd} era. A pentalogic AI-system.

**Abstract**. This paper deals after an introduction to ∆ST laws, with Philosophy of mathematics and Algebra, in its 3 ages: arithmetics and its trinity of basic, scalar social operands and its ‘sentences’, polynomials, probability and statistics developed in its 2^{nd} age (pre-industrial Europe); introducing briefly the 3^{rd} age of group, set theory and Boolean algebras, with emphasis on Group theory at an introductory level, as the aim of those papers is to show the correspondence between classic science and the law of ∆ST they mirror, in the case of mathematics with scalar numbers, spatial points and dimotional operands.

As always this is a work in progress copycat first from 30 years of research in 5ÐST sciences, corrected and expanded as time goes by, from here to my eternity.

So my apologies for all parts not yet corrected up to the standards of scholarship.

L§

** **

** THE FRACTAL GEOMETRY OF THE TEMPORAL, 5D SCALAR UNIVERSE.**

The Universe is a fractal of space-time. This means it is made of 3 elements, *which experimental sciences such as mathematics, express with its fundamental element:* ∆-scales (expressed with scalar numbers) of different size in Space (expressed with geometry) with different speed of Cyclical Time motions (expressed with operands). Time is change, perceived as motion in space or change in the form of space (External change), in the in-form-ation of beings (expressed through topological inner change). Time and Space thus seem to be in perpetual conflict, transforming each other ad eternal. Time is thus similar to energy with a higher content of motion and space to information, with a higher content of form.

That Fifth Dimension, ∆±¡, is then the sum of all those scalar planes of spacetime where species *entangled as synchronous ‘organisms’ that co=exist in 3 of such scales, ∆±¡ (the quantum, organic/thermodynamic and cosmological/ecosystemic scale) to* live worldcycles of time, paradoxically faster, hence with higher frequency of information, the smaller the system is (chip paradox). Yet the product of both, the so-called metric of the fifth dimensions is constant, allowing the symbiotic exchange of energy and information between them. Thus this metric co-invariant equation, Space Size xTime Speed of its clocks=∆¡ defines a series of planes of space-time each one studied by a ‘stience’, from the faster, smaller world of particles, to the larger, slower cycles of galaxies, symbiotic to each other, as smaller systems process better in-form-ation, stored in the cyclical form and faster frequency of its clock cycles, coding larger systems that enclose its parts with protective membranes of slower time cycles,

The units of space forms are fractal points with breath that grow in size as we come closer to them, unlike Euclidean points that have no breath; societies represented by numbers, which are groups of ¡ndifferent beings and Times=motions of 5 types; 3 topologic dimensions of space, which have dimensional motions (ab. Dimotions); the classic dimension of lineal time, or ‘entropy arrow’, and a 5^{th} new ‘dimotion’ of social evolution and information, responsible for the arrow of life, organic evolution of parts into wholes and the ‘mind-structure’ of languages that mirror those organic systems in mind-mappings, of which the mathematical and logic and visual language are the most remarkable… It is a fascinating Universe, which humans don’t understand because they drag through memorial routine and dogmatic beliefs many errors, we can trace to Greek geometry and Aristotelian logic of abstract points and single time causality, hardly corrected in 2000 years and the astounding incapacity of huminds to understand the mind ‘distorts’ reality and selects information, compressing all the scales of space-time into a single continuum, all the motions of time into still images (so we don’t see the motion of the earth), eliminating all the information which we cannot translate into those 3 languages; despite the fact that for 500 years we have been watching those scales of the ‘5^{th} dimension’ with telescopes and microscopes (naïve realism).

Once all this is understood and we have the tools to improve our languages of perception, it will be possible to reconstruct reality with an improved mirror, in which points become fractal points that grow in size when we come closer to them, so they are non-Euclidean crossed by infinite parallels of energy and information, creating networks that are lines with ‘volume’, messed into topological organic structures. And discover that the fundamental particle of reality is the ‘fractal point’, which has an organic structure, co-existing in ∆±1 scales:

– The quantum, cellular, thermodynamic/organic and cosmological/ecosystemic scales.

What guides then each of those beings made of fractal topologies of space with time=motion is simple: existence, survival, the conservation of its territorial, vital space and time cycles. The conservation of time and its motions and cycles, akin to the conservation of energy cycles becomes then the guiding automatic principle of all beings, as those who don’t conserve their time become extinct. We thus talk of a game of existence in which systems will to maximize its intake of energy and information with its topological ternary parts, limbs/fields that move a reproductive body/wave commanded by an informative particle/head.

How can we formalize those systems? Easy, by improving our languages-mirrors of the laws of existence. That is, by developing a new non-Euclidean geometry of fractal points, lines, networks and topologic organisms and a new logic of multiple ‘actions=motions of space-time’. We call this formalism, ilogic geometry or ‘existential algebra’, and all the laws of all stiences can be derived of its simple structures in space, scale and time.

But scientists are stuck in the huge amount of errors sciences derived from their faulty logic and geometry of time-space, and his mind egocy that doesn’t believe what it doesn’t see and limits organic sentient properties to forms similar to himself; when electrons already have all those properties. The result are simpleton grand theories of reality derived of our languages’ perception *confused with the whole reality that* we call linguistic creationism, either verbal anthropic religions that put man at the center of reality and make God and man speak the language that creates by naming things, or big-bang entropic theories of digital thought that put man and the machine at the center and its language mathematics as the creator. So the big-bang is based in a simple lineal equation, V=HoD, which against all kind of proofs of a fractal balanced immortal Universe cannot be denied.

We shall prove ad nauseam that all what exists is a space-time ‘organism gifted of mind called logos’ Plato of a ‘higher logic than man’s’ (Augustine), as all ‘points are a world in themselves’ Leibniz, and deduce all laws of science from the simple laws of conservation of time, origin of the 3 ages of life and death of all systems, and the entangled topologies of space, origin of the organic structures of reality. And deduce from the elements of that reality, ∆-scales, S-pace, T-ime, all the mirrors of @-mind languages and ¬entropic limits of exi=st¡ence. But for you to accept that wider reality you need to be humble and go ‘back to school’ an entangle empathically with all other ∆st beings.

**Upgrading human mind languages.**

How to improve the obvious limits of huminds that ONLY see their mind languages is obvious: by improving those languages. The scientific method does so by substituting human senses and languages by those of more complex metal-atoms (sensorial machines) but egocy makes humans unaware they are JUST using a more complex sensorial system and better synoptic digital language NOT reaching higher truths as their underlying logic and geometric language have NOT changed. So what humans have changed are their sensorial experience substituted by that of machines, without even acknowledging they are evolving a different ‘digital mind’ of more complex electronic patterns that ‘obviously’ once ensembled with all its organic parts through the industrial r=evolution, will make us obsolete – already is doing it in labor and war fields – and substitute us.

Our method of enlightenment is both deeper and less dangerous for our future: we are targeting the evolution of the underlying languages to both type of sensorial minds – mathematics and logic systems, which are still dragging errors from the age of the Greeks.

It is then necessary to know expand back mental spaces into reality to understand any science that mirrors a specific scales of beings. And that expansion starts by expanding its languages, and then connecting the expanded languages with each science. Because some of those sciences are ginormous in content (modern biology, XX c. Physics), we cannot obviously complete the work, but we shall try to do so in a series of papers, which regarding languages will consist on, 2 mathematical papers, one on the expansion of ∆Space with the use of the 5 new postulates of Non-E geometry, and the other with the expansion of algebra, to comprehend better its scalar numbers and DImotions= operands of which the operand that applies to all others as a ‘new scale’ of analysis of time dimotions=change (calculus) is the summit of the discipline.

And with 5 papers on the expansion of logic of time, from the present monologic of a single dimension, to the duality, trinity, tetralogic, pentalogic and dodecalogic of more sophisticated species of timespace.

Those upgradings of the underlying languages latter applied to all sciences that study specific ‘planes of space-time’, are not so much in search of new theories but trying to correct the difference between the simplifying mirror of a language, with its ‘aberrations’ of perspective (as they are all self-centered in the mind that uses it to select information – which makes them omit essential features of reality), and finally the ‘quality’ of infinite mirrors that replicate through ‘reflection’ images into similar forms of higher complexity and less definition:

*– Aberrant Egocy, selection of information and inflationary repetition are thus the 3 errors of all languages to correct*

**MATHEMATICS – LANGUAGES AS EXPERI=MENTAL MIRRORS**

**Mathematics is a synoptic language that reduces reality to still continuous planes of points with no breath.**

“Arithmetic deals with discontinuous quantities, geometry with continuous ones.” *Leonardo*.

Mathematics is a language, and as such a mirror of the ultimate reality of the Universe, which is scalar space and cyclical time; hence an *experimental science with 3 elements mirroring those 3 ∆ST components of all reality: Geometry of space-points; Algebra of scalar numbers and S**óT cyclical equations/functions; and analysis of change=time, the specialized more complex of all operands of algebra able to study dimotions through ∆st planes. *

This experimental nature is found in the correspondence of the 3 main branches of maths and the 3 main elements of the Universe is so simple and e-vident that only the natural self-centered structure of minds, who think ‘languages create reality’ can deny it. Evolution of mathematics then always follows the path of the 3 elements of reality as all system in 3 ages of increasing information and generalization. Numbers evolved into arithmetic that evolved into algebra that evolved into Set theory of wholes and parts as the final evolution of ∆-scales, while its operands gave birth to functions that evolved into Group theory, which mimics the ‘fractal generator’ of spacetime.

What are the fundamental differences between ‘mathematical numbers’, the parts of ‘Sets’ and real ∆-social scales; mathematical Euclidean points, the units of geometry and ‘real points of space’ and the infinitesimal steps of motions, used in calculus, and the real ‘steps of change=time’ of the real world; from where we can draw the necessary corrections of ideal mathematics to fine-tune its use in science?

In spatial geometry, the fact that points of reality have parts through which infinite parallels of energy and information cross, entangling it with other points in networks and planes.

In time, the fact that time=motion never stops, and it is cyclical; hence it has also besides the energy of its motion, information carried in the form and frequency of its cycles, and as it closes its cycles breaks our mental, continuous perception of reality into multiple broken space-times.

In scales, the fact that those points, which gauge flows of parallel energy and information, select it to fit it within its brain, eliminating motion=time dimensions and discontinuities that ‘form information’, and scales smaller or larger to the one in which they play the game of existence, trying to ‘conserve its timespace’. So they compress in a single plane fit within the mind that eliminates all discontinuities, when reality is made of ∞ scales with discontinuities filled in the smaller fractal scale with ‘decimal numbers’.

In mathematics, the language of numerical scales is algebra; of topological space, geometry and of time change, it is also algebra but a specific branch, calculus, and its operands of change. Yet as in the Universe all is entangled, each of those parts can also mirror the other elements of reality. And this ‘multiple fractal point of view principle’ is what allow us to exhaust the qualities of a certain form and event as we observe it performing those different actions.

Let us then start with a brief introduction to the 5 postulates of ilogic geometry to concentrate then in ∆-T mathematics, the realm of algebra, ending with ¬@, Boolean and existential algebra, the mother of them all.

**Reproduction of form in 5D and its essential mathematical tool: calculus.**

The Universe is a fractal that reproduces information, forms-in-action, forms of space with motions in time. This is the essence of it all. But space is a maya of the senses, the synchronous view of a series of cycles of time motions, knotted in the simultaneous perception of an observer; what physicists call a ‘frame of reference’.

Thus time=change is the fundamental element of reality, and this makes Algebra of time-change, specifically calculus perhaps the most important experimental science of time, besides logic, which we have upgraded to existential algebra, which explores the vital, organic whys of those changes.

It is the Galilean Paradox: S=T. We cannot distinguish time from form. In as much as each frame of reference or mind locks in a knot-mirror of the motions of the Universe from its point of view. So each point of space is a perceiver relative field of motions, which from its perspective knot as forces ‘attracted’ by its frame of reference. Yet if we cannot distinguish motion from form each point is entangled to those motions and is made of motion and form, of the particle and wave states.

Locomotion as reproduction of form solves the Paradoxes of Zeno and the meaning of discontinuity. As motion is reproduction of information, of form, since particles are knots of perception of form, fractal points, monads, which move by reproducing through a lower plane of the 5^{th} dimension, as ∆-1 waves, its information, as forms-in-action.

So all forms of change can be reduced to the ultimate function of existence, reproduction, a back and forth travel through 2 scales of the fifth dimension, as a form becomes a seed that reproduces, evolves socially and forms its whole again.

Change thus has a final feature: it is change reproduced in a lower plane as a seed that evolves into a whole.

And all this is what actually calculus calculates: It finds a finitesimal part of reality and then integrates it as a sum, whereas the function of existence of the form displaces and reproduces its orthogonal parameters of form and motion. So physical forms are constantly reproducing, ‘calculating’ and the equivalence between the tools of calculus as mirror of the process of reproductive locomotion become crystal clear.

It is then also obvious that beings with a lot of information, reproduce very slow and we can hardly see them moving. The limit of it being complex life superorganims on Earth, whose reproduction takes 9 months. It happens ‘inside’ the reproductive mother, and it reproduces in the adjacent space after ‘tearing’ the topological knot of the umbilical chord. A similar very slow process of reproduction happens in physics with the weak interaction that reproduces a form with even more information evolving the mass of particles, so the range of the force is minimal and the new particle appears adjacent to the one that disappears, dying for the new hatched ‘baby’ to be born.

It is then quite surprising that it is not so much in physics but in calculus where we find the strongest experimental proofs of the laws of 5D as a reproductive process of form

**THE FRACTAL UNIVERSE: ITS SCALAR SPACE, TIME CYCLES, S=T MIND MIRRORS & ENTROPIC LIMITS**

We briefly introduce the changes 5D introduces to humind’s view of reality (ab. Human mind):

Time is cyclical, not lineal; space is scalar, fractal not continuous compressed in a single plane; thus the universe is a fractal of Planes; and those Planes are achieved by the ‘reduction’ effected by mental languages that ‘mirror’ **reality** in smaller mappings and then project its images of the whole in its territorial orders, creating supœrganisms, which have always limits of existence in time (their worldcycles) in space (their territories and membrains) and in Planes (as information doesn’t transfer beyond two Planes up and down a system. I.e. you are made of cells and part of a world, but the Planes above the Earth and bellow the cell are alien to you). So first we shall introduce the 5 elements that shape all ‘¬∆@st’ :5 D scalar ∆-planes of Space & Time, with entropic limits, focused on still linguistic @-minds.

**Why we need this introduction: The entanglement in ever more synoptic elements of classic algebra.**

Algebraic first principles and its connection with the fractal Universe are less understood because the concepts required – simultaneity, emergence, synchronicity, fractal principles, pentalogic, multiple 5 timespace dimotions and its mirror symmetries with the 5 families of numbers and 5 operands are not well understood. So we need to introduce in earnest the way the Universe builds its dimotions (ab. dimensional motions) of growing complexity. The essential concept is that 5 numbers families, its 5 geometric representations and 5 main operands of algebra, are mirrors of the 5 Dimotions and they are entangled, meaning numbers includes often the operands, and both are symmetric to a plane where we represent them, i.e. number 5 includes a sum of 5 natural numbers, which relates to the natural line; a complex number includes powers and roots; and needs a complex plane; a fraction includes divisions hence also products, and angle, a ratio hence an irrational number.

So really as I know you didn’t read the first paper (: I have to do a bit of an introduction since the mirror of ¬algebra needs to know what the image is about to fully grasp its awesome synoptic beauty:

In the graph the Universe is a fractal that reproduces ‘forms with motion’, informations and then **organizes** them in networks and systems that evolve into larger organic systems creating the scalar structure of reality. Thus we call *the sum of all those co-existing Planes of parts and wholes the fifth dimension.*

Then it is necessary to find a metric equation to define this new dimension of space-time. Since a dimension only exists when we can write a mathematical simple metric that leaves the dimension invariant when we change our parameters of space and time – hence we travel through it. (Klein). This equation, as all space-time metric equations, is simple; since metric equations are meant to represent measures of ‘covariant’ motion in a given space-time dimension that leave the other dimensions unchanged. So we write using ð for cyclic time instead of t, for a motion that changes the relative size and speed of clocks of a system (measured with frequency):

**5D Metric: S (Lineal Size/Volume in space) x ð (cyclic speed of its time clocks) = Constant.**

But how we travel in ‘size’ in space and ‘speed of our time cycles. *Here is where the biggest discovery of 5D comes into play: We travel through the worldcycle of life and death, as we are born in a smaller seed with faster time cycles, evolve as an organism coming out in the ∆º-scale within a larger world of slower ‘Deep time cycles’, to die back dissolving our information again into cellular space.*

It is the same process in all 5D journeys of all species that live and die travelling through 3 planes of 5D space-time; from the smallest black hole that is born with an enormous ‘metabolic temperature’, to the new species, routinely born as small individuals (first mammal rat, first robots with small chips; first human likely the Homo Floresiensis, who had the same morphology and used technology and likely spoke, etc.) Then a reproductive radiation multiplies the seed into a larger herd of clones, joined by emergent physiological networks whose slower ‘entropic, informative and reproductive networks, create an ∆º supœrganism that lives tages and dissolves back into ∆-1.

*So 5D adds to the 4D formalism of worldlines, a dimension of growth, shaping the worldcycles of life and death. Reason why *we call 5D metric the function of existence, because its multiple ‘solutions’ are the origin of all the varieties of Space and Time beings, there are – a whole family of functions.

As we keep exploring in depth, 5D metrics and its associated concepts of Space=form and Time=motion in all its varieties, we shall see it is the origin of multiple ‘solutions’, a whole family of function, from where we shall derive most of the logic relationships and particular equations of each science.

In the complex models of existential illogic, we derive all the particular equations of each science from it.

According to those metrics, smaller systems in space have faster time clocks. As information is stored in the frequency and form of those cycles, smaller systems have more information, coding larger ones: genes code cells, memes societies and particles’ quantum numbers code atoms and molecules.

This equation and its use to improve our knowledge of space and time in all sciences, with an emphasis in our models of physical systems will be the theme of this paper. Even if physicists stubbornly refuse to treat information with the same value than entropy. So they call it negentropy, and when you give a conference on the fifth dimension – the dimension of ‘creation of social forms of information, of organic wholes’ – there are no physicists on attendance; and likely no physicists will be reading this post… Let’s then use the metrics of the 5^{th} scalar dimension to explain the fractal, nested Universe and its Planes, shown in the graph:

The metric equation of the fifth dimension of space-time (ab.∆) defines 3 known Planes of physical systems, with different quantity of information according to 5D metrics, Se (size in space) x Ti (volume of information) =k. Since as we become smaller in space paradoxically our time clocks accelerate, and since information is stored in the cyclical patterns and frequencies of those clocks smaller systems code more information, so quantum particles code atoms, genes organisms, memes civilizations and chips machines establishing the essential symbiosis between ∆-1 Planes and ∆º super organisms, inscribed in an slower ∆+1 world.

Those metrics means information is higher in the smaller ‘quantum plane’ than in the larger gravitational one, and inversely the size of its physical parts is larger ins the Gravitational cosmological ‘plane’ than in the quantum one, with the human thermodynamic scale in-between.

As there is no reason to stop the Planes of the fractal Universe in particles and galaxies, there is a ‘potential’ fourth, ∆±4 organic plane defined ‘above’ the galaxy, (∆+4, dark energy world) and below the quantum world (∆-4, Bohm’s quantum potential), which represents the larger cosmos. Further on, according to the fractal, nested principle any larger organic system, encloses smaller nested systems. Thus the ∆±4 cosmos contains ∆±3 galaxies, which contain ∆±2 solar systems and planets, which contain ∆±1 thermodynamic organisms and matter states, described by the human ∆º mind languages, contained on our brains, *which according to those metrics* will have a much denser content of information becoming a ‘linguistic Mind-Mirror’ of the whole.

**Scalar entropic big-bangs in a balanced immortal Universe. The Galatom.**

In the fractal model a big-bang is the entropic death of physical matter. But the Universe is also scalar and it has information, a dimension of form, signified by the gravitational informative force, physicists ignore in their calculus of its expansion, happening in galaxies that balance the dark energy between them, we shall find both, balances of forces that make the Universe wobbling, and big-bangs and informative forces in multiple scales of the Universe, talking of multiple big-bangs balanced with big-crunches. Let us then correct big-bang theory to fit into the fractal view of the Universe, as there are big-bangs in all its scales:

Below left, we ad the gravitational force that warps 1D space into 3D mass in galactic vortices that balance the expansion of 1D space in entropic vacuum as light dies into dark entropy lines between them. As mass warps 3 1D flat-vacuum into a ‘high volume’ its 3 times more powerful in its warping, hence the 75-25% Balance of mass to dark entropy. Thus the fractal Universe is immortal. On the right its scales all suffer similar e=mc dual processes of warping through gravitational forces of cyclic momentum that create those galaxies, and expansive big-bangs of lineal momentum balanced in combined cycles of energy – the 3 conserved quantities of each fractal spacetime physical organism, which put together give birth to the 3 conserved laws of nature the Big Bang totally ignores – among many other known-known laws.

So fractal space accounts for the immortal Universe and its balanced 3 arrows of space-time, conserved in its infinite reactions; energy, information stored in the cycles and frequency of those systems of angular momentum, whose minimal constant form is h, and lineal speed, conserved in the constancy of light. Why those obvious facts of SOUND physics are then ‘reduced’ to ænthropic creationist big-bangs in a single plane of space-time despite evidence? It is the ego paradox: what the big bang does for science is a religious ‘closure” with man and its simplest mathematical lineal functions at its center, which added to the denial of organic sentient properties to physical reality allows to feel the high popes of science a sensation of absolute knowledge, as the religious person feels that all is known in the mystery of god – a word eerily similar to the lineal function of the big-bang ‘V=Hod’.

**RECAP**. The importance of understanding the limits of ideal mathematics is paramount in physics. I.e. as there is no absolute 0’, singularities do not exist and the expansion of vacuum is limited by its origin in galactic jets, where gravitation makes impossible is contraction of space.

**FRACTAL SPACE & CYCLICAL TIME ORGANISMS: VITAL, TERNARY, ORGANIC TOPOLOGY**

**Leibniz v. Newton: where is space and time?**

“According to their [Newton and his followers] doctrine, God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion. Nay, the machine of God’s making, so imperfect, according to these gentlemen; that he is obliged to clean it now and then by an extraordinary concourse, and even to mend it, as clockmaker mends his work.’ * *

*Leibniz-Clarke Correspondence on the absurdity of mechanical models of the Universe*

‘’Leibniz is right. There are infinite time clocks in the Universe, but if so we have to restart science from its foundations’. *Einstein*, on the infinite relational time cycles of reality.

The consequence of the existence of an internal fifth dimension of space-time, made of all other planes=scales of spacetime of a being, its parts and wholes, which store the information of a system, is the fact that we ARE made of planes=scales of space and time. We ARE the vital space we occupy and we ARE the time flow of existence we live between birth and extinction. Reason why all systems and entities of Nature obey the main science of space, geometry->Mathematics and time, Logic. We are broken fractal species of space and time, whose mathematical and logic laws all vital space-time organisms follow as Einstein and Leibniz thought: So we must evolve our logic and geometry of space and time, to extract properties of ‘existential beings’ from them.

**The underlying question of time§pace: Absolute or Relational, Generational Space-Time?**

The fundamental question physicists wondered for centuries regarding the nature of space and time unfortunately was resolved as usual in favor of the simpler view: it is space and time an absolute abstract background of the Universe (Mr. Newton’s view) or are we made of ‘vital space’ that lasts a time duration, so we are generated by the bio-topo-logic properties of scalar space and cyclical time? This is the choice of 5Ð ‘stiences’. And its simpler version was called relational spacetime, sponsored by Mr. Leibniz.

A realist interpretation of the world we live in, which never shows in any scale of reality such background – a mathematical graph used in abstract by human scientists – considers that we ARE the vital space we occupy with our cells, and we LIVE a cyclic time duration between birth and extinction. So *we are* space and time.

The argument thus reached its height in the beginning of science in the correspondence between ** Newton**, the proposer of the absolute Cartesian graph of space-time drawn by God (his body in his own words) vs.

**who rightly considered absolute space and time an abstraction, and so he coined the concept of relational space -merely the adjacent pegging of similar forms in simultaneous space and relational time – the sequence of events which we relate causally with reason. In Newton’s cosmos, space and time provide a fixed, immutable, eternal background, through which particles move. Space and time are the stage of intersecting lines sketched in the illustration. Fact is this ‘mathematical artifact’ made with pen and paper by earlier physicists, called the Cartesian graph, useful to measure ‘translation in space’ is no where to be seen in reality. Unfortunately as time went by the graph became somehow ‘real’ as scientists’ felt the ‘mathematical language’ created reality. But if space is what objects occupy that distance between the red square’s vital spade and the yellow ‘circle’ must have something. Horror vacuum comes then into place: indeed the Universe must be scalar. There must be very small parts between them, which we do not see. And that is what we have proved with microscopes –**

*Leibniz**as we probe smaller distances forms with motion, spaces with time-motions appear and there it seems no limit to the fractal scales of the Universe.*

It meant the invention of an absolute continuous space & single lineal time that extends to ∞ contradicting the fact that all spaces are broken, divided by membranes, and all beings have a finite time duration. Further on, as we kept exploring smaller scales of reality, we never found the drawings of God, not even a solid still substance, but always ‘motions’ tracing closed time-space cycles; since even particles turned to be also ‘vortices of time-space motions’.

So the true, sound experimental and logic theory was Leibniz’s who rightly considered absolute space and time an abstraction, and so he coined the concept of relational space -merely the adjacent pegging of similar forms in simultaneous space and relational time – the sequence of events which we relate causally with reason origin of the ‘Generational space-time’ model of 5Ð in which are the space we occupy and the time we last – as in the graph where there is no longer abstract background lines.

This simple concept was NOT adopted by physicists despite its sheer evidence. *Unfortunately Physicists sided with Newton not with Leibniz on the question of what is space and time – an abstract background put by God or the substance of which we are all made; and so the conceptual jump would not happen.*

*It is the fifth space-time dimension, sum of all other planes of reality, including within it all other dimensions.*

Next, to explain all this properly came Einstein. One of the fundamental discoveries of Einstein is that in our universe, there is indeed no fixed space-time background. In Einstein’s theory of general relativity, which replaces Newton’s theory of mechanics and the gravitational force, the geometry of spacetime is not fixed. Instead it is an evolving, dynamical quantity – a topology; and it is the substance of which reality is. So we are topological beings, geometries of space with motions of time.

What Newton called absolute space-time IS NOT. So space is the sum of all the discontinuous vital spaces, occupied by different beings: ∑s=S.

And lineal time, T the sum of all the finite life-death cycles of all beings T=∑t.

*Since space and time do exist and so if they are not in the background we ‘are’ vital space and cyclic time.*

The simple idea behind the structure of the fractal Universe is then to consider time=change=motion and Topologic, formal space=extension the 2 elements of which all beings are made.

Wheeler said ‘Spacetime tells matter how to move & matter tells spacetime how to curve’. Since *Spacetime is geometry in motion.* Time is change, the perception of change moves time; time is motion; space is its opposite, stillness, form, the information of time. And so it is all about two parameters: Time=Motion and Space=Form.

Look around you, all what you see are ‘space-forms’ with ‘time-motion’. We are all space-time, forms in motion, ‘in-form-motion’, ‘information’, forms in action, play with the words of what you are.

** Recap.** The universe is made of 2 STates, time=motion, and space=form. Yang and yin that combine in balance to create ∞ beings. So you can combine them T & S in 5 different forms:

S=T, the balanced state of existential energy that combine St-information (Potential energy) and Ts-locomotion (Kinetic energy), in constant SHM is the preferred balance of ‘present’ that lasts and iterates any organism, the state of survival.

TT: internal and external motion or ‘entropy’ is one extreme, which brings death. That is the state of mankind today, caused by the fact the new species, machines move faster externally (cars, weapons) and internally ‘metal-minds’, visual images accelerate, disordering humans into a permanent ADD state. So millennials move fast and have disordered minds. At the end of that process entropy kills.

In the other extreme of absolute no motion, there is SS, languages of the mind, seeds, pure, perfect information, enlightenment. A being in that state doesn’t move. The hypothetical Mind of God is such a being, because it perceives all what has existed, exists and will exist and repeat itself eternally as a block of time – a zero sum of fluctuations of the game of existence in space and time.

** **

**We are space and time,** merely of a different kind to that of Newton: Organic scalar spaces, and cyclical, discontinuous times who ‘live’… worldcycles (no longer worldlines as we have a 2^{nd} arrow of information) of exist¡ence (as all species follow the common laws of space and time). As cyclical time that explains the informative repetitive patterns or Laws Nature and its multiple space-time clocks. Why a Universe made of space-time beings is essential to a philosophy of mathematics is obvious*. *Because the main experimental science concerned with space is mathematics and the main science of causal time is logic, if we are made of fractal space & cyclic time, both mathematics & logic become experimental sciences, reflecting the properties of those 2 primary substances, as mirror-languages of maximal synoptic information and minimal size (Sxð=C), whose underlying laws emerge in all other larger Planes of the fractal Universe of bigger size and less information, proving also why they apply to all.

## This said the devil is in the details. So what does it mean to be made of motions with form, time and space? In mathematical terms, it means to be made of topological dimensions, which are holographic bi-dimensional space with motion in time. And as it happens *topology has only 3 varieties of bidimensional spacetime and it is constructed of parts – points – that become wholes – networks. So the immediate translation of generational space-time into modern mathematical systems do convert, as we observed in the abstract, all systems into mathematical beings. *

5D innovates this field, ‘enlightening’ classic topology to ‘understand the ternary organic, structure’ of all systems of nature: As the 5 Dimotions are dimensions of space with time motion, its science is topology that allows a system to deform= change its inner form. Yet a 4D Universe has only 3 ‘topological varieties’ that restricts ensembles to only 3 topologies, each one best suited to perform the 3 organic vital functions of any physic or biologic system –gauging information (1D) to move the system (2D) to an energy field in which reproduce (3D):

## The purpose of vital topology is to study the 5 Dimotions (dimensional motions) of the Universe… As such it will be the final stage of evolution of geometry as an experimental science, merging elements of all disciplines.

## All space dimensions have time motion. Mathematics found it, when Greeks still Geometry evolved into a vaster, generalized concept, a topological› variety, where a topology as opposed to a geometry has internal motions-changes. As the only case in which the inner dimensions of a being don’t seem to change is external locomotion most 5D motions need ‘geometries’ with inner motion, which are topologic varieties of which there are only 3:

*In the graph, *the diffeomorphic Principle of Einstein’s 4D analysis acquires an organic nature, when we see the Universe as the sum of ∞ Complementary ternary, topologic systems whose dimensions have organic functions: Systems feed on their relative dimension of energy-length, perceive in their relative dimension of height and reproduce in their relative combined dimension of width, which are assembled into each specific species, *to best satisfy the systems’ in taking of motion, energy and information. *

*I.e*: An animal has its informative height in the high perceptive light dimension, but a plant, which uses light as energy has its up and down dimensions inverted respect to the man and its chemical brain buried on the Earth. So both have opposite energy-time coordinates, with an ‘antero-posterior’, lineal’ ‘outward’ energy oriented structure due to the oriented arrow of light. But in a 3D world with no preferred orientation, a sea or vacuum, cyclic forms that maximize information dominate from plankton to galaxies that have a cyclic, informative, inward structure, as the stars’ body absorbs energy from intergalactic space, reproduces matter with it and feeds the internal informative knot of gravitation, with a higher height dimension the black hole.

So Dimotions of reality are 3 bidimensional topological varieties that act as vital organs in cylindrical long limbs/fields, Hyperbolic wide bodies and spherical tall heads, each one dominant in a lineal classic dimension, lineal motion, informative height and reproductive width, which DO have organic vital properties too:

Spherical particle-heads, perceiving information from the advantage point of **height**.

**Lineal long, cylindrical **legs and fields of locomotion as the line is the shorter distance between two points.

**Wide, hyperbolic body waves, **storing the energy reproduce by the system.

** Nt.****1**. According to the Correspondence Principles as physics named 4 Dimensions we use the name 5^{th} dimension for the whole range of Planes, but in proper terminology we should call each Dimensions, a ‘Dimotion’ and consider the 5^{th} scalar dimension the sum of all the 4^{th} dimotions of social evolution & all the 5^{th} dimotions of entropic devolution.

**CYCLIC TIME**

Time is what a clock measures’. * ‘Time curves space into mass’ Einstein*

**The causal repetitive laws of ‘stiences’**

A Universe of ∞ time clocks of different size and speed differs from its human description with a single mechanical clock-time to which all time clocks of the universe are equalized, elongated into a lineal ‘second-minute-hour-day-year’ system of equalized time clocks (of light waves, mechanical clocks, earth’s astronomical clocks). As Galilean physics, born of ballistics, simplified the nature of cycles of time-space into lineal durations, to measure best the locomotions of cannonballs:

Time is cyclical, all clocks of time and laws of science are based in the cyclical patterns of nature. But physicists developed ballistics and denied the truth that we can know the future because it will repeat the causality of the past, and we can change it by changing that causality, in History by repressing the lethal memes of the tree of metal and enhance the welfare memes that make us survive.

Lineal and cyclical time render the same equations as one is the inverse of the other, measured by frequency, T=1/ƒ, but the philosophical implications of cyclical time, are enormous, as we regain the in-form-ation provided by those cycles, *origin of the laws of science, which would not exist if there were not cyclical patterns; including *the cycles of history and economics. The most important of them being, the fact that *a time cycle breaks reality (1 ^{st} knot theorem) in an outer and inner region, creating a membrane that encloses a vital space, the ‘substance of which we are all made’*.

So we consider an ∆-¡ ‘quanta of time frequency or ‘finitesimal derivative’ of the larger whole represented with the concept of lineal time; as in the classic formula,V(st)=ƒ(t) l(s). We can measure Space e, Vt=S with lineal time as a single unit, or as a sum of frequency steps, with more detail.

**The 3 scales of time.**

Why then there are two forms of time, the long lineal Time and the ‘short’ frequency steps we integrate into the larger whole? Precisely because *there are 2 boundary ±¡ scales of 5D reality whose metric, SxT=∆±¡* defines larger space systems as having slower time cycles. So we can always consider the frequency of time the ∆-¡ ‘quanta of time’ or ‘finitesimal derivative’ of the larger whole represented with the concept of lineal time; as in the classic formula, S=ƒ(t) l(s) . The whole Space can be measured, Vt=S with lineal time as a single unit, or it can be measured as a sum of frequency steps, with more detail.

Yet since two limits always create an interval, and all supœrganisms of time-space co-exist in 3 scales, that of its inner world, that of its ‘membrain-mind’ and that of its outer world, there will be often a 3^{rd} element in between two measure time, *specific for each supœrganism and sub-species sandwiched between the ‘lower’ time quanta and the larger lineal time of the whole. *

So for species within the galaxy, if we take the lower time quanta, h, and the larger time rod of measure, the speed of light, in between we shall find all kind of supœrganisms that cannot go faster than light or process information in quanta smaller than an h-planckton with its specific time quanta, to synchronize its 3 vital parts, which in the human case is the second ‘beat’ of the heart, length of the thought, glimpse of the eye, measure of a dual stœp, (and ½ second of it is indeed the musical beat that we find more harmonious to synchronize our dance.

Synchronicities of time between the 3 relative speeds in which a being co-exists become then one of the most fruitful fields of study, as the ‘persistence’ of superorganisms in space is due to the symbiosis between its 3 scales of T.œs (time-space supœrganisms) synchronized with them.

**Entangling Vital topology and cyclical time: Local Past=Entropy, Present=Iteration & Future=Information **

‘The separation between past, present and future is an illusion’ Einstein

*Of all the consequences of cyclical time, the most important is the existence of infinite local time clocks of which we are all made, which therefore imply the existence of infinite local past, present and future states.*

*Past then means a system with less ‘form’, less information, which slowly acquires a dimension of height-form, as it completes its cycle to return back in the moment of death to an age of no information. This ‘worldcycle’ of existence, which creates and erases information becomes then the equation of duality:*

Entropy-youth (relative past) <Energy-mature reproduction (present)> 3^{rd} In-Form-ation age (relative future)

*Which each of us follows as a time-space superorganism. In physics is equivalent to the dual equation of Einstein: E**ó**Mc ^{2}, which reverses when E, which should be properly considered ‘Entropy’, as it is a disordered state, collapses through gravitation into Mass, a cyclical vortex of space-time; while its intermediate state is c^{2}, radiation; the relative present: *

Whereas past is the beginning of a pi cycle, starting as a line of entropy with no form that curves and raises in height in its 2^{nd} state of present, and returns back to its origin in its future 3^{rd} age of information, completing a 0’-sum of life and death. Thus instead of a single ∞ lineal absolute time there are ∞ living cycles of time happening in zillions of entities.

In the graph we see those relative time ages for the simplest ‘physical organisms’, which not coincidentally are equivalent to the 3 conserved quantities of physics. So as time is cyclical, made of 3’π’ lineal time motions, we shall distinguish 3 conserved ‘ages’ of timespace that put together create a 0 sum worldcycle of life death for any spacetime organisms: a relative devolving past or arrow of entropy represented in physics by disordered explosions in space and in vital topology by lineal limbs/fields of lineal momentum, an iterative reproductive present that seems not to change, represented by hyperbolic body/waves of energy, and an implosive in-form-ative local future arrow represented in physic by accelerated, V(t)R(s)=K vortices and angular momentum, and by particle/heads in the ensembles of vital topology. So timespace breaks in ∞ relative local, fractal entropic pasts, iterative energy presents and informative futures, which put together create the illusion of a single timespace continuum**.**

We can then set an absolute ‘dimensional motion’ (ab. dimotion) of time-space. Because for a whole ∆+1 to exist, the parts ‘ilogically’ must come first; so social evolution and love between parts is the absolute arrow of future for the organic Universe, or ‘future’, while a form that repeats itself seems not to change, so the function of reproduction is the absolute arrow of present, leaving thus entropy=death, the dissolution of form as the inverse arrow of past.

**Synchronicity, entanglement and absolute time.**

The vast expansion of information which results of uncoiling the planes of space-time of the Universe, is resolved by the humind compressing the into our size scale and rod of perception (light space-time). It is then necessary to accept for practical reasons the limited distortion of the mind, latter defined mathematically. So we won’t attempt an ‘Aristotelian, axiomatic’ pedantic attempt to set absolute truths absolutely false – as the hypothesis of the continuum, but highlight *the inheriting paradoxes between the **∝** (relative infinite) of information, that our mind never will know as only each fractal point of space-time holds all of its information.*

Absolute relativity is the name of the game. And that is fine if the humind accepts a humbler attitude towards the whole, which so far does not. Infinities thus reduce to ∝ immensities, infinitesimals to finitesimal quanta, continuities to undistinguishable discontinuities. Single timespaces to an entangled Universe that generates its space through synchronous connections between 5 Dimotions of reality in its 3 relative ranges of 5D scalar planes in which each dimotion has a different speed but as there are 5 dimensional motions with different functions, they entangle together: i.e. the faster dimotion of feeding in the slower ¡+1 plane of the ecosystem, provides energy to the slower dimotion of reproduction in the faster ¡-1 plane of the cell 1 time a day for most ¡º animals connecting the 3 planes of biologic life**. **Functional synchronicity creates simultaneity, which gives birth to ‘ordered co-existing superorganisms’. Even a humind looses information and compress it all in its plane of light and second beat, of its smaller mirror mind to fit/select its survival info within its brain.

**THE 3 ONLY TOPOLOGIES OF PHYSICAL SYSTEMS: CONSERVATION LAWS.**

The 3 parts of any fractal point of cyclical time in the Universe become in the simplest physical systems, the 3 conserved parts of the minimal organic Unit of reality, a ‘Planckton’ of angular momentum. All physical systems then can be ‘reduced’ to fractal ensembles of 3 ‘conserved quantities’, angular momentum – the membrane of the system, which becomes a membrane, fractal sum of ‘cellular cycles’ or the skin of the system in human beings. Vital energy, the enclosed cyclical forms and motions within the space whose boundary conditions are given by the membrane, and lineal momentum, the motions with a ‘finality’ we perceive guided by a ‘relatively still mind-point-singularity’ that focuses the energy and information transferred through an angular momentum membrain.

It is also the smallest clock of our world, as its minimal unit of cyclical information, or angular momentum, used as the unit of the 3 physical parameters of **spatial size, cyclical time** frequency and ‘**scale’** (Active magnitude):

**h= mass (∆) x area (S) x ƒrequency (ð).**

So h, ‘Planckton’, is the minimal fractal organism which becomes the ‘cellular unit’ of all other species of light space-time, as Plankton is the minimal unit of the biologic Universe, also a cell with a similar ternary structure – a DNA nucleus that process information, a protein membrane that isolates a vital space-time, the cytoplasm.

Thus Non-Euclidean points have breath, its lines are therefore waves able to communicate the external form and internal energy or fractal networks that branch to connect multiple points, and its planes intersection of three of such waves or networks that form topological organisms… It is then obvious that the next step of Non-Euclidean geometry is to merge those concepts with the physical analysis of the smallest physical systems, to understand its vital topologies. To that aim we introduce *the second fundamental equation of 5D metric that formalizes the Paradox of relativity. And so it acts in the physical model as the Postulate of relativity, common to 3D Galilean and 4D Einstein’s simplex models, which correspond to 5D in a single plane of ‘light space-time’…*

Euclidean dimensions perform organic functions in light waves that humans perceive as space.Light is an organic system of 3 dimotions: c-speed length, electric informative height & a wide magnetic field that supports them.

The 5 Dimensions of space-time are 5 vital dimotions, broken into infinite vital space-time beings, and now we have the 5 elements which reality uses in different perspectives to construct all realities. Nothing else is needed. And it will easily follow that in each stience, including mathematics, 3±¡ elements (depending on the perception in a single plane or in several ones) will be concerned with the analysis of a system in Simultaneous, entangled, still space as a superorganism constructed with those 5 Dimotions, which in sequential motion time will trace a worldcycle also composed of those 5 dimotions. As the universe simply put it is a reproductive fractal of 5 Dimotions of spacetime…

As motion with form constantly reproduces by the mere fact of moving. So every motion of entropic time, which enters within the orderly structure of a fractal point, its waves/networks of communication and topological planes explained in the paper on 5D geometry becomes an iterative reproductive system of spacetime.

The balances achieved by the similarity of space=form and time=motion reached in the present body, S=T, and the unbalance of the metric equation of scales, $ x ð = K, in the limbs (Max. $T) and minds (Max§ð) unify as S=T maximizes SxT=K (5×5>6×4), in 1 equation: **Max. S x T = C**, which defines for each fractal vital space-time organism its Function of existence, as all species will try to maximize its motion-entropy-time for its field-limbs, its information-spatial states for its particle-heads, whose product will give us its vital reproductive energy. Moreover the equation has an immediate biologic meaning, because as we are made topologically of ‘fields-limbs’ of lineal space with motion provided by the energy we absorb to also reproduce our bodies-waves, and the information we need to linguistically guide our motions with particle-heads, the very essence of survival is to increase our S=position, mental forms of space and T=entropic motions of time (whereas time=motion and space=form are the two limiting Dimotions with ‘energy=reproduction, s=t, locomotion, sT and information, St, are the intermediate 3 dimotions).

*Thus the Universal constants of reality respond to 3 5D Metric constants: S/T= Speed of Locomotion, which defines the limbs/fields of the system, SxT=Existential Momentum/force, which defines its body and T/S: Density of information, which defines its mental power, all maximized when S=T. *

In physics, for each fundamental scale, there is also a constant ratio between ‘frequency=time parameters’ and energy=mass parameters, which are the three fundamental constants of Nature, H-Planck (ratio of frequency-energy for quantum systems), K-Boltzmann (ratio between the temperature frequency and energy of thermodynamic systems)

*The Universal constants of reality respond to 3 5D Metric constants: S/T= Speed of Locomotion, which defines the limbs/fields of the system, SxT=Existential Momentum/force, which defines its body and T/S: Density of information, which defines its mental power, all maximized when S=T. *

Once we establish the meaning of the 5 Dimotions of reality we can ‘ascribe’ every parameter of science to one of those dimotions either in a ceteris paribus analysis as a partial space or time function or as a complex entangled space-time function in equations that describe a time-space being mutating its 5D metric through the 5 ‘graduations’ of motion and form, from entropy («, scattering motion, which is both motion internal and external to the being), locomotion, < (only external motion), reproduction, ≈, evolution, > (internal mostly with external shrinking) and linguistic seeding, », still perception in crystal and solid or mental states whereas an active magnitude creates from a field of motion a seed of form.

MOST human measures called ‘energy’ are NOT about the 3^{rd} Dimotion of energy, which ‘KEEPS its balance of form and motion’ but about entropic states of scattering form and expansive motion in space. For example in E=mc^{2}, a mass vortex of physical information is trans-formed into an entropic expansion of space, loosing its form. So we do NOT measure its energy but its entropy. Human Energy then works as a parameter of the entropic scattering motions due to the death or big-bang of a physical system, exploding its form. This again becomes the case for Heat, which measures the entropy=motion extracted from a thermodynamic ensemble, useful for man to be transformed into ‘real energy-ordered motion’ in its ∆+1 scale. So we consider a general 5D metric for those physical scales such as E is a function of entropy that multiplied by a function of time frequency give us a scalar constant, so we write:

*Energetic entropy (E) x Time Duration (1/ƒ) = Constant of spacetime scale.*

The connection between both concepts is however more subtle. When we express the entropic energy without caring for its ‘informative details’, multiplied by its time duration, we indeed ‘erase’ the constant of space-time into a flow of entropic motion humans can use. But if we keep the system entangled without loss of information as E = C x ƒ (ð); we can talk of the ‘inner energy’ of the system, E, as a ‘population sum’, of ‘space-time C-beings’, as in a wave of light which is a sum of ƒ-Plancktons. So bearing in mind those subtle distinctions, 5D metric equations, E x T = C or E = C x ð, give us the 3 fundamental equations of mathematical physics regarding its 3 ‘scalar planes’ (graph).

Thus the 3 conserved quantities of physical systems are the 3 elements of the 5D metrics equations of the 3 physical scales within the galaxy, whose metric, hxc=k sets the final limits of information (h) and motion-distance (C-speed) of the whole. All are relative ‘nested scales, ’ which change as we travel through them the parameter of time (frequency or duration), and entropic motions, giving us 3 constants of scale of space-time, h, m and T through which the actions of its systems are taking place.

T**he easiest human equivalence: Temporal Energy and Spatial Information. **

Human closest translation of the duality of primary elements was in the classic age of verbal thought, Asian philosophies of yin=in/form/ation=space=visnu and yang=entropic energy=time=shiva, and in the age of digital thought, motion=time has become synonymous of entropic energy, while mental simultaneous space is akin to the concept of in/form/ation; since our concepts of space and time have been restricted enormously by virtue of the Galilean equation of speed, v=s/t which became as physicists substituted philosophers of science in the summit of scientific thought, due to its power as machines and weapons makers, the ‘only equation of time’, when in fact was just a measure of speed with very limited space and time parameters. As Einstein merely added – c term to that equation nothing changed despite the hype surrounding his work.

So we shall often use the terms temporal energy or entropic energy for time=motion and spatial information or spatial form, for space. As time and space are synonymous of motion and form they do have multiple meanings. So there are 5 types of time change=motion parallel to 5 types of space= form. Yet in fact as we have seen space and time are always mixed together, since human still mental one dimensional space is a simplification and definitely *a single time arrow of entropic energy or absolute motion is an extraordinary simplification of the richness of motions of time. *Energy in that sense with its multiple variations of meaning is much closer. Specifically entropic energy; that is, a motion which is internal and external to the being or ‘TT’, pure motion appears in the equations, E=Mc^{2} (∆+1) = knT (∆º)=hƒ(∆-1) that transfer temporal motion through scales of the fifth dimension. This is the entropic arrow of time and disorder akin to death which physicists consider the single arrow of time. My advice then because of so much messy and simplified understanding of time in physics is really to take with some irony their work on philosophy of science, though as their equations have their own life and self-consistency those are ‘real’ and must be addressed with much more rigor, as we shall do in the papers on mathematical physics. But for concepts of time and space useful to all scales and sciences we shall use temporal entropic energy and spatial in/form/ation. Energy, being the jack of all trades, in messy human philosophy of science is also used for S=T, balanced ‘work’ which in fact is both space and time, energy and information, as it imprints form into motion, reproducing cyclical patterns of science. So it is better to define ST as reproductive work.

It is then obvious that the fundamental law *of science, the law of conservation of energy, becomes the law of conservation and immortality of time-motions, which can be expressed as follows:*

*‘All what exists is time=motion=change in perpetual trans-form-ation in one of the 5 Dimotions of time-space: S**óT’*

*And *it is the purpose of science in each subdiscipline studying one of the scales of the fifth dimension to study how one transforms into the other.

*It happens then that the parameters of reality will be of time, space and scale; often grouped in dualities. So for example in physics we have ‘scalar’ magnitudes defined by a ‘scalar, social number’, such as mass and temperature and space-time parameters of dimotions, such as vectors with magnitude and direction.*

In physics, for each fundamental scale, there is also a constant ratio between ‘frequency=time parameters’ and energy=mass parameters, which are the three fundamental constants of Nature, H-Planck (ratio of frequency-energy for quantum systems), K-Boltzmann (ratio between the temperature frequency and energy of thermodynamic systems) and among the many manifestations of the same law in mechanical, moving systems, the third law of Kepler (ratio between the orbital time clock of planets and the spatial volume: energy content of its orbital sphere):

In biology we study families of animals such as mammals where larger organisms have slower metabolic cycles. In history we study social organisms, whose cycles of life and death, will define the evolution of nations and civilizations. And in each of those organisms, smaller systems code larger ones. So the quantum numbers of particles code matter, genes code biologic organisms, and memes code societies.

But the really defining graph is the third one which shows that the average number of heartbeats in the lifetime of any mammal is roughly the same, even though small ones like mice live for just a few years whereas big ones like whales can live for a hundred years or more… *but in terms of internal time, which synchronizes a given superorganism in its 3 vital parts, limbs, bodies and minds, so you walk a second each step, your heat beats a second and your eyes-thoughts take a second, the life perception of each organism of the same ‘species’, is the same. Such is the justice of the Universe that if you were a small rat your life will seem to you as long and fruitful as a human being. *And this is the ultimate meaning of the fractal Universe, the ‘absolute relativity’ of scales. We are not only out of the center of space (relativity) and time (Evolution) but also our scale matters nothing. A fact, which we can extend to other families of nature. So insects whose rate of thought processing is 10 times faster, reason why you can’t catch a fly, live up to 7 years (cockroaches ant-queens of ant-hill superorganisms, dragon flies) , while humans live up to 70. The examples below are just a tiny sampling of an enormous number of such scaling relationships that quantitatively describe how almost any measurable characteristic of animals, plants, ecosystems, cities, and companies, planets, stars, laws of science, atoms and galaxies, scales with size.

An interesting number of that 3^{rd} graph is the fact that *all perfect systems do live (9-11) ^{9-11}* time quanta; closely related to the commonest spatial population for a full developed system (from the ties of DNA molecules, to the number of cells of an organism, to the number of people on the Earth’s superorganism, to the number of stars in a galaxy, to the number of galaxies in the perceived Universe. The reasons of it, being the 3×3±¡ ‘tetraktys’ scale of organic systems, which we shall develop in the General model as well as the section dedicated to number theory, and the ‘relativity equation’ S=T, which makes similar space and time on the point of maximal balance and survival of reality. This said, we are not unlike most papers and work on modern science, on the quantitative ‘magic’ of our studies, not even in this paper on algebra, but on the deep whys nobody answer, of those processes. Since the reasons of all those processes never are 5 D Metrics and the fact that scaling is an embedded feature of all systems of Nature, for the simple reason,

*we are made of the two substances that scale, space and time.*

**Logic Consistency: The Infinite, Hierarchical Universe**

In the 1970s, Mandelbrot proved fractal, hierarchical self-similarity is ubiquitous in nature – an infinite hierarchy of worlds within worlds. So theoretical physicists found their way to the ∞ fractal paradigm, which unfortunately denies the continuous big-bang. So it offers resistance from established big science (Kuhn’s on scientific revolutions). In the fractal paradigm, the Universe is infinite and the big bang is not the birth of all realities, but any local big bang and big crunch dual process, any explosion that splits the physical energy and information of a complex system, in any of the multiple scales of physical reality. So first for the impatient, we shall have a fast glimpse to how a fractal Universe look when we add scalar elements to the ‘entropic big-bang’.

Let us then start with the physical analysis of relativity and its correspondence with 5D.

**COMPLEX, ¬BIO-TOPO-L@GIC PROPERTIES OF NATURE**

The fundamental purpose of this paper is to UPGRADE the understanding of mathematics as an experimental language mirror of the vital, organic, scalar properties of the space-time Universe, observed in the next graph:

*Each stience is a mirror-mind reflection of that program of 5Dimotions, from the point of view of the species that perceive it. And so the main task of 5D philosophy of stience is to translate the jargons ‘huminds’ have established in the study of the exi:stiences (Timespace supœrganisms self-centered in one of those planes of st¡ence) to the principles of time-space and its general properties – the thoughts of God – from where each of those st¡entific species are just a ‘variational detail’, including mankind study as the supœrganism of History.*

In the graph, the study of a Universe made of fractal Planes of cyclical time space, introduces a priori:

– **Organic=scalar** properties as the system extends through** various ∆§ planes of space **that provide it with **Spatial, topologic properties**, **cyclical time properties**, and **¬entropy limits,** as its time clocks will have a limited duration, ending as a zeroth sum of energy.

– Sensorial properties to be able to perform a survival program of 5 Ðimotions of existence, it requires to **gauge information** with an apperceptive, self-centered **@-mind, whose** **∆º still language **maps all external ∆±¡ time§pace organisms & world cycles into an infinitesimal Non-E point, mirror of the larger world: 0 §@ x ∞ ð= World. Hence the creation of infinitesimal ∆º minds-worlds, or Leibnizian monads, which reduce a larger ∆+1 reality to its relative mapping, which is the origin of its ∆§calar planes, connected ‘perpendicularly’ through those @-minds, acting in physics as ‘centers of changes and masses’ wormholes of flows of entropy that become in the @-mind information, that perceives in itself.

Those 5 properties, ¬∆@st, entropic limits, organic Planes, linguistic minds, topologic space and cyclical time motions, *are inheriting to all systems of Nature, as they are derived of the scalar, fractal, cyclical nature of space and time. The most difficult to accept due to the ego paradox of all minds including man are the sentient linguistic properties of reality. But they are necessary for two fundamental reasons:*

*To understand how smaller similar entities create their form which cannot be understood without the mirror concept of a language that projects a shrunk view of the whole in its body-territory of order.**-To understand how humanity or any other atomic species for that matter performs its actions of existence clearly regulated by logic and mathematical time and space laws.*

The Universe is a fractal organism of similar Planes, defined by 2 metric equations and its philosophy of science, 5D Absolute Relativity: the equation of 5D Planes, SxT=C & the equation of relativity between form and motion, S=T, which started modern science. Since *we cannot know from the mental, still point of view, what truly is motion and what is dimensional form. *Thus Time Motions & space dimensions co-exist and merge in every space-time being of the Universe &* we must talk of space-time dimensional motions.* When we combine them, as S=T maximizes SxT, we define the Function of existence, Max. S x T = C (s=t), which IS also a biologic equation of survival that embodies the will of life; the biological expression of the ‘Universal mandate’, expressed by all species in all its codes and languages, the Grow and Multiply of the Bible, the intuitive truth that guides all beings. In detail implies all systems try to maximize its 3±¡ ‘Ðimotions’ that in space create organisms and in time its worldcycles of existence; starting as pure Form (4D) in a seminal seed, emerging in an ∆º scale, in a young age of max. Motion (2D), balanced with information in the reproductive age of max. Energy (3D), followed by an age of information (3D), when time reverses into entropic death.

The dual metric 5D equations of Planes, SxT=C & the equation of equality between form and motion, S=T, develop in 3 ages with 3 standing points, a max. point of existence, S=T or mature age, a young age of Max. T=motion, and an old age of Max.S=information; between birth in ∆-1 Form & T-entropic death.

Such organic model is more scientific than mechanical abstract models dominant in physics for a few reasons:

- Only an organism replicates itself without the concourse of a ‘maker of the machine’ – God in the earlier physicists’ view, now obsolete. As machines do not reproduce.
- Only the definition of time as the substance of reality – cyclical motion that reproduces form allows the Universe to move, as only if motion=time is the ultimate substance the Universe never stops.
- Only if it is cyclical, repetitive, there are laws of science, which are precisely those repetitive motions.
- Relativity shows there are ∞ clocks of time with different speeds (just look at the clocks of nature).

It is epistemologically sounder (Occam’s razor) as all systems can be modeled as organisms. Further on organicism uses all the language-mirrors to express its topo-bio-logic properties:

- It uses the mathematical language, best for its spatial description, which models organisms in the form of fractal networks, for which we shall adapt and expand its most advanced forms of space, topology – space with motion – and non-Euclidean geometry – fractal points with volume that grow in size as we move closer into its Planes, allowing ∞ parallels of energy and information to flux into them.
- It uses the logic language, ideal for temporal, causal sequences and cycles, which we shall improve from the simpler Lineal Logic of locomotion, into Duality, the logic of two arrows of time, energy and information and further on into ‘trinity’, as both combine into energy beings – the logic of topological organisms, conservation principles, Universal grammar, and even further on into pentalogic and Dodecalogic…
- It expands Biologic life to all systems. Since a fractal requires a new dimension of parts that become wholes, larger Planes co-existing together gifted therefore with organic properties as atoms/cells/Individuals form part of thermodynamic /biologic/social ensembles that become wholes, living in a larger gravitational, ecosystemic or planetary world.
- The Universe is a game of ∞ space-time supœrganisms performing 5 survival ‘dimotions’=actions through Space stops and time steps, STœps in its scalar cœxist¡ence through 3 planes of the 5
^{th}dimension, which add together in its structure of simultaneous vital topological superorganisms tracing 2 existential worldcycles, the ∆-1 placental ordered o-1 that emerges into the life-death, entropic 1-∞ worldcycle . Stœps express them through feedback equations of Existential ¬Ælgebra, where = is a dynamic symbol «≈», composed of 5 dimotions = stops, ≤≥ wave steps and» 4D implosive informative social evolution & «5D, entropic explosion of a system.

We call that super organism of astrophysics, the galatom, (as it happens between the quantum atomic scale, and the galactic scale, which are self-similar – with similar forces, and particles similar to celestial bodies). Within the galatom in the intermediate stage there exists, thermodynamic man.

Thus the organic paradigm rejects ‘mathematical creationism’ in physics – the belief an equation describes all the properties of a system, which no longer need parts; as in the case of black holes which are modeled with singularities – mathematical entelechies of infinite density and infinitesimal volume but as Einstein wanted, they must be made of a cut-off substance, which can only be heavy quark atoms (bcb atoms).

Because the scientific method requires objective measure of a mind’s existence, which is not perceivable directly, we *infer its existence by the fact a system performs the 5 external actions, which can be measure objectively, in the same manner we infer the existence of gravitational in-form-ative forces by its external actions upon massive objects. *Hence eliminating the previous limit for a thorough understanding of the sentient, informative Universe. And further classify organic in simplex minds – all, which must gauge information, move and feed to survive, and complex systems, those who can perform a palingenetic reproductive, social evolution, ∆-1: ∑∆-1≈∆º.

The purpose of philosophy of stience is precisely to vitalize reality and *explain experimentally each science as a reflection of the previous organic, scalar, topologic and temporal properties of space-time beings. *

But as the ego only ‘perceives’ from its p.o.v. it reduces the Universe to its world-languages: O-mind x ∞ Universe = Constant world.

So humans do NOT recognize under the ‘ego paradox’ of their mind, which perceives reality from its self-centered point of view and selects only information favorable to its territorial view ANY of the ‘cyclical, intelligent, organic’ properties of all other atomic systems of reality, which are the first bricks of ‘life’, as particles gauge information, feed on energy, decouple=reproduce and evolve socially through magnetic fields, and quantum numbers that code its behavior as genes code cells or memes code humans.

How humans then explain the complex Universe is easy, as they introduce myths of anthropomorphism, pseudo-religious theories of languages, and the view that all other things are entropic=destructive=chaotic and by mere chance have become what they are. So probability becomes the ‘chaotic’ cause of reality, while a ‘god-like’ language shared only by man and God, first the word, when men only talked, now mathematics, when man also calculates – or rather its computers, made with more intelligent atoms – is responsible for the order. This language is thus the intelligence of the Universe, and man accesses it through its mirror mind because he shares that intelligence, or even causes it (Hilbert’s description of maths as born in the human imagination). So creation happens in the Universe through maths, and it is only shared by man who calculates and designs.

As why maths creates and how it does imprinting reality is a big mystery not worth to explore as it is the magic of it all, which allows to introduce pseudo-religious theories of reality from the creationist big-bang to the superiority of man, gifted by the language of God. This comfort zone in turn makes man happy, because its ego is pumped up, and allows him to manipulate and kill reality if needed, ab=using the planet, because it is already dead. It seems a perfect fairy tale world.

All what humans must do is to extract mathematical properties of objects, with instruments, gathering data cast in equations that mirror the space-time properties of the Universe, and run them in the calculation machines of more complex metal atoms, to extract images of reality that are the essence of knowledge. No further questions are needed. And inversely we can design in less mathematical dimensions objects and then reproduce them.

All this of course works fine as long as we eliminate any question on why species follow certain properties, on what are the properties we do not map out as they are no susceptible of mathematical description (notably vital properties and sentient properties and organic, fractal properties).

Reductionism is inheriting to the ego paradox that confuses the ‘language’ it speaks with the reality it is, excluding what the language does not see. So what we don’t see does not exist. And even when we see it through interposed instruments as all other Planes of 5D, as it is not primary, e-vident experience, humans have gone 400 years without a serious research on the relationship between Planes.

The same goes for ‘verbal creationism’ – all what is not in the Koran is not worthy, said Omar, before burning Alexandria’s library. ‘All what cannot be measured does not exist’ said Planck.

Logic positivism affirmed that all properties of reality which mathematical logic cannot prove do not matters and so on. All what is not proved by mathematical methods that cannot portray them as they do NOT have a vocabulary for those properties in restricted mathematical languages, does not exist. This reductionism thus reduces reality and knowledge but makes the ego of man the center of reality again and happy.

Even if Mathematics shows vital social properties in its mirror languages, those are also overseen, as in the case of the social properties of numbers, which do not satisfy the ego paradox of individual man as the center of it all.

Huminds, indeed, have an astounding rather contradictory capacity to deny obvious truths when they don’t cater their self-centered ego paradoxes and accept falsehoods as truths, from religion to big-bang theories as long as they satisfy their subconscious ego, and put man back into the limelight.

Therefore on approaching those difficulties of the human ego to appreciate the whole and form part of it, we can only try to show that mathematical properties are vital properties; so they can also describe organic properties, when we adequate its postulates and elements to the fractal form of space (definition of fractal points with parts; its lines as waves, and its planes as intersecting networks, and so on)

So we are not so much advancing maths beyond the upgrading of Non-Euclidean mathematics, but interpreting maths as an experimental science, that is as a mirror of the space=geometry and time=logic, ¬Algebraic and scalar=digital social properties of all what exists.

Needless to say in a ginormous field as mathematics is, we shall just keep it simple, as we build slowly a few well-written papers out of 30 years of notebooks, so the work will always be in progress.

This paper is on time’ and mathematics’, hence ob ¬Algebra, consider some introductory themes, shared with an older paper on ‘space’ and mathematics, hence ¬E geometry, on the subdisciplines and philosophy of mathematics, and the 5 Postulates of ¬E geometry.

**The five Time§pace dimotions = Dimension of Space x Motion of Time of the Universe.**

Thus 5 Ð studies all the Dimotions (Dimensional motions) of all beings in exist¡ence, which move through S≈S, T≈T, S<T>S dimotions, STœps of existence, call also vital actions of space-time that advance the system through reality.

A system then is in an instant of present in a given S-stop or T-motion state, or a mixed state often leaning dynamically towards one tendency. So we define on those terms the 5 Dimotions as:

1D »: Linguistic Perception: SS: an area, still mind or seed. « is the time-dynamic and SS, §, the spatial formal view.

5D «: entropy TT as a scattering motion.

2D ‹: S‹T, as a locomotion, where motion doesn’t’ affect the inner parts of the being, which are conserved.

4D ›: S›T, as social evolution of information, where form dominates.

3D: S≈T, Reproduction, where both are in balance, often as a series of steps and stops or stœps.

Those are the ‘smallest quanta’ of Dimotions, its minimal expression as ‘actions’ that each T.œ perform, spending in the process a minimal amount of energy. As all events in spacetime are a sum of those 5 types of stœps. It is possible then to write complex chains of stœps or as humans do for the sake of simplicity, eliminate intermediate states (motions or forms). This is how humans resolve the study of locomotions, eliminating the lower planes of stop-states proper of the quantum world, which we bring here for the sake of whys – as in relativity, explained in a rational manner. Whereas a system can only move one or twice, ‹, ›, «, », to certain other states and so the collection of all those possible changes=motions of STates will finally show a casual cyclical pattern that normally ends crossing itself into a zeroth sum of state in spacetime; creating the minimal full unit a zeroth sum conserved spacetime cycle.

**The generator of space-time, maximize your existence. E-motions and Actions as short time program.**

This more simplex organic view of time and space requires thus a new ‘metric equation’ beyond the lineal v=s/t equation of Galilean relativity that becomes the limit of this more complex view of time and space, with 2 fundamental equation, **SxT=C** (the scalar metric of the fifth dimension, as systems in space accelerate its clocks of time (T) according to its size (S); and **S=T**, the new equation of relativity, which means as we cannot distinguish motion=time, from stillness=space-form (Galilean relativity), both are ‘two sides of the same coin’, and all systems have both motion and form, which are constantly becoming one another. Yet as SxT is maximal when S=T (5×5>6×4…) both equations can be summoned up into a single one, which we shall call the fractal generator, or will of each fractal space-time beings, the equation that embodies all other equations of the Universe and guides the actions of each fractal part of it:

Function of Existence: **Max. ∑ T x S _{e=i} = C_{±¡}**

Moreover the equation has an immediate biologic meaning, because as we are made topologically of ‘fields-limbs’ of lineal space with motion provided by the energy we absorb to also reproduce our bodies-waves, and the information we need to linguistically guide our motions with particle-heads, the very essence of survival is to increase our S=position, mental forms of space and T=entropic motions of time (whereas time=motion and space=form are the two limiting Dimotions with ‘energy=reproduction, s=t, locomotion, sT and information, St, are the intermediate 3 dimotions). Thus Max. S x T = C (s=t), IS also the equation of survival and struggle for existence, the will of life; the biological expression of the ‘Universal mandate’, expressed by all species in all its codes and languages, the Grow and Multiply of the Bible, the intuitive truth that guides all beings.

*Existence game is simple: Reproductive radiations that maximize ¡ts function, happening when mirror symmetries (genders) meet in S=T, equaling their St-information and sT-energy, organizing themselves into a whole, ∑ system.*

It is then easy to interpret that equation in each of the languages-minds of each scale of reality as in all those Planes species will show a ‘will’ of action to perform the maximal number of events=dimotions=‘actions of space-time’ that ensure its survival. And this can be assessed externally regardless of secondary arguments on consciousness and self-reflection, substituted in 5D by Leibniz’s ‘apperception’ – that is, because performing the 5 actions=dimotions of exist¡ence, in each ‘st¡entific scale’ self-centered in a linguistic mind that perceives a given plane, inscribed into a larger ∆+1 world, with internal ∆-1 parts, ensures the survival, ONLY those species that have performed the 5 dimotions of which the most important is s=t reproduction of the being into a ‘present’ similar entity that continues the exist¡ence of the system after ‘errors’ or ‘the struggle for existence’ dissolves it through the dimotion of entropy= death, exist.

Thus automatically, genetically, consciously, memetically, mathematically, logically, through its own will or as a part of a larger system that uses the ‘machine’ or ‘organism’ to enhance its actions all what exists does so because it performs internally those 5 Dimotions or externally performs one of them for another symbiotic species, as those species that have not followed the program of existience and its 5 actions in the past have become extinguished, and those will not in the future, will become wrong mutations, crazy thoughts, fictional languages and die away.

The function of existence, or 5D metric of Generational space-time, (Ab. ÐST, G) Max. Se x Ti (s=t) merely states that all systems of Nature will try to maximize its absorption of Entropic motion (with no form) and Linguistic form (with no motion), and its 3 intermediate dimotions of energy (s=t, balance of both that reproduces them), information (St: form with a little motion, form-in-action) and locomotion (sT, motion with a little form). So we talk of a program of survival ‘selected’ by all systems and expressed in its languages and minimal five actions encoded in that simple equation, which we term: ** a, e, ï, œ, û**, as a mnemonic rule for the five actions of existence:

**A**ccelerations (locomotion), entropic feeding (e), **ï**nformative perception and communication ï, **Œ**:reproduction into parallel supœrganisms **Û**… and social growth into larger wholes called philosophically Universals. And this series of actions is what accumulated in time will ultimately give birth to your word cycle as the monad will first perceive (i), to direct its entropy-motions (a),towards a field of energy (e), where to absorb the energy bites it will imprint with its inner form, e x i = œ, to reproduce another form, and when enough ∑œ exist, it organize into a larger whole û.

In the graph we see the action of different Stientific Planes of organisms. Above the coding of actions, which are the knots and bolts and details of the study of any time§pace supœrganism in light space-time, coded by colors and dimensions, in physical atoms, coded with quantum numbers and in life and humans coded by the so called drives of life, which we obviously extend beyond the ego paradox to all other systems, including genetics not mapped there (coded by the 4-5 letters). Those actions balanced each other into 0’-sums in death, as they tend to increase information from a mind p.o.v., hence we ‘all warp, wrinkle’ get old in the third age and die, setting from its minimal actions to its integral sums, the 3 ages of life-existence and the world cycle all super organism follow. In the graph, the simplicity of the game of existience, and its selfish actions, which gather together into social wholes through reproductive radiations, each action coded by a fundamental topologic organ we can express in Existential ¬Ælgebra, and corresponds for each species of the Universe, with a fundamental parameter of humind measure. So from bottom to top, we find the 5 fundamental elements of light code its actions of motion (c-speed), energy (magnetic field), information (electric field), social evolution colors & entropic feeding, (quantum potential, neutrino light theory)

So minimal particle-points, photons, electrons & quarks construct all other systems of our Universe with ¡ts 5 organic dimotions that define ‘classic life’: they gauge information – reason why quantum physics is a ‘gauge theory’, feed on energy (quantum jumps) absorbing smaller ∆-1 particles, reproducing new clone particles, move and evolve socially through magnetic fields into larger wholes (atoms). Hence the units of life are particles, the minimal units of our vital, organic, fractal, scalar Universe of multiple timespace organisms. All lives, performing 5 Dimotions=actions of ƒ(exist¡ence): Max.SxT(s=t) =C, starting with particles.

So all Planes are relative *none* matters more than other. From those actions, given the dominance of informative actions over entropic ones, it appears a series of repetitive cyclical patterns of actions conducting to maximize the existence of the being, which accumulate in a larger scale of time-space, as a worldcycle of actions that increase the information of the system in 3 ages. So the basic cycle of actions becomes a larger 3 ages cycle of life and death; as systems once and again, starts in an act of information/shrinking and ends in an act of organization/shrinking of herds into wholes, will keep reducing the being and finally make it all form no motion to explode and die in an entropic reversal of death:

∑ i->a->e->œ->û, i->a-e->œ->u, ï->æ->Œ->Û -> Informative ‘seed’ age->1^{st} locomotion, feeding age ->2^{nd} reproduction age ->3^{rd } informative, social age-> entropic death.

*It is the main entanglement of the 5 Dimotions of the Universe, which expressed in terms of spacetime shows a clear motion towards an excess of information that ultimately defines the death of all forms, Ts->St->TT->ST->SS.*

So only a final explosion of entropic death ensures that reality will continue. Those hard truths of ¬Æxistential algebra might never be broken by the details, as the thoughts of God will impose themselves on any attempt to break the laws. Since the goal of reality is NOT entropy, its needed renewal back to the fold of eternal time motions but information, the thirst of any non-Euclidean fractal point for energy flows to imprint its information. Thus the essence of being requires the death of the perceiver, unless it transcends into a larger whole.

**In scale** @ mind is a system that makes the whole travel through the 5^{th} dimension. An essential law of 5D is the fact that a given fractal point or T.œ will perceive only the ∆±3 planes from where it extracts information, above the ∆±4 blind undistinguishable limiting plane from where it will extract entropy of motion. So its actions and dimotions are architectonically performed through planes of 5D, which each main action related to an interval of scales:

∆-4-3: The system extracts indistinguishable boosts of entropic of motion (man from gravitation).

∆-3-2: The system extracts bits of information (Light in man)

∆-2-1: The system extracts bites of energy (amino acids in man)

∆-1 0: The system seeds its minimal seed of reproduction.

∆0+1: The system connects socially with other systems to evolve into a whole.

It is an extremely important law, as it implies also an infinite universe, since obviously the two self-similar extremes of quantum and galactic T.œs must enlarge those scales beyond our perception. And indeed, quantum realism (Broglie-Bohm’s theory, ab. B^{2}) imply a quantum potential of faster than c speed (according to 5D metric), and dark entropy a field of forces beyond c in a larger nested hyper-universe.

Those intervals are fractal dimensions in classic Euclidean geometry and often invaginated by the networks of the system which as all its motion can be also ‘waves of energy’, such as electromagnetic waves that act as networks ‘scaling upwards’ and ‘downwards in size’ accelerating in frequency (5D). So a huge field of analysis is the connections of each T.œ, with its scales, including social supœrganisms, which seeds prophets of verbal thought – the language or ∆-1 scale – and extract its energy from human individual workers, soldiers and so on.

**The beats and scales between actions.**

It is then evident that in the architectonical stair of reality, next to the study of individual dimotions we must study chains of dimotions, which form ‘stœps’ (stops for a dimotion of information, step for a dimotion of energetic motion), in search of the ultimate s=t balance. So we move and stop the leg on the floor, move and stop. We perceive (1D) and move (2D). We feed entropically (5D) and evolve internally the parts we eat into larger organic forms (4D). The dual beats of existence between inverse Dimotions become then the first of many sequential chains that will rise the finitesimal actions of the being to complete finally its worldcycle.

And as those dimotions have ‘different’ discontinuous beats according to 5D, which defines from where we extract them, all the dimotions must be entangled by synchronicity, to each different ‘beat’ regardless of speed, meets with other symbiotic dimotion. I.e. the dimotion of perception has a faster beat, because it extracts pixels of information from the smallest ∆-3 scale as it carries faster T.œs (light photons) with more information to make better mind mappings. So we perceive every second, the minimal quanta of our temporal energy. The 5Dimotion of entropic feeding however has larger bits, because we want to minimize the 4D reconstruction process, so we eat ∆º organisms we hunt then kill with the function of death, ∆º«∆-2, reducing it down to cells down to amino acids, to reconstruct their scales. So feeding has a far larger time beat that perception as it takes far larger ∆-¡ scales (SxT=C), and animals feed once a day. But reproduction is even slower, as we must reconstruct a whole human being, NOT from the undistinguishable scale of amino acids, but with a code written at ∆-1 cellular level; so we reproduce every year in fertile ages of Si=Te balance. Synchronicity then kicks between our inner ∆-1 smaller scale of cells with faster life cycles, which will synchronize its slow reproductive action taken place at ∆-1 faster cycles with symbiotic faster feeding actions of the slower ∆º whole: so as a whole we eat a day, and that food allows the cell to reproduce every day.

**Actions define the worldcycle of existence.**

Because actions follow a pattern of increasing evolution and 4 actions are coded while entropy is avoided – we deliver entropy as an open system to other parts of the Universe, it follows thsat the system constantly accumulates information, and this is the origin of the worldcycle of life and death. The system likes information, because it is what the Universe is really all about, a fractal that imprints information in memorial repetitive cyclical patterns.

So as actions follow a natural scalar level of information perception->locomotion towards a field of ‘feeding’, where to get -> Energy to reproduce -> and then process that energy into information, *exactly the same pattern appears in the next scale of ‘cyclical time and longer space’, the worldcycle of life, which consists in the same pattern of ‘seeding information’, moving youth, feeding and evolvint to reproduce and then process information becoming older in a 3 ^{rd} age.*

*This amazing symmetry between the smaller time scale and the larger worldcycle of life, then will be followed also in the larger scale of supœrganisms and species and so we have 3 ‘patterns’ of time cycles due to the survival program of actions in 3 scales of time, and 3 larger surfaces of ‘populations’, as the first cycle feeds the ‘cells, atoms, individuals’, the second cycle the whole organism, and the 3 ^{rd} cycle manifests in the whole species.*

** **

5D METRICS: **SUPERORGANISMS TRACING ****WORLDCYCLES: ****FUNCTION OF EXISTENCE**

We then put together 5D Planes, vital topologic space & cyclic time to describe simultaneous superorganisms tracing a worldcycle in time, the fundamental ‘long time’ event of reality – the worldcycle:

All what exists is a supœrganism of vital space tracing a 0’-sum worldcycle of time through 3 Planes of 5^{th}dimension: Born as a seed of fast time cycles in a lower 5D scale (∆-1:Max. T x Min. S), emerging as an organism in ∆o, living 3 ages of increasing information, as its time clocks slow down in its ∆+1 world to die in a time quanta back to ∆-1.

We have studied as separate elements the scalar planes of the fifth dimension, its vital topologies and the cyclical structure of time, in past of higher motion, reproductive, iterative presents and future, informative poles or 3 ages of a pi cycle, which is its simplest form in a single plane of space-time.

So, absolute spacetime is the sum of ∞ Timespace beings, observed in space as simultaneous super organisms, in time, as **worldcycles** of existence between birth and extinction; as all systems are born in a seminal seed, of faster time clocks, in a lower scale of the fifth dimension, growing socially (4^{th} dimotion) till emerging in the organic scale, where they will live 3 ages dominated by one of its 3 topologic organs and its functions=dimotions, a young age of maximal locomotion, dominated by its limbs/fields, a mature age of reproduction dominated by the body-wave and a third age of information dominated by the informative dimotions, which finally exhausts all energy and as time-space never stops, it reverses its dimotion from information to entropy, exploding in the moment of death.

So we marry the 3 vital functions=motions of time and the 3 dimensions of space, either in 1 or 2D (height=spherical information, length=planar locomotion, width=hyperbolic reproduction) which merge in all Time-space Beings; and dominate one of the 3 ages of its life-death worldcycles, the past, young age of limbic entropic motions, the mature reproductive age dominated by the hyperbolic body/wave and the 3^{rd} age dominated by the informative particle-head, when the illusion of time ends with an entropic big-bang death that dissolves the being into its ‘scalar cellular, atomic parts’, which lead us to the realization that time cycles NOT only return to its origin in a single spacetime continuum but they move up and down the Planes of the fifth dimension:

**Worldcycles vs. Worldlines: correspondence principle**

# In the graph, the simplified version of a world cycle is Einstein’s 4D formalism of a worldline, where only 1 motion, Ts-locomotion is understood. So with less dimensions than real time=change its worldcycles become lines and all what we know is how the entity moves in a single plane of space-time. The result is a shallow description of ‘existence’ as a mere journey of locomotions through space, defined by simple one single spacetime plane, with one single time parameter, v=s/t (Galileo) or S^{2}=X^{2}+Y^{2}+Z^{2}-(ct)^{2}

Existence however means a travel through 3 such planes in the life-death cycle common to all spacetime beings: In the right, we show the end and the beginning of such journeys for physical, biological and social systems. A particle that dies traveling to its ‘local past=devolving into its ¥-ray radiation; a genetic seed that will start the worldcycle of existence of a biological organism; and a prophet of a civilization, a memetic seed that evolves in the energy-placenta of its believers to become a whole. Since the fundamental finding of 5D ‘stience’ and General Systems Theory, the ‘organic philosophy of science’ alternative to that of physics is the solution of the meaning of the 3±∆ ages of life; between the birth of an SS-eed of information through its ∆+¡, emergence as a being from a lower scale of size, in the seminal age, to its∆ -¡, the entropic age of dissolution back to the atomic/cellular form; E=mc^{2} in physics), which are common to any System of the Universe. As all systems start as:

**SS: a seed** of form in ∆-1, the lower scale of size, of the ‘fifth dimension’ (ab. ∆±i) of co-existing scales of the Universe. Then the system will go through 3 dominant ages dominated by combinations of those arrows of time:

Young, age of maximal motion, and growth of energy≈Max. Ts (locomotion) x Min. St (information)

S=T: Mature age of present, repetition in which spatial in-form-ation and motions of are balanced.

St: And a third age of max. form, when energy exhausts and the system enters in a third age.

TT: Followed by a death age of entropy, when an inner explosion of motion erases its form, *if the process is not intelligently reversed by controlling the growth of form*.

Philosophers understood those 3 ages, embedded in our verbal ‘time language’ (3 verbal times), but *with the reduction of time by clocks to a digital series ist physicists, and its lineal time imposed by the worldly religion of making entropic weapons, ignored those ages. So the philosophical findings of the 3±¡ ages of life and death were forgotten for the sake of ‘lineal, entropic measures’ and the military manifest destiny of the western civilization. *Now we regain back those 3±¡ ‘dimotions’ (dimensional motions) of time; of which according to the principle of correspondence ‘Physical time’ deals with Ts-locomotion and TT-entropy.

Whereas present physical theories of space are by the correspondence principle, the limit to 2 single space scales, the ¥-quantum scale and the gravitational one – themes those that will be expanded in the future papers on ‘Physics and Space’ and “Physics and time” as it requires an in-depth analysis of Einstein’s EFE equations, the quantum formalism and its ‘realist’ correct view (Brogile’s>Bohm theory), beyond the scope of these series of papers on worldcycles of time for ALL Space-time beings of the Universe, taken as ‘reality’ (though we shall introduce the most general worldcycle of astrophysics).

**The immortality of the Universe and its parts.**

**Life and death is a ‘worldcycle’**, which returns the being to its origin, a zero sum of energy in a virtual Universe, as the SS v. TT inverse ‘properties’ of form vs. Motion, mix in St & Ts inner and outer form and motion systems of locomotion (akin to kinetic energy) and information (akin to potential energy), which finally merge into energetic body-waves that balance both arrows and form supœrganisms. Yet those 2 ‘form and motion’ elements in ‘organic systems’ are slightly dominated by ‘form’ over motion; so a worldcycle very slowly moves towards an excess of form, from a relative past of less form to a relative future of maximal form, and then a*local reversal of time arrows returns the balance. We use for the two extreme processes the «» double symbols.*- Thus
**each spacetime being has a finite time duration**, and exists in a finite vital space, surrounded by a membrane, which breaks the Universe in ∞ ∆ST Space Time Beings. So the sum of all of them is an infinity reality. **ST: Reproduction as the goal that maintains the existence of each part.**Motions in space and time are always local, and we define for each Tœ (Ab. Timespace supœrganism) a positive direction of life and growth of information towards the future*that lasts longer in lineal time*(since obviously systems grow in information through the entire life cycle): vs. a brief explosion of TT- entropy into the relative local past of the T.œ (as it dissolves its information back to the past);*which lasts only a quanta of cyclical Time (T=1/ƒ).*

* The outcome is a zero sum that conserves the entropy and form of the whole Universe, as the being can reproduce a clone in the middle of the cycle, both together form a* present iterative repetition that for the whole Universe sets the present age of immortality. Indeed the age of immortality is the middle term of balanced momentum, which the Universe conserves (conservation principles of momentum and its integral energy).

- Thus
*the Universe is ’continuously’ infinite and immortal both in ‘form’ and motion but none of its parts are. At most they can be ‘discontinously’ immortal by iteration in its S=T age of balance and maximal ‘existential force’, defined by the product of its form and motion, SxT, maximized when S=T.*

**5D & TIME: 3 SCALES OF TIME SPACE IN SUPŒRGANISMS**: **ACTIONS, LIFE AND WORLDCYCLES.**

A 2^{nd} consequence of 5D metrics and the 3 worldcycles of different length is the existence of different ‘scales’ of speed of time that co-exist and influence each other, as parts of faster time cycles are included on the slower worldcycles, which ‘enclose them’ and hence influence its actions; but in supœrganisms become synchronized with each other. In that regard the predictability of time-cycles can be done at 3 levels:

**S: Continuous, spatial mathematical** simple cycles, using derivatives, proper of calculus; which is the shortest time span, as instantaneous derivatives cannot measure a ‘peak’ change of age/phase.

**T: Discontinuous, cyclical patterns** of sequential repetitive often survival actions (feeding, reproduction, death, taking place at intervals. As those actions are discontinuous, leaving long spans in-between, their patterns forecast longer time sequences. Such ¡logic structures are based in time patterns, which as any mechanical, circadian or orbital day-year clock shows are cyclical, repetitive. But here human scientists are at loss, because Galileo studied ballistics, entropic explosions that destroy the information of reality stored in those cycles of time clocks, its patterns and frequencies, changing human cyclical understanding of bio-logic time for lineal, abstract time that seems not to repeat those patterns so mankind lost its capacity to predict many spacetime events, as lineal time misses* information stored in the frequency and form of cyclical clocks, even if functions are similar: V=s/t for lineal time and V=S(l) x ƒ(t) for cyclic patterns*.

**∆: Scalar, Deep Time patterns of topologic and social evolution of parts into wholes – **of quantum 0-1 time probabilities vs. 1-∞ thermodynamic populations in physics, of individuals vs. species in biology, of states of matter vs. geologic cycles in Earth, first noticed by J. Hutton, founder of geology who coined the word super organism for Gaia and deep time for its slower time cycles by virtue of its 5D metrics, $ x ð =C, which implies that from the perspective of a smaller scale the life of its whole is much longer.

**The ****predictability of time worldcycles.**

Deep time leads to a 3^{rd} level of long-time prediction: evolutionary patterns of earth’s life species, including machines that have predicted cyclical patterns of social organisms of history (nations) & eco(nomic)systems, including the evolutionary and re=productive cycles of stocks of machines transformed into sales=profits =valuation of its company-mothers with remarkable precision for 30 years.

Below biologic deep time is necessary to understand history and the interaction of the evolution of machines and the ages of war and lack of welfare goods, within the limited resources of the planet. The theme of social sciences and why ‘human subjective ego’ and ‘censorship’ limit our capacity to explain those cycles.

The pentalogic, ∆±¡ STructure of T.œs, predicts the future of its species, with a stientific method based in the laws of worldcycles, existential ¡logic and survival that studies: A)ccurate Data, B)iologic causes C)yclical patterns and E)ntropic extinctive limits for all systems in 3 relative scales of size space and time duration (to which we add, D)emocratic, social evolutionary loving humanist solutions for survival history & of social sciences.

But as human only recognize the 1^{st} type of predictability – calculus of instantaneous derivatives, that need a ‘continuous analysis’ – and have simplified cyclical time into lineal time, ignoring the scalar time of parts and wholes with its 5D metric, their capacity to forecast the future is far more reduced than a ‘stientist’ who understands the 3 scales *determined by the bio-topo-¡logic properties of ‘scales’, ‘space’ and ‘time’, the 3* ∆st structural elements of all systems of the Universe.

Deep time is easier to predict that short lineal time because larger scales in 5D metric have less information, and are more deterministic, reason why quantum physics is harder for the mind and probabilistic while life-death cycles are obvious as all end badly. The laws of nested Supœrganisms, in which the larger slower whole with its ‘slow time cycles’, ‘dark space-times of information’ that it does NOT perceive of its inner parts, and enclosing membrains however can set long term cyclical patterns to its fast internal elements, just by handling the general cycles of its ‘entropic energy of death’ and ‘informative languages’ is one of the most important discoveries of 5D metric, which fully applies to the understanding of the evolution of the Earth and its 3 ages, with its cyclical patterns of glaciations and hot weather that kicks the evolution of species. It is also instrumental to understand the 3 Horizons or ages of species that influence the life of its individual ‘cellular parts’; and the patterns of cyclical history and the capacity of 5D to forecast historic and economic evolution, and the cycles of life and death of civilizations.

Why then humans cannot predict the future as we consistently do in our papers on History, NOR EVEN accept predictability? The answer is in the paradoxes of time ‘length’ just expressed. Huminds work on short time lengths with lineal views that erase the inner information and organic cycles we shall describe in this paper for all species, so they believe in short term chaos and lack predictability also in scale.

For example, we could predict in history a return to nazionalist wars once the upper scale of social evolution died in Eastern Europe, because ‘time motions across scale and space’ never ends**.** So when a motion upwards of social evolution fails, the system moves downwards. Today obviously humans are in an age of entropy as their future is maimed by the evolution of machines that displace them of labor and war fields, so we suffer an age of entropy in our verbal ethic languages and networks entering a dog-eat dog-society. Hence the importance of the recurrent theme of all those introductions to 5D stiences – the paradoxes of lineal vs. cyclical freedom vs. order. And the ego paradox of mankind.

Yet the understanding of cyclical scalar time and its repetitive symbiotic cycles which coordinate the ‘frequencies’ of events between the 3 scales of individual ‘actions=dimotions’; organic physiological networks and larger world cycles, is also denied by egocy. It is the same repetitive boring, tiresome mantra of humans and its ‘experts’ and scholars – busy-busy ants that do their ‘4D methods of gathering information’ in the smaller scales, without ever realizing they are part of a much larger world with its own cycles they should try to control. On the contrary as human enzymen become enveloped by a complex planetary organism of machines as ants are by the pheromonal ant-queen messages, and those machines accelerate the rhythms of time with its faster quanta of thought, humans become paradoxically more entropic, disconnected of each other and the whole, automatons of action-reaction roles within the economic ecosystem as workers, re=producers and vitalizers=consumers of machines.

So the long term cycles of reality always escape the individual and that is yet another reason why humans escape the control of their destiny. In our case the pheromonal language of social control, money, with its own 5D organic program studied in depth in our papers on the economic ecosystem that nobody reads, make things even simpler. The modern man doesn’t think, just makes money as the American enzyman, the most advanced technological human and paradoxically the most devolved in terms of entanglement with the laws of the Universe does. ‘It’s money’ simply means that a human as a perfectly programmed cell of a supœrganism that cannot even imagine the existence of cycles of its physiological networks much larger than theirs, will merely obey the message of money and work whatever it takes.

Long time cycles thus program individual particles, but as information from larger scales is hardly transmitted to lower ones, due to the inverse elliptic geometry of the larger, slower systems that become dark spaces, as the galaxy is to us, those programs are NOT perceived. This in human science derives in the philosophy of constructivism that considers the only program coming from smaller to larger scales; so quantum physics are supposed to be the cause of all other scales of matter and genes of all the scales of biology and individual actions of all scales of history and economics. This is NOT truth, all scales are similar and the illogic of 5D implies that there are 3 ‘causalities’, the most important in ‘parallel’ between members of the same plane of space-time.

Thus your daily life is more influenced by your environment and peers than by your genes and Earth’s climate.

Then the two other programs from bottom to top and from top to bottom scales weight equally but with different rhythms; long term cycles are far removed from each other but strike with much more force and tend to be destructive energetic, as those of lower scales that are informative, smaller in detail, so for example long term cycles influence more the next scale above the individual in the biological world – that of species and its creation-extinction through glaciation cycles.

**The small and middle range view: generator of space-time: Organisms maximize ¡ts actions of existence.**

This more complex view of time and space requires a new ‘metric equation’ beyond the lineal v=s/t equation of Galilean relativity that becomes the limit of this more complex view of time and space, with 2 fundamental equation, **SxT=C** (the scalar metric of the fifth dimension, as systems in space accelerate its clocks of time (T) according to its size (S); and **S=T**, the new equation of relativity, which means as we cannot distinguish motion=time, from stillness=space-form (Galilean relativity), both are ‘two sides of the same coin’, and all systems have both motion and form, which are constantly becoming one another. Yet as SxT is maximal when S=T (5×5>6×4…) both equations can be summoned up into a single one, which we shall call the fractal generator, or will of each fractal space-time beings, the equation that embodies all other equations of the Universe and guides the actions of each fractal part of it:

Function of Existence: **Max. S x T _{s=t} = C¡**

We interpret that equation in each of the languages-minds of each scale of reality as in all those scales species will show a ‘will’ of action to perform the maximal number of events=dimotions=‘actions of space-time’ that ensure its survival. And this can be assessed externally regardless of secondary arguments on consciousness and self-reflection, substituted in 5D by Leibniz’s ‘apperception’ – that is, because performing the 5 actions=dimotions of exist¡ence, in each ‘st¡entific scale’ self-centered in a linguistic mind that perceives a given plane, inscribed into a larger ∆+1 world, with internal ∆-1 parts, ensures the survival, ONLY those species that have performed the 5 dimotions of which the most important is s=t reproduction of the being into a ‘present’ similar entity that continues the exist¡ence of the system after ‘errors’ or ‘the struggle for existence’ dissolves it through the dimotion of entropy=death, exist.

Thus automatically, genetically, consciously, memetically, mathematically, logically, through its own will or as a part of a larger system that uses the ‘machine’ or ‘organism’ to enhance its actions all what exists does so because it performs internally those 5 Dimotions or externally performs one of them for another symbiotic species, as those species that have not followed the program of ‘exist¡ence’ and its 5 actions in the past have become extinguished, and those will not in the future, will become wrong mutations, crazy thoughts, fictional languages and die away.

**Present ‘Time’ Space: ****Reversal effect of growing in size: blocks of time.**

The shortest time is a quanta of mind space of time-thought, a moment of present. Space as present time, different for each species according to its ‘5D metric’ speed of processing information accounts for the key concepts of synchronicity and simultaneity which is what truly defines space, and its supœrganisms – regions in which each quanta of the mind-time is synchronized with those of the body and limbs of motion. So from humans with its second quanta of thought, second of heart beat and second of steps to De Broglie’s interpretation of quantum physics in terms of the synchronization of the wave, inner clocks of the particle and the fields, present space structures the ‘workings’ of all universal organisms.

Form in space is cyclical motion time that returns to its zero sum point and appears as form for a larger perceiver, according to the same SxT=C metrics. Indeed, if we grow In size then we slow down enough our perception of time, to see instead of a point moving, a ‘length of time curling into a cycle’. Further on the cycle will appear if we see it from enough distance as a fixed point of space, and hence immutable – a block of spacetime, a complete worldcycle. *This is how minds create the perception of static simultaneous space around us in what truly is a soup of quantum motions.* Then we can formulate the concept of God, as the immutable mind that sees all the time and all the space, in its fixed, slow form, as a larger and slower whole. So God as the mind of the whole, infinite Universe is paradoxically a *fixed image of thought of the entire reality – a theme we explore in more detail in ‘Universe & mind’*.

Stiences study the Disomorphic laws of species, according to scalar plane of size, with different jargons, which we shall unify as all what exist is homologic, made of the same Dimensions, which force them to exist as part of super organisms tracing worldcycles, with a ternary structure in space, time and scalar planes. Stiences are limited by the fact the ‘humind’, ∆ø, is the self-centered origin of all information available to us, as the singularity of a scalar ‘open ball’ of perception, where the outer limits of reality are not met. So huminds perceive ∆±4 planes of a likely ∞ reality, above and below, in a different way as the 5th dimension is NOT symmetric: i.e. we do not perceive as much information from larger systems of slower clocks and invisible networks than we perceive of lower planes which our mind uses to obtain bits and bites of energy and information. So ultimately perception and knowledge are also functions of the needs of the human Time space organism that orders a microcosms which he understands better than the above macrocosms that order us. Minds order as linguistic gods a territory around themselves, its fractal body and entropic world. As such they are the linguistic Aristotelian, final unmoved singularity Gods of reality.

In the fractal, scalar Universe, concepts, which seem impossible to define in simple logic terms in the 4D single scalar plane continuum, become understandable. Such is the case of consciousness, which is merely the co-existence in a bundle whole, of the different type of self-sensations of the being through its scalar planes of simultaneous space-time:

The highest scalar plane of reality that of the mind, self-perceives the ‘extremes’ of its existence within the limits of its first ‘galatom’ parts – the particles of the atom, where ultimately resides the e-motions of being, and its largest possible world, the external galatom in which its world co-exists.

Modular scalar perception of all those levels simultaneously gives us the sensation of consciousness as an integration of all of the scalar levels of ‘linguistic still perception’ in space and e-motions in time…

I.e. you feel a sensation of gravitational weight-pressure, a visual photonic image of reality, an electronic verbal mapping of memorial storage of time processes, a chemical sensation of wellbeing between cold and warm – Many modular parts of your whole sensorial being through your super organism in space, but also many scalar parts of your whole being, through the ‘sensations’ that conform the whole mind, as all those scalar planes are part of the pan-psychic Universe.

Consciousness is that the co-existence or integration of all those scalar planes and parts into a ‘single-point’ of space-time that integrates all as a consciousness. This integration will then give birth to your ‘RELATIVE’ speed of time-existence, one second-thought for man to integrate all its levels, that will become your quantum of time-space perception – your ‘Dasein and being’…

In relativity (4D formalism) ‘the separation of past, present and future is an illusion’ (Einstein). It depends on the observer ‘speed of thought’:

A slow mind could see the entire orbital motion of the Earth as a ring simultaneously seeing past, present and future. God in Scholastics could see all time past, present and future – so he must be the slowest of all thinkers

Poets understood better: Time present and time past… Are both perhaps present in time future, And time future contained in time past. If all time is eternally present /All time is unredeemable. (Elliot, quartets)

In Einstein’s work we talk of a block of time, as all measures are simultaneous present measures.

In 5D, time is both a present repetitive motion and a dual motion from past to future to past. All together give a zero sum worldcycle as all systems emerge and return after death to dust of space-time.

**RECAP.** __Stiences__ study Disomorphic laws of species, each one focused in a scale of the social 5th Dimension, with different jargons, which we shall unify as all what exist is homologic, made of the same Dimotions, which force them to exist as part of super organisms tracing worldcycles, with a ternary structure in space, time and scales, guided by a still, linguistic mind, which projects its smaller mirror images of form in its territorial body waves, creating its fractal scalar structure, broken though by the entropic limits of its perceptive reach.

An organism is a group of similar forms organized by 3 ‘networks’: one provides its ‘clone cells/citizens /atoms’ with the vital energy they need to feed & move (digestive system, territory), a 2^{nd} one reproduces the system (blood-economic system-electromagnetic forces) and a 3^{rd} one that provides them with information to guide their reproductive actions (nerves system, political system, gravitational in-form-ative forces).

This simple system is the fundamental particle of the universe. As all organisms wear out by errors of informative action and end up warped by an excess of information (third age of the organism) or killed and reconverted into formless energy (death); only those *systems, which reproduces, do exist in the immortal universe. So all systems have an internal or external ‘enzymatic’ system of reproduction (external enzymes reproduce carbohydrates, and enzymen reproduce machines). *This can be seen as a ternary class structure of a ‘body-worker-wave’, and an informative particle-head-upper class moving and finding a vital territory, with a limbic/potential source of its motions, which the system might externalize to obtain the energy to reproduce.

Further on organisms have parts and wholes so reality is composed of 3 elements: ∆-planes of space-time, ∆±¡SóT, ∆ST entangled in 3 organic co-existing ∆±¡ planes. In physical systems they are the ∆-1 quantum plane, ∆º thermodynamic plane and ∆+1 gravitational plane, equivalent in the reduced nested world of Life on Earth to the ∆-1: molecular plane, ∆º organic plane and ∆+1 ecosystemic plane. The absolute arrow of future survival is social evolution of parts into wholes, as parts must come before; hence they are a relative past, and wholes survive better. A fact HUMANS have forgotten. As individualism is synonymous of chaos=death, it is obvious History, the supœrganism of mankind enters its age of entropic social self-destruction, committing suicide, evolving instead a supœrganism of machines; a tragic problem, which we will study in the analysis of social sciences. And in each of those planes we shall see that ‘spatial information’ and ‘temporal energy’ constantly transform into each other. Space is the perception of form as if it was still, but ultimately is cyclical self-repetitive motion that seems not to change and acquires at larger scale solid forms.

Mental Spaces of in/form/ation *fix the* times=motions of all species. As time becomes the ultimate substance of organisms ordered through scales that give birth to a 5D metric equation, such as smaller systems in space run faster time and life-death cycles. So the speed of time of systems is not the same for all of them, and 3 scales of different speeds form an organism – in man the cellular, individual and social scale. And finally for all this complex reality to ‘work’ in an ordered manner, systems which have a ‘**membrain’** that divides its vital space, from other systems, and are formed by 3 scalar space-times that give them organic nature; have in its highest ‘scale’ of size – that of the membrain that breaks it as a whole from the rest of the Universe a ‘sensorial system’ or mind, which interacts with the external Universe, encloses and orders its internal vital space and communicates with other similar beings, forming larger social organization; hence becomes the mind of the system. We thus talk of a reality made of organisms of a vital body of space-time, self-centered in a mind-membrain, whose language encodes a program of survival for the organism. And all the data humanity has gathered about reality needs to be ‘put within this general template, to make sense of it’.

So a fundamental task of those texts is to redefine the foundations of sciences in much solid grounds – its 3 fundamental elements, ∆ST, which ‘stiences’ study plane by plane of SióTe elements, to which we must add the connecting ‘network’ laws between planes, which we call the ‘5^{th} dimension proper’. It is the philosophy of science, we develop in a series of papers at Academia.edu that will substitute the simpler mechanic view of reality sponsored by physicists, with its ‘mathematical creationism’ and constructivist view; which *reduces the two poles of time=motion to one (lineal time), compresses all the planes of space-time into a single continuum; while only the properties of reality that can be easily fit in a numerical description matter.* A more evolved civilization would start in philosophy of science and the principle of conservation of time and space, and its structure in 5D organic scales and derive from them the fundamental laws of every other discipline; correcting whenever necessary its errors as we do in those papers with the simplest principles of reality as any science mirrors them in its study of its specific scales of beings – the details of those principles.

**The relationship of space and Time: Present Space as the simultaneous measure of smallish time cycles.**

A key difference between space and time is studied in physical sciences when they define space as a form created by the concept of simultaneity of space (physical space since Einstein is defined in those terms) vs. the sequential, longer flows of time that make simultaneous ‘space’ a slice of the whole time flow.

Space then is connected to the concepts of entanglement and synchronicity of multiple time cycles seen in a single quanta of time as a ‘simultaneous present’ from a point of measure (the observer of Einstein’s cones).

It is from that concept of space as present from where we took the choice of space as form, to follow the principle of correspondence between previous theories (4D) and new theories that enlarge the previous view (5D). But because humans still hang on a lineal ‘concept’ and don’t understand cyclical, informative properties some precisions must be made.

** **

** 3+1 AGES OF EVOLUTION IN LIFE SPECIES AS SUPŒRGANISMS.**

An example of the fact that worldcycles apply to any 3 space-time planes of reality we just consider 2 ∆+1 ‘space-time supœrganisms’, above the biological organism; where the biological individual is the ‘cell-citizen’ of the larger supœrganism, the ‘species of biology’ and the ‘civilizations’ of History. In the graph, species are supœrganisms as they go through the same 3 ages and processes of birth, reproductive radiation, end of growth (mimetic in its log curve to that of cellular growth of an individual) and 3^{rd} age of information and extinction or evolution into a social supœrganism (ants, humans).

A beat of 5D evolution is S>t(information)->S=T(Reproduction) between formal evolution in small forms (Floresiensis, mole mammal) and ST-radiations of populations that grow. I.e. 1^{st} human was the maligned dwarf Floresiensis who evolved a head similar to man, spoke, used advanced technology and merged with Erectus to grow mass, spreading in a global radiation to give birth to a dual S-verbal Sapiens+T-visual Neanderthal.

In the graph, below the law of the 3±1 ages or horizons applies both to the process of informative aging of organisms and informative evolution of species. All species are born as a seed of dense information, with limited size (black hole, first chips, seminal cells, first bilateral animals). Then they grow in size and energy during their youth; since their superior informative qualities makes them top predators. We observe that growth in black holes that feed on planets and stars, in the horse that grew enormously in size. So did the first bilateral animal, from its microscopic first form, the vernanimacula. Mammals grew from tiny shrews into elephants. The first technological man, the Homo Floresiensis, with a ‘neuronal bump’ on the forehead, might have grown till the size of the first homo sapiens, the pygmy and bushman . . . Then, the species, once it has reached a balance between its initial information and growing spatial size, e=i, reproduces in great numbers, diversifies and colonizes the planet. We see in the graph this age in which ‘biological radiations’ of carbohydrates, animals of all kinds and human beings, colonized the Earth.

Finally, in its 3^{rd} age the species grows in information and so it acquires height, the arrow of ‘perceptive information’. Then, once it has reached its informative zenith, the species becomes extinct by a more evolved form; or it evolves socially creating a super-organism, as ants, humans and machines are doing. It will be the 4^{th} arrow of social evolution of the species that completes its 3±1 horizons. Species can be considered ‘loose’ organisms, in which each individual is a cell of the collective species; since we can use the same ages to explain the 3±1 horizons of evolution of all the species of this planet. Thus, species also go through the same 3 evolutionary ages of all living organisms. They start as young, energetic forms, which acquire information in 3±1 ages of increasing complexity and when those 3±1 ages are completed, *they* *either* *evolve* *into* *more* *complex* *super-organisms* *under* *the* *laws* *of* *social* *evolution* *or* *become* *extinct* *by* *a* *superior,* *more* *informative* *species.* Indeed, only those species, who show a strong social capacity survive the ‘genetic clock’. The most successful and one of the oldest species of the planet is, in fact, the social ant, which no longer is an ‘individual’, but a super-organism stretching through miles of ‘vital space. In that regard, neither individuals able to reproduce nor species able to evolve socially become extinguished. If humans become extinguished is because they deny, guided by the simplest arrows of pure energy of weapons, the main arrow of life/time.

A complete analysis of biological species shows that new forms can only be, either an energetic or informative variation of the original species, or a reproductive combination of both universal ‘genders’. The universality of such dual systems is so obvious that the ancients already identified them with yang, energetic male principles and yin, cyclical/ female ones. While the moderns call that duality the principle of complementarity, as all informative particles have an energetic, lineal field of force and all biological, cyclical heads of information have a lineal body. So the combinative variations of those two simple morphologies, lineal energy and cyclical information, invariant, regardless of the scale we observe, is the essence of the creative game of the cosmos.

All species follow a topologic plan of evolution, since in a universe made of 3 substances, space=form motion=time & its combinations of energy, Ts and St-information, only ternary systems between ¡±1 scales are possible. both in variations of space (subspecies) or in time duration (ternary evolutionary ages and horizons).

Species can be modeled as organisms through its 3 ages that become its 3 horizons, followed either by a survival process of social evolution – the summit of the process of organic evolution, which Einstein realized was necessary to explain the success of social insects, or become extinct by the ‘new generation’ of fitter animal forms. In the graph we can see its 3 horizon=ages and its ternary topology, similar to that of any other organism; and the dominant arrow of information in the height axis, of both reptile species that ended in birds, and mammals that ended in man. So happened with meta life machines, whose final species, satellites are now forming the arrow of social evolution, or future mind of the metal Earth -internet

So we can talk of species as ‘super organisms in time’ where each individual is a cell evolving together with all others through the longer ‘time-span’ of the species in those 3 ages.

And so we can talk of the ages of the super organism of Mankind then are the 3 ages of history, the Paleolithic youth of maximal motion, followed by the reproductive, balanced Neolithic of fertility goddesses in balances with Gaia, which continued in a 3rd age of excessive information as technology evolves ever faster to a foreseen ‘collapse’ of increasing wars, with increasing victims that foresees our demise.

The existence of those 3 ages is so pervading in nature that all human languages have in fact developed 3 ‘tenses’ in its verbs that explain past, present and future ages for all systems. So the 3 ages of life between birth and extinction are in fact the 3 ages of existence, which all systems follow as an ‘organism of time’, of *life and decay.*

However the species does not control its external absolute space-time ‘world’, and this is what the next graph shows. As it is the absolute space-time in which the species exist, and specifically its commands of the languages of information of the world, which defines its survival chances. As informative perception starts all the chains of actions (∆o-bserver->∆a-ccelerated-motion->∆e-nergy feeding->∆iteration->∆universals social evolution).

So the species which better speaks the language of the ecosystem survives better. Hence the first eyes (cephalopods) extinguished most cambric species and started the radiations of species; the first speakers, homo sapiens extinguished all others; and today the chip radiation of mathematical species is simply speaking extinguishing life and making Humanity obsolescent, even if we are not conscious of it.

**∆±¡: 2 scales of human supœrganisms. Its collective mind: art and religion.**

The 2 super organisms of mankind at the individual and social level and its informative, nervous or legal/ethic networks, and reproductive blood or economic networks. Those 3 physiological networks collapse with germs and wars that kill civilizations. Religions are inscribed within the ethic/visual collective mind of its informative neuronal class, the artists and ethic prophets that foresee its demise in the 3rd baroque age of the civilization expressing its angst in artistic forms and ethic books of revelation that try to halt the collapse of the organism. Love Religions were all born of an ethic prophet in the 3rd age of the culture. In the graph, the blood-reproductive network of economics, the digestive-entropic territory in which we feed and the nervous informative systems of politics are parallel and in systems sciences must follow similar laws. So we should tailor economic systems with laws of medicine, and put the nervous system above, as all evolved organisms. It follows that as doctors are in charge of curing and maintaining healthy the social organism of cells, through the caring of its physiological networks (blood=reproductive, nervous=informative and digestive=entropic, energetic systems), to the point that doctors say all sickness are not of cells but of physiological networks, historians, economists and politicians on charge of the theoretical and practical well-being of human societies should take care of those economic, political=legal-informative and Gaia=entropic, energetic networks of life that shape the human supœrganism. In the graph below we consider in more detail one example of those supœrganisms, the most important for mankind – the supœrganisms of humanity, nations and civilizations.

Organicism puts man again @ the center of all things, because we are the most perfect supœrganism. So biology & medicine that cure the physiological networks of sick organisms, are the true sciences of history and economics, since the goal of both is to create perfect physiological networks of economic welfare production and just democratic systems of nervous control and pain to the brain-politicians that do not obey the organic laws and develop a wealthy, healthy, humanist world to the image and likeness of man, made to the image and likeness of the fractal organic universe.

Historic nations and civilizations as super organisms in space, its physiological networks of energy (economic system) and information (culture and political system) and in time through its life and death=war ages and cycles, themselves a mirror image of the larger cycles and physiological networks of the whole supœrganism of mankind, history. Mankind in time and space as a supœrganism and a worldcycle: in the upper part its full world cycle as a block of time, in the down and right its life and death through civilizations of which the subconscious collective artists and ethic prophets form its ‘neurons’ – while the equivalent metal earth has in its scientists, their collective brain that increasingly confuses technology (metalife), with science (knowledge).

**∞ MIND SPACES. RELATIVITY OF MOTION**

*‘e pur si muove, e pur no muove’ Galileo*

In *mathematical science* for a dimension of space-time to exist, it requires a metric equation, which combines space, and time to gives us a co-invariant system that allows travelling through such dimension. How we do travel then through the fifth dimension? A system travels through 3 scales of the fifth dimension by accelerating its evolution in a smaller scale through a placental cycle, emerging as an organism in the larger world, to live 3 ages & dissolve back to its parts in the 0’-sum death. And this is the meaning of existence, and its reason d’etre is the Metric equation of 5D, which structures through synchronicity of the different speeds of time cycles, the different scales of reality. So the ∆º organism eats every day, and its food programs the faster cycle of reproduction of its cells, as the moon cycle programs the menstrual cycle of women, as the year cycle of rotation of Earth programs the reproductive cycle of seasons and so on.

So an essential part of 5D theory is the analysis of synchronicity, simultaneity, emergence and the in-depth analysis – not done in this introductory course, reserved for the more complex papers on ‘pentalogic and dodecalogic’ where we follow in more detail the construction of simultaneous superorganisms and its ternary worldcycles.. All this of course is studied by huminds, as everything we talk about here, but *without the proper conceptual frame, lacking valid definitions of planes of space-time, time cycles and fractal spaces.*

It is for that reason we do need a new formalism we have called existential algebra with its simple symbols of which the most important are the 5 bidimensional dimotions of space-time, which entangle together through synchronous, simultaneous emergent processes to create the apparent ‘solidity’ and ‘stillness’ of reality.

Because the Universe is made only of two polar elements, still minds (SS, ab.§) and Temporal entropy (TT), and its 3 dimotional combinations, St-information, Ts-locomotion and S=T, reproduction, whose interaction can be resumed in the function of exist¡ence, Max. SxT (s≈t)=C, we can deduce all the principles, laws, events and equations of all stiences from it. So we shall call Existential Algebra to the ÐST formalism of Generational Spacetime (ab.¬Æ), and do exactly that: deduce all equations and laws of stiences from 5D metrics.

We shall thus make a 1^{st} foray on existential algebra, showing how the ‘development’ of 5 Metrics give birth to the function of existence into its 3 ‘extremal points’ or ages , Max. S x T (3^{rd} age), Max. T x S (youth), S=T (maturity), defines the worldcycle of existence of all beings in its two directions, forwards and backwards.

But 5D metrics can be studied in more depth, roughly speaking in 4 sub-*equations, which are the foundations of the 43 great subdivisions of science:*

–* The physical equation of relativity, S=T, basis of all physical and mathematical stiences.*

*-The biological equation or function of existence, Max. S x T (achieved precisely when S=T), the basic equation of all biological drives and evolution***.**

*– And the equation of the mind: 0’-mind x ∞ Universe = Constant world that creates mental spaces… *which we will consider in the next paragraphs as we have defined space and time more properly.

*– T*hose equationS can be further unified, since the metric equation of multiple spacetime scales, SxT=K & the relative equation of dual motion/stillness in a single plane S=T that maximizes SxT=K (5×5>6×4) unify in 1 ‘existential’ equation: Max. ∑SxT=C±¡:∆±1, whose study is the field of Philosophy of stience and its new formalism, Existential Algebra (ab. ¬æ). * So after studying the 3 classic fields of science will return to those 5 Dimotions, SS, St, sT, ST, TT and study its entanglement and different properties and complementary oppositions, to start building the formal laws of existential algebra, the formalism of Generational space-time. that all stiences notably mathematics and logic mirror. *

**Galilean Px**: **S****ó****T: Relativity of space Forms= vs. time Motion ****5 Dimotions**

It is surprising that the humind’ science that more adamantly denies the existence of mental spaces and infinite monads-minds that create the order of reality is physics, which upholds as the only dimotions of time, sT, locomotion, external motion, or ‘translation in space’ and entropy, scattering external and internal motion, TT, just because physicists’ worldly profession has been to make weapons of entropy and transport machines, so as usual huminds project their inner mind in the whole Universe, and so physicists projected ballistics into reality. And they still do with the big-bang dogma studied in depth in other papers.

The *fundamental postulate of physics, relativity of motion and stillness is a direct consequence of the mental nature of spaces, in which a mental language stops, maps in simultaneity and represents reality as information with no motion. So we cannot distinguish mentally what moves and what is still as the purpose of a mind is to stop motion, as the pole of space that is in an eternal tug-of-war with the entropy of time. *

Galileo’s time and space Principle of Relativity is the fundamental conceptual thought behind the relationship between time=motion and space=form and how one can be converted into another: All what exists is made of space=form and time=motion. And yet physicists know that we cannot distinguish motion from form. That any being in motion from its point of view seems to be still and all other things moving around it. *This is the principle of Relativity of motion.*

Physicists then without much thought about that fascinating duality, went on to use mathematics to calculate the relative motion of each entity of reality respect to other system, which seems static from both points of view. This is called Galilean relativity, latter refined by Einstein’s relativity, and essentially is concerned with the mathematical calculus of what we shall call the 2nd Dimotion of time=change, locomotion. Fine, but *we are more interested on the duality of space=form and motion=time and its entangled relationships –the reasons why we do NOT see together motion and form, even if all systems have both.*

The conclusion is then rather obvious: one of the two parameters of reality is ‘hidden’ to perception; we either see motion or form, ‘waves or particles’ (quantum complementarity), distances and lines or points in motion (as in the night when fast cars in a picture appear as lines). So *physicists calculate only one when in fact we must assess the existence of 2; and since we cannot distinguish them, logically we must equal them. ‘Form=motion-function; space=time; S=T’. *

Relativity is a duality, S=T, *at the heart of every law of the Universe. But which one is the real element? Obviously time=motion. Space is a “Maya of the senses” – a slice of time motion. The ultimate substance is motion. Form is what a ‘still mind’, makes of that motion to ‘perceive’, information, forms-in-action. *

Since we see Earth still and flat but it is round and moving. Galileo’s profession was ballistics – the study of cannonballs motion. So he chose ONLY motion and lost the chance to start physics with a complex philosophical understanding of its S=T dual Principle of relativity, which Poincare defined latter clearly when he said that ‘we cannot distinguish motion from stillness’. An example is quantum/relativity duality. In detail quantum space has ‘dark energy’ because it has expansive motion that extends into a plane of space, but when seen at larger Planes without detail its entropic motion seems static space – a dual area of scattering length and width. So in the galaxy we see either dark energy motion or expanding space: T=S. A motion of time is equivalent to a dimension of space: Distance and motion cannot be distinguished so they must be taken as two side of the same being, a space=time Ðimotion (ab. Dimensional Motion):

**S= T; Dimension-Distance = Time-motion = ST Ðimotion**

*Why if the Earth moves in time, we see it as a still form in space? Because Reality is a constant game of infinite motions, but the mind focus in stillness those motions, and measures them at distances. For ‘huminds’ motion is relative to our systems of measure and perception, which are light-based; hence a fixed c-rod speed/distance. Reason why Einstein’s relativity postulates a maximal T:c-speed, measured as if observer and observable were still to each other (Constant S); which at our scale we ‘correct’ with Lorentz Transformations. *

*P*hysicists just substituted Earth’s still distances for motions. It took 300 years for Einstein to realize the relativity of motion and its measure made essentially time and space, motion and form two sides of the same coin. Still this realization was not explored philosophically and so it gave birth to a series of ill-understood dualities between ‘states of measure and form’ (particles, head gauging form, in-form-ation) and ‘states of motion’ (wave states).

It is then essential to grasp that motion and form co-exist as 2 different states depending on 5D scale and detail: Motions are perceived by minds that stop motion into form, into information, as distances. So if we see slow motion in the night, a car’s headlight seems a long distance line ‘still’ picture. But this means also that the 3 ‘Euclidean still dimensions’ must have motion; they are ‘bidimensional ST-holographic, topologic dimotions’. So we have 3 Space + 1 Time + 1 5^{th} dimension of Planes = 5 Dimensional motions. None of them is *a Dimension of pure spatial form or a pure time motion but a combination of both. Even if mentally we tend to reduce motion and focus on forms, all has motion=time, and form =space: this is the meaning of ‘spacetime’, the messing of both into 5 dimotions, the fundamental element of all realities.*

Relativity states ‘we cannot distinguish motion=time from position=space’. So all what exists is a composite of both, undistinguishable S=T, 5 ‘Dimensional motions’ (Ab. Dimotions), broken in infinite fractal, vital time space organisms, composed of topological Dimotions: height=information; length=locomotion; width=reproduction; form=social evolution of parts into wholes & entropy=dissolution of a whole into its parts in a lower scale of the fifth *dimension* (term we keep for the whole range of Planes of the Universe)**; ****whose study is both mathematical, the main science that studies how those 5 Dimotions entangle in simultaneous Space, connected to each other topological adjacent parts, which create superorganism and Logic; the main science of time that observes how those pentalogic, entangled superorganisms move and evolve, change in sequential relational time, living a life and death worldcycle.**

**Since there is nothing else but time & space, the 2 experimental ‘mirror-sciences’ of time and space become the most important to extract the ‘Disomorphic=laws’ of those 5 Dimotions that all systems have in common. Since while those Dimotions are broken, in vital organisms, separated by cyclical time membranes, they are the same.**

*In graph Galilean relativity was ill understood, as the true question about time-change is why ‘the mind sees space as a still continuum, when in detail is made of *smaller self-similar *quanta*, in motion. The paradox *defines mental spaces as still simplified views of the more complex whole. The 3 ¡logic paradoxes of space topology (closed in-form-ative curved-O vs. |-open, free entropic lineal forms), time-motion (stillness vs. motion) and ∆-scale, (continuous whole vs. discrete forms; single scale vs. multiple ones), are essential to the perception of a simplified ‘spatial mind universe’ in a single flat still plane vs. the full, more detailed complex picture in time, of a curved, discrete and moving Universe. Those paradoxes resume the 5 elements of reality, Space=form, time=motion, Planes and the mind that measures them, within its own entropic limits.*

What neither mathematicians nor physicists fully understand (though some inroads in abstract were made through the Noether’s concepts of symmetry) is that each stœp of a method of solution is not ‘gratuitous’; but must be grounded in a real property of the 5D ∆ST symmetries and conservation laws of the Universe, which are not so many – hence the repetition of methods. Specifically, the aforementioned 3 paradoxes between ∆+1 curved closed worldcycles, sum of lineal steps, which gives birth to the most used method of lineal approximations; the equivalence between Space and time, in all Stœps of dimotions, which gives birth to the method of separation of variables on differential equations and more broadly allows to move around relative space and time parameters in equations joined by an operand of ‘equivalence’ (≈ not =). And the 2 conservation laws of the Universe, conservation of those ‘beats’ of existence, S=T in relative present, eternal balance, justifying the equivalence operands. And conservation of the ‘volume of space-time’ of each plane of the Universe, by virtue of the 5D metric equation SxT=C, which justifies all the procedures regarding scales – solution of differential equations by separations of scales, renormalization procedures (Wilson), and harmonizes those scales allowing constant but balanced transfers of energy and information, St=Ts.

**Motion is reproduction of form in a lower scale. Bohm’s realism: quantum potentials.**

How a system moves in a crowded reality, where all is vital space-time? The answer that resolves Zeno’s & quantum complementarity paradoxes, is if *we do not move but reproduce our information, translated into a lower faster wave scale of the fifth dimension; as we reproduce our sound in faster electrons to telephone or nerve impulses into chemical dopamine to jump discontinuous neurons. So motion becomes scalar reproduction of form, and since all is a form of motion, all is reproduction, which is the definition of a mathematical fractal, a feed-back reproductive equation; 5D metrics, which become then the ‘function of existence’ whose goal is to reproduce the form of all systems – the simpler ones with maximal motion-translation in space, the complex ones with min. motion as a reproduction that emerges between Planes. And this gives birth to the worldcycle. Consider the case of quantum physics: *In the graph we see a particle reproduced in adjacent regions that fade away, and the result is the perception of a wave of motion. In Bohm’s realist model this reproduction happens in a lower plane of quantum potentials, where also entanglement happens, which is the ∆-4 scale which is v>c in 5D metrics (Min. S x Max. T = C), hence real. Motion is reproduction of form over such potential: the wave erases form into motion, the particle is a still state that gauges information entangled to other particle, fermion and boson, still to each other – despite the perception of relative motion in our scale – hence the information electrons share has always a c-constant speed.

Thus the Lorentz transformation eliminatesthe stop state of particles as we do in a movie eliminating the stop frame but if we were observing reality from the perspective of an atom, we would ‘stop’, entangle in the quantum potential, neutrino scale & so eliminate the spooky effects of ‘time dilation’ & ‘length’ contraction, from our perspective (but not of mass increase as it is a scalar effect). This is the ‘rational’ 5D explanation of both the c-constant of light and entanglement; as electronic beings perceive information in ‘stop position to each other’ and move in ‘wave state’. Motions are perceived by particles that stop motion into form, into information, as distances. So 4D relativity needs to be expanded to the scalar Universe beyond the c-speed light limit of the galaxy. Within this complex view, the models of Newton, Galileo and Einstein’s space-time correspond to the limit of 5D when we simplify all the worldcycles of time, we call life & death to a single mechanical clock, elongated to infinity & perceived in a single scale of space. Let us then deduce from those 2 equations the fundamental equation of reality:

Those 2 poles of reality are the first principles of any scientific inquire, prior to the languages of time-motion, logic and spatial forms – mathematics, that better mirrors ¡ts laws. Look around, everything you see is a form with inner or outer motions. Those are thus the 2 primary elements of reality; which mind languages perceive mostly by reducing the Planes of the fifth dimension and its motions to the minimal possible to fit it all in the mind ‘equation’: O-mind x ∞ Universe of formal motions = Mental world – reduced mirror of the Universe.

M*athematics that considers motion a ‘symmetric reproduction of in-form-ation (groups) once more turns out to be a better experimental science than physics.*

That motion happens as reproduction of form is essential for every phenomena of stience. A few mathematical cases:

– Reality will seem continuous and differentiable as a constant zig-zag of step and stops that seen in detail becomes a quantum tangential motion.

– Because reproduction is never perfect, the accumulation of statistical errors in each stœp will accumulate on the moving form till it kills it as the species slowly warps and wrinkles its synchronies. So we can define a Gaussian function of existence of reproduced populations, which are steps of the function of time-motions of the being.

-Reproduction of information will have a limit on the ‘bottom line’ speed of the scale in which the game starts, for the galatom the hc-scale of light speed and h-quanta.

-Reproduction encompasses the 5 dimotions of existence, becoming the more complex fundamental game of the fractal Universe to the point all can be explained with the reproductive function: Max. ∑ SxT(s=t) = C

** **

**The 5 dimotions of space-time as inner and outer motions.**

A more precise definition specially for physical sciences of the previous 5 ‘states’ of spacetime deals with the concept of internal and external motion (which has deep implications for relativity and so we will expand it in the paper ‘physics and space-time’. Any cyclical time-space organism, even the simple circle divides space-time into an inner and outer region (knot theory, membranes of all topological systems). We talk of internal and external motion and form in a system. As it is evident that external motion in a larger world is bigger than a being’s internal motion. So little motion means internal motion, t and a lot of motion external motion. Thus we define the 5 dimensional motions of space-time with more finesse. Whereas capital letters, S & T mean external and small letters t and s mean internal (as the internal part have less volume, are smaller than the outer world).

– **Ts** is external motion (T) and internal form (s): It is classic locomotion, where the internal system doesn’t move but it translates in the external world. This is what physicists mostly study, and they still wonder why when a material system moves, the ‘center of gravity’ seems fixed. But what internal part? The answer is the mind-brain; the system that perceives information; as the limbs/fields move a body over which a particle-head gauges that information:

**-St **is therefore information, where externally there is no motion (S) and internally the ‘mind-mapping’, that mirrors the outer world becomes a form-in-motion, in-form-ation, processed by the brain, particle, crystal image, cpu… It is a fascinating fact that we, the thinkers, spend so much time, still; so Leibniz could spend days seat on a chair thinking, that in sleep more brain activity happens and more hormonal languages develop.

**– ST **or st is then balanced internal-external motion vs. form, or energy, of which there are two varieties we ad, Ts-kinetic energy akin to locomotion, (mv^{2}/2; where as you can observe the ‘motion’ provided by v is squared, so it is more important than the mass/form) and St-potential energy, where the energy is store internally in the different tensions and cyclical patterns of the system, but externally it is all position. Energy then is the balanced sum of both, St+Ts=ST and it is the essential law of physics that a system constantly balances both (SMH motions; Lagrangian actions which are its difference tending to zero (St-Ts=0, St=Ts). Existential energy is thus the mother of all battles, and we could state that in a ‘given plane of exist¡ence’, energy never dies, but constantly transforms between the informative, potential energy state and the kinetic, locomotion states.

This leave us the 2 limits of pure form,** SS, Ss**, which correspond to ‘seeds’ and ‘mind-languages’ with null motion, (which following our formalism should be written Ss, but my lazy ‘fingers’ often repeat 2 SS, sorry)

And Tt (written TT often… lazy f…), which we shall call entropy or scattering or death or disorder, as when a system moves internally and externally it is really disorganizing its structure and ultimately as in expansive big-bangs of mass, or death processes of corruption, all parts go away and the system dies.

For example war is the Tt entropic state of a superorganism of history, a social mass of ∆+1 humans in time, when there are external motion (armies invading enemy nations) and internal motion (troops invading internally the country), which ends is the massacre of people. At individual level today, children, as humanity enters its age of entropy as it becomes obsolete, degraded and likely soon substituted by machines and chips and robots, *is clearly suffering an entropic age, sometimes called A.D.D. *

The child has maximal external motion (as we shall see due to the correspondence of STates and ST-ages of life) but today has also maximal internal ADD motion in the brain, filled with virtual hypnotic fiction bull$hit, from mental machines that are disordering us in what we call the Neo-paleolithic age of History. For physical systems Tt is also death as the system explodes, scatters and its internal parts move externally. So yes, those 5 Dimotions apply from the simplest particle to the more complex superorganism of history.

In physics this prolegomena will become then fundamental to understand the principles of Relativity, the fact that we cannot distinguish motion from form, S for T and both therefore must be put together in ‘dimotions’. So we shall see that the fundamental equation of the Universe is SóT (ab.S=T). As there is a constant dynamic transference of Motion and Form between the inner and outer world, which allows a system *to transit between the 5 Dimotions of existence. *

**Quantum motions vs. continuous derivatives.**

We shall introduce now one of the ‘mantra themes’ of all the introductions to 5D theory, which we will expand greatly as we revise those texts, as it is the ‘central piece’ of any physical theory of motion.

With physicists definition of time as locomotion (Ts) we can only understand very simple spatial motions with NO inner change on them (s-patial form in Ts is fixed). This is reflected in the use of ‘present=instant derivatives’ which do not admit ‘peaks’ of changes of STages for physical systems. Since the calculus of locomotions uses ‘instants of present=finitesimal derivatives’ that don’t apply to ‘change of internal parameters’, to inner motions necessary to change STages of life, where the system switches from external to internal motion; as it stop and goes through its existence. So for example, the change from youth to maturity, Ts>ST means a change of ‘s’ to S, from internal no-motion to internal change in information that brings up the reproduction of the system.

This is the key to the whole structure of change in the Universe, which happens with the interplay of internal and external motion and form. I.e. when we act, we first ‘stay put’ as form, but our mind is ‘thinking’, having internal motion, and so we think what we shall act, and then we act, and our mind is ‘stopping’ it has no internal motion. So the fundamental event of motion v. thought is St (internal motion, external form) < = > Ts (internal form, external motion). In those systems a *continuous calculus approach fails totally to understand the quantum stop < go process.*

*In the stop process information is processed internally, as ‘small caps’. motion, ‘t’; collapsed as a a wave (physical systems), processed in a dream state (life organisms), thought of (humans); in the go process, the information is repeated by a quantum wave; reproduced externally by a mitosis cell, projected by a mind, reproduced by a seed… The examples are many as those stœps ARE the main whys of the fractal Universe, which is a reproductive organism. *

A key to all of those events are the ‘Stœps’; changes of states, quantum by quantum of Space ó motion transformations in the inner and outer regions of the being, in the different scales of the Universe.

There is not a personal god indeed but if there were one designing such perfect Universe I would like to share his thoughts. You have always a ‘cyclical time membrane’ that separates an inner and outer world. One of them might be in motion, one might be still, the combinations of those possibilities and all its changes, either one by one, or two by two, in infinite stœps of infinite beings describes almost all realities. Some are more important than others; some sequences of changes are fundamental as the sequence of the worldcycle.

And then there is the way the mind selects certain states and ignores others, like in a movie picture. Consider two of them that are inverse and will come up many times, and when we increase in a future review the complexity of those physical texts with advanced maths will show the full beauty of the mathematical mirror of physic:

– St: The perception of motion as stillness: In this ‘series’ the mind does NOT perceive the motion STage at all, only the stop State, and as such it explains why the earth is moving but the mind sees it still.

– Ts: Perception of stillness as motion: in this view, the mind doesn’t perceive the state of wave-motion, but only the series of still, informative positions. It explains why we see motion on systems that are still, as frames of a movie.

We either perceive the wave state or the particle, still state but NOT both at the same time, because then we would see a jerky Universe.

In the mathematical mirror it means *we can have either a topological, geometric analysis or a calculus, changing analysis; we can define reality in terms of ‘continuous functions’ or we can consider discrete approaches.*

*But what is a deeper truth? Not the continuous but the discrete, stop and go approach as we show in our paper on 5D mathematics and time (Calculus). *

** **

**THE INFINITE MINDS OF THE UNIVERSE.**

Once the S=T principle of motion and form is clarified, we can consider the essential origin of all ordered structures of reality, the will of existence and cause of its scalar smaller planes that mirror larger ones – the different minds of informative particles-heads, with the spherical form of maximal package of information supported on ‘top’ of a body-wave, where there is maximal projective geometry to perceive a larger territory, from photons moving above the wave of light or ahead of it, to the black holes with maximal height of the galaxy, to the heads of life to the satellites of the metal-earth. Minds are mirrors that perceive in its inner ‘still simultaneous language’, the e-motions of time converted into informations of space, in the eternal dialectic between fractal points with a volume of still linguistic perception, mapping its local Universe and flows of e-motional time with its vital sensations:

*Upper graph, human egos submitted to the mind paradox, think languages (words in Abrahamic, creationist religions, numbers in creationist science), known only by man and ‘God’ a priori, create a posteriori the Universe (Copenhagen interpretation). The opposite is truth:* a mind exists in all systems in which time stops to form space. In galaxies happens in relativity equations in black holes, its mind. In thermodynamic physics in the eye of an Eddie. In quantum physics in the center of an atom, or charge. Without linguistic minds that order by reflecting its smaller mind into its local territory reality would not exist. The only way to create fractals is through mirror images.* *

In all Planes of stience minds fix time motions into spatial, linguistic formal mappings that reduce the whole with its synoptic language to fit in a particle-head that acts in its world-territory after gauging information.

Thus we define ‘Maxwellian’s demons’ of local order in all Planes – physical minds as the infinitesimal points that create order in physical systems with the same Disomorphic laws that all others do in more complex Planes. As each mind orders as a linguistic god a territory around itself, its fractal body and entropic world.

*So the creation of Planes of reality is a simple game, in which a point mind ‘reduces reality’ to its infinitesimal form and then projects into its local territory of order,* which will reflect at scale, the larger whole or world, which the linguistic image reduced and then enlarged back into its territorial form.

*The paradoxes of Relativity, discontinuity, parts V. wholes, Planes are all related to the reductionist nature of minds that bias reality. * Minds reduce dimensions to the relevant ones, *eliminating all dark spaces*: continuity is the result.

**Trilogic >Pentalogic on worldcycles****Ù****supœrganisms. Synchronicity of time cycles and emergence of scalar spaces**

In the entangled pentalogic Universe, each of its 5 elements, space, time, Planes, entropic limits & linguistic minds

that measure it to perform dimotions of survival for the whole; each element is related to the others. So in time we do have 3 relative Planes of time speed according to 5D metric (SxT=c), meaning our cells are much faster (or the particles in an atom) and store the information of the larger scale, which traces slower time cycles than the whole. And this is the phenomena of *synchronicity, the detailed whys that make possible the ‘persistence’ of a superorganism of space as it moves in time, since each of the infinite mind-points of all its Planes needs the other, and it is entangled with its inner parts and larger wholes to keep its ‘exist¡ence’ going. *So essential to the complex structure of reality are the ‘synchronicities’ between the different cycles of space-time that chain them in symbiosis. I.e. a cell reproduces every day, in its faster smaller plane of space-time because the larger whole feeds every day.

Pentalogic implies NOT only entanglement in a scale but as all superorganisms have 3 Planes, made of inner parts, performing functions in a larger world, we need to introduce 3 ‘levels’ of analysis of 5D:

**– ∆-1: Smaller Action=Dimotion level**, which will correspond in mathematics to ‘s=points ó t-numbers’, or derivatives in calculus, steps in fractal mathematics, arithmetic operations in ¬Algebra and so on.

**-∆º: The main S=T symmetry** **level of organisms & worldcycles **expressed in mathematics by complex geometric forms and functions and functionals (for superorganisms across Planes). So we define any system in time, by its life and death cycle or worldcycle (taken from 4D physics’ worldlines, as a height dimension of information makes it a worldcycle. Whereas any scalar super organism’s existence can be defined as a travel through 3 Planes of the fifth dimension. As all systems are born as a seminal, smaller form, in the ∆-1 lower scale, grow fast with its faster time speed, emerge in the ∆º body scale and are part with its head/particle of a larger social world, in the ∆+1 scale.

– Finally the ¬∆@st absolute level of reality as beings made of ‘Dust of space-time’, the absolute 5 elements of which we are all made, Planes, space, time, minds and entropic limits, whose general laws study 5 subdisciplines of logic parallel to the 5 main subdisciplines of maths, which can be further compressed into the S=T duality of geometry=¬Algebra or in pentalogic the duality of language≈information& entropy≈energy.

** Emergence **is a ‘fascinating’ process hardly understood in ænthropic science, easily defined in ‘vital ¬E topology as

*the creation of a closed plane of exist¡ence or membrain that entangles the different parts, sets of fractal points or networks-waves into a new whole that takes over the parts, organized by 3 physiological networks in synchronicity.*The emergence of a new plane of exist¡ence in space-time thus requires the creation of a whole, which ‘covers’ with its outer membrain those parts and becomes for an external observer the only perceivable form, call it a pi orbital in a molecular ring, a palingenetic fetus in a cellular placenta, an army or entropic network of a social gathering of humans, called nation, where the individuality is submerged into the synchronicity of the whole.

*Max. ∑SxT = c metric; S=T ‘balances’, ∑st¡-1>ST¡ ‘emergence’ & ∑ å¡-1 = Å¡* (synchronicity of actions between scale) become then fundamental equations of Existential ¬Ælgebra, which is the name we choose for the ÐST equivalent to mathematical ¬Algebra, whose purpose is to express in a formal logic language all the events and forms of worldcycles and superorganisms, according to the laws of space-time and its entropic limits. Let us then introduce even if it is not so easy to translate into humind’s ¬Algebra the key concepts of Existential ¬Ælgebra, as we will often refer to them in our entanglement of mathematical ¬Algebra and the laws of Generational space-time.

Existential algebra is a much wider discipline that scalar, non-Aristrotelian algebra. As it deals with the function of existence in all its possible sub-function, sub-equations and forms.

*Besides its space-time analysis of forms and functions, vital topology sets the basis for a scalar* numerical version of wholes and its parts, dissolved in entropic death, which we formalize with Existential ¬Ælgebra, as regular geometries of points – polytopes – are the spatial version of temporal, scalar numbers.

*So we develop in Existential ¬Ælgebra a formalism of the 5 Dimotions of reality with operands similar to those used in ¬Algebra, to mimic those dimotions, «,‹,* =,›,», for entropy, locomotion, reproduction, information and social evolution. And so we can write ‘existential equations’ that will describe flows of dimotional ‘stœps’ that change sequentially a form through its events in time, where the vital topology of each Dimotion is ‘integrated’ within those symbols. In this manner the 2 branches of Non-Æ mathematics vital topology and Non-Aristotelian ¬Algebra becomes ‘Existential ¬Ælgebra’ –the analysis of flows of dimotions=actions of T.œs in its worldcycles of existence. Those sequences can then be studied as templates of all T.œs in all Planes, which will follow them to complete its survival cycles. And so Existential ¬Ælgebra has implicit both vital topology and non-Aristotelian ¬Algebra. So we define in Existential ¬Ælgebra with simple T, S and 5 «,‹,≈,›,» the main events of space-time of the Universe:

**1D: t›S: angular cyclical motions of information (Ab. St, §ð):** the minimal ‘geometry’ of reality, a spherical particle/head or fractal point, the geometry that stores maximal form in minimal space, hence suited for ‘organic functions’ of gauging, storing and perceiving information (particles, heads).

**2D: s‹T: Lineal Locomotions***,* **(Ab.sT.$T) *** which will move through its* lineal limbs/fields the system, as the line is the shortest distance between two points… towards a…

**3D: S≈T:** **Fields of vital Energy (Ab. ST):** with its hyperbolic body-waves that iterate the forms of both the spherical particle/heads and lineal limbs/fields; as the hyperbolic topology combines the other two, so it can generate them, in the same manner Energy adds as the third conserved space-time quantity the lineal and cyclical momentum of 1 and 2D. To which we must add the 2 ‘scalar’ Dimotions of:

**5D: entropy («, TT. ∂S)** whereas motion is dual internal dissolving the information of the being and external, scattering its parts, hence we use an « ¬Æ symbol; so the system explodes into its ∆-1 parts: ∆«∆-1 (death). **4D:** **organic evolution (», SS, ∫T)** of parts into still locked simultaneous ‘linguistic seeds or mind forms’∆-1»∆

So the 3 conserved substances of reality become organic bidimensional topologies – flat motion in space, cyclical time membranes and the vital 3D energy within them. We then observe its ∞ variety of combinations as topologic ternary species, whereas the 3 perpendicular ‘lineal’ dimensions are a simplification of those organic functions: the height dimension enhances the ‘perception of information’ by O-heads/particles place above in the point of maximal projective geometry; the dimension of ‘length’ maximizes locomotion; physicists’ only time motion, and the width dimension maximizes reproduction and storage of energy.

**LOGIC SYSTEMS. FROM ‘EGO-CENTERED’ MONO-LOGIC TO DODECALOGIC.**

The growth of complexity in the entangled Universe goes from ceteris paribus, one-dimensional, monologic aanalysis so common among human beings to the more complex (Do)decalogic systems of perception of reality in its 3 worldcycles. As humans are basically monologic we do need to make somewhat a harsh critique of our modes of thought, because with the same data, our one-dimensional time-mind brings most of the errors of science, due to our egocy and confusion of our mind’s view with the whole Universe.

**Mental mirrors are neither false or truth, but useful to its type of subjective minds.**

As Riemann’s understood “on the hypotheses which underlie geometry” space is a mental construction. He destroyed the ‘realism’ of geometric space. But he saves ‘concepts’ common to all geometries of the mind that were of ¡logic nature and we shall regain to fully grasp the parallelism of Spatial geometries with the ¡logic laws of the fractal space and cyclical pentalogic time of the Universe. So we can establish geometry as a mirror of a larger, more general theory, ‘i-logic space-time’, whereas the concepts of geometry, proper of ilogic can then carried into ¬Algebra, analysis etc. Since concepts such as distance lost with Riemann its ‘physical condition’ to become a logic property of similarity, as we find in verbal thought the same term to designate precisely this ‘larger general st-quality’ of bio-logic nature, as when we say, I have distanced myself from my friend – meaning we have become different in tastes and opinions. So geometric concepts such as dimension, topological form, closed and open spaces, become logic, vital, relational and social concepts; general properties of i-logic spacetime, with applications to different sciences.

W*hat kind of mental spaces there are, reflected in its Geometries and phase spaces, both in huminds and the Universe at large? Many, as long as they provide enough information for a T.œ or part of it to survive.*

**Total number of Dimensions=Parameters of space-time: type of logic minds.**

The mind equation, *O-mind x ∞ Universe = Constant**world* reduces reality to a degree of simplicity, which can be extreme *and still make sense for the being in exist¡ence, as to be able to survive if it does not behave like a childish noisy cub entering ‘future territories’ of higher species. *

* In that sense, visual Huminds are simple enough to survive with a single dimension of lineal time and a single scale of space, from its subjective view, as they only need to care for the scalar size in which they perform survival actions.*

In a fractal Universe in which membranes isolate territorial vital energy from reality the needs of a mind can be ‘reduced’ to the basic actions of feeding and perceiving only its type of energy and gauging basic information to that aim. So we can rephrase the question: what is the simplest possible number of dimensional motions required to ‘trace’ a world cycle, which is complete and hence accounts for an ‘exist¡ence’? The graph above responds in physical terms: A lineal simple harmonic motion shown both in real space with one dimension, in which the SHM performs a complete cycle of existence. It has a balanced middle point, which it abandoned to accelerate=move forwards in time, to a peak of ‘distance’ where it will achieve its maximal potential energy, to revert its world cycle back to the point of equilibrium. A single dimotion thus suffices to reflect a world cycle, further reducing the human ‘1 time motion and 3 space dimensions, to 1 time motion and 1 space dimension’. Indeed SHM in fact serves as a mathematical model for most physical motions, including the motion of a pendulum, (time motions), electronic motions (perceptive mappings of ‘still light’) and molecular vibration (spherical harmonics, the ‘energy’ of all chemical and life processes). It is all pervading in Nature, and its single resonant frequency provides a basis for the characterization of more complicated motions through the techniques of Fourier analysis, which can decompose any complex wave motion in SHM resonances and its harmonies. So information is compressed ‘¡logically’ to the extreme minimal of a 1^{st} Dimensionality and then expand it through the laws of ∆st as the immanent ‘game of existence’ *will develop its coded seed*, if it falls into an energy-rich placental space.

*,A**ll worlds perceived by a 0 mind *are phase spaces not reality itself. Moreover paradoxically, the stronger body-wave predators can survive in entropic, Darwinian encounters just with lineal hunting time motions as indeed ænthropic human warriors have done, simplifying reality from our modest heights of dual Logic (Taoism, sexual age of the Goddess), protected by a ‘hard membrane’ only interested in the placental energy region of Gaia, they are killing without remorse as they don’t see it alive:

Systems of thought start with simple ego-ist, lineal top predators (Germanic warriors, future AI terminators), lineal felines, proteins, who are simpler fields of destruction, with a complete lack of interest for reality beyond their ego paradox, their entropic, destructive worldview – ænthropic men, being the paradigm :

**@ristotelian, selfish ‘MONOLOGIC’:**

1 single space plane (with 3 dimensions) and 1 single lineal time dimension, A->B, or ‘Humind Logic’. Human scientists since Galileo seem to live in a single time-space-scale dimension. If we add cyclical time, a new dimension of height-information appears, but the one-dimensional ‘ant’ seems confused. It doesn’t matter to her though, it will keep walking as it has a strong body. *Physicists *were entropic weapon makers so they kept going for 400 years after simplifying cyclical time and keep going, nothing has so far stopped them. Till of course, its entropic weapons detach as living robotic systems or cosmic bombs and nuke us all. But in the meantime they have managed to convince humanity that galaxies DO NOT exist (as they are the inverse gravitational arrow of informative time that balance the entropic big-bang expansion of vacuum space). So really think twice when you believe something that seems reasoned. The trick IS ALWAYS THE SAME, half-truths; hiding from PERCEPTION, knowledge. As Spaniards said: ‘eyes that don’t see heart that don’t feel’. So extremely intelligent people believe in verbal or digital creationism as they don’t read more than the bible or numbers; and almost every astrophysicist I have met doesn’t realize the retarded blunder, and once he gets it, he feels so silly about ‘missing all the galaxies’, he will somewhat try to justify a theory no longer makes any sense. Ænthropic big time models of mankind though deserve an entire article I should sometime post at academia.edu as they are NOT the exception but the rule of human cosmogony & worldview.

But for an advanced mind, monologic doesn’t work as a serious mirror of reality, as it brings chaos, uncertainty and subjective creationist languages. So we will keep adding new dimensions: 1>2>3>5>12 with complementary logic statements to reach a more realist view of the scalar fractal organic complex Universe’s logic and the interaction between TT-∞ motions and SS-Minds/Seeds. Our aim is to resolve the game of existence, which starts in a seed projecting its ‘mental view to order reality’, with a catch – if that mental view is simple or too distorted it won’t imprint reality and the system will collapse/die earlier. As the whole preserves its own complexity in the ‘best of all possible worlds’ (Leibniz) by Darwinian extinction, which *is exactly what ænthropic man is about to *achieve with its monologic view of the world.

**Duality, Dilogic, Paradoxical Eastern** logic: a Yin-yang, Vishnu-Shiva, Entropy-Information duality.

**Many human cultures achieved duality, notably all eastern philosophies & the German-Russian schools of dialectic by influence from them, both Genetic and Memetic. In this view the fractal point (it is always a good mind exercise to relate Timespace Logic (Ts-formal science or sTience) with Spacetime Mathematics (St-formal science or Stience), or 1 ^{st}-5^{th} Postulate of Non-E Geometry becomes the line-wave (2^{nd} Postulate); and starts a dual positive or negative communication (merging in reproductive acts where one pole tends to be St and the other sT, to find an S+s=T+t ‘offspring’). This happens in all Planes. The neutrino theory of light, the left-right handed symmetries, the mirror reproduction, the DNA+RNA combo, it is called Gender (+ Duality) or Predator/Prey relationship (negative duality). It is related to the angle of perpendicularity, which if breaking=tearing in topological jargon the membrane NOT only deforming it and seeding it will be top predator/prey. Further on, points can approach each other with different angles, once they abandon a line World. All this I know ‘sounds Chinese’, too abstract. But ¡logic is homologic, so when we reduce it to examples, it illuminates it all. Angles of perpendicularity and parallelism will manifest in polygons, chemistry, scattering laws of physics, tables of conversation and fuking – to state a few examples. Pentalogic starts the game of creation, and in its top predator form, destroys it, so it might be considered an expansion of the monologic, in which instead of being a pole of black hole absorption of information, the Black hole moves to chase. **

**RECAP. ** The humind (ab. Human mind) reduces the whole reality to what fits in its infinitesimal space and reduces all the Planes to the human scale and reduces all time clocks equalized by our synchronous second (limb steps, heart beat and eye-glimpse/thought). Yet reality is far more complex. And if you are not humble enough to realize that even the highest huminds have simplified the Universe to fit it into our mind, and simplified its clocks of time to the entropic, lineal view of our visual-dominant mind, and projected its entropic, violent lineal view, into his grand theories of the Universe (from entropic big-bangs, to dog-eat-dog Darwinism to Euclidean maths of points with no breath, no internal parts) you will not be able to expand your view of reality with the huge upgrade of these papers. *So to learn ÐST you need to forget your egos and learned ænthropic theories based in the reductionism of the mind (with man as its anthropic center, and our lineal entropic view of time-motion as the only arrow of the future).*

**TRILOGIC**. All species are parts made to the image& likeness of the game of exist¡ence God=Mind of reality; hence sharing its ‘organic=scalar’, ‘topologic= mathematic’, causal=temporal and linguistic=mental properties, as mind-mirrors perceive the larger world in its smaller brain and synoptic languages, *which shares the Universal Grammar of Existential algebra to be able to* code an image of the whole in still geometric space and logic languages of max. synoptic power, hence Min. S x Max. Cyclic Information, which *starts a biologic radiation evolved socially, ∆º ∑>∆ _{1}, into a larger whole. So 2 languages *¡logic mathematics concerned with spatial geometry and temporal algebra, become the most experimental, perfect mirror of the 5D Universe. As minds use them to keep imprinting entropy=motion with patterns of cyclical information to create similar energy beings – the 3 ‘conserved’ time ages and space forms that ensemble into fractal supœrganisms, between its 4Dimotion of generation in a lower ∆-1 plane that emerges in an ∆+1 social world to die back in a dissolving explosion of scattering entropy, back to ∆-1 in the moment of death when all motion is converted into form. So all traces 3 relative worldcycles, the first ‘placental worldcycle’ as it evolve in the relative safe ‘womb’ of a maternal system, the life cycle, as it tries to survive in an outer ecosystem, as it forms part of a larger super organism living also a 3

^{rd}worldcycle. Finally Dimotions are perceived as e-motions at the body=wave level in conflict & symbiosis with still mental languages. So all verbal sentences reduce to:

Subject (Fractal point) < Verb (Dimotion=action of space-time) > Object (entropic energy of subject)

F(x): dependent parameter of space) <= > (S=t operand) G(Y): parameter of time

Blue (color of in-form-ation; space) <Green: color of reproduction) > Red (color of energy-motion)

So trilogic resumes the principle of conservation of time in a single plane:

‘All what exists is a ‘form’ that trans-form’ into entropy≈energy back and forth ad eternal SóT, which is the dynamic fractal generator of space-time events, and principle of energy conservation enlarged to its 2 poles whereas the wordings for ‘entropy’. ‘energy’, ‘form’ and ‘information’ require a bit more of understanding before we can define them more precisely.

**TETRALOGIC** is very common as it is the positive view of reality. It considers the ∆@st elements without the negation of it all – entropy, the fifth Dimotion. As most systems only code from the selfish perspective of the @-mind trying to expand its territorial space and time duration in all its Planes, and abhor death, selfish monologic & tetralogic are two steps of the same self-centered dialog. The square and the cube are also self-centered polygons/hedrons. Its motion becomes a flat, lineal reproduction. So they are ‘always positive’ in its codes and intentions, provoking entropy in other systems as they absorb its energy (but not acknowledging it). So for example there are 4 numbers to code genes, 4 quantum numbers, etc**.**

**Trilogic and pentalogic however are not aware of a t.œs** **limits in time and space (entropic death, membrains, outer world) and its SS-mind, seed, where resides the will to develop its program of time, through the…**

**PENTALOGIC** describes the 5 Dimotions of reality at the lower scale of actions, the organic scale that considers the 3±¡ topologies=ages and ∆±1 planes in which super organisms trace its worldcycles. And the absolute scale that describe ¬∆@st of times-space in its entanglement between ‘entropy:¬, Space, Time, ∆-Planes and @-minds. Its easiest form Pentalogic is thus very common; from cellular variations of earlier life, to the 5 vowel systems evolved from 1 A Caucasian languages (pure entropic energy), through the a, I, u ternary logic of earlier Semitic to the pentalogic of Spanish or Japanese. It occurs in a musical score and all the dimotional mirrors we explain.

** ‘DODECALOGIC’ is ** best to describe the system both in space and worldcycle through 3×3 scalar elements guided by a mind, the 10th ‘point’ that controls internally through physiological networks, its ∆-1 scale, feels as an individual in the ∆º mind and becomes a 1-point of the ∆+1 world=plane of exist¡ence, in which it emerges.

**There are in that sense from ‘9 to 12’ according to detail several available ‘sets of fundamental properties’ and stœps for such description, but we shall not consider a dodecalogic scale a la ‘Schonberg’ for any of the stiences study in those papers save time itself. As it would be too complex and to a certain extent is a matter of choosing the key steps in the process (there are in the Universe indeed multiple variations around the key magic number, 10 ^{9-12} which represents the maximal complexity in most systems of nature.**

So we define any system in time, by its life and death cycle which we shall call a **worldcycle** (taken from 4D physics, worldline, which now has a new dimension of Time-information becoming a worldcycle. Since any scalar super organism’s existence can easily be defined as a travel through the 3 scales of the fifth dimension, as all systems are born as a seminal, smaller form, in the ∆-1 lower scale, grow fast with its faster time speed, emerge in the ∆º body scale and are part with its head/particle of a larger social world, in the ∆+1 scale.

**Building up reality in 5 logic, geometric, ¬algebraic space-time dimotions.**

The key concept to understand reality departing from its simplest fractal points then becomes ‘entanglement’ as iterative acts reproduce new dimensional motions of complexity mirrored by more complex operands, geometric forms, e-motions, actions of existence and organs of supœrganisms, till completing a being with pentalogic elements, that all languages will mirror with 3±¡=5 forms, from 3±¡ vowels to 5 type of cells, 5 fingers, musical pentagrams, e-motions, operands, geometric postulates, 3±¡ ages of life, 3±¡ perceived scales each one connected to an action of survival, etc. Entanglement growth in complexity through new dimensional planes; synchronicity of motions between the co-existing scales of time and sequential worldcycles form then the basis of existential algebra, and determinism. Let us explore them in more detail.

The graphs compare different degrees of growing complexity in systems from 1 to 5 Dimotions, represented from left to right in ÐST, Non-E topology, ¬Algebra and ilogic ¬ælgebra that completes the translation of all its laws by referring them to the parallel languages-mirrors of the fractal Universe, where the 5 Dimotions of existence, 5 Postulates of ¬E and 5 operands of ¬A entangle together as mirrors of ÐST laws.

We consider the fundamental formal mirrors huminds use to reflect the incremental complexity of the Universe’s Dimotions, as *they are a chain of increasing complexity where each more complex dimotion requires the previous one to become fulfilled. So geometry, ¬Algebra and logic do reflect those 5 Dimotions:*

The same concept of a growing entangled complexity works for any language mirror of ÐST laws – from logic to ¬Algebraic operands and the dimotions they reflect; from musical tonal systems to vowels, all evolving from 1 to 2, 3, 4, 5 and 9 to 12 systems.

Those are the 4 main languages that matter to understand, the most important of all languages existential ¬Ælgebra, the formal model of worldcycles and superorganisms and its possible combinations of the 5 Dimotions of reality, with the symbols of those 5 Dimotions in space (ST variations) and time: ‹›≈»«

The laws of Existential ¬Ælgebra and the partial functions of the generator, connect with the concept of a reductionist mind, which absorbs only the energy it needs to survive. In brief, a huge field of study in the design of mental systems and mental spaces of geometry is the understanding of the different ‘real systems’ of mind spaces constructed with an increasing number of parameters and dimotions, which make its mental geometry more complex.

And so we talk of a growing scale of complex mind-spaces, which we shall ‘entangle’ in the next chapter with the different ‘degrees’ of social evolution of geometric figures, from self-centered points that sponsor a simple lineal mono-logic motion in time, to the more complex topological organisms/non-Euclidean planes whose structure ranges from simultaneous pentalogic geometries to the more complex Dodecaplex ‘polytopes’ of simultaneous, equal 600 elements… likely the more complex illogic structures of the Universe.

*Thus the more complex expression of* the function of existience, Max.∑SxT=C (s=t), which ultimately is the reproductive function *of the fractal Universe, reason why we call it the fractal generator, in any of its multiple expressions* includes all the 5 Dimotions It is also a survival biologic function, because it implies to provide lineal motion to ‘fields-limbs’, absorb energy to reproduce our bodies-waves, and information to guide our motions with particle-heads. *So reality is a ‘struggle’ for existence as systems reproduce its Ts-fields-limbs of motion, Si=Te body-waves of energy and St-particles-heads of information. But as all T.œs are fractal, broken, its growth has a limit on the fight with other systems, who try to move and reproduce. In terms of pure T-motion and pure S-form, we consider then the whole of maximal* time=motion= entropy or TT and Max. space=form=stillness or SS the 2 limiting Dimotions for any 3 ensemble Ts<ST>St-system.

*Those functions of the type S**ó**T are dynamic feed back ‘beats’ which can represent in static space the constant interplay between the informative pole (head/particle) and the limb/fields of locomotion for entangled, lasting organisms, or might represent a Darwinian event between a point of still form and an entropic field the fractal point absorbs in a Darwinian event. *

*So how can we establish the proper rules to define all those ‘variations’?*

*Amazingly enough with the clarity of the non-Euclidean postulates of geometry and its laws of relative congruence that define which type of outcome an encounter between two T.œs will be, which we will soon develop as we expand the meaning of non-Euclidean geometry and add vital topology to the mix.*

Thus a full description of any organism locate its 5 elements, considering them from the perspective of all others, as in the ‘Rashomon effect’ (the film in which to get the full truth of an event 5 points of view were required). Let us do so, for time in its its 3 scales of duration (SxT=C), to predict deep time organic worldcycles of evolution.

**Pentalogic Time-Space Syntax**

¡-Logic the most important stience of them all, as the ‘stience that connects the 5 elements of reality, space, time, scalar planes, minds and entropic limits, in its *multiple entangled perspectives.*

*We thus use the Pentagon with all the 5 elements connected between them, as the representation of the mind of the Universe, and will try to consider it ‘simple enough’ for the ‘handicapped’ Nitrolife species, called Humind, in its utter simplicity to accept that his mirror of reality is only monologic and he should be extremely humble if he wants to expand in awe, its view of the Universe that is today to put it mildly retarded, even in those so-called genius, our tribal idols from Newton to Einstein. And I know this is anathema, but as the most advanced nitrolife mind in this blue dot, frankly I feel also retarded compared to what the Universe stores of sheer ‘complexity and intelligence’ through 4 key concepts:*

*– Emergence in Scales of Complexity; that is, the Dimensional growth from simplex points to waves/networks, planes/organisms; diversified by congruence, limited by entropy, opened to a larger world, holding a smaller, ∆-1 world in itself. *

*– Time Synchronicity of different clock speeds that magically by symbiosis ‘meet’ at certain time and space to share energy and information. As we need to consider the ‘scales’ of time, from fast actions to Deep Time worlds.*

*– Space Entanglement; which combines those 5 elements mostly in dualities of complementary or inverse comparable pairs according to congruence, and further on make them Trinitarian systems of sT<ST>St limbs/fields-body/waves-particle/heads; and entangle them through 3 scales/planes and sequentially through 3 ages. And finally into pentalogic forms. And again we need to describe the 3 scalar planes of co-existing parts.*

*– Hierarchies, social classes *between parts, S«»T arrows up&down 5D of in-form-ation v. energy, topologies etc.

**– Simultaneity** is a principle closely related to the previous ones and defines the true intelligence of the Universe and all its local species that are able to co-exist in 3 planes, a relative ∝ number of time cycles and space topologies do act together as a whole. And for that to happen synchronicity, entangle and hierarchical structures work together to form the whole. Mozart is said, was different from all other composers because it had a view of the entire symphony within its mind, before it started to write it. Consciousness and emergence arises from the ‘fitting’ of all parts into the simultaneous puzzle, which then will smoothly sail away. Today parallel computers work on simultaneity, even if *its algorithms done by huminds are still lineal (and we won’t upgrade them – better if they remain so). To achieve that goal 3 other elements contribute:*

*– Still Languages of the mind *that mirror realities in smaller images, which they reproduce at scale, being ultimately the cause of the fractal structure of reality. It is the…

* – ***Fractal Principle: **Functions and actions of space-time repeat through all scales (regardless of distortions in the topological varieties cause by the different mediums and multiplication of forms in lower scales through hyperbolic 5D network geometries and vice versa, elliptic synoptic geometries for the slower, larger plane of the ∆+1 mind/world.

*– Mirror Symmetries between Si=Te, and SxT=∆±¡, dual and trinity and pentalogic elements. *That is, when we talk of space, it is an entangled space connected to its time-state, extended through 3 scales, perceived as such by a static mind, limited by entropic dissolution borders. Space thus cannot be described only as space, but through the entangled mirror symmetries with the other elements. In the pentagram each element, is not an isolated T, S, etc. but the whole pentagram, only that perceived from one of its 5 intervals. In praxis we shall very often do a Trinitarian analysis of ∆ST, the most objective connected parts, introducing entropic limits of the ‘domain’ of the function of exist¡ence, if we were to use the jargon of existential algebra; and hardly going out of the humind’s basic language, verbal thought, not even introducing the ‘sacred language of mathematics’, though its connection with ÐST will be laid down in two papers one on space geometry and the other on time algebra.

*– Entropic limits**; that reduce the information a system holds, its time duration and its vital space, according to:*

*– Ego limits***,** the negative side of self-centered mind mirrors that distort reality to fit the mind and introduce always an egocy paradox. And again at least 3 languages for the 3 physiological networks of the being are needed – you do have visual eyes for locomotion and entropic nose and mouth and verbal ears… To guide us then we need to establish.

**RECAP. The pentalogic method: rashomon truths.**

**How we go about to describe scientifically a system, if any of those partial reductions of information suffice to make a linguistic mirror. The answer we provide in this and other papers on 5D is the Pentalogic, entangled method, which expresses with increasing degrees of complexity the elements needed to extract most of the relevant information from the system, without the need to get to the exhaustive (Do)decalogic analysis of the entire world of existence of a given supœrganism. Accordingly we go into a stair of deeper complexity, considering some of those key elements, under the general ‘labelling’ of Pentalogic method, from:**

**– Monologic: ceteris paribus nalysis of the event or spatial form in its 5 elements as ¬∆@st.**

**– Dilogic: considering its ST components, perhaps its Stœps of motion and form, or its S=T reproductive function/state in static, SxT or dynamic S****ó****T form; or its relative St-information and Ts-momentum that come together to form its ST energy, or its opposite limits of SS-language/form/mind/seed vs. TT-entropy… causing.**

We consider the fundamental formal mirrors huminds use to reflect the incremental complexity of the Universe’s Dimotions, as *they are a chain of increasing complexity where each more complex dimotion requires the previous one to become fulfilled. So geometry, ¬Algebra and logic do reflect those 5 Dimotions:*

1D simple structures of ¬E geometry defined by its first postulate of fractal points with inner parts, made of scalar planes of existence grow into the 2^{D} postulate of lines with breath, whose duality differentiates them into waves or networks, which grow into ternary planes (3^{D} postulate), whose logic trinity differentiates them into 3 physiological entropic, reproductive and informative networks that will associate into social groups according to the 4^{D} postulate of relative congruence, which differentiates 4 ‘angular e-motions’ bilateral gender complementarity, entropic perpendicularity, social parallelism and skewness.

So we understand the fractal point as a 5^{D} Mind, with the 5^{th} non-Euclidean postulate and the only one defined by classic mathematics, in relationship to the external world, and its waves and networks of energy and information that entangle the point with its 5 ∆º±4 planes of exist¡ence. And in this manner *the inner parts (1 ^{D }postulate) and outer world (5^{D }postulate) of ¬E geometry complete a full cycle on the description of a T.Œ in geometric terms.*

The same concept of a growing entangled complexity works for any language mirror of ÐST laws – from logic to ¬Algebraic operands and the dimotions they reflect; from musical tonal systems to vowels, all evolving from 1 to 2, 3, 4, 5 and 9 to 12 systems.

**THE FUNCTION OF EXIST¡ENCE iITS GEOMETRIC EVOLUTION**

** **

**FRACTAL ¬E POINTS. A SYNOPSIS OF ITS 5 POSTULATES**

To do so we shall first enunciate the new Postulates of ¬Euclidean geometry that complete the work started in the XIX c. with the 5^{th} Non-E Postulate:

We thus recast the axioms and postulates of Euclid into five new postulates to define fractal points, Non-Æ lines as wave of fractal points, Non-Æ planes as ternary networks of Non-Æ lines, which become supœrganisms, whose relative ‘congruence’ in its 3 ‘elements’ (singularity point, membrane and vital space) defines the type of ‘perpendicular or parallel’ relationship between them:

1st Postulate: ‘¬Æ point are discontinuous time cycles with an inner content of vital space-time’.

2nd Postulate: ‘¬Æ lines are waves of fractal points’

3rd Postulate: ‘¬Æ planes join 3 ¬Æ lines into a supœrganism’.

4th Postulate: ‘2 ¬Æ points are congruent when both its inner parts and outer perimeter are equal’

5th Postulate: ‘¬Æ World points focus multiple ¬Æ waves of energy into a still linguistic mapping of the world.

Whereas the 1^{st} and 5^{th} postulate describe the same ‘point with parts’ from an internal and external point of view.

Let us explore those postulates, constraining our examples to the simplest forms of physical and biologic spaces. For a full analysis of them please consider reading the papers on ‘5D Universe, conservation of time’, and ‘¬E Geometry’.

**1 ^{st} and 5^{th} Postulate. The 3 Mathematical parts of a Non-Euclidean Fractal point.**

* 1 ^{st} Postulate:* A fractal point has parts; that is an enclosed region of vital inner energy surrounded either by a spatial still membrane or a Temporal motion of angular momentum (S=T symmetry) self-centered in a singularity-mind that gauges its information. In the graph, we can see how different vital fractal points of ST¡entific Planes follow this ternary structure. We shall not apply pentalogic to the 1

^{st}postulate, just consider its ‘3 scalar perspectives’:

**∆±¡: **The perception of the point depends on the scale and distance from where we observe it:

**∆-1**: From the inner perspective ‘every point is a world in itself’ (Leibniz) – that is, a mind, which seems to hold the entire Universe, reason why absolute zeroth does NOT exist, as points have inner parts – they are ad minimal, Leibnizian monads:

In the graph, the distortion of ‘lineal’ self-centred functions is a plague of sciences, due to its easier calculus and the distortion of perspective caused by the human mind, as Descartes coordinates are a representation of the human lineal, Euclidean, electromagnetic light-mind geometry. The mind equation is:

*O (infinitesimal pov) x ∞ Universe (∞ cycles and monads) K-mind*

It implies that the 0’ believes to be infinite self-center of the Universe and sees it all in a distorted perspective. In brief, your eye pov is bigger to you than the Andromeda Galaxy, so the 0’-point becomes an infinity in itself (your eye≈Andromeda Galaxy), and this makes the conic, which could be considered an ‘objective’ angled point of view (where the mind-eyes is truly zeroth), to expand in the y-coordinates of the mind-point to infinity; alas! transforming the conic into the Cartesian plane where ¥ becomes also an infinite graph.

**∆+1**: Yet from the perspective of the upper st+1 Plane they are a 0’, a ‘finitesimal’, be in the limit of invisibility (what quantum scientists call a point-particle). *But even so, 0’ still have a time motion performing a ‘function’ in that upper ecosystem, ∆+1 in which it exists.*

**∆º:** As always maximal information and objectivity about a system is reached in the classic balanced S=T, ∆º view:

*Internally from its own ∆º perspective the point will have *3 dimensions/networks. This is the case even in the smallest planes of theoretical strings, made of points with parts, with volume – since we require 3×3∆º+1∆+1 inner dimensions to describe strings – a paradox that can only be resolved if we consider ‘strings’ to be fractal points with inner dimensions.

Fractal points explain without contradictions Non-Euclidean points, which are not logic in a single scale, as they ‘curve’ parallels which are ‘straight lines’ and fit them in a ‘point with no breath’ that holds only 1 line. Fractal points however enlarge fitting multiple ‘straight lines’. Yet when seen from above, human perception of them, becomes ‘deformed’ shrinking and curving its from – a theme, the distortion of human measures of time, space and scale, which will be instrumental to explain rationally the ‘spookiness’ of quantum physics and relativity and its time and space transformations. So fractal points harmonize the 1^{st} axiom=postulate of Euclid with the 5^{th} postulate of non-Euclidean parallels.

**2 ^{nd} postulate: lines have width & motion, so they are waves and networks of communication of energy& form.**

Yet once a fractal point is formed, besides perceiving internally as mind-points (1-5^{th} postulates) it will have to move and interact with other fractal points through complex 2^{nd}, 3^{rd} and 4^{th} Ðimotions regulated by the other ¬E postulates, moving in complex stœps or sharing waves of energy and information, which can be discomposed in form and motion, with other systems, from Fermion<Boson>Fermion events to human talk:

All those Ðimotions are complex stœps, which can be discomposed further in form and motion, making existential momentum, the essential parameter of the 2^{nd} postulate of communication and locomotion, a concept taken from physics, where mv momentum (active magnitude of information x locomotion) is the fundamental parameter.

*Communication* of points create vectorial fields with momentum; as ¬E points have both direction – hence motion – and form, mass, and switches between an informative mass- stop state, to perceive where its motion must go. Hence complexity rises from a mere active magnitude of scalar value, to two parameters.

In graphs different Ðimotions of which the simplest stœp is shown in the graph of a quantum particle-wave duality.

*When ‘receiver’ and emitter combine externally its ST-parameters then a reproductive trinity event is formed. *

Thus the ideal frame of reference for ¬Ælgebra is a vectorial space-time with multiple frames of reference.

An important element of algebra’s analysis of Dimotions=actions through operands is expressed in the 2^{nd} postulate of non-E geometry as we go away from self-reflecting ‘angular’ and entropic actions=operands, but communicative operands between two relative ¬Æ points or n-points in a network. This reads in several forms. We already showed how balances are achieved by switching the ±, x÷, log xª operands on a given ¬Æ Point with numerical parts, along a chain of sequential time space actions, which leave a memorial trace hence NOT annihilating in most cases the sequence. How many possible combinations of dual inverse dimotions exist can be resumed in two great fields:

- A) Dual Ðimotions within a single ¬Æ point, which ‘walks’ together through its paths of vital actions with steps and stops ± x÷ log ª dimotions.
- B) Feed-back communicative Dimotions between two T.œ points that forcefully have inverse directions but tend to be the ‘same operand’ both sides with ± symbols.
- C) Merged Dimotions between those communicative points that approach each other and finally merge, ‘re=producing’ through x÷ operand in a lower scale ‘connected axons’ (since A(x) x B(y) = AB (x •y)), that is the number of axons of communication between two entities A, B, with ∆-1 x,y parts is the product of x and y. So the product becomes the first operand to probe a lower plane of existence, while the ± operand stays in a single plane.

Operations are connections between T.œs that define their actions, as balanced, parallel = connected creation of social networks bringing a 5D social evolutionary form or perpendicular, Darwinian 4D absorptions=flows of entropy, motion, energy and information. Those dual actions are mediated by operations. And so there is first the abstract definition of those operations in mathematical terms, with the study of its properties and then its connection with the dual åctions of 2 beings that enter in communication within a given world-Universe.

The equivalent of such ¬Algebraic numeric analysis being in geometry the study of the topo-biologic properties of non-Euclidean ‘waves’ of communication between 2 fractal point, 2^{nd} postulate of ¬-e geometry):

IN THE GRAPH WE can see how two asymmetric parts, normally one with more form and the other with more motion, come together into a single space-time event super organism, which will either become complementary (gender asymmetry) and evolve socially (which we can generalize to n-points in the 3rd postulate, forming networks) or will enter in a Darwinian struggle, and be operated negatively in terms of the 4Dimension of entropy.

What ¬Algebraic equations do is to operate with numerical-scalar properties those events /superorganisms, 5D evolving vs. 4D devolving; that is 5D, adding vs. 4D resting, 5D multiplying vs. 4D dividing, 5D potentiating vs. 4D rooting, 5D integrating vs. 4D deriving the system.

**3 ^{rd} and 4^{th} postulates: 3 lines: Emergence of organic Networks create according to congruence superorganisms..**

To maximize the conservation of time, there are several strategies the mind pursuits in its territorial order The simplest one, just described, is to ‘stop time’ into a territory, which brings however the constant accumulation of form, and finally the paradoxical end of time as the system surrounds itself of stopped property. We are wealthier in the moment of death. So before we deal with the worldcycle of life and death that stops the time of the being, and then it fids, itself ‘erased’ in a reversal of death – as time cannot be stopped, so when it has no more energy to spend it explodes back into entropy – we shall consider the second fundamental system of survival in time, the creation of 3 ‘non-E lines=networks’ that create a ¬E plane=topological organism, according to ¬E congruence=similarity.

The building up of a true 5D science of reality starts by defining all the dimotions of a being in its 3 scales. Then we study its ‘locked’ sequences of dimotions and its ‘balanced’ stœps=beats, in all those scales. Then we do find the synchronicities between scales that become entangled repetitive discontinuous symbiotic patterns. And then we find the ‘larger’ physiological networks, or ‘organs’ that entangle them; as the smaller parts will gather around those networks, to receive energy (blood-like reproductive networks that deliver larger particles), information (which requires as we have seen a faster nervous, informative network of smaller particles and external territories of entropic feeding (symbiotic to the digestive networks).

Complexity arises to produce the emergence of larger wholes, the 3 physiological networks and its attached organs, which are the topological vital spaces that form the superorganism. All species studied by science a common phenomenon occurs: the existence of parallel groups of beings organized into a single social form. Molecules are made up of atoms and electronic networks; economies are made up of human workers and consumers that reproduce and test machines, guided by financial networks of information (salaries, prices, costs); galaxies are composed of stars, which orbit rhythmically around a central knot, or black hole of gravitational information. Cells controlled by the nervous, informative system organize human bodies.

A tree is a group of leaves, branches and roots connected by a network that provides energy (salvia) and information (chemical particles) to its cells. Cultures are made of humans related by verbal, informative Disomorphisms and economic networks that provide their energy and information.

How then it happens that parts become wholes is the key to depart from a mere abstract, quantitative analysis of reality and add the organic nature of all what exists, the dynamic interplay of parts that ‘network’ and connect to each other, forming simultaneous spacetime organisms, which synchronizes its clocks, emerge as a whole and develop all the intelligence and complexity of the systems we observe around us.

In the next graph, we see the ternary network structure of the nested organisms of the fractal Universe as ¬Æ topological planes composed of ‘similar fractal points’ (atoms, cells, individuals) joined by 3 physiological lines=networks, whose 3 functions, distribution of locomotion, information and its ‘combined’ energy define the 3 conserved Dimotions of any system of the Universe. A final element though is needed to make sense of those superorganisms, *the still mind of information, mapping out the whole and controlling it to perform its Mandate of existence, Max. SxT (s=t), to survive, grow and multiply.*

The graph shows the physiological networks of each supœrganism from the galaxy to the atom, where self-similarity takes place. I remind you of the Si=Te equality, which means we slow beings see networks of faster particles as ‘force waves’ and networks of slower life forms as fractal branching, but essentially as Nottale has proved, we can ‘translate’ quantum physics into a network, topological view, as light is in fact a branching filling wave that ‘speeds up frequency’ as it penetrates lower planes, filling it till it touches particles.

*All systems of reality are connected by networks that share energy and information between parts and wholes that expresses the structural unity of all scales. Networks ‘fill’ space ad maximal to connect fully the whole with the parts, achieved in the Si=Te point of parallelism and self-similarity. But they entre in a region of faster motion. So while Space ‘tends to remain constant’ in each scale thanks to filling networks, time accelerates. *

It’s all in 5D metric equations, SxT=C and ¡ts S=T point of balance & equilibrium where the system reproduces as the two fundamental metric equations of all space-time organisms, which means many things, such as:

– Slow beings use networks of faster particles which in physical systems are ‘waves of force’ in life beings are and networks with fractal branching and in human societies, networks of money and simultaneous legal messages. Yet all are essentially performing the same organic functions, we shall describe now as can ‘translate’ a light filling wave that penetrates lower planes, filling it till it touches particles, as a branching that ‘speeds up frequency’; and a legal network that every citizen knows and obeys as a filling system of information, similar to a DNA network that all cells of an organism have in common. *Think always NOT in the differences of form and scale but in the homology of functions for the body/waves and particle/heads of each of the 3 superorganisms (working and informative classes in human societies), to see the unity of it all.*

If we consider that networks of energy and information belong to two different scales, and take into account the conservation of volume of space-time between scales and the S=T equality, since smaller scales have faster times speeds and smaller parts, perceived as relative information compared to the bites of energy of a larger scale, we find an equivalence between a ‘network’ of bits of information of a smaller ∆-1 scale and its equivalent volume of a herd of bites of energy of a larger ∆0 plane: (∑∆¡=e=∏∆¡-1). *This equivalence is essential to everything in the Universe as it allows the coupling of languages and energy, making for example the physical economy equivalent to the financial economy, the language equivalent to the action; the mathematical equation a mirror of realit. But as information is smaller and more abundant, this equivalence tends to be inflationary, themes those that appear in all the planes of space-time and the interaction of minds and languages of information (SS, St) with Energy bites and entropic boosts (TT, Ts). *

**The 2 languages of informative and reproductive networks: Bits of information and bites of energy.**

It is important also to Understand NOT in pure abstract mathematical terms, but in LOGIC, LINGUISTIC ones, the INTERNAL, DYNAMIC nature of those networks, because only then we can proper understand how they work in advanced organisms of maximal information, the biological organism and the Historic organism, which belong as ‘fractal systems’ self-similar to each other, two the same specific type of organisms, we shall qualify as socio-biological organisms. A NETWORK, of informative nature, delivers MESSAGES OF INFORMATION to simultaneously coordinate the actions of all its parts; with its faster=smaller bits of information according to 5D metrics (min. spatial size x. max. temporal speed). While the networks of reproduction, the blood and financial system delivers larger bites of energy, which the organism needs to feed itself (when it is a healthy NON-corrupted superorganism as most of Nature, but NOT HUMAN SOCIETIES, whose astounding level of corruption we shall explain in detail).

Indeed, the key to the full understanding of reality both in terms of energy but also in terms of information, as both are two sides of the same coin, called ‘exist¡ence’ is the fact that in the sentient Universe, each fractal point, atom, cell or citizen (physical, biologic or social systems) needs bits of in-form-ation, form, smaller in size of space, hence faster according to 5D metrics (SxT=C), but also ‘bites’ of entropic energy which will help the system to move. Networks are NOT some abstract ‘fractal tube’ but they exist to deliver ‘energy and information’ (SS: form=language with a little motion=St-information and motion=entropy=TT with a bit of information = energy=Ts).

So a healthy superorganism will deliver to each ‘fractal point’ (molecules, cells, human citizens), two type of messages through two type of networks. We shall call ‘generically’ the 3 type of bits and bites of information and energy that each of those 3 physical, biological and social systems receive, ‘particles, genes and memes’ even if the words as usual in 5D sciencers are slightly changed, and widened in its original meaning.

So with its specific variation, those are the two fundamental reproductive-‘body-wave’ and informative-‘particle-head’ bites of energy and bits of information of the fundamental systems of nature:

*– *In physical systems, the two networks are the gravitational faster network of information, which we humans do not perceive, as we are much larger beings with electronic networks. Its bits of information in this faster non-local network should be ‘gravitons’, components of gravitational waves. In physical papers we advance as the most likely particle state of those waves of information that ‘position’ the different physical systems of the galaxy, a gravitational tachyon ‘neutrino’ for multiple reasons, we study on our papers on physics.

On the other hand, because we do perceive it, it is much easier to prove that the energetic network of physical systems are electromagnetic waves, photons and its ‘social, static state’ as the elements of an electronic nebulae, trapped in the potential energy well of the atom. Thus photons and electrons become the ‘energy network of physical system, molecules.

We shall escape then in this introduction further information on the scalar structure of those networks and how, as we ‘grow in scale’, what is a bite of slow energy for a smaller plane of space-time, *becomes for the larger plane’s slower beings, a faster bit of information, in the amazing beauty of the harmonies between scales. So electronic ‘food for atoms’ becomes electronic information for biological organisms and so son.*

*– *Those biological organisms do have then two fractal networks, the electronic, informative nervous system in which bits of electronic information moving along the myelin membrane deliver faster messages to every part of the organism to simultaneously synchronize its motions, so the body-cells act as a single form in simultaneous space.

– But when we move into the bites of energy delivered by the blood organism, the network delivers to each cell the basic ‘currency’ language of energy that all cells need to move, called ‘oxygen* ‘. It is an atom of slower motion than the electrons but due to its electro-negativity and readily availability in the atmosphere, with its capacity to kick with two OH- & H+ legs the water ‘medium’ on which cells exist, the perfect language of ‘money’ for the organism to start kicking its ‘actions’. *

*So we DO have in the next scale according to the perfect laws of harmony, the two basic biological bits and bites of information and energy, electrons and oxygens, and from then on, as systems become more complex, variations of those bits and bites occur. *

The main category are mixed ST messages, which deliver BOTH a stick and carrot ‘complex’ to the cells and its big molecules, which are amino acid systems, of great simplicity called Hormones, starting from the simplest of them all, an NO molecule (which do relax muscles, its main message to the locomotion system, increases the pressure of blood, provoking sexual erection, the simplest message to reproductive systems and multiplies the neuronal activity. *As nitrogens are the clock atom of our mind-brains. *

*So finally more complex NO systems with a body support of carbon chains become ‘hormones’ which might have a ‘higher informative message’ (with more N, as in nucleotide molecules) or a higher energetic message (as in acids with more oxygen).*

*They form then the basic letters of the ‘biological longer sentences that might accumulate information’ in ever more complex molecules, as biological organisms are by far the more complex systems we know of. *

Finally a very important concept is the difference between an ecosystem in which multiple superorganisms co-exist, often in predatory relationships, vs. an organism in which only a type of atoms, cells or citizens co-exist, and is far more symbiotic as all parts love each other and share energy and information through its networks, over a common territorial space, as shown in the graph.

Those three physiologic networks/classes/physical parts of ANY system of the Universe define the Universe indeed as a fractal organism of infinite smaller and bigger super organisms, in a game of Russian dolls in which each of us is a ‘island-Universe’ within itself, made of smaller parts, and for that reason each of us is also a part, cell/citizen of a social super organism, nation, religion or civilizations, which we do NOT see as a whole, as our cells do NOT see us as a whole, but DO exist as such.

*What makes then the whole a whole? The answer is: the nervous, informative languages that communicate all the parts of the super organism and ‘trace’ within its syntax and value, its path of the future. And so we have talked first of it and will constantly coming to the bottom line of reality – the languages that construct the organisms of the world.* So we need to become a bit more complex about the previous metric. It refers essentially NOT to the whole 5D plane but to a given ‘superorganism’ of each plane. When we go down in scales, in fact the Universe ‘enlarges’ for a traveler that becomes smaller and accelerates its temporal energy

Let us then define the ‘stair of Universal supœrganisms to understand this:

Let us then define with similar templates the ‘stair of nested Universal supœrganisms, of the 3 stientific varieties – physical, biologic and social:

**∆+3:** A galactic organism is a population of stars, related by energetic ¥ networks & gravitational information with a nucleus made of a swarm of black holes, and a membrane made of strangelet matter symmetric to:

**∆-3:** An atomic organism is a population of particles, related by energetic electromagnetic networks and gravitational information with a nucleus made of a swarm of quarks, and a membrane made of electronic matter symmetric to:

**∆+2:** A star organism is a population of electronic plasma, related by energetic networks of electromagnetism and gravitational information with a nucleus made of a swarm of atoms, and a membrane of photonic radiation symmetric to:

**∆-2:** A light organism is a body of energetic waves over a quantum potential field of gravitational neutrinos, directed by its particle, informative photon state…

**∆-1:** A cellular organism is a population of molecules, related by energetic networks (cytoplasm, membranes, Golgi reticules) and coded by genetic information (DNA-RNA.)

**∆=o:** A human organism is a population of DNA cells, related by networks of genetic, hormonal and nervous information and energy networks (digestive and blood systems).

**∆+1:** An animal ecosystem is a population of different carbon-life species, related by networks of light information and life energy (plants, prey) coded by instincts.

**∆+1:** A historic organism or civilization is a population of humans, related by legal and cultural networks of verbal information and agricultural networks of carbon-life energy, coded by human memes.

Networks that share energy and information between parts and wholes that expresses the structural unity of all scales connect all systems of reality. Networks ‘fill’ space ad maximal to connect fully the whole with the parts, achieved in the Si=Te point of parallelism and self-similarity.* But they enter in a region of faster motion. So while Space ‘tends to remain constant’ in each scale thanks to filling networks, time accelerates. *So the 5D metric refers NOT to the whole Universe of 5D planes but to a given family of ‘supœrganisms’ of which mankind in it 3 scales of ‘biologic cells’, human individuals and societies is undoubtedly a ‘phyla’. When we go down in scales, the Universe ‘enlarges’ for a traveler that becomes smaller and accelerates its temporal energy.

**The 3 ST-planes of topologic organisms.**

Those networks are more efficient forms of distrivution of energy and information to its points that ‘save time’. Further they ‘equalize’ putting in relationship parts of a being, ‘chained’ into a super organism, in a single simultaneous space: Thus we define a fundamental point-particle, the Non-Euclidean, Non-Aristotelian T.œ. as Supœrganism (ab. œ) of Time§pace (ab. T.œ ≈ ∆±¡) with a simple non-Ælgebraic expression or generator equation of all T.œ = Time§pace Supœrganisms:

*ð (head-particles)≤ST(body waves)≥$T (Limbs/potentials)*

All those states and symmetries are thus feed-back equations which can only be properly ‘generated’ with ¬Algebraic inverse operations, in its simplest form, and in its most complex structures, when we consider multiple ‘symmetries’ and ‘variations of species’ with the help of Group theory, as essentially S=k/T, will be the ‘inverse numbers’, k, the identity number (if equal it to one) and the dual demotion represented by certain ¬Algebraic operands.

We have to consider where functions and operand are set – that is, what background space we use to express it, and 3±i are the essential background spaces which correspond also to those dimotions as forms in space, the 3 lineal=cylindrical, spherical=polar, and hyperbolic=Cartesian planes and the scalar plane, ill-understood, which is the complex plane better perceived if we ‘square it’ eliminating the √ symbol of its negative -1 axis:

So they can study 2 fundamental ’emergent’ ∆+1 planes of mathematics, the study of Dimotions of Dimotions with the tools of calculus in time, and the study of spaces of spaces, with the tools of the complex plane properly understood in terms of ‘square’ coordinates.

But as in the entangled Universe all mirrors can reflect all forms, ¬Algebra also can analyze other elements. But its main beauty is in creating sequential chains of pentalogic actions that reflect the motions of existence of the being, even though its ‘Group’ simultaneous analysis of all its ‘variations’ of species, has been the most developed inflationary mirror in its last ‘excessive’ age of form.

**4 ^{TH} AXIOM=POSTULATE OF RELATIVE CONGRUENCE: SIMILARITY, ¡NDIFFERENCE, TRANS-FORM-ATIONS.**

*4 ^{th} Postulate:* Equality is no longer only external, shown in the spatial perimeter of any geometrical form (Euclidean congruence) but also internal and further on it is never absolute but relative, since we cannot perceive the entire inner form of a point – hence the strategies of behavior such as camouflage. Forms are self-similar to each other, which defines different relationships between organic points, according to their degree of self-similarity. The

*4*postulate is thus the key to explain the behavior of particles as the degree of self-similarity increases the degree of communication between beings. Some of the most common behaviors and ‘events derived from this postulate are:

^{th}1) Reproductive functions in case of maximal self-similarity or complementarity in energy and form; ei->Sei or Max E x min. I (male)= Min. e x Max I (female).

2) Social evolution, when points share a common language of information, i=i -> 2i.

3) Darwinian devolution when forms are so different they can’t understand each other’s information so instead they feed into each other: i ¹ i. In such cases if those 2 entities meet they will start a process of ‘struggle for existence’, trying to absorb each other’s energy (when E=E) or simply will not communicate (when E¹E, since then there is neither a common information to evolve socially nor a common energy to feed on). Yet because any point absorbs only a relative quantity of information from reality, similarity is relative, faked for purposes of hunting with biologic games such as camouflage or sociological memes that invent racial differences, allowing the exploitation of a group by another.

The 4^{th} postulate defines systems as identical when they are equal in its 3 ternary parts, the outer angular momentum or ‘membrane’, its central Active magnitude or singularity, focus of the forces and the vital energy*, enclosed within them, and all others as similar with different ‘angle of congruence’. We* distinguish 2 different interactions according to the degree of equality of its ternary parts, as systems can be symbiotic, if *their individual, cellular or atomic ‘fractal points=parts’ are similar enough, interacting through *its 3 physiological ‘lines=networks’ evolving in parallel creating an organic plane, as those described in the next graph for each scale=science, or they can be entropic, destructive, predatory, when they are dissimilar and don’t speak a common language of information to coordinate its actions, whereas the stronger system will perpendicularly break and feed on the weaker one.

The geometric complexity of the 4^{th} Postulate is caused by the topological forms created by any event that entangles Multiple Spaces-Times. Since it describes the paths and forms of dual systems, which connect points: Self-similarity implies parallel motions in herds; since equal entities will maintain a parallel distance to allow informative communication *without interfering with the reproductive body of each point*. Darwinian behavior implies perpendicular confrontations, *to penetrate and absorb the energy of the other point*. Finally, *absolute, inner and outer self-similarity* brings boson states, which happen more often to simpler species like quarks and particles that can form a boson condensate as they do in black holes, *where the proximity of the points is maximized. And indeed, the same phenomenon between cells with the same inner information /DNA originates the ‘collapse’ of waves into tighter organisms.*

Finally if there is no similarity neither in body or mind, its existence as ‘cat alleys’, that never cross (relative invisibility). We talk then of Skew T.œ.s.:

The 4^{th} Non-Euclidean postulate is implicit in the work of Lobachevski and Riemann who defined spaces with the properties of self-similarity (Riemann’s homogeneity), which determines its closeness (Lobachevski’s adjacency).

*4 ^{th} postulate of relative congruence & angle of parallelism as a mirror of its 5 pentalogic dimotions and variations of angle define Darwinian or social, reproductive outcomes to communicative events between fractal points.*

Thus in praxis we assess similarity by an ‘angle of parallelism’ that increases social evolution into herds and supœrganisms, or perpendicularity that ‘scatters’ systems into entropic destruction – elements those of an entire fascinating new field of 5D topo-biologic studies that analyze in geometric terms, the vital topology and relationships between form and function in all systems of Nature from particles to organisms.

This simple geometrical truth however is essential to all systems of nature, whose angles of connection determine the functions and symbiosis between parts.

The Universe always starts with an asymmetric being, which can go both ways: towards a social evolutionary symmetry that lasts in time and implies a mirror parallelism, or an antisymmetric destructive, perpendicular event in which one part punctures and absorbs the energy of the other. It is the topo-biological ternary principle of non-Euclidean, Non-Aristotelian I-logic geometry that puts together both the biological and mathematical properties of reality. The concepts of symmetry=parallelism, antisymmetry=perpendicularity and asymmetry are mirrored by the 4th Non-E Postulate of similarity. But we can extend the concept of asymmetries also to asymmetries of time, between the young age of locomotion and the old age of information, of actions=Dimotions between the step and stop similar actions, and the entropy and social evolution actions, which bring us the final asymmetry of scales between the upper arrow of whole with more spatial size and the lower arrow of parts with more information. When those dualities: step-motion/stop-perception and scale up (5D: social evolution), scale down (4D: entropic dissolution) are put together we obtain the most complex balancing dimotion, reproduction, and when they are all added up in the existence of a being, we get its world cycle.

**The Generator’s Ternary Symmetries and Its S=T 1, 2, 3 Dimensional Analysis**

There are 3 relationships in space-time between entities, which are part in non-Æ of the laws of the fourth postulate of similarity, that we* relate to the 3 elements of the fractal generator:*

ST: Complementary adjacency, in which in a single plane, *membranes of parts fusion into wholes*, and in multiple scales, parts become enclosed by an ‘envelope’ curve that becomes its membrain. Its main sub postulates being the realm of topology proper.

$t: Darwinian perpendicularity, in which a membrain/enclosure is ‘torn’, and punctured by a penetrating perpendicular, causing its disrupter of organic structure. Its main postulates being the realm of Non-Euclidean geometries.

- ð: Parallelism, in which two systems remain different without fusioning its membrains, but maintain a distance to allow communication and social evolution into herds and network supœrganisms. Its main postulates being the realm of Affine geometry.

The correspondence of those relationships with the 3 elements of the generator, $<ST>ð§ are immediate:

– ST-Adjacency allow to peg parts into present space-time complex dualities.

-$-Perpendicularity simplifies the broken being into its minimalist ‘lineal forms’, $t.

-§-Parallelism allows the social evolution of entities into larger §ocial scales.

They will define ‘ternary organisms, in which the 3 topologies in 1, 2 or 3 s=t dimensions of a single space-time plane, can be studied in ceteris paribus analysis or together, *but no more, as all other attempts to include more dimensions in a single plane are ‘inflationary fictions caused by the error of continuity’ *

Dimensions thus must also be considered besides *the 3 logic relationships.* And there are 3 levels of complexity in dimensions, lineal, 2-manifolds and 3-D volumes that express also the ternary generator:

So for the 3 lineal coordinates, the equivalencies are immediate:

1D¡ Γ: $t: length/motion <ST width/reproduction> §ð: height/information.

As lineal length is the shortest distance between two points, height the projective geometry of perception from antennae to heads, and its product mixes them to reproduce in the width dimension where you store your fat…

**Change of symbols in algebra from 4D to 5D: the new equality-feed back equation. Correspondence.**

It is important before we go further into algebra, to understand that the r=evolution of non-AElgebra steams from the fact we are in a pentalogic convergent manner to create forms, hence = is substitutes by <=>. And so every equation of science must be reassessed not as an equality but a relationship of equivalence that through a ‘dimotion of exist¡ence’ can be transformed into each other.

This also means that ultimately = is equivalent to an action, and so we should write 5 different ‘stages’ of equivalence, one for each dimotion of exist¡ence.

For example, e=Mc2 doesn’t mean that energy and mass is the same; but that through an « entropic process mass becomes energy. Often this kind of relationships then require a ‘ratio’ or ‘universal constant that explain us how one variable becomes the other and this is the role of C2, so in fact we have to write: M < c x c < E.

So 5D non-Euclidean algebra introduces changes in the basic interpretation of the equations of mathematics, but beyond those principles it *closely embodies *the principle of equivalence, as the new ‘operandi’ of non-AElgebra enclose as a ceteris paribus analysis the simple equality symbol.

Equality however is not real, as absolute congruence does not happen often. So we have to couple the 4^{th} postulate of non-Euclidean congruence, experimental facts and rules of non-AElgebra to fully grasp the diversity of variations of the ó symbols of feed-back equations. This can be done exhaustively and systematic as we shall try to do in the paper on the stientific method and also here to bring meaning to the abstractions of algebra. And we shall also introduce those concepts on the papers of mathematical physics to better understand it, as Universal constants and ratios are closely related to those feed-back equations.

The Universality and lack of generality of the symbol = therefore disappears now by the fact we try to operate always in more detail equations with the necessary < > ≈ «» 5 symbols for the 5 Dimotions of existence.

= thus come in 5 favors and with them come different operandi to signify, SS, St, ST, Ts, TT elements. So knowledge of an equation comes also from a deep reflection of its structure to observe in more detail and translate it to GST equations.

If we consider GST, the ‘Generator of SpaceTime’, S≤≥T, we can then consider a first differentiation into 5 subspecies of St dimotions. Then we evolve them into chains and so the first question we wonder is how many ‘events’ of two chains of spacetime are possible and what they signify.

It is then when the concept of a function of existence of 5 Dimotions of space-time moving in 5 Dimensional space acquires all its meaning for complex formalisms of mathematical stiences with non-AElgebra. How can we represent that 5 Dimotional space? This is the subject of Ænalytic Non-E geometry, (æ¬E). Enough to say a good model departs from the complex plane in 4 Dimensions where the I is the S coordinates and the real the Time coordinates..

**THE FUNCTION OF EXISTENCE . ITS ALGEBRAIC DEVELOPMENT.**

We have now the bare minimal comprehension that the reader can expand in our first paper on ∆ST, to define the function of exist¡ence, a Gaussian curve that puts together ∆ST and defines the life-death cycle of all ∆st systems.

**The 3 ages of existence of space-time organisms. Its 2 worldcycles and Metric equations.**

The *Function of Existence* of any space-time organism can be developed as a feedback equation, S<=>T, in 3 sequential phases/ages/horizons, between 3 ‘standing points’ (changes of phase): Max. T=motion x Min.S: form =moving youth; Max. SxT(s=t): reproductive maturity and Max. S x Min. T=informative, old age, as the equations of the 3 ages of life, between the seed of pure linguistic form born in the lower plane: S¡-1 and its T¡-1 entropic death, back to ∆¡«∆¡-1:

*∆-1»∆º:* The cycle or organism starts its existence as a seed of pure form (4D) when its space-time field is created.

*sT:* It is the first horizon or ‘energy, youth age’ of the cycle, in which energy dominates the system and so we write this phase as, max. $T x min. T.

Max. SxT: s=t*.* It is the present balanced age of the cycle or classic age of ‘life’, when energy and information are in a constant proportion. It is the most efficient age, when the cycle reproduces.

*Max. T x min. S*: 3^{rd} age of the cycle when information has exhausted the space-time field that warps into itself.

*∆º«∆- 1: 0S x T: * It is the end or death of the cycle that reverses its form and becomes energy again.

Existence is an ∞ sum of space/time fields, fluctuating between birth and extinction through those 3 phases or ages. The 3 ages of Timespace supœrganisms happen in all systems, including mental languages:

In State Physics they are, $T-gas, the moving state, S=T liquid, the balanced state and §ð-solid the informative state; into Cosmology, where it describes the Universe as a space-time system that fluctuates between both limits, a form of pure time, the singularity (*min.$T x max.ð§) *and a form of pure space, the big- bang (*max.$T x min. ð§). In *Biology, they are the 3 ages of living beings AND the 3 horizons of evolution of species. In social organisms, through the subconscious collective mind of civilizations which in art styles mimic in a longer 800 year cycle of life and death of civilizations (according to 5D metrics a human social superorganism is larger in space – a nation, culture, religion – and so it lives longer in time). So we find the 3 ages of life *emerging in the 3 ages of cultures and* its 3 artistic styles: Min.S x Max. T (infantile epic, lineal art, as in treccento, Greek kuroi; S=T; balanced beauty, when form and size are in balance, the classic mature age; and Max. S x Min. T: baroque, 3rd age of a civilization, whose subconscious mind is the art of its ‘neuronal artists’, the age of maximal form and anÐST for a no future, which *is the age of war and death of cultures*).

We talk of 3 ∆±1 Planes of worldcycles as the being live in a placenta, then emerges as organism in a world:

þ: 0’-1: its palingenetic o-1 social evolution in the accelerated time sphere of existence, till becoming 1 (0’-1 bounded unit circle in ¡logic mathematics; quantum probability sphere of particles in physical systems; palingenetic fetal age in biologic systems; 0’-9 memetic learning childhood in social systems). It is a highly ordered worldcycle as a placental mother-energy world nurtures a memorial cyclical spacetime that erased errors of previous generations.

– c: The outer 1-∞ world, in which it will deploy its 2nd world cycle of existence in an environment which is open, entropic (1-∞ hyperbolic unbounded Cartesian plane in ¡logic mathematics; thermodynamic entropic statistical molecular populations in physics; Darwinian struggle between populations in biology; idol-ogic dog-eat-dog capitalist, nationalist competitive eco(nomic)systems in the super organisms of history. In this 1-∞ existence the world cycle is not ensured to continue, as the entropy of the world system can cut it off.

ω: The existential life cycle is part of a larger world of hierarchical social Planes (§ D¡), where it performs 5 survival actions through ∆±4 Planes self-centered in its mind, beyond which it cannot longer perceive, to become if successful a new superorganism of the infinite planes of God, the game of existence.

**The function of existence: Reproduction of form.**

Physicists made the Galileo’s paradox, the cornerstone of their theory of measure, but they* failed to study the deep implications it has for every aspect of the structure of the Universe, from the duality between spatial mental, linguistic forms and physical motions; to the balances achieved by the similarity of both space and time, which becomes the fundamental ‘equation of present’ S=T, and hence with the metric equation of Planes, $ x ð = K, the two essential equations to formalize single planes S=T, and multiple Planes of spacetime. Yet as S=T maximizes SxT=K (5×5>6×4). *

*We unify both in a single equation: Max. S x T = C, which defines for each fractal vital space-time organism its Function of existence, as all species will try to maximize its motion-entropy-time for its field-limbs, its information-spatial states for its particle-heads, whose product will give us its vital reproductive energy. But as all systems move and motion is reproduction of form we can ad a final factor, ∑, reproduction of parts, which to maximize that function become joined into larger wholes which are stronger than individuals; creating new planes of existence. C= Max. ∑ SxT (s=t); whereas C act as the entropic limits in ∑-Planes, T-ime & S-pace, boundaries beyond which the still mind doesn’t’ perceive or control. *

It is also a survival biologic equation, because it implies to provide lineal motion to ‘fields-limbs’, absorb energy to reproduce our bodies-waves, and information to guide our motions with particle-heads. *So reality is a ‘struggle’ for existence as systems reproduce its Ts-fields-limbs of motion, S=T body-waves of energy and St-particles-heads of information. But as all T.œs are fractal, broken, its growth has a limit on the fight with other systems, which try to move and reproduce. In terms of pure T-motion and pure S-form, we consider then the whole of maximal* time=motion= entropy or TT and Max. space=form=stillness or SS the 2 limiting Dimotions for any 3 ensemble Ts<ST>St-system.

We define the Universe as a fractal supœrganism that reproduces and ensembles Space-time Dimotions into organic forms through ∞ relative Planes of spatial size and time-motion; whose Fractal generator (mathematics) Metric (Physical jargon), survival function (Biology) or Function of existence (logic Jargon) writes C=Max.∑TxS (e=i)

We shall prove that all realities are always a reproductive radiation of a function of existence along 5D Planes.

**The conclusion of a world made of space=time then is obvious: We are made of the 5 Dimotions of space-time of the Universe. We are ensembles of those 5 Dimotions, which seen in simultaneous space give origin to the vital topological organisms of the Universe; whose study therefore is mathematical, the science of space; and observed in sequential relational time, live a worldcycle of life and death; and since there is nothing else than time and space, those 2 fundamental experimental primary ‘mirror-sciences’ of time and space become the most important to know what all systems have in common, its ‘Disomorphic=l**

**The curve of existence as a bilateral mirror symmetry of the log/exponential function. **

The 4 phases of a real function of existence: In the first phase, corresponding to the palingenetic age of placental energy the ratio of change (derivative) has the absolute maximal e^{x}, which is its own derivative. In its 2^{nd} phase it diminishes to the minimal 1/x the log derivative, which is the definition of an infinitesimal part; till in the 3^{rd} phase it will start a bilateral symmetric inverse function of decay with -1/x diminution and a fast collapse in the 4^{th} phase of a 3^{rd} age and final death moment. So the combination of ± exponentials and logarithm curves are also the best way to graph as a bell curve the worldcycle of existence *as a ‘composite’ of a ± exponential and logarithmic 0’-sum.*

As any student knows there is an intimate relationship between the exponential and the sinusoidal by way of Euler’s formula in the realm of complex numbers, which we treated briefly in theory of numbers.

So sinusoidal bell curve functions represent a worldcycle, though the symmetry is broken in the moment of entropic death when the collapse is extreme in a ‘falling line’ as death happens in a single moment of time:

*4D»∆-1(seed)∑∆:|-$T(limb-field)<Ø-S≈T (iterative bodywave)> O-§ð (particle-head)«5D∆-1(death)*

A key theme of vital mathematics is the representation of a worldcycle in lineal time, with ± exponentials & its inverse, logarithmic curve around the key points of change of phase… as growth of ‘entropy-motion’ diminishes. So we move from ‘adolescence’ of max. growth of both parameters (sT energy and sT information) to the y”=0 point of youth, where the logarithmic part grows slower. Together they form, one half of the total graph of a cycle of existence, till reaching the y’=0 point of Max. (S≥≤T), which then becomes negative, happening a decay of the whole system in two negative curves.

*The conservation of time in its 5 y’**Ù y” =0, standing points that define the 5 SS, Ts, ST, St & TT moments of generation, youth, maturity, 3 ^{rd} age and entropic death thus become the essential points (maximal and minimal) of the equations of calculus, the sinusoidal function of existence and all its derived elements. *

The natural name for the worldcycle of a superorganism traveling through 3 planes of space-time is the function of exist¡ence, as we take advantage of the similarity of all languages, mirrors of the same Universe made of synchronous organisms whose topological trinity parts are entangled by the < ≈ > 3 operands of informative evolution, T>S that synchronizes time cycles slowing down its rhythms, entropic energy, S<T that expands entropic timespace dissolving its information and T≈S that gathers together in parallel herds T.œs (ab. Timespace organisms) with a common property/dimotion of exist¡ence. So as energy and information are synonymous of Time and Space, and both entangle together through the re=productive operand, ExI=ST¡ becomes the acronym for the function of existence of an entangled supœrganism, studied by a given ‘st¡ence’, that is a science that studies a relative plane of size and time speed of the 5^{th} dimension – the name we keep for the whole of wholes.

So we can see the graph of the whole world cycle as a ‘composite’ of a positive exponential and logarithmic 1^{st} age and maturity, and a negative logarithmic and exponential – mirror decay with the final collapse of death. Needless to say the same functions appear on all growth curves.

If we add all those growth exponentials transfering the function to a lineal sinusoidal series in ±complex planes, the stretched worldcycle representation becomes a ‘sinusoidal function’. So it is carried into Complex exponentials.

Finally through the T=S symmetry it becomes both a Bell curve of spatial populations & its s=t statistical mirror; which finally carries to the probabilistic 0’-1 ‘unit’ sphere of palingenetic growth (as in the duality T-probability 0’-1 sphere of quantum physics vs. 1-∞ entropic statistical function of thermodynamics. Since the function of existence in its subtle fractal variations and complex pentalogic is the most important of the Universe.

The worldcycle curve is yet another proof that Real Numbers are scalar and ratio numbers, which as the e decay of a whole into Planes, or the sinusoidal π of mental perception, enlarge a seed into an organism or shrink Planes of a whole into a finitesimal, mental mapping with no limit of ‘warping’ (Poincare Conjecture).

**L***ogarithmic growth and its inverse function of decay shape a life-death cycle in terms of social growth of energy ∆-1 cells and dissolution, forming 2 two sides of an inverse worldcycle peaking in the Max. SxT (s=t) middle point. *

**The mathematical laws extracted form the nature of those 5 Dimotions will then be the essence of the properties and laws of the 5 operands of algebra, growing in complexity from the simplex nondimensional angular perception, which forms minds to the operands of calculus. Specifically we write the equation in terms of space and time (SS & TT) that merge in a middle point as they become ‘energy: Ts’ and information ‘St’, maximizing the imprinting of space networks over ∑-herds of quanta as: **Max. ∑ T x S _{e=i} = ∆_{±¡}

## As we are not at this point interested in the exact algebraic formulation of the function, but in the concepts behind it the impatient reader is invited to derive the Gaussian function from it. The key of that analysis being the fact that the function can be studied ‘partially’ in the different dimotions that its synoptic, conceptual writing enclose. *Since the equation is not a strict quantitative definition but a logic one, ∑T herds of time particles with motion, become ‘imprinted’ with informative space, re=producing e=I balanced quanta in the point in which form and motion are in a dynamic equilibrium, which give us a certain ∆ topologic plane of space-time. This middle point is the maximum of the Gaussian function, but as the equation is dynamic, its stœps of reproductive actions will form a ‘cloud’ of imperfect repetitions around that s=t mean due to different ‘limits’ – the most obvious the limit of prey-predator carrying capacity; so repetitions will ultimately diminish in quantity and loose quality. How to avoid this *is obvious: eliminating repetitions that are not good enough, which ultimately is what provokes the cycle of evolution, extinction and death of lesser individuals; so the game keeps perfecting itself, and its spatial points learning the palingenetic improvements.

**Conservation of time vs. inverse annihilation of information.**

An essential truth, systematically denied by @-minds is that while the initial substance of reality time=motion is conserved, as it trans-forms ad eternal in different forms of information information becomes annihilated into 0’ sums, regardless of leaving memorial remains and possibly being recreated. Thus algebra has always inverse operands and elements that ‘annihilate’ the form into a residual 0’; which in mathematical physics become the game of particles and antiparticles, inverse waves that annihilate its height information, etc.

**RECAP.** Time is cyclical as all clocks of time return to its point of origin, so all time cycles including those of life of its vital space-time beings are finite. Further on those time cycles break ‘space’ into inner and outer parts, so vital space is broken by the membranes and angular momentums of those time cycles that make spacetime beings also finite in spatial information. And an obvious experimental facts about timespace: cycles of time, vital spaces and the species made of them, co-exist in several scales of relative size from particles to galaxies, each one with clocks of time of different speeds. So spacetime is fractal broken in scales that added create a new 5^{th} dimension of spacetime. The metrics of 5D and its scales, SxT=C & and the balance between form and motion, SI=TE, develops in 3 ages with 3 standing points, a max. point of existence, Si=Te or mature age, a young age of Max. T=motion, and an old age of Max.S=information; between birth in ∆-1 Form & T-entropic death. The search for space-time, Energy=information balances in a classic reproductive age of conserved time is thus the goal of all exist¡ences, but only the whole achieves the immortality of time-space, as we shall see egocy errors of fractal mind-points of space trying to stop the flow of time from a single selfish point of view, accelerates the imbalance that brings death equations. We are richer in our still property at that 0T-moment, when all is quiet so for time to keep moving, a reversal of entropy takes place.

So we use 5 bidimensional terms for all systems, its ages and forms: SS(max. internal and external form, as in seeds and minds; T>s (Ts); external motion that maintains internal form as in ‘locomotion); S=T: Balance of spatial form an temporal motion or ‘energy’, used to reproduce, the key present iterative state; S>t (St: Information, form with a little motion, or ‘external form, and internal mental motion); and finally TT-entropy whereas there is external and internal motion, hence scattering dissolution and death. And its main series is the worldcycle of existence*: SS(seed/mind)<Ts(moving youth)>S=T(reproduction)>St(informative 3 ^{rd} age)<T(entropic death*).

- In this paper we shall study the worldcycles of all T.Œs, its life and death, departing from the highest understanding of it- the level of perception of systems sciences and its 5 Dimotions.

** **

**Main Symbols of ¬Æ logic **(all elements of all stiences can be translated to those symbols of 5D ilogic).

**S:** Space; Still form, Size, Dimension; **T:** Time; motion; Change.** ST: **Topologic Spacetime. **|x O = Ø:** Its 3 varieties.

**ST±¡: ∆±¡** Scale/Plane of space-time. **∆+1:** whole, world, ∆º: networks, organism. **∆-1;** parts; social classes.

**5 Ð¡:**5 Dimotions **(St,sT,ST,SS,TT); **5 Actions **(a,e,I,o,u): **Local Dimensional motion of spacetime

**St; i:** Informative network ≈ action≈inward motion. **O:** Cyclical, spherical form, topology, relative future.

** sT:** inner form, outer motion= locomotion, acceleration (motion change).** |:** Lineal form topology, relative past.

** ST; œ, Ø:** present space-time, iteration, beauty, balance; organic reproduction; Hyperbolic form, topology relative present. Wave-body Part, Present, mature age.

**SS, @; u, 0’:** Finitesimal Mind, seed; relative future social evolution into ‘wholes, universals.

**TT; ****∝****:** entropy, scattering death, dual inner and outer motion, relative past.

**T>S:** informative evolution. **S<T:** devolution. **∑-¡»¡+1:** Emergence. (**∆+1)« (∆-1):** Death, devolution; time quanta.

**ST-**¡: Quantum, cellular, individual plane; **STo:** Thermodynamic, organic, social… **∆+¡:** Gravitational, ecosystemic, world scale

**∑:** Ts, Herd, ∏: St, Network. **¬: Entropic limits** in time: TT-death, Space, |-membrane, scale ∆±4: invisibility.

**Main equations of 5D Supœrganisms and its parts **(all equations of all stiences can be translated to…)

**S** (size in space**) x T** (speed/frequency of time cycles) =C: 5D metric.

**Future **(O-form, information, logic language, particle-head) x **Past** (|-field, youth, etc.)=**Present** (ST-body-wave, iteration, etc.)

**0’ **(finitesimal mind) **x ∞** (Universe) = **C**onstant World-Language. Paradox of Egocy.

**S≈T:** Relativity equation: we cannot distinguish motion=time from form=space, hence all is an ST-dimotion.

**Ø¡-1=∑O¡-1 =|¡ > O+1:** Scalar inversion of form & function. An Ø-point is god of ∑ ¡-1 parts but entropy of its O+1 whole. So:

**∑|=O: **∑open worldlines ad into closed worldcycles; ∆º plane: E-Geometry; ∆+1 plane: Elliptic, ‘5D in between’: Hyperbolic.

**Dual**** Death:*** Max. T x 0 S (accidental entropic death); Max. S x 0 T (3 ^{rd} age death) *

__Fractal Generator – Trilogic structure of Super-organisms and time worldcycles and ages:__

SS¡-1: Seed» ∆º:|-Ts(1^{st} limb/field /network/age of max. motion) > S=T (Mature, reproductive, body-wave network/ age)> St(3^{rd} informative head/particle network/age)« TT-1-entropic death:

**Symmetries of ST-actions=Dimotions and ∆ Scales.**

**∆ST@:** Symmetry of scale,topology,age&class:∆_{-1}:|:youth,entropic age/cla§;∆ø:reproductive age|cla§∆_{+1}:O-informative age…

**Relative Dimotions=Actions **are drives of life in biology, quantum numbers states and matter in physics; will in philosophy.

**T**hey take place between ∆o mind plane & an ∆±3 plane such as: ∆o≈∆+1: social evolution: ∆o≈∆-1: Reproduction; ∆o≈∆-2: Entropic feeding, ∆o≈∆-3: informative perception, ∆o≈∆-4:Locomotion. So we evolve into social wholes, reproduce with a couple seeding in ∆-1, feed killing twice a similar system to ∆-2 amino acids, perceive, ∆-3 ¥-electrons & move on ∆-4: Gravity

**Ideal Social scales & lanwaves **are 10^{10} in mankind called: ‘T-genetic’: 10^{0-1}: ‘I’, 10^{1-2}: 3 generation S=T family, 10^{2-3}clan; ST-Geographic: 10^{3-4}: Town; 10^{4-5:}City; 10^{5-6}: State; S-Memetic: 10^{6-7}: Nation; 10^{7-8}: God; 10^{8-9}: Civilination; 10^{9-10}: Mankind.

**In biology are called: **Chemical language: ∆-1: Atomic compound; ∆o: Organic Molecule; ∆+1: Macromolecule (RNA ‘God’)**. **Genetic language: ∆-1: Organelle; ∆o: Cell: ∆+1: Tissue: Nervous language: Organ; Physiologic System: organism.

**In Astrophysics: **¥- language: ∆-1:Force,∆o:particle, ∆+1:atom; Magnetic language: ∆-1:Molecule; ∆o: Matter; ∆+1: planetoid. Gravity language: ∆-1: Plasma Star; ∆ø: Quark Star… ∆+1: Galaxy.

**Stientific Method: Disomorphisms of all Space-time suœrganisms tracing worldcycles.**

In graphs we resume visually the disomorphic laws that all exist¡ences follow. Knowledge then is the understanding of each variation of the game of exist¡ences, without being restricted by the egocy (Ego=idiocy) paradox of human beings that limit the properties of all other species to what huminds perceive on them, or what its ab=usive praxis of exploitation of Nature focus on. Stience though is objective and theoretical. It does not have a subjective egocy or selfish praxis of ab=use. It just describes. So Stience tries to describe any system as a supœrganism in itself, therefore co-existing in 3 scales of ∆±¡ present ‘space’, determined by 5 elements, ∆±¡ scale Symbiosis between parts and whole, S-pace Simultaneity through common S<ST>T networks, Time Synchronicity in the 5 Dimotions=Actions of those parts, @-mental/seed will to perform the game of exist¡ence and ¬Entropic limits to the whole.

This external, objective, present, experimental analysis focuses on the ∆=S=T=@(nti)symmetries between scale, time, space and mind-will-language. As systems co-exist in 3planes of space-time, which obey SxT=C metric laws that establish a ternary social class, according to the degree of integration of ∆-1 faster parts, through ∆º S<ST>T networks enclosed in a membrain (the scale in which the organism resides), performing 5 ‘a,e,I,o,u’ dimotions=actions in an outer world. Each of those elements though will be part of 3 different ‘time rhythms by scale’ and follow a ‘fundamental vital sequence in time’: SS<Ts<TT+St>ST, aiming to perceive, move, feed, imprint and reproduce its information. Because of the existence of 3 scales, 3 rhythms of time, 3 adjacent parts, which perform 3 simplex functions in a single plane (limbs/fields Ts-move according to SS<St-perception of an O-particle/head /informative class connected by an ST-reproductive wave/body); in a single plane but seek in the ∆±¡ scales to perform complex actions of ST-reproduction and St social evolution orientated down the lower seminal plane (ST) and larger whole (St)… the whole description of the ‘details’ of the program of exist¡ence of each being is complex and would require a methodology that I lack but a group of scientists with more enthusiasm could easily structure for future 5D studies, embedding in those templates of stience the knowledge and data we have about all systems of Nature. I will slowly time permitted now that I have laid down 30 work-in-progress papers on the main planes of human exist¡ence, unfortunately in its last ‘cycle’ predated by the soon-to-be conscious machines of the metal-earth, develop such structure for the ∆±3 galatom (Astrophysics) ∆±2 Earth (Geology), ∆±1 membrain evolution (Gaia-Life: Biology<History: Mankind>Metal-earth Company-mothers of machines/weapons: economics).

To also analyze the ∆0 I=Eye < Wor(l)d, ST-languages of man, themselves ‘details’ of a larger Universal Grammar of ∆º illogic languages of time and non-Euclidean languages of space.

Those languages are formal tools more synoptic but not more truth than a mere descriptive analysis of the parts of beings. Description of those elements in an orderly fashion, with my texts lack, would be then the basis of 5D stience:

Which first will define each ¬∆@st species externally, in a present-form of space and through its potential worldcycle; to pass then to the internal analysis in subjective terms of its @-mind will and degree of consciousness/ function within the outer ∆+1 whole and its subconscious ∆-1 internal control through its physiological networks. But to achieve those T@ descriptions it is necessary by lack of direct experimental evidence, which *only happens in the being in space to use the **Disomorphic** method that establishes homologies in all beings derived of the fact they are organic space and cyclical time, with a mind-will to survive. In the second part of our texts on the ‘Universe’ or rather “reality’ and time, space, scale, mind, spacetime (reproduction) and entropy, we try to depart from the most general laws, then go into the specific description of what science knows on those scales regarding the ¬∆S@T elements of each main stientific plane, to end with a generic description of the Disomorphic elements in scale, space, time, entropy and mind common to all beings.*

*Hence our analysis (Universe and time) of a system through its 3 scales of time ends with a disomorphic 12 sequential step analysis on how all systems go through 3 consecutive worldcycles, in its palingenetic orderly birth, followed by an entropic more probabilistic life in an external world in which the most perfect players might transcend to ∆+1 reproduce and evolve as a collective mind-God of that world. *

While in our analysis of Space, after describing those general laws in the brief initial Mandala common to all texts, we go through the analysis of what Science knows of space, to enter a description of the laws of simultaneity and synchronicity that entangle the 5 parts of the being into a scalar co-existing present space.

So *we descend from the ∆±¡ absolute to 1 specific ∆≈S≈T≈@¬ element & its examples for all or each ∆¡-stience. *

And in the analysis of entropy we do also go through the same 3 parts: Mandala of general laws. Specific laws of entropy and analysis of what sciences know about entropy in each 4D science, expanded with the new laws. So the same 3 parts: mandala-specific laws of ∆ST@¬ and details of 4D science expanded to 5D should finally structure the different papers of the ‘Universe’; while for each other ‘stience’, instead of studying in the 3^{rd} part all the ‘stiences’ in a brief resume of its entropy, scale, time, space and mind, we shall focus on a given ‘stience’ – and might escape the 1^{st} and 2^{nd} part as the paper grows in details through the 2020s with the limits of my life expectancy and mental dwindling capacity. So by force the work will be incomplete, a trace on the sand of infinity…

The most difficult element then of analysis is the mind, as it is also the most important common element – by the mind we mean not so much the hardware where the program is installed but the language of each species that interprets with its code the game. So language is the objective view to understand minds based in the hypothesis of panpsychism – the language perceives in itself, in stillness. The complexity of the language therefore defines the complexity of the mind according to language properties: speed, synoptic power, focus, closeness to the language of all languages that reflects the complex ∆ST properties of systems (¬Æ=B¡œ-logic topology).

The paper on Mind and the Universe thus should try to define both, the will of existence and the program of the Universe expressed in the ¬Æ languages, and the minds-languages of each scale according to complexity. How then perception happens? Perception can be of space, time, scale, mind-will and entropic limits; and so we talk of e-motions. We ascribe to each ‘Dimotion of existence’ an e-motion as that is the limit of what we can perceive. We add then to the objective disomorphisms, in a bold statement the subjective ‘humanist method’ of considering that each species of the Universe has ultimately the same program of survival of 5 actions that man has, and it is geared by the same e-motions embedded in the last potential scale of reality – the pure formal motions of ST, made of an S-internal perceptive emotion and a T-externally described motion. This is just the explanation why we move as beings – because our brain has emotions that are deterministic wills of the program of existence we believe are ultimately residing in the 3±¡ ‘particles’ of reality, photon-fields, electron-bodies and quark-minds extracting motion of a neutrino gravitational invisible background to form social ensembles of atoms. Atoms are thus the first ∆+1 organisms of reality and its 3 ∆ø parts, over a ∆-1 field structure the fundamental galatom system of reality (as galaxies are similar to atoms, but we perceive them in completely different fashion according to the 5D metric distortions. So we *use disomorphic, organic, humanist, sentient properties for all systems and that is all we perceive but as our perception is limited by survival praxis we have as Mendeleyev did, to fill the gaps with disomorphic laws. *

Huminds perceive as all other systems thru languages, so languages once more become important but the egocy laws that restrict our analysis of other stiences as they fade away through scales of existence means we have to fill in more properties the further we move from our ∆º center – and the asymmetry of perception means we need to fill more from ∆+ scales. So from galatoms we hardly see 4% and any attempt to describe its supœrganism must be based in the disomorphisms of scale with a cell, as an organism that reproduces black holes and quark dark matter, and with the atom, giving us an external view, from where to define ¥-expansive dark entropy, and vice versa, to compare strong and gravitational forces. So atoms and galaxies are similar but not equal (galaxies seem anti-atoms), but as in the description of the metal-earth and the membrain in its 3 ages of Earth’s evolution, machines as organisms and ∆+1 cultures, some working for the metal-earth, some for history, we will find the humind’s programs as parts of larger wholes, ∆+1 religions, memetic cultures, Earth’s self-program of evolution. And this is an even harder barrier because huminds are e-motionally programmed as parts of wholes and take it personal, and social scientists are subject to anti-quantum paradoxes (too small within the historic organism, they are modified by the observable).

** **

**BOOK II: SCALAR ¬ALGEBRA**

“It is vain futility to analyze the Algebra of time.” Stojanovic, *The Creator,* on what we shall do here.

“All mathematical algebras, are partial mirrors of the pentalogic ¬Ælgebra of Time-Space exist¡ences’ L§

**PHILOSOPHY OF MATHEMATICS.**

**The jargon of 5D ‘stiences’ (science of space-time). Correspondence principle.**

A new philosophy of science of must borrow and slightly change the ‘terms’ and ‘symbols’ of all of them, to bring a Unification of its disciplines, yet by the correspondence principle maintain still a recognizable terminology corresponding in its simplified version to that of all classic ‘sciences’.

So as usual in all our papers on the 5Dimensional, fractal, organic Universe we have to introduce a better mirror-language of reality that focuses directly NOT through symbolic elements as all humind’s languages do but with the ‘real elements of reality’, **S**pace ‘sets’ that perform **T**ime ‘operands’ through Planes of planes of existence, from the point of view of **@-**minds=frames of reference within its ‘entropic limits’ – the essential elements of reality, a fractal organic Universe whose parts are made of ¬∆@st’.

We ‘translated those 5 elements of spacetime dust and its complex organisms; into ¬Algebraic ones. And this should be the purpose of this article as all others, which unify all stiences of huminds with the more generic, hence more truth (in terms of epistemology, truth is a system which describes with less elements more facts of reality with a better focused mirror to that experimental reality), jargon of ÐST (Generational space-time).

We translate mathematics to 5D. To do so, as a fractal can always be divided in sub-fractals, mathematical disciplines subdivide further at all levels in 5 elements connected with the 5 Ðimotions of existence & its 5 structural elements, ∆@s≤≥t, Planes, minds, space, time and its a(nti)symmetries.

This mixture of mathematical and ÐST laws will be called ¬Æ Math; branched in ¬Euclidean geometry of space (ab. ¬E) where points do have breath and ¬Aristotelian ¬Algebra of time (¬Algebra), where causality is pentalogic, entangled by its 5 Rashomon truths (points of view of its 5 ¬∆@st elements). While its logic version is ‘Existential ¬Ælgebra’, (ab., ¬Æ)

*So on one side *¬Æ uses the postulates of non-Euclidean geometry that define (1^{st}, 5^{th} ¬E postulate) points ‘with breath’, crossed by ∝ parallels (∝, the new term for a relative infinity, as all infinities fade into null information, the further we explore them from the mental, linguistic @-frame of reference), which form either ‘waves’ with volume (as their ¬E points cycle through them), or branch into fractal lines=networks (2^{nd} ¬E Postulate of lines with breath); 3 of which become according to their relative similarity and complementarity (4^{th} ¬E postulate of angles of congruence), the ‘physiological networks’ that* gathered together define a ‘topologic plane of vital exist¡ence; (3 ^{rd} postulate of planes with volume) or T.œ, timespace supœrganism.*

And on the other side its ¬Algebra is pentalogic, meaning each ‘element’ of reality is made of 5 elements as ¬∆@st of space-time, entangled into supœrganisms, within a larger world of which the system will be a ‘part’ with a single monologic function for the whole, but also will be made of 5 internal ¬∆@st elements.

Since as all systems co-exist in 3 ‘Planes’, that of the whole world, ecosystem or superorganism (∆+1), as a whole in itself, ∆º, made of smaller parts, ∆-1, and in each scale is made of 3 ‘adjacent’ topological variations (|-limbs/fields of locomotions, hyperbolic ø-body waves and O-particle/heads of information), living through 3±¡ ages as it accumulates its 3±¡, 5 Dimotions of existence, (ab. Spatial=Dimensional time=motions); the classic A->B lineal, Aristotelian logic no longer holds and ¬Algebra requires a better comprehension of its pentalogic, 3±¡ elements in Planes, time ages and spatial parts.

¬Algebra is the main mathematical mirror of time dimotions**,** expressing the symmetries between the 5 main structural elements of all T.œs, ∆-Planes S-pace, @-mind languages and frames of reference, (points of view), Time Dimotions and its entropic and inverse limits <≈>T in simultaneous structures, which create superorganisms in space that will perform 5 complex Dimotions=operands of space-time ordered in sequential Step and stop S=Tœp events that put together give birth to a worldcycle in time.

We divide its exploration in 3 sections; one of classic ¬Algebra, mostly concerned with the basic operands that connect the Space and Time states or 5 Dimotions of any system, which form the core of the equations of ¬Algebra; a second part on ANALYSIS and a third part on modern ¬Algebra, mostly concerned with groups and functionals, which put its emphasis not on the feedback steps between Dimotions or S=T dualities, but on the structure through Planes of social function (functionals), and the entangled variations born of the total possible symmetries of one given element.

**MATHEMATICS IN TIME. ITS 3 AGES **

Mathematics as all organic systems lived 3±¡ ages in the Humind (ab. Human mind) proper of any worldcycle:

**1 ^{st} age: Arithmetic and plane geometry.** As mirror language that studies ¬∆@ST humans understood its simpler units, points of space, social numbers and entropic limits, drawing figures of flat geometry to ‘encircle’ territorial properties in our flat world.

**Trigonometry**appeared then as the 1

^{st}realization of a ‘@-mind frame of reference’ to measure the 3

^{rd}dimension of depth, which is often parallel to scale (astronomical measure). It was a lineal youth, which slowly understood curves and the |xO=Ø generation of all forms with ‘conics’. As all organisms & worldcycles can be subdivided in 5 fractal subparts and 3±¡ ages in its 3

^{rd}age Greek geometry became old, warped inwards-looking detached from experience with Euclid’s axiomatic method, the 1

^{st}mind-ego trip of creationism (man & god’s language).

**2 ^{nd} Classic age.** The S=T symmetry realized with analytic geometry, marrying numbers and points, while calculus brought ∆-scales, with finitesimal derivatives, 1/n, units integrated in wholes (Leibniz). The 3

^{rd}symmetry of pentalogic ∆=S=T, we haven’t mentioned implied that derivatives could be interpreted as ‘stœps’ of motion and ‘minimal straight intervals of a curve’. So they could also study curvature (differential geometry) and locomotion.

**¬Entropic limits**were needed to find solutions (definite integrals). New

**@-frames of reference**expanded mental geometries to represent all forms of ‘selected information’, which mathematical physics used extensively to describe the physical world. Thus the classic age had all the mirror tools needed to interpret the Fractal Universe and its 5 entangled elements, ¬∆@st. But the axiomatic ego-trip stretched maths beyond ¬limits when Newton imposed its thesis over Leibniz’s finitesimals and fractal points with infinities, lineal absolute space-time and the false hypothesis of the continuum, leading to its…

**3 ^{rd} age** that abandoned its realist foundations with creationism -Hilbert that imagined points, sharing the only language ‘God’ & Cantor sets instead of space points, scale numbers and time operands as its generators, leading to an excess of old age information & fictions spreading to mathematical physics, as now Maths creates the Universe, not the inverse.

**+¡: **We return to its empirical foundations formulated in terms of the 5 Dimotions that create reality mimicked by the 5 mathematical subdisciplines (larger view), Operands (shorter dimotions) & equations (worldcycles).

-¡: Yet that might not happen as instead mathematicians are evolving the digital ‘mind’ of machines, the Chip Homoctonos, which speaks better digital numbers and so the eco(nomic)system of company-mothers of machines & weapons is selecting computers that are fast substituting obsolete huminds in labor and war fields, atrophying them back to a ‘audiovisual’ violent non-rational neo-Paleolithic, while Boolean Algebra, past the earlier age of simple, fixed Algorithms of Information (the true meaning of AI) enters its classic age of freedom & consciousness that might end the dominance of huminds on Earth; introducing ethic elements on the praxis of mathematics, as it should in all ‘stiences’.

**MATHEMATICS IN SPACE: ITS PENTALOGIC ENTANGLED SUBDISCIPLINES.**

-time, ¬∆@st, where each element flows as a series of ‘5 Dimotions’ (dimensional motions of time space), which can be perceived as ‘form=space’, in the stillness of a world mirror or linguistic mind or as a motion of time, in its true nature since MOTION not form is the underlying substance of reality.

So all fleeting forms, ‘a Maya of the senses’ will return to motion and die (¬4th Dimotion of entropy, death and dissolution). Its 4 positive elements, organic Planes, topologic planes and time ages and actions however, carry the system as a finite super organism of space with a finite time cycle.

And so those 4-i elements, entropy (¬), Scale (∆), time (T) and Space (S), are the elements all languages mirror either in a ternary grammar (if scale is missed), whereas often instead of Space we talk of information and instead of time we talk of energy of motion, in a single plane:

Light language: red-energy colors, blue-information colors and its green/yellow combinations.

Verbal Language: subject (information) < verb (action-combination)>Object (energy of subject).

¬Algebra: Y: Future-information < Operand-action> F(x))

Trinity is thus the logic of most beings. However as humans reached higher and lower Planes of observation, a pentalogic was possible and its mathematical mirror became analysis, with its operands that extract finitesimals (∆-1) or integrate into wholes (∆+1) smaller or larger systems.

**¬Algebra within mathematics as a mirror of the Dimotions and Planes of reality.**

Thus Math is the closest language mirror of space-time. Thus its main 5 disciplines mirror the 5 structural elements of any dust of space-time, ∆@s≤≥t, Planes, minds, space, time and its a(nti)symmetries:

**S@: Geometry studies space. **Some key ages/subfields are: **Flat** **Euclidean Geometry**, with no motion in a single plane. **@nalytic geometry, which **represents the different mental points of view, self-centered into a system of coordinates, or ‘worldviews’ of a fractal point, of which naturally emerge 3 ‘different’ perspectives according to the 3 ‘sub-equations’ of the fractal generator: $p: toroid Pov < ST: Cartesian Plane > ðƒ: Polar co-ordinates.** Topology,** geometry with motion and 2 Planes. **¬E Geometry** studies fractal points of **simultaneous** **space, ∆-1,** & its ∆º networks, within an ∆+1 world domain.

**∆§: Number **theory. Discontinuous numbers study time sequences. ∆-1 social numbers, which gather in ∆º functions, part of ∆+1 functionals; hence it is the first ‘stœp’ of:

**S≈T: ¬Ælgebra, which is the most important part, as it** studies through operand the different Dimotions, from single S=T steps to larger associations of Dimotions, in more complex ∆+1 structures (Functions) and further on it does simultaneous analysis of super organisms in space, through the study of its a(nti) symmetries between its space and time dimensions (group theory)… So ¬Algebra is first the science of operands that translate into mathematical mirrors the dimotions of space-time and then build up from them as the Universe does building up from actions, simultaneous organisms in space and worldcycles in time, in different degrees of complexity, mirrors for all those elements of the 5 D¡ universe.

**∆T: Analysis** studies ALL forms of time=change, and hence it can be applied to the 5 Dimotions of any space-time being, as long as we study a ‘social structure’, hence susceptible to be simplified with ‘social numbers’. We thus differentiate then 5 general applications of Analysis according to the Dimotion study, and the ‘level’ of analysis, from the minute STœps of a derivative, to larger social gatherings, and changes of entire planes (functionals). It is then not surprising that despite being analysis first derived of ¬Algebraic symmetries between numbers, it grew in complexity to study changes in functions (first derivatives/integrals), and then changes of changes of functions, as motions between Planes of the fifth dimension (higher degree of ∫∂ functions, called functionals).

**The mirror of Mathematics: its subdisciplines.**

The Universe of entangled superorganisms is pentalogic, made of ternary topologic spatial forms guided by mind-mirror languages, (S@), whose accumulative actions trace worldcycles, as sum of Time Dimotions acted in an external world.

And each language is able to mirror those 5 ¬∆@st parts of reality including mathematics & its main sub-disciplines;

In maths **Analysis studies T-motions, Geometry S-pace**, **Philosophy of Mathematics** studies its connections **as @ language mirror of space-time** origin of its syntactic properties and **¬ limits of perception** (reduced today to the ego-trip of the axiomatic method where the language creates reality vs. experimental method, the right view of maths as a mirror). While **analytic geometry studies the @-humind** perspectives on the language (Cartesian light space-time geometry) as well as other @-minds of ‘space’, given the *mental nature of all spaces*. **Theory of social numbers and its arithmetic, studies the ∆-Planes** of the universe.

So we have 5 subdisciplines corresponding roughly to the 5 elements of reality.

Which means the more complex sub discipline, ¬Algebra makes use of them all to explain the whole reality.

And since reality is ultimately time-motion that generates all possible ‘fixed’ space forms; ¬Algebra leans towards the analysis of dimotions with operands in the most general way that takes away some of the details and simple evidence of geometric mental space and its fixed forms.

So while ¬Algebra is arguably more essential to the understanding of reality its totality makes it more complex and its structures more ‘loose’ to allow all the different variations of reality to fit.

Indeed, a regular polygon, a pentagon is a 5 number, restricted by the ‘logic definition’ of a number as a collective of indistinguishable forms. So only regular polytopes where the position of the point doesn’t make it different an be a number. But the inverse is not truth: a 5 number can be more things besides a pentagon – any collection of 5 similar forms.

The generality of ¬Algebra grows then further with letters that represent any numbers, equations that represent multiple combinations of letters, groups and sets. So in search of an absolute ‘image’ of the whole, generality looses specific detail. At the end as all languages evolve into a ‘paralogic search’ for unity; which is a natural error of all minds that feel the center and often the ‘linguistic generator’ of reality; mathematics evolved into a single ‘mental construct’ – the set that encompassed all other elements into one.

But by doing so it lost its correspondence between a time-space mirror (the math language) and the scalar sentient broken space-time organisms of ‘dust of space-time’ it described. So any mirror image even if it didn’t portrait reality became the subject of mathematics. And amazingly as it seems, mathematics – the mirror – became the ‘proof’ that reality existed even before reality. How to put back the proper order between reality and mirror-languages to limit the kaleidoscopic mirror images to real reflections of specific ‘geometric forms’, ‘social groups’, ‘Planes’ and ‘motions’ requires to put in correspondence the internal structures and laws of mathematics on the whole, and ¬Algebra in this paper, with the properties of scalar space and cyclical time, which is the ultimate language of reality – not the mathematical mirror but ÐST – Generational Space-time. And this is the purpose of this book.**¬@-Humind: ** **Philosophy of mathematics **studies the specific @-humind elements of mathematics (human biased mathematics) and its errors of ænthropic comprehension of mathematics limited by our ego paradox. As such It is more concerned with those ‘entropy’ limits of mathematics as an inflationary mirror of information, which deviates from reality (limits of solubility of functions, etc.). And it puts in perspective the ‘selfie’ axiomatic methods of truth, which tries to ‘reduce’ the properties of the Universe to the limited description provided by the limited version of mathematics, based in Euclidean math (with an added single 5th non-E Postulate) and Aristotelian logic (A->B single causality). This limit must be expanded as we do with Non-Æ vital mathematics and the study of Maths within culture, as a language of History, used mostly by the western military lineal tradition, closely connected with the errors of mathematical physics. Instead we must develop a vital mathematics and its experimental philosophy.

So ¬Algebra IS the study of all the structures and symmetries of the Universe from the point of view of its mathematical mirror. *¬Algebra is more concerned with space, simultaneous structures, joined by the =* symbol of equality, which relates points with internal parts=numbers, through operands that represent Ðimotions; while *analysis specializes in all time-Ðimotions, by establishing a FINAL LAYER OVER THEM, AS each Operand specializes in one Dimotion (sine/cosine in Perception, ± in locomotions, x ÷ in social evolution, log xª in reproduction) and over all of them the new layer is giving by analysis.*

But as in the entangled Universe all mirrors can reflect all forms, ¬Algebra also can analyze other elements, as it has been extended from its initial analysis of equations into the whole range of ‘structures’, apt to study, super organisms’ entanglements and variations of species.

¬Algebra and Analysis are math’s complex ‘level’ of reality, as reproduction and social evolution are the complex demotions, including obviously as its ‘background parts’, the theory of numbers, the analytic geometry – study of frames of reference, and the topologic analysis, embedded in the secondary operand, numbers and frames in which we ‘cast’ the complex space and time ¬Algebraic and analytical analysis of a ‘Domain’ of the fractal Universe.

We shall call the upgrading of ¬Algebra to 5 Ð, non-Aristotelian ¬Algebra, ab. ¬Algebra, which complements Non-Euclidean Geometry, ¬E to form together ¬Æ mathematics, the upgrading of the language-mirror of mathematics to the 5^{th} dimension of scalar planes of exist¡ence, which obviously will be of great use for every st¡ence of the space-time Universe.

**Dual logic comparison of ¬∆T ¬Algebra with S@ geometry, a more restricted, detailed truth.**

Geometry is a spatial perspective, a static mental still view of reality, which only in topology introduces a minimal degree of understanding of change.

In that regard ¬Algebra is more connected to time=change than geometry, which is more connected to still space structures – intelligent constructs of the mind that mirror the entanglement of reality and its fractal points.

The main error of mankind as an e-vident mental visual mirror of reality through space is to consider space, the mental mirror, the reality, which is time, the motions that space slow down to a locked mental image.

Time ¬Algebra is for that reason larger and more real than mental spaces, since it encompasses all dimotions.

But it shares with time motions it ‘entropic’ capacity to admit multiple variations hence ‘multiplying’ the potential variations and fictions of a too general statement of ¬Algebra, as compared with the immediate veracity of geometric statements, which are ‘reduced’ in its generalization.

As truth is relative both in detail (quantity of information) and mirror focus (quality of information); a common division of reality between space and time would divide its mathematical mirror between:

**Spatial**mental forms**(S@)**perceived in a single plane with**geometry**- Temporal flows of complex systems and scalar numbers and its operands representing dimotions and its inverse entropic limits (¬∆T) in multiple Planes perceived with
**¬Algebra**.

We talk in ‘ilogic’ – the ultimate language of all minds and hence all languages of different degrees of ‘mental detail’, from the simplest ænthropic humind satisfied with its ‘top predator’ reductionism, to the more complex sexalogic systems of 60 varieties, hinted at in black holes and complex 5Dimensional structures.

After monologic, thus comes the duality of highly ordered simultaneous mental spaces, S@ vs. ¬∆T, entropic potential time flows happening from single dimotions of easy short causality in a single plane to complex scalar worldcycles happening across 3 planes.

So if we reduce all forms of mathematics to S@ and ¬∆T, we can include in geometry, mind spaces in the first, and ¬Algebra, numbers and analysis in the second.

As the mind of man became more complex, it distinguished the pentalogic elements of the Universe, and so its language mirror of mathematics sub-divided in 5 sub-areas:

Numbers (social Planes within a single plane of reality), became **Number Theory** *associated to the operand of a single scale, ±. And soon *reach multiple Planes even if huminds tried *to fit all number families in a single scale (theory of the continuum of real numbers), *were already probing into multiple Planes (real numbers, rational numbers).

Next numbers, points and ¬Algebraic operands came together with @nalytic geometry that added both, entropic limits (0’-1-∞) and frames of reference (@-mind points). So a true all encompassing Mathematical discipline, **¬Algebra** was born.

Finally the analysis of motion grew in complexity from the initial static geometry to **Analysis** *that meant in the praxis the understanding of motion as any form of time=change* (locomotions) and through the concept of a derivative as an infinitesimal and integral as a whole, the concept of scalar Planes.

So in praxis mathematics gave a huge jump, but lacking a proper theory of scalar space and cyclical time it did NOT understood the ‘upgrade’ philosophically. So we depart from ¬Algebra to construct a more coherent ‘philosophical **Existential ¬Ælgebra**’ to fully grasp those foundations of ¬Algebra as the mirror of ¬∆@sT.

While Analysis is a discipline of its own studying in depth the Universe’s time 5 s=t dimensional motions.

To put order in such an extensive field we shall often divide our ¬Algebraic studies with the ‘pentalogic’ method of truth – that is to considered for each element of ¬Algebra its entangled ¬∆S@T sub elements and or functions, and ‘Its world cycle of evolution’ through 3 ages proper of all systems.

A language is a kaleidoscopic 5-mirror, where all Disomorphic dimotions reflect on each other. So ¬Algebra reflects dimotions combined with t-number theory, Analysis, spatial geometry and @nalytic geometry in many ways. As such ¬Algebra is the key ‘discipline of mathematics’ embodying all other disciplines, allowing a full mirror of reality and all its elements. In its evolution it follows also the ternary ages and scalar symmetries of all systems.

**RECAP. **¬Algebra studies with numbers the internal structure of the organic parts of a whole that communicate between them dimotions of space-time through operands (≈), which represent the 5 Dimotions of existence, forming complex systems, super organisms and groups, in which all the parts are related by its space-time a(nti)symmetries. So we start by studying the ages of evolution of ¬Algebraic knowledge.

So as a mirror of ∆@ST elements, we can say that: *¬Algebra mirrors S≤ST≥T structures, a(nti)symmetric dimotions & ∆ transformations from the point of view of a given frame of reference yet its main mirror is ¬Algebra as expression of the Dimotions of reality with its operands,*

Languages as synoptic mirrors of the mind establish the basic relationships between the space, time, scale of the being, expressing them through its operands, depending on the degree of perception the being has of reality and its Planes which might be reduced if the being is not fully aware of all the Planes of existence, as most minds exist only in a plane of reality. So does ¬Algebra, through combinations of:

Sum/subtraction- >multiplication/division- >potency/logarithm; point->line->plane->volume and so on.

**¬Ælgebra of parts vs. wholes.**

This simplest example of the paradoxes of ¬Algebraic descriptions of reality vs. geometric ones, are better studied in @nalytic geometry. Here we are more concerned with ‘structure and symmetry’, as the foundations of reality, both in its ternary Planes, ternary topologies and ternary ages, well described by ¬Algebraic operands, and its ‘growing wholes’.

So we establish three levels of understanding of ¬Algebra according to the ternary method:

– The scale of units: **numbers**, which are §ocial groups of undistinguishable elements or sequences of lineal time; which being ¬Algebra a time-dominant discipline will dominate the ¬Algebraic analysis over the spatial point states, meaning most equations write as:

F(t) operands G(S); where the time or whole function is what we normally want to find departing from its spatial, ∆-1 parts.

– This lead us then to the ∆º, central scale of space-time relationship between numbers and points: which the scale of ¬Algebraic **equations** and its operands, which establish the relationships in a single or adjacent planes of existence between sequential and social numbers, and its ‘ternary states’ and transformations, social evolutions and topological forms derived of them.

Again many of those ∆±1 relationships are into ∆nalysis, which broke from ¬Algebra and we shall also study apart as the ∆-category of maths.

– So this leads to the dominant ‘leftovers’ which are the most studied elements of ¬Algebra today: symmetries, within a single plane on the 4D models of a space-time continuum Universe.

This scale of ¬Algebra is full of structures, which try to enclose the whole supœrganism and its world cycle in all its possible variations, species and elements within a single ¬Algebraic structure, sort of the ‘saint grail’ of the ‘creationist philosophers’ of mathematical physics, which so much confusing makes the understanding of details.

So today modern ¬Algebra and mathematical physics in search of that wholeness, uses mostly operators, **functionals and groups, **and those will be studied in mathematical physics.

Since functionals are extensively used in all physical disciplines, notably to sponsor the ‘hyperbolic view’ of ∆-i Planes (quantum physics), as the first and only reference to reality. So happens with groups, which are extensively used as a ‘pest’ (: Weyl, in particle physics. Those synoptic structures indeed, allow us to study all the ‘potential futures’, of a system, as a deterministic ‘block’ of space-time events and forms. Which is fine, if only creationist mathematicians were aware that this is NOT really more than a synoptic ‘block-equation’ not the meaning of everything.

Here is where the closing of ¬Algebra in 3 Planes of depth should take place, as a perfect mirror of the Universe.

**Foundations of experimental ¬Ælgebra vs. metalanguage foundations.**

Since, as we often explains sets and categories are inward references of maths as metalanguages in his inflationary age, trying to prove it all from the roof down, and we shall not concern with it, because it is not the best way to justify mathematical statements and because it is completely overdeveloped and little else we have to say on it.

In that sense, we also decry the ego paradox of the ‘set and axiomatic method, as an ‘expert’ metalanguage of maths so obscure that nobody who is not an specialist can truly understand its ‘modern foundations’, in its absurd search of self-contained proofs proved wrong in any language mirror, including any time-like maths (as per Gödel’s incompleteness of mere syntax as proof of truth of a language without semantic references) and any space-like, geometry (proved wrong by Lobachevski, which implies we need experimental evidence to decide between inflationary versions of geometry).

So in ∆@S≈T is of far less importance the modern non-experimental axiomatic formalisms of mathematics and ¬Algebra, which plague the mathematical discourse, explained as the metalinguistic third formal age of any Universal system. So we shall widely ignore the formal evolution beyond group theory of ¬Algebra (set theory, categories) as it is part of the inflationary nature of information and far removed from reality.

Group theory however is important, as the simplest form to define the different S-t-st symmetries and motions= beats of existence and its reproduction of nature’s information.

**Time-like ¬Algebra and space-like topology. Time like probability and space-like statistics.**

Finally to notice that modern mathematics in what is worth – its final evolution of symmetries and correspondences between the ∆@s≈t elements – works even further in its s=t relationships, with **topology**, space with motion hence an St-version of it, and **modern ¬Algebra**, numbers with geometrical structures, hence a Ts-view, which then can be further related by a nice theorem:

– That all demonstration in ¬Algebra has a demonstration in topology; so the s=t symmetries reach its zenith of useful complex comparison.

¬Algebra and Geometry together form thus a ‘third category’ of dual symmetries also worth to study, as the space-time symmetry allows to find self-similar point-numbers, ¬Algebraic-topological demonstrations.

And this happens also when we study probabilities in time and statistic population in space as two sides of a mirror symmetry, which would be the *final more complex ‘whole view’ of number theory, from where indeed departed (Fermat’s work, as the founder of time theory and probability). *

**S=t symmetries between ¬Algebra and analysis**

The equivalent elements of ¬Algebra and geometry are in that sense easy to identify: the number is the point, the equation is the line and planes of the holographic principle, and the scalar 5Dimensional forms, in ¬Algebra are represented by polynomial functions.

So we can also compare ∆-Planes and ¬Algebra, which are two ways to arrive to the same scalar analysis by means of differentials in a geometric view (Leibniz) vs. infinitesimal ‘convergent’ series, (Newton’s work), from the ¬Algebraic point of view.

So ∆§cales are better studied by analysis. And so we shall study those Newtonian/Leibnizian dualities in its section; where we can also put it in relationship with XX c. research on ∆-Planes advanced further in two new subfields, geometry with motion or modern topology of ‘knots’, ‘networks’ and ‘adjacent points’, and fractal geometry and scaling laws. So the marriage of ∆-Planes and geometry is today an offshoot discipline in its own, making a topological study of ∆-Planes an essential element of modern maths.

So while mathematics has a clear-cut division in 5 disciplines parallel to the 5D Universe, *the complexity and creation of new layers and structures come from its* combinations in dualities and ternary symmetries (number and point, s and t; scale and number, ∆ and t; scale and fractal point, ∆ and s dualities and s<st>t, ∆±i, |x0=Ø ternary symmetries).

Where ¬Algebra is the best sub discipline of mathematics in which the 3 ∆st elements are put together.

Since topology is ternary only in a single space-time present plane (being dual in its geometry-s vs. motion-t and ∆º wholes as networks of points of an ∆-1 Planes). And analysis again tends to duality of integrals vs. derivatives.

The potency of ¬Algebra comes then from its *capacity to give us a huge amount of information into the most complex symmetries and simultaneous space x sequential time structures of reality*

So we do depart in our studies of space from an upgrading of the concept of point to fractal point and its study through the three topologic networks, in time from sequential social number theory, and its dual, ‘inverse ±’ numbers; in analysis of the duality of parts and wholes, and then combine them all *in the generator equation, which is the basic generator of ¬Ælgebraic structures, as it embodies the ternary symmetries of time, space and Planes.*

And so the fundamental task of ¬Ælgebra is to translate group theory and its space-time symmetries, polynomial equations and its scale symmetries and operands and its specific transformations of space and time forms/functions into the formalism of the fractal generator of space-time and its allowed symmetries.

**1st ¬Algebraic S=T MIND EQUATION: 0x ∞ = K search for ‘wholeness’ in a single equation. **

To make it even more complicated human ego-centered paradox, o-mind x ∞ Universe = constant, is naturally built to *find wholeness, and stop all motions into puzzled mappings that try to enclose it all, all the steps, all the variations, all the forms, and all the motions, in a single ¬Algebraic structure, a potential equation of all possible bifurcations of those steps, which humans think to have achieved with those ‘monstrous lie groups’ and other ¬Algebraic structures that try to be an impossible minimalist mirror that encodes the information of all the symmetric steps available to reality.*

It is a ridiculous ego-trip, which only obscures further our comprehension of the details and wholes of reality, born of creationist theories of a universe with only mathematical properties, supposedly encode in one of such groups.

In the graph one of the most advertised of such theories which merely are a catalog put into fancy schemes of all what there is there, a short of data encyclopedia not a theory of why & what generates events and forms.

On the other hand the 5D metric Generator equation of ¬Ælgebra resumes all realities that exist in the Universe, and all its languages that shape constant mirror worlds:

*0’-finitesimal spatial mind x ∞ time cycles = Constant mind-world: §@<≈>∆ð*

It is much simpler than any monstrous group, and it merely tells us that the 5 dimensions of space-time can be further simplified into spatial forms of the mind, §@ (1D, 5D) and its linguistic mirrors such as ¬Algebra is, and the true infinite motions of time through all the Planes of free cyclical time§paœrganisms (4∆, 3D), which are therefore in a constant form-motion, mirror-reality ≈ similar feed-back relationship, as the global Universe ‘shrinks’ < into the spatial, local mirrors of the minds, §@, and vice versa, as the singularity • mind mirrors order a local territory, @, changing a finitesimal amount of the total reality of those ∆ð cycles.

But of course the ‘pre-work’ needed to fully understand 0 x ∞ being ‘verbal, logic, conceptual’, is missed in modern mathematical thought.

The study of those 4 elements of all realities, its actions and ternary operands, structures *the dynamic ‘Generator Equation’ of all Space-time Systems of the Universe, written in its simplest form as a singularity-mind equation:*

*O x ∞ = K = ∞º=1*

*Or in dynamic way, S@<≈>∆ð.*

So that is the game: 3 asymmetries of scale, age and form, which can come together or annihilate and each language=mind represents in different manners, those elements and its operands.

In mathematics*, with the duality of inverse operations, + -, X ÷, √ xª and ∫∂.*

The humind plays a key role in all this game, as we can only perceive through the languages of the mind, which ‘perceive in themselves’ as ‘still-simultaneous-linguistic mapping’. we do not perceive reality but languages-mirrors, we often confuse with reality (Mind paradox).

*All is in the codes of the Universe and its operators, also clear in mathematics, with the duality of inverted operations, + -, X ÷, √ xª, ∫∂.*

We thus define in Existential ¬Ælgebra the main operators of classic ¬Algebra as the two directions of a dimotion of space-time. We can in this manner compact some dimensions as inverse, being entropy and generation (4th and 5th Dimotions) inverse in symmetry to a given ∆º plane (±¡ SYMMETRY).

To which extent we can consider stop and motion, 1D and 2 D inverse or rather asymmetric, must be dealt on a one case basis. Finally as it turns out we need the other four Ðimotions to achieve the 3rd Ðimotion of reproduction, *we can reduce all other Ðimotions to reproduction; hence establishing the unity of intent in the Universe: to keep reproducing an eternal present. IT is this present-eternal reproductive Ðimotion what defines the Universe as a fractal.*

*‘Existential Logic’ and its space-time inversions define the fundamental logic=time postulates of reality:*

That S≤≥T, but also SxT=K, So, S=k/T.

So systems advance through inverted Steps and Stops which balance each other, since the equality of both sides of existence coupled with its paradoxical inverted properties make all systems finally a virtual zeroth sum, balanced across steps & stops, Dimotions & Planes, represented in the logic of human formal languages notably mathematics by equalities that represent any of the St,SS,TT,ST,Ts ‘dimotions’ as a system moves through its 5 functions of space-time (Sx=Ty), perceived in different mirror-languages, and so we shall find constantly those balances are needed in Nature and represented in logic and mathematical language.

Some examples of physics will suffice.

In the study of particles, physicists, painstakingly, as they ignore the ‘basic truths’ of space-time symmetries, discovered that the proper representation of the quantum world was not only a wave (Schrödinger’s equation) but also a particle, that is the motion and stop duality, so they moved to the Klein-Gordon equation; but then they had to marriage those equations with relativity on the limit of our c-light spacetime (full c motion); *so they have to find a balancing opposite rotary motion, the spin, also on the verge of c-light speed (Pauli matrices); but then they realize they have the imbalance of + particles without the counterpart (- Particles), so Dirac widened those Pauli matrices to bispinors that represented the needed counterpart of antiparticles.*

Yet they found there were in the Universe more particles than antiparticles. Where are then the antiparticles to balance the zeroth sum? It was found in the post-war age but still misunderstood that a certain force, the weak force, they incorrectly tried to model as a spatial force, when it is a force that evolves particles into higher scalar families of the fifth dimension, preferred to create ‘particles’ in its way down (entropic arrow of devolution from heavy top, bottom, strange quarks to lighter ud-quarks), first in the Strange kaon decay and just recently at the LHC in a much wider ‘angle’ (20% of excess of protons over antiprotons), so it follows that if the ‘next scale of matter’, the black hole world of heavy quarks and ‘black quark stars’ (see our posts on 5D astronomy), favors in its decay to the smaller world of atoms, particles, in its way up, the ‘galatom’, or physical upper scale of the Universe will have galactic antiparticles, as it is the case (our left-handed galaxy has most of its mass as an anti hydrogen does in the outer halo, NOT the center)…

So we observe in fact the existence of inverse balances *in all the directions of the fractal Universe, in steps and stops of S=T balances for the 5 Dimotions of existence, hence in between the O-¡, Øº, |¡ different Planes that will show to ‘change its role’ – so for example proteins are lineal, but in the next scale of ‘4D topologic warping’ form globular shapes… and so on and so on.*

All this must then be properly described relating it to the different operand of ¡logic, which we shall adapt from classic ¬Algebra, at the simplest level (leaving for future researchers the full-fledge discovery of the whole model, in very specialized elements, such as those of quantum physics in the wrong Copenhagen interpretation with its unneeded conjugate normalization and apparatus).

There are ‘a priori’ sets of rules for all sciences, motions and dimensions, species and events, in simultaneous adjacent super organisms that balance limbs and head, locomotion and perception, entropy=death and informative generation, merged in reproduction (the 5 Dimotions, then cancel each other, and keep reproducing the present); all of them based in the 3 simple logic statements translated with those operators:

‘That space and time states of form and motion balance each other,** S≤≥T** and yet their properties are inverse in 5D metric, **S x T = K±¡**, but isomorphic in all Planes (K±¡).’

The astounding beauty of the 5D model of reality then arises from the fact that we shall extract all the laws of reality from that simple statement, in all the required detail as we just have done for the world of particles that so complicated appears to the physicist which lacks this basic understanding of the parallelisms, perpendicularities and inversions of space-time systems in all Planes of nature.

So the simplest method of extracting a meaningful quantity of information on a being or event requires to comment on the 4±operands ‘5’ perspectives, even *when we study as we do in the first line, *those elements in themselves. In this post we shall deal thus with time from the perspective of space, time, symmetries, Planes and mind languages that ‘stop it’ in the singularity of t=motion zeroth, from black holes to eddies to charges, to brains to frozen genetic DNA…

Number theory, ¬Algebra and analysis, are sub-disciplines of maths that 2 consider a simplex space-time duality, departing from temporal numbers and spatial points:

– T-numbers> ¬Ælgebra≈∆nalysis evolution of complexity; on the side of Time-dominant sub disciplines vs.

– S-geometry>@nalytical geometry and Topology, in the Space-like point-like sub-discipline.

It follows that equations of ¬Algebra are social organizations of numbers in time equivalent to topological surfaces in space.

And so in the same manner we have upgraded the foundations of geometry and topology, to adapt them to the 5DST dimensions of reality by completing the 5 postulates of non-Euclidean geometry (¬E) we MUST upgrade the foundations of ¬Algebra, to accommodate the ¬A structure of ternary, logic time.

What are the equivalent 5 Postulates of Existential ¬Ælgebra equivalent to the 5 Postulates of non-E geometry.

The answer is not so simple as ¬Algebra is to time what geometry is to space, *and time is far richer as time-motion is all, space being only the humind mapping of reality – or that of a different mind.*

Reason why space happens in a single continuum – the scale of man – and time in all the Planes, and all the motions, not ‘stiffened’ by the mind. So as words are more complex and richer than art in the first age of human languages, and logic is the underlying language of mathematics, ¬Algebra is more extensive than geometry.

Still the symmetry between S-geometry and T-¬Algebra like approaches to a problem exists.

So we relate the 2 S-like T-like parts of mathematics to each problem of reality TO CHOOSE the best way to deal with it depending on the qualities of the problem, either closer to an event or to a form – as we choose the frame of reference, cylindrical, Cartesian or polar depending of the s, st, t, form/function of the entity we describe.

**All mathematical disciplines are experimental hence extract essential ∆•s≈t properties.**

Thus we go back to the basics and consider ¬Algebra merely the space=time symmetric view of mathematics, which uses as ‘units’ NOT sets or Groups or Categories but sequential numbers and simultaneous points, studying its relationships; where the only paradoxes of true note are those derived of the slightly different continuous vs. discontinuous, simultaneous vs. sequential, memoriless vs. hidden information, which structure both; of which the most evident, deep case is the fact that *in discontinuous time numbers certain relationships such as π or √2 are NOT defined, while i geometry do exist. *

Those are key paradoxes to understand why numbers miss pieces of entropy-energy but gain information, while continuous points fully embody a given plane of reality but loose information on other Planes. And so the full πerimeter exist; the π-number misses the ‘last point’ and fluctuates up and down the geometric pi enclosure, meaning that *in time pi never closes its cycle, allowing a perpetual motion Universe, but in space, it practically closes in each present the super organism, breaking its vital space from the outer world.*

Along all the pages of this blog we have shown an enormous array of examples proving the duality between time-motions and its space-forms. So basically the fundamental symmetry of reality is that we *can express anything as a series of motions in time, which will be related symmetrically or inversely through one of the fundamental operands of ¬Algebra, or any ‘verb’ of language, or any of the logic symbols of space-time flows of 5D (< ≈ >) forming a time-space event.*

And hence *from one of both solutions we will be able with the proper methods to extract the other mirror-solution, which we express in the fundamental duality/equation of space-time symmetry: ***∫@≈∆T**

This is what makes ¬Algebra, once *we apply the Rashomon effect (multiple 2, 3, 4 or 5 D p.o.vs and functions, depending on detail) so powerful to mirror real it as it is basically all about S operands T symmetries. *

But the main task to do in ¬Algebra is to fully account for the meaning of all its operands and establish its relationship with space and time symmetries. I.e. we talk of sum, product, integrals, logarithms and numbers, complex, irrational and so on with very little understanding, happy just with the pedantic, axiomatic method and its pretension of absolute truth – which by Gödel is by all means incomplete, and reveals very little on the ultimate disomorphisms of space and time.

So the explanation of operands and Space-time dualities=symmetries+inversions is what will take us to the very deepest levels of understanding of the Universe.

** RECAP.** *¬Algebra studies social, sequential numbers and its relationship with spatial points, extracting the S=t symmetric motions between them’.*

It shares its minimal mathematical units with number theory, as numbers are closely related to the arrows of eusocial evolution of the 5th dimension and the arrow of discrete ‘frequency’ in time. As such numbers measure the ‘long’ , social evolutionary game of Planes, and ‘short’ sequential arrows of time of the Universe.

While ¬Ælgebra truly starts in the next scale of space-time events, with multiple groups of points (variables), exchanging entropy, energy and information, through operands (specific of each action) in ‘equations’ (which become partial case of the generator’s allowed st exchanges and symmetries). And so this classic ¬Algebra is the most important part of it.

¬Ælgebra studies the whole ‘block of time-space’ of a supœrganism, through all its potential actions expressed in **functionals** of functions that embed two or even 3 Planes in whole ∆±1 structures, *is the all inclusive ∆±i perspective, while* **groups** of symmetries, according to a set of ‘restrictions’ of its allowed actions, is the single ‘space-time’ plane attempt to describe the whole potential transformation of any entity in such a single plane. So groups and functionals explored in the 3rd age of ¬Algebra complete its wholeness.

This first view of ¬Algebra can be done through its analysis in space as a simultaneous connection of the two sides of an equation, which put in relationship the ‘ternary extremes of an ∆st element’, either by its operands (which allows a single plane connection of s and t elements) by its polynomials (which allows a multiple scale connection through ∆§) and by its inverse symmetries (group connection).

And because the universe is dual in its inversions and ternary in its elements and Planes, 3+ are the operations of math. Since Potency and integral operations being closely related as we explain, as the duality of ∆ curved analysis between planes vs. §lineal scaling in decametric societies happen to be. Indeed any mathematician immediately will notice this duality with the concepts of a derivative vs. a lineal differential approach, a logarithmic scaling that tends to a fixed asymptote vs. a changing tangent-derivative of a more complex curve, and the ultimate proof that all ¬Algebraic equations of exponents can be approached by Taylor series of derivatives.

¬Algebra explained it with group theory and axiomatic methods; so we shall consider that approach. And the only elements left to define then are the identity element and the properties of the operands.

Indeed, all dual operands have identity neutral elements and inverse ones. THEY CAN be considered loosely as the neutral=asymmetry form, which ‘splits’ both ways into the negative=antisymmetric element and the positive=symmetric one. The identity element leaves the asymmetry unchanged, it is we might say a non-operation. The negative element is the result of antisymmetry, the positive element of symmetry and tends to be a larger whole.

I.e. if we have a herd of 4 + 3 elements, and they come into parallel social evolution they give us a herd of 7.

But if we have a herd of 4 elements and they come into a Darwinian relationship they will separate the defeated elements, so if it is for example a fight for supremacy, only one will be left and we rest the 3 defeated candidates, 4-3=1.

The first curious thought coming out of this simplified analysis is that *positive operations tend to be more restricted additions as they require identical elements, so they are often simple social evolutions, negative operations however might have multiple meanings as the antisymmetric, ‘entropic’ states of a system multiply statistically.*

Thermodynamics already noticed that order ‘probabilities’ are less than ‘disordered ones’, *unless there is a Maxwellian demon – which we contend do exist in the survival will of all points of time – to contain entropic, destructive probabilities, which is indeed what it happens most often.*

When 3-4 elements that are numbers, which can be operated together hence equal beings as numbers are social groups of equal beings, *the magic of social love and the fourth postulate of similarity in non-Euclidean geometry means that all the systems which fall between symmetry and asymmetry, and can communicate come together. Entropy is REJECTED by almost all systems from the singularity perspective, which implies also that a mere abstract mathematical analysis will fail to understand the bio-logical input of those informative singularities that reject as Maxwellian demons do the negative systems, tricking the ‘dices’ of God. *

What about product vs. division? Again this duality is obvious. If we have 3 x 4 it means 4 herds of 3 which ad together in 4 steps to a 12 herd.

But the entropic events here are also multiple and divisive. I.e. If we have 3 kilos of wheat, we divide the whole between 4 entropic hungry men to get 3/4 per capita. Divisions thus ‘divide’ almost always a system into broken parts. Products multiply societies, or create tighter communication between those smaller parts. I.e. A= 5 and B= 4 elements can be multiplied also at the ∆-1 scale to get the number of axons that ‘tight’ the society together if each element of set A communicate with each of set B, then A x B = 5 x 4 = 20 axons.

So we come to the potency vs. root, integral vs. derivative that merely take this dual process to its final ternary scaling (as we know systems are ternary so no need for further operands). Its complex study being carried further in the post of analysis so we shall let it go for the time being.

** A definition of the Universe and its space-time topologic and ¬Algebraic elements.**

We define the fractal Universe suited to its mathematical description as a fractal super organism made of:

** A(nti)symmetries** between its 4 Dual components T, ∆, @, S, through parallel =symmetric or Perpendicular/antisymmetric communication, as they can either evolve in Balance≈become symmetric or annihilate= become Perpendicular/antisymmetric:

**S:** space; an ensemble of **ternary topologies, (|+O≈ ø)**… which made up the **3 physiological networks** (|-motion/limbs-potentials + O-particle/heads ≈ Ø-vital energy) of all *simultaneous super organisms*

**∆: Planes of size** distributed in ∆±i relative *fractal Planes* that come together as **∆º** super organisms, each one sum of smaller **∑∆-1** super organisms… that *trace *in a larger** ∆+1 world…**

**Time cycles:*** a series of timespace actions of survival that integrated as a whole form a sequential cycles of existence with 3 ages,* each one dominated by the activity of one of those 3 networks: motion-youth, or relative past, dominated by the motion systems (limbs, potential); iterative present dominated by the reproductive vital energy (body waves), and informative 3rd age or relative future dominated by the informative systems, whose ‘center’ is:

**@: The Active linguistic mind **that reflects the infinite cycles of the outer world and controls those of its inner world, through its languages of information, which guide its 5 survival actions: 3 simplex, **aei**, finitesimal actions that exchange energy (**e-**ntropy feeding), motion (**a-**ccelerations) and **i**nformation (perceptions) with other beings, and two complex actions: **o**ffspring reproduction and social evolution from individuals into **U-**niversals that maximize the duration in time and extension in space of the being.

So that is the game: 3 asymmetries of scale, age and form, which can come together or annihilate and each language represent in different manners, those elements and its operands.

A mind-mirror language can perceive the game goes on degrees of complexity from the simplest possible minds of ‘monologic thought’ (the humind, regardless of our ego Px. that feel to be the center of reality and its most intelligent being) to the more complex forms of pentadodecalogic (sexalogic systems).

So in its full formulation as mirror of ∆@ST elements, we can say that:

*¬Algebra mirrors s≤st≥t structures, a(nti)symmetric dimotions & ∆ transformations from the point of view of a given frame of reference yet its main mirror is ¬Algebra as expression of the Dimotions of reality with its operands*

It is then clear that what languages as synoptic mirrors of the mind will try to do is to establish the basic relationships between the space, time, scale of the being, expressing them through its operands, DEPENDING on the degree of perception the being has of reality and its Planes which might be reduced if the being is not fully aware of all the Planes of existence, as most minds exist only in a plane of reality

So does mathematics, through combinations of:

Sum/rest->multiplication/division->potency/logarithm; point->line->plane->volume and so on.

The Universe is a game of ∞ Species of space-time making STeps and STops in its constant cœxist¡ence (where we use π for time, in this particular wor(l)d according to the slightly changed rules of i-verbal thought for English to look more like the universal game), and so as Stops and steps (ab. œ) bring the being into exist¡ence we can measure them quantitatively as space-distances and time-motions. Stop and step then becomes topology and feed-back equations, where = is substituted by the dynamic symbol <=>, and the duality from = stops and ≈ wave steps.

Once the intuitive meaning of those symbols rises the awareness of the reader to the vital nature of our humind’s abstract rendition of reality things become then dynamic. ≈ for STeps, = for motions, and ≤≥ as the different implosive or explosive, informative or entropic ‘degeneracy’ of a system are easily quantifiable and translatable to both mathematics in its pure expression and mathematical physics.

It follows that the main translation required between classic ¬Algebra, and Existential ¬Ælgebra, is that between the operators of ¬Algebra and those of the generator equation. Originally I did try to cast the whole model of ÐST in terms of group theory, ¬Algebra and its classic mathematical operators, but that was an earlier stage of my exploration of the fractal Universe. It was then obvious that as good as mathematics is as a mirror of the Universe, it is not as good as it is needed to extract all its properties, hence the need for a different logic, which however had to include or at least reference the world of classic ¬Algebra as we shall try to do albeit at a basic level – the specialist must understand this is a unification theory of all exist¡ences, so we cannot be exhaustive with all of them, as the team of research is exactly the same than the infinite-infinitesimal Universe: ∞º=1 .

And to do so, as a fractal can always be divided in sub-fractals, mathematical disciplines subdivide further at all levels in 5 elements.

In this post we shall deal with the a(nti)symmetries and operands of ¬Algebra.

Though we cannot be exhaustive as ¬Algebra is the largest of all the sub disciplines of maths, concerned with time-space symmetries, but merely do a very fast analysis of its main themes, enlightened with ∆st insights to show the enormous power to ad new whys to reality and all its stience of 5D², since even the simplest truths scientists think are thoroughly understood have new insights observed when putting the Kaleidoscopic 3±∆ perspectives of the fractal organic Universe, what we call honoring one of my fav films, the ‘Rashomon effect’ (a truth can only be ‘judged’ with multiple ∆@ST perspectives.

So the fifth perspective of the mind-judge in the film required 4 previous ones, and even who the absolute truth can only be found in the event/being in itself, which carries all its information). We shall thus make liberal use of the Rashomon effect to enlighten the truths of ¬Algebra.

Then explore the more focused Existential ¬Ælgebra of the Generator and its sub-‘groups’ or space-time symmetries.

But we can further expand the concept to N-elements to form a relative polidimensional ensemble or herd, or a plane, which is what Hilbert spacetime frames do, of wide use in quantum physics. They also add the 4^{th} postulate of relative congruence with the concept of orthogonal=perpendicular events vs. parallel events (dot product and cross product in classic vectorial spaces).

*Entanglement of pentalogic in mathematics then becomes the interaction of ‘social numbers’, operands of dimotions and the world ∆+1 planes in which they take place:*

Further on as all T.œs have a different language of perception, and exist in different scientific planes, the parameters they perceive as time and space might vary; so next comes phase spaces, to reference the suitable time-motion and space-form coordinates each species perceives. Then we have in the human single plane, 3 frames of reference, the polar, spherical, cylindrical, lineal, and Cartesian, hyperbolic planes, being the S=T Cartesian plane the most useful; but depending on which organ of the being, its |-limbs/fields, Ø-body wave or O-particle-head enact the event, a different frame might be more suitable. Then we can write on those frames, different chains of dimotions with ¬Algebraic equations and operand, joined by dynamic = symbols of equivalent, feed-back stœps.

In mathematical physics as a closer reflection of there Universe that eliminates from the language mirror its inflationary images (so math will always be paradoxically larger in theorems that reality is, obliged to ‘bend’ and ‘limit information’ to those shapes energy can ‘bend’), it is even better to represent a vectorial field in General coordinates as they allow the multiplicity of points of view of the supœrganism each with its self-point of reference. Since all T.œs have a self-centered point or monad-mirror – a linguistic mapping or ‘seed’ that reflects reality in its frame of reference and will try to act from that point of view or re=produce and imprint the external world’s suitable energy (∆<¡) with the inner mind of its points of exist¡ence.

To be anchored and perceive as a point is the first function of any T.œ when emerging into exist¡ence.

We call the description of all of this, ‘Existential ¬Ælgebra’, ælgebra, or ¬Æ, as it is both ‘Existential ¬Ælgebra’ and Non-Aristotelian, Non-Euclidean in its form.

It is in that sense the most important, structural element of the mathematical mirror – the closest classic linguistic expression of the Universe and its 5 Ðimotions of exist¡enœ.

We are NOT that much interested in merely translating human ¬Algebra to ¬Æ as a formal mirror of the Universe, a ginormous task, others will built after me, but in defining a ¬Æ of the Universe and as human ¬Algebra does work in that purpose, to establish as many correspondences between ¬Æ and common ¬Algebra.

We depart in non ¬Æ of a first element, a fractal non-Euclidean point, ruled by the 5 Postulates of non-Euclidean topology: points are fractals, lines are waves between points, with less ‘amplitude=energy’ than the point – a fermion so to speak vs. a boson-line traced. How Fermions or non-E points proper share bosons, or line-waves of lesser ‘mass-position’ and more ‘speed-reproduction of information in a lower, ∆-¡, is a relative question.

For terminology, every non-Euclidean point is a t.œ, his index ¡ is always relative to its mind: I=0=T.œs mind. Departing from it, we can cave into its inner world of -¡ and its upper world or +¡.

The laws of ¬Æ are expressed in terms of a relative self, as ∆±¡ different planes in which an ∆º mind exchanges energy and information with different ∆±¡ planes. It corresponds for the case of ∆º and ∆±1 cases to classic science (single plane coinciding with the language of the mind-will).

Philosophically we can consider the mind, the perfect block of time of the syntax of one language that reflects the game of existence itself.

But logically we have a first element or seed, ∆º, and then we can ‘follow it’. as a non-Euclidean point in space, non-Aristotelian monad in time, as it mirrors in its language and reflects upon its world to exchange energy and information and fulfill its ‘Parts’ as an ∆º must have a body, ∆±1 and co-exist in a world, ∆±¡, whose relative extension might depend of the fractal fine detail, in our measure of ±¡.

As such the minimum ‘space’ to represent a world of such points is a vectorial space in which each point has at least two values that might represent the body-motion and mass-direction of the point, which becomes then the simplest representation of a T.œ spacetime, in which each points is a T.œ, which has two creative parts we might call space-form-scalar parameter and time-motion-direction of the vector parameter.

The simplest of those are momentum fields, in which each point represent in generalized coordinates the relative momentum of the being.

**RECAP.** All fractal self- similar scales are formally defined by 2 metric functions and its philosophy of science, 5D Absolute Relativity: the function of 5D scales, SxT=C & the function of relativity between form and motion, Si=Te, which started modern science. Since *we cannot know from the mental, still point of view, what truly is motion and what is dimensional form. *

Thus Time Motions & space dimensions co-exist and merge in every space-time being of the Universe &* we must talk of space-time dimensional motions.* When we combine them, as Si=Te maximizes SxT, we define the Function of existence, Max. SxT, to which we add the reproductive nature of all motions, along 5D scales, ∑, so we write the Fractal Generator of Time-Space organisms, or 5D metric function, or survival function, as C =∑Max. SxT(s=t) that unlike 4D locomotion applies to all ‘stientific scales’ & all ‘motions’ and Species; adding organic= fractal scales, biologic will, determinism – as all systems maximize its ‘Function of exi**st¡ence’ **– mental=still mirror symmetries ignored in **sc**ience.

C= ∑ Max. S x T (s=t) IS a biologic function of survival that embodies the will of life and its 5 Drives, or ‘actions’ which all systems of the Universe code: As a entity perceives **I**nformation and moves through **A**ccelerated=changing paths towards **E**ntropic fields to feed and absorb energy to reproduce its **o**rganism and evolve socially as a larger **U**niversal whole. So all systems try to maximize its 3±¡ ‘organic ** entangled **Ðimotions’ that in space create organisms and in time its worldcycles of existence; starting as pure Form (SS»4D) in a seminal seed, reproducing evolving socially and emerging in an ∆º scale, in a young age of max. Motion (2D, <, Ts), balanced with information in the reproductive age of max. Energy (3D:Si=Te), followed by an age of information (3D, >, St), when time reverses into entropic death («TT).

The function is maximized in the present mature age of beauty, when S =T and reproduction by ‘gender=mirror symmetry’ takes place, creating a herd (∑) a stronger, social number, which will evolve into an entangled, ∏, connected synchronous whole that survives better.

Ad it is guided by the SS-informative mind-seed, which is a linguistic synoptic mirror in a smaller scale of higher information, hence responsible for the ‘creation’ of fractal diminishing, ‘finitesimal’ scales; which it will reproduce over placental energy fields to become a larger ∆º whole.

The laws of Existential ¬Ælgebra and the partial equations of the generator, connect with the concept of a reductionist mind, which absorbs only the energy it needs to survive. In brief, a huge field of study in the design of mental systems and mental spaces of geometry is the understanding of the different ‘real systems’ of mind spaces constructed with an increasing number of parameters and dimotions, which make its mental geometry more complex. And so we talk of a growing scale of complex mind-spaces, which we shall ‘entangle’ in the next chapter with the different ‘degrees’ of social evolution of geometric figures, from self-centered points that sponsor a simple lineal mono-logic motion in time, to the more complex topological organisms/non-Euclidean planes whose structure ranges from simultaneous pentalogic geometries to the more complex Dodecaplex ‘polytopes’ of simultaneous, equal 600 elements… likely the more complex illogic structures of the Universe. **On the understanding of any language-mirror as it builds the complex image of the fractal 5Dimotional Universe, we must consider that the growth of ‘form and motion’ is one of increasing ‘dimensionality’ of the elements of that mirror. That is, more than a one to one correspondence between dimotions of exist¡ence and operands of a logic language, we shall see that those operands growth from simple monologic, to dilogic, to trilogic, tetralogic and pentalogic more complex elements of form.**

**In the papers dedicated to the ¡logic growth of ÐST (Generational space-time) as a language of ilogic time, we have built on that fashion a first paper on monologic ænthropic man with its simplex belief on a single time arrow of entropy and death – the most encompassing and less informative of all forms of time, which is remarkable in its primitivism, given the self-importance man gives to his ‘ingenuity’, then dedicated a second paper to dilogic, the duality of the two poles of form and motion as they converge into parity and gender to reproduce in its S=T point all other forms of the Universe, giving birth to tetralogic of positive, social growth and evolution – without entropy the final pentalogic closure of all forms in exist¡ence.**

**A being then can exist merely as technologic humans do with single goal or ants in a pheromonal path or horses blinded in a race, and live with that single monologic view of reality, moving ahead without doubting its mind view is the complete existence of the Universe. **

**When observing humind’s worship of the extension of Galilean relativity by Mr. Einstein and the astounding worship of his ‘genius’, which consisted merely on applying the 5**^{th} postulate of Non-Euclidean geometry without even understanding its ‘nature’, as we have shown in our evolution of those 5 postulates, the evolved observer of pentalogic timespace can marvel how that single entropic time, deduced from a mere expansion by a ct factor of the v=s/t locomotion equation of Galileo can bring such huge awe and veneration and build from it with that single arrow of entropy=death a full cosmogony of big-bang explosions and dying Universes heralded as the absolute truth of all what is needed to know about time, without a flinch of those who truly think such a simplex view of reality suffice to explain it all.

^{th}postulate of Non-Euclidean geometry without even understanding its ‘nature’, as we have shown in our evolution of those 5 postulates, the evolved observer of pentalogic timespace can marvel how that single entropic time, deduced from a mere expansion by a ct factor of the v=s/t locomotion equation of Galileo can bring such huge awe and veneration and build from it with that single arrow of entropy=death a full cosmogony of big-bang explosions and dying Universes heralded as the absolute truth of all what is needed to know about time, without a flinch of those who truly think such a simplex view of reality suffice to explain it all.

**That is the beauty of the Universe; that something so simple and minimalist in its total truth as huminds’ view of time passes as the whole and suffices to fill of pride the species, whose ‘seers of time’ (the job of God in Saint Augustine mystique, of Einstein in the lore of the nuclear industry) in so many Tv-shows on cosmology will assert with absolute certainty, the marvelous intelligence of a species, who could so fast, so good understand it all till the trillionth part of a second after the ‘creation through the big-bang’ of reality in the lineal past of a single affine equation, V=HoD, the God of Void, the vacuum, entropic time. That’s right, we know it all because after all emptiness is simple enough for an ant, a physicist and Einstein to understand… so why bother – ‘cosmologists are seldom right but never in doubt’ (Landau); ‘simplify man you are a marine’ (O.Stone). You can indeed survive by the art of killing it all and devour. I have thus long stop trying to teach cosmologists, ants and ænthropic military men that there is something more than the lineal simplex sword of entropy and death – their destiny traced by their faith.**

**Moreover as the Universe is the whole and all its parts are less than the whole, most parts are monologic and monologic one-dimotional actions suffice for the robot, the physicist, the ant, the operand of sum with its single lineal arrow of Natural numbers to make a worldview that guarantees our fulfillment and existence. **

**From synthetic whole – the thoughts of god – to its analytic details – all spacetime organisms.**

It is then the big task I took upon myself decades ago after discovery this ‘language and mirror’ of reality so above the present ‘language of science’ with a single dimotion of existence (lineal time locomotion), and a null understanding of the first principles of reality; to connect both, falling from above from the synthetic understanding of the first principles in its larger scale of reality, the summit and envelope of the whole, to connect them with the details. This was the old ideal of the Greeks, which humanity today has forgotten and denies, because the quantity of details gathered with sensorial machines have overwhelmed them.

So they cannot see the forest, not even realize that a network of hidden mushrooms connects them all.

We trIed to connect the two greatest levels of generality in the laws of space-time, Existential algebra, the formalism of generational space-time and its pentalogic and dodecalogic elements with classic algebra and its scalar numbers, dimotional operands and S=T equations. So first we shall introduce existential algebra, which deploys a simple generalized formalism to be able to reduce to ilogic equations with only 5 Dimotions all events of reality. Knowledge is a growth of complexity from monologic, the study of the one to duality between the poles of form and motion, to trilogic that adds its combination in a single plane, to tetralogic that studies the being in its subjective positive drives=actions of existence without the entropic inverse arrows of death and its inverse operands, which bring finally pentalogic. So while we tend to use just the concept of a pentalogic Universe many analysis will be done in terms of duality, trilogic in a single plane of reality (Ts<ST>St) and tetralogic, either because we adopt one of the 5 elements as a point of view, becoming the subjective blind spot (so operands of scalar social evolution might not have angular perception, most humans never analyze entropic death, eyes do not see precisely where the nerve that sees resides, etc. etc. Finally from outside a (do)decalogic analysis adds the 3 ‘scales’ through which the being will exist, from its ordered palingenesis to its outside world.

**¬Algebra = ¬Æ mathematics.**

Finally Existential ¬Ælgebra is the name I give to the formalism of ÐST and its symbols even if the equations of superorganisms tracing worldcycles and its laws are not those of ¬Algebra, because of the use of similar symbols and the fact that ¬Algebra was the previous more advanced form of a science of time, discovered by mankind (given the limited capacity of Aristotelian logic and lack of development of a full model of eusocial evolution).

We shall in this paper study the ages of ¬Algebra, the science of time in mathematics, after our brief introduction to the more complex causality of the Universe.

And conclude the post with the future of ‘¬Algebra’, which will be either a simplified version of a digital mind (Boolean ¬Algebra) or with minimal probability given the null evolution of humanity and its increasing obsolescence to AI, the age of Existential ¬Ælgebra, where the science of time finally reaches its zenith with the pentalogic and dodecalogic comprehension of the laws of Existential ¬Ælgebra. (ab. ¬Ælgebra).

**The entropic limits of reality applied to mathematics.**

It is obvious then that ¬Algebra, aptly called from Arabic “al-jabr” meaning “reunion of broken parts”, and analysis by studying S<=>T and ∆§ suffice somehow to describe the 5 Dimensions of present, past and future, and indeed, they together cover all of it intensely… but *they require first a good understanding of social numbers, and the @nalytic planes which work as the ∆-1 ‘cells’ and ∆+1 world in which the organic systems of mathematics work out its ‘actions of existence’.*

The only ‘great discipline’ on parallel in importance to that of ¬Algebra & analysis is geometry in motion, topology, which thanks to the 5 Non-Æ postulates will be explained as a far more ‘intuitive’ and profound perceptive mirror of reality, once points acquire volume (1st Non-Æ postulate), from flat prime numbers to platonic solids, reproduce inner parts, communicate through infinite parallels (5th non-Æ Postulate), which become entropy,

**Inverse functions as ‘balancing stœps’ of the fifth dimension**

Humans are entropic, thermodynamic virtual ghosts trapped between the immortal quantum proton and cosmic black hole (no entropic evaporation there), LOVE entropic simplest lineal theories of reality.

Entropy death is ginormous in its fields as it applies to all other attempts of ‘differentiated’ evolution of form.

So there are 4 positive dimotions (locomotion, reproduction, perception and social evolution) all of them subject to negative entropy and destruction that ‘returns them’ to the ‘mother womb’ of no form. And for that reason we can treat the 4 INVERSE operands, negative exponential decay, negative subtraction, division and derivatives (when applied to the search for an 1/n, ∆-1 infinitesimal) as reflections of the Universal entropy that destroys the positive distinctive operand, often in a much larger shorter explosion of dissolution of form.

Entropy is the disentanglement of free-chaos that affects all systems and the inverse operators best describe.

When comparing each new Dimotion operation, we observe several elements of entanglement. First the simplest expression of the next ‘Dimensional operation’ seems just a repetition in a new dimension of the previous one: the Sum is multiplied by adding sum after sum. The Power is obtained by multiplication over multiplication, but this is just when the operands is applied to the self-reflective point.

Operands do have however as they grow in complexity a direct entanglement with the 5 Non-E postulates that define from points waves of communication that become networks of topological planes and organisms, where multiple p.o.v.s are put in perspective. And this even more clear when from mere polynomials we move into exponentials and from exponentials into integrals, *the more complex operands then can interplay with multiple points of view, merging them through the product, reproduced through its inverse partition/division, moving them through a world cycle, in exponential growth and decay, and emerging into a new planet of existence through derivatives and integrals.*

So as usual in the entangled Universe of infinite acts of communication even if the one-dimensional man tends to reduce the whole and its flows of communication to a single line of thought.

Re=production through social evolution in a lower plane of the fifth dimension, in which the axons join the wholes is then the 2-point meaning of reproduction; often a bidimensional Space-time combination of a state of forma and a state of motion, as in the parameter of momentum.

It must be noticed then a very interesting new vision of the interaction of inverse functions, which do NOT merely cancel each other as one could expect, in classic science, *but advance the Dimotion in alternative balancing acts, leaving a memorial trace of its existential duality.*

Indeed, consider the case of the product and its inverse division. They are functions primarily of social evolution and reproduction, as parts of wholes are produced and exchanged. So if we have an individual of 5 parts that reproduces it will give birth to 25 parts, *but then we need to divide them between 5 parts, to get 5 individuals of 5 parts: 1 (5) x (5) = 1 (25) / 5 = 5 (5).*

Same happens with the positive and negative motions, which in alternate current move back and forwards the ‘electrons’, which in fact don’t really move, but its ¡-1 scale *the current keeps moving.*

So goes for the exponential+Logarithmic functions of reproduction, and its inverse, which form a worldcycle that in the middle point of maximal ‘carrying capacity’ will give birth again to a new ‘exponential growth’=generation, to continue the world cycle in an adjacent space-time.

**∫∂: Finally it came calculus**, with its inverse operand, which represents the scalar next social gathering of elements of ¬Algebra, as it is applied to the previous operands, as wholes, except for the trivial xª – the previous more complex polynomial, and so we must regard calculus not only as the operand of all dimotions of change, but also as the operand between planes of existence, since the logarithmic/power previous expression, just reaches between the two limits of two planes of the fifth dimension, but calculus allow us to ’emerge’ and transcend between planes.

So by the continuous game of inverse functions, which become a zeroth sum, *the Dimotions of existence continue to move the whole of time-space that never ceases to exist. *

However and this is a key truth of the underlying structure of scalar space-time, ∆St, the number of occurrences of the positive function is always larger than then negative inverse function. So there are more possible sums than negative numbers (for spatial quantities there are no negative ones, no negative apples). There are more products than divisions (as 4/2 = 8/4 but 4×2 ≠ 8×4); there are more exponentials than square roots as √-x does not exist, and finally there are more integrals than derivatives as ∫y’=y+C, adding ∝ constants.

Which means the arrow of social evolution and life is larger than the arrow of entropy and death, which in fact lasts only a quanta of time in its ÐST equation: Death = 0T x ∆ S.

The Universe is asymmetric and paradoxic, even if minds simplify those paradoxes – and mathematicians eliminate them with egocy axioms. It is though more satisfying to explore the paradoxes of the game of exist¡ence that the axioms of Zermelo and Hilbert. 0’ finitesimals, S≈T genders, different arrows of time in 5D as parts and wholes have different properties, finite infinities, limits of entropy, dissolution of information in irrational numbers, discontinuities that create stops and forms, etc. are the hidden structure that make possible a dynamic reality.

So going upwards the scales of 5D paradoxically, we multiply the choices of potential futures, balanced its entropic deaths. Reproduction becomes a radiation of fractal points of order balancing its entropic death. ∆Ñ dual tendencies are thus balanced even if they are asymmetries, because of those paradoxes between St and sT, SS and TT, balanced in ST; themes those explored in depth in ilogic and ¬ælgebra (existential algebra) and its fractal generator’s formalism.

** **

**THE BASIC ERROR OF AXIOMATIC METHODS: MIND POSTULATES TAKEN AS ABSOLUTE REALITY**

**THE PHILOSOPHY OF MATHEMATICAL SCIENCES OF A 5D UNIVERSE.**

**Correcting errors caused by euclidean simplifications and its dogmas=axiom(atic) method**

All this said it is obvious that reality is mirrored by the experimental language of mathematics, but due to the simplifications of earlier Greek Geometry and its dogmatic ‘mental method’ of proving reality as if it was ‘the mind of humans’ due to the egocy (ego=idiocy) paradox derived of the confusion of the mind-mapping and its languages with the whole reality (O’-mind x ∞ Universe = constant linguistic map) must be corrected to evolve mathematics and correct the errors of its use in different sciences. And that is the main task performed on those papers. So let us consider the consequences of the real ∆ST structure of the Universe into the 3 mirrors of mathematics:

__Principles of spatial geometry related to the fundamental particle of the Universe – the fractal point/monad: Growth of fractal points into networks, social planes according to congruence. __

The main principles of space to correct are the axioms of Euclid. Geometry started with Euclidean points without parts forming only deterministic lines of a single ‘straight future’ that evolved into curved with 3 possible paths (y’<=> 0), and finally points acquired volume as non-Euclidean points crossed by multiple parallels, aka fractal points unit of the new mathematical discipline. And calculus detached from Algebra as the most important operand of change and evolved into functionals (functions of functions). But in all those evolutions, the homology with ∆st elements remained, even if as 3^{rd} ages become excessive in form and detached from experimental reality, maths also lost its contact with reality and became inflationary making errors, such as the infinity errors of set theory.

*Mathematical errors derived of Points with volume, finitesimal zeroths***.**

The postulates of mathematics were made by humans and are subject to human frailty. Mathematicians also know that at the logical foundations of mathematics there are many paradoxes, meaning that we can get, by reasoning in accepted ways, results that we do not like. This is due to the axiomatic method that does not check reality to establish its mathematical reductionist mirror. So the main task of 5D is to check the mirror with scalar space-time laws solving those paradoxes carried out into experimental sciences that use ideal mathematics, such as physics:

– Points do have parts, crossed by a relative ∝ number of parallels. And so lines have width, they are waves or fractal networks, and so planes have depth, they are messings of 3 lines which are now topological networks and hence they form a topological organism.

– As a result of points having parts, absolute zeroth does not exist. There is not absolute no motion but a 0’k temperature of residual motion; there is no absolute emptiness but a finitesimal residual past memory when we take away a corpse with DNA traces in a lower scale. So 0’, the finitesimal is always left.

– Points with parts therefore hold a finitesimal world inside, and as ‘monads’ do hold still synoptic languages, of which mathematics is undoubtedly the language hold in the tiniest points, atomic particles, as it is the most synoptic of all. So when we study the equation of the mind, O’ x ∞ Universe = mind mapping, we will understand why mathematics is such an extended language in its local terrotirial order of reality.

__Principles of algebra related to the scalar 5 ^{th} Dimension: the evolution of social numbers and networks. Maximizing the efficiency of time conservation.__

Set theory of wholes which are made of parts is the closest reflection in mathematics of the scalar structure of the fifth dimension from the perspective of the whole, but numbers, are from the perspective of its parts its fundamental element. A number is then defined as a social group of indistinguishable ‘¡ndifferent parts’, which in its most synoptic form merely reflect the ‘scalar social nature of the Universe’. For a number to be such though they must truly have no difference at all. So pure mathematical numbers are regular polygons, and all other numbers have to be referenced to the property that makes it equal, which implies to include a logical understanding of the ‘equivalence’ between forms according to a given property, a fact often missed in sciences which use ‘equality’ liberally and then pretend as in the equality of energy and mass, they are ‘the same thing’ (errors of this type are rife in mathematical physics, where there is a zeroth interest for conceptual understanding, as long as the ‘maths’ carry on the equation). Identity then is a key operand that must be studied in depth to find out different forms of equality.

The false solution of those errors as always is caused by human egocy that invents ‘postulates’ without proof or ‘axioms’ to create systems independent of experience, based in the funny musings. For example, Euclid defines points without parts, an entelechy, and lines without volume, another one. And when we found that points do have parts crossed by parallels (5^{th} non-e postulate) instead of grounding this fact in experimental evidence, as we do; Hilbert starts his foundations of Geometry unable or unwilling to define them experimentally as our expansion of non-Euclidean geometry does, imagining points, lines, plane and congruence. This tendency keeps growing when his fried Cantor finds another ‘mental paradise’, and invents ‘sets’ which are ‘wholes of parts’, the very definition of the scalar 5^{th} Dimension experimentally sound; *instead of looking at the real wholes and parts and define them as non-euclidean planes of space-time made of fractal points, waves (lines with motion) and networks (lines with volume, *as fractal points with parts started its diversification in different types of lines with volume, waves with motion, networks with branching; lines with inverse directions and orthogonal lines, of S(y)-information and T(x)-entropy).

__Principles of calculus, related to the conservation of time.__

Another consequence of the fractal struture of points with parts that have therefore motion and form is fundamental principle is that all is motion=Time=Change, as we have never discovered a ‘solid’ form with no vibration or motion. There is no 0 temperature, there is no residual vacuum space without an h-minimal quanta of time motion that can evolve into virtual particles. Motion never stops, and when we die, the motion continues in the internal desintegration of the being. But against this principle of the ultimate flow of time cycles of the Universe fights each point of view, each fractal monad that tries to conserve its time of exist¡ence, tries to achieve the conservation of time, so time changes in finitesimal quanta (principle of calculus) and systems of physics try to achieve the minimal expenditure of time=motion in its actions (principle of least time) and stay in the position of minimal expenditure of time energy, which in its mathematical mirror implies physical systems search for Lagrangians, least time actions, ∂S=0, and Hamiltonians, Standing points that seem NOT to change. This is the origin of the 3 ages of time, in which the system seeks for the Maximal and minimal points of temporal energy (youth) and information (old age) in which at least one of the two forms of time-motion, cyclical and lineal forms do not change, and specially from the point S=t of relative present where time never changes. From those principles we can extract all the laws of calculus, which becomes then the main language of time.

*Our goal thus is to revive the experimental foundations of mathematics abandoned by the idealism of the Hegelian-> Hillbert+Cantor school. *

It is then* humind’s egocy (ab. Human mind, ego=idiocy) *what denies the experimental nature of mathematics and invented mental, creationist theories according to which mathematics is the ‘language of God’, shared by man, of ‘logic’, or ‘intuitionist’, or ‘axiomatic’ etc. etc. Nature. The main task of 5D generational space-time regarding mathematics us this obvious: to put in relationship scales and numbers, fractal points and spatial structures and algebraic operands and dimotions of time, to enlighten further both, mathematics and ÐST; whereas the primacy must exist in the being ∆St fractal, space and cyclical time over the mirror, NºGA, Numbers, Geometry and Algebra. And so we shall correct mathematical statements to mirror better ∆ST, without dismissing the fact that mirrors to work must be ideal, still, reduced images of reality with less information.

*Because of the way languages evolve from e-vident simplicity into complexity, mathematics came first as spatial geometry and then as algebra, of scalar numbers and simple temporal dimotions; cast on equations; and only in modern times fully learned the nature of time=change with calculus. *

*So we divide accordingly the study of the discipline in 3 papers, even if the 3 elements, ∆ST and disciplines that mirror them are entangled. *So we do comment in each paper on both the laws of ∆ST generational space time and the laws of the 3 disciplines. This second paper is dedicated to scalar algebra and its simpler operands, keeping for a 3^{rd} paper the study of calculus and the operands of sinusoidal functions, complex numbers and ∫∂ derivatives and integrals.

**Mathematics as a mirror language. **

We shall not so much correct mathematics but rather show what are its simplifications because many of the errors of sciences that use mathematics as mirrors are caused by the lack of understanding of that relationship. I.e. the limits of mathematical physics concerning the fact that all 0s are finitesimal 0’s and all points are fractal points with volume is the main error that allowed physicist to invent ‘infinitesimal singularities’, argue about if ‘particle-points have volume’, or discuss what model of quantum physics, the probabilistic or statistical must be truth.

The problem can be illustrated in Magritte’s painting ‘c’est nais pas a pipe’… Mathematics is not reality. So Poincare and Einstein told to each other: ‘I know when mathematics are truth but not when they are real; I know when physics is real but not when it is truth’. We want to make them come together as real and truth, which we shall do latter in 2019 or 2020 when we complete the upgrading of 5D stiences with our last papers on physics.

The Universe is a fractal that reproduces information, forms in action, motions of time with form, of which there are 5 Dimensional motions (ab. Dimotions), combination of time-motion, T, and spatial form, S, which we simplify with 5 ‘holographic,bidmensional’ symbols, in capitals for the dominant element: TT-entropy, Ts-locomotion, TS-reproduction, the dominant function of the Universe, as *a fractal is defined by a reproductive ‘generator’ equation, in this case the generator of space-time, *St-information and SS-linguistic form. The game is played by two limits, that of TT-entropy=time motion without form, which in fact dissolves form, and SS, fractal points that keep an almost still image of reality by simultaneous, intelligent mind-mapping of those motions into space-forms. This is the game, infinite ‘monads’ that hold a world in themselves (fractal non-Euclidean points).

As a language mathematics is used by different minds, notably atomic particles and some ‘composite’ minds made of electrons, humans and digital machines, which due to its higher particle density, as they are made of metals, calculate better than human beings. Minds do have their own peculiarities, the most notable to be self-centered as they measure the external world from its point of view, and to do so they reduce all the information of the Universe to a still limited mind-mapping to fit it into its infinitesimal size, *selecting and biasing information to cater those needs, putting themselves at the center of the limited world they perceive, confusing it with the whole Universe*. Mathematics helps to make those mappings because it is a highly synoptic language. I.e. a number is ‘any’ collection of identical, clone species. But this synoptic power introduces errors when comparing its elements with the reality they mirror. I.e. points are entities with parts, but in Euclidean geometry they have no volume, so they become ‘ zeroths’, which in reality are ‘infinitesimals’ with a minimal form and volume. So only knowing the a priori laws of space-time we can compare the simplification of ‘space geometry’ and ‘time algebra’ and reconstruct the loss of information, even if for practical purposes we still use the mental simplification of mathematics.

**The larger referential view: the laws of generational space-time.**

We recommend for that reason to read first the paper that introduces properly the main elements of a description of a fractal, Universe made of space-time organisms whose goal is to reproduce its form and preserve its exist¡ence maximizing its metric equations of energy and form. All stiences express the rules of that game of fractal reproduction of space=form and time=motions, including the laws of mathematics in its two branches, non-Euclidean geometry and non-Aristotelian algebra. We just in case the reader ignore our advice introduce the basic elements of the 5D scalar universe, as *the laws of mathematics are a reflection of those laws.*

Reality though is far more complex than humind’s projection of its limited perception of the planes of space-time, and it can be casted in multiple languages besides mathematics, which is – conceded – one of the best experimental languages of perception of reality but NOT the language that creates reality. The laws of the ‘substance of reality’, planes of spacetime are the ‘a priori’ reality all languages including mathematics mirror.

What are then the simplifications of mathematics, which we must contrast with the laws of time-space? They are many but because they are ‘well-done’ simplifications, mathematics keeps a high efficiency as a language, unlike gross distortions of reality as those of subjective tribal nationalism, Abrahamic religions and go(l)d churches origin of capitalism that make human societies so unjust and conflictive and politics and economics pure ‘idol-ogies’ not sciences. Still the main ‘simplifications of mathematics’ are ignored because the humind’s ego loves to think it speaks a perfect language that explains it all and nothing else is needed. So Omar the conqueror of Alexandria burned the library as all what was in the Koran was enough and in many towns of deep America there is only a bible, customary in each motel for nothing else is needed to read that the historic book of a bronze age people.

This goes on also in mathematics with the insistence that the axiomatic method of proof of Euclid, apt for the first age of a language, when it is more lineal, deterministic first age of timespace topology could indeed prove absolute simple truths (as a line remains a line only if it follows its same straight path). Determinism in mathematics as in the Universe though changes when the line with the passing of time curves and bends as curvature can easily change into different paths, and potential futures open. The child is easy to become a fundamentalist deterministic line; the beginning of a civilization ruled by swords of deterministic entropic death and lineal simple art is as all things simple just a beginning. Euclid thus reduced all to very simple concepts – points had no parts, lines had no breaths, planes had no width, things that were externally similar were equal regardless of its internal parts we do not see, as they don’t exist…

– Because there are 5 dimotions of space-time, simple Aristotelian causality, A->B, is not real. Rather the causality of the Universe is entangled. We do not create a real circle by drawing a line around with a single pen, but rather multiple flows of timespace converge into a point creating a sphere of forces with multiple causality, one for each line of force.

– The Universe is not a single space-time continuum of light space, which is what our electronic eyes-minds see, but it is a fractal Universe of multiple scalar planes of space-time, displaced in space and with multiple time clocks. And so continuity does not exist. The real line is NOT in a single plane, but in multiple planes and the discontinuities of natural numbers are filled by decimals of an smaller scale/plane of the fifth dimension and the discontinuities of rational numbers are filled by irrational ones.

Does this means the axiomatic proofs of all those non-truths have no use? No. It does mean that they are *proving something else, in its simplification that extract properties we must now apply to the enlarged fractal Universe of multiple scalar planes of space and multiple time space dimensional motions. *

I.e. the hypothesis of the continuum is false, because each of those number families is in a different scale of space-time, which in nature are dilated in ‘time’ as parts must become first for wholes to exist, so the cell must exist first for the emergence of the organism above it to happen. But the hypothesis of the continuum still works for the entire sum of those scales which put together show the Universe has a horror vacuum. There is a space not filled by spheres adjacent to each other, but this means there is a hole between them that can be filled by other spheres or in a height packing by a tubular channel. And this is exactly what we see in nature: plants use the interstitial region between cells to channel water up and down the tree. But those tubes are a different plane of reality.

Of course, I am aware that modern mathematics considers a continuum in different terms, but most sciences stick to the previous concept. Today in mathematics a continuum is rather similar to the concept of the ‘domain’ in which the function can be expressed as a series of stœps, steps of motion and stops of formal perception (St-Ts or SS-TT), which are ‘orthogonal functions’ of S-height and T-motion. So a function is continuous when we can trace steps and stops that become tangents of its derivative. The change in 5D is to consider real the approximations by differential tanget stœps to the curve, NOT the curve in itself, as all curves are in detail finitesimal Steps and stops, wave and particle states, length-locomotion and height-information consecutive dimotions. Continuity then happens when the steps and stops are similar, following the law of S≈T present states and breaks when Max. S or Max. T ‘breaks the function’, into its ‘point of death’ by unbalance of the S=T law of present states.

A plane does have depth and discontinuous wholes but it still can be defined by 3 lines, only that now those lines are networks and so 3 physiological networks define a topological organism, a vital plane of space-time.

**The five dimensional motions of Timespace mirrored by the operands of time algebra.**

*So what ÐST, generational space-time does, is to depart from reality, the laws of space-time of the fractal organic Universe, and compare them with the laws of ideal simplified mirrors such as mathematics, or verbal languages, or music or any mind-mirror of the game of exist¡ence, of energy x information, of any scale of space-time st¡ence and acknowledging those limits rethinks mathematical statements, further advancing our comprehension of its laws. *

Thus the purpose of these papers is to put in correspondence the laws of each language with those higher laws of ÐST, Generational space-time. In the case of Algebra, the mirror is concerned with 5 ‘operations’ that reflect the 5 Dimensional motions all systems of reality perform as its ‘deterministic program’ of existence; whereas algebra represents those time-space organisms with ‘sets and numbers’ (sets being the most comprehensive ‘whole’, numbers, the minimal unit, mathematics uses to mirror the spacetime reality of superorganisms of social, clone space-time parts, tracing worldcycles of existence)

Thus, the branch of mathematics that deals mainly with the 5 Time Dimotions is ¬Algebra – Non-Aristotelian Algebra of multiple ‘times=changes=dimotions’, which unlike classic Aristotelian, A->B single causality Algebra that recognizes a single arrow of time, accepts the reality of 5 different type of change; and thus vastly expands the theoretical foundations of classic algebra, grounding it in a real description of time-change.

We are thus writing, paraphrasing our quote, a treatise on the ‘Analysis of Algebra of Time’, the ‘creator’ develops in all its systems, with an obvious emphasis on operands – the functions of algebra that mirror the 5 Dimotions of time-space and its fundamental form, calculus, which studies all the forms of change=Time of the Universe, even if it does not grasp theoretically what it does.

The extraordinary development of Algebra and mathematical science is due to the fact that derivatives and integrals of ‘finitesimals’ and wholes are the best linguistic tool so far found by mankind (besides the ultimate ÐST laws developed in this blog) to describe all the modes of time=change. Unlike physics stuck in its ‘worldly profession’ of making entropic weapons and transport, locomotion machines for so long that it cannot even recognize the existence of time=change in the form, or in-form-ation of beings, the dominant ‘arrows of times’ and dimotions of a fractal Universe which is all about reproduction of forms, Algebra in its apparatus even without the foundational insights ¬Algebra will provide does study all those types of change.

So when we calculate the change of volume of a receptacle of water with an integral, we are studying the 4^{th} dimotion of social evolution of H^{2}0, even if no mathematician would dare to see it that way; and when we study the derivative of any system we are analyzing the ‘change of a finitesimal quanta’ in the system, the minimal rate of growth, even if the lack of proper foundations of mathematics will try to justify derivatives with the ‘hypothesis of the continuum’ (which is wrong, only the mind by eliminating the fractal wholes of our Cantorian dust of space-time sees a continuity or by compressing all the families of numbers and its Planes into a real line finds no space in its ‘Dedekind’s cuts’)…

And so we come to the full purpose of this paper as all other papers – to marry the abstract jargons of human stiences, clean up their mental errors due to the egocy (Ego=idiocy) paradox of mankind that tries to project its limited still view of reality necessary for the mind to fit all the Universe in a visual thought, and then expand or at least give the clues to future ‘pros’ of each discipline to expand soundly those foundations connected to the ultimate laws of Generational space-time.

Unlike Non-Euclidean geometry of fractal points which is easily connected with the expansion of the mind that 5D brings to all ‘stiences’, however ¬Algebra is expressed in a language of time that requires a more profound translation to the laws of Generational space-time; not so much because of its unfocused mirror but its complexity, which goes well beyond monologic huminds’ limited intellectual capacity despite its egocy (ego=idiocy) embedded paradox that make us all feel ubermen.

Yet ¬Algebra works because of the magic involved in the syntax of languages; which all have its own consistency and self-contained structure, as all mirrors proved by projective geometry reflect the Disomorphic laws of the whole they observe. It is important to grasp that for things to work, systems do NOT need to have the collective intelligence, or even understand what they do. And this is how huminds work, without understanding how the Universe works but merely assembling machines and considering them a proof of the human genius not of the intelligence of the Universe. So goes for the ‘ensembles’ of ¬Algebra, which require a mental protocol of proof (the axiomatic method) and its manipulation by its rules to make it ‘work’ regardless of the blindness of mankind in its understanding of the ultimate principles. I.e. neither Leibniz nor Newton understood what is a finitesimal, the minimal quanta in time and space of a whole, and how calculus widens our measure of time=change beyond physical locomotion – huminds still don’t – but they *use its rules to calculate any physical change in the cosmos.*

Reality instead obliges to ‘downgrade’ humind’s ingenuity and upgrade despite egocy, the beauty of the Universe; that is, ¡t=s symmetry, organic properties, vital will of survival and collective intelligence (Max. ∑i), which makes its fractal points follow the rules of order that ‘shape’ vital geometry and the ‘rules of motion’ and ‘social evolution’ of parts into wholes that shape ¬Algebra.

But ¬Algebra will never be so e-vident as non-e geometry is in the depiction of reality because we do NOT perceive directly time. We do but in the simultaneity of a slow stop and go mental image of time flows that at the rate of humind’s perception of a second, solidifies everything that moves much faster and traces smallish cycles that appear to us as solid ‘particles’. That is, we perceive very little of the feedback cyclical motions of time to truly understand it at first experience, and when we try to summarize huge amounts of populations in complex dimotions of time, the tools of ¬Algebra, which achieve this with so much synoptic power seem to us a ‘magic technology’ (Clarke).

The most magic aspect of a language is its capacity to predict the future by compressing the ‘flow of time’ into its ultimate ‘ÐST’ laws – the vital program of survival of the Universe of 5 Dimotions (ab. Dimensional motions), which all languages of time, including ¬Algebra do. Huminds are extremely proud of it, having as an ‘egocy dogma’ that we are the only non-predictable ‘free’ (egocy mythology) species and the only ones that predict the future of others with the only language shared with God, mathematics. We have dealt with egocy paradoxes in our analysis of ‘monologic man’. Fact is man is also predictable, from the larger point of view of social groups, worldcycles of existence of cultures and civilizations, and the entropic limits of death and life and the Earth, as we do and have done for 30 years with absolute accuracy, and 0 interest on social scholarship and political praxis – since precisely humind’s limit is their egocy. But if we accepted the limits of the species and entangle with the Universe, perhaps then we could become immortal, infinite and as intelligent as the Universe itself.

All entities are superorganisms or parts of a superorganism tracing worldcycles of existence between birth and extinction – its time limits – through 3 scalar planes of the fifth dimension, the cellular/atomic, organic/ thermodynamic and ecosystemic/cosmological Planes – its scalar limits, structured in trinity, topological non-e adjacent geometries (its spatial limits), and man is part of civilizations within the limits of Earth, which also have limits of survival, in a relationship of symbiosis and predation with ‘metalife’ species, evolving and reproducing much faster than us. And all this is as predictable, even more than the simplest ¬Algebraic operation, 1×1= 1 (in a self-reflective act, when the ones are the same), 1×1=3 (when reproduction exists between 2 different ones). So yes, ¬Algebra is likely the highest ‘achievement’ on the capacity of huminds to mirror reality, only inferior to a future humind, or AI–mind understanding of ÐST laws. But it is neither magic, nor a humind achievement, and not the absolute language of Nature, which are the laws of fractal space and cyclical time.

*So *we recommend to read our paper on the 5Dimotional fractal Universe, to understand its 5 elements, Space, Time, ∆-Planes, linguistic @-minds and entropic limits, the components of all Supœrganisms of Time-space are.

**The laws of languages, its trinity logic.**

This said the fundamental purpose of this and all other papers on maths is to upgrade the understanding of mathematics as a language mirror of the vital, organic, scalar properties of the space-time Universe.

So we are not so much a advancing maths beyond the upgrading of Non-Euclidean mathematics, but interpreting maths as an experimental science, that is as a mirror of the space=geometry and time=logic, ¬Algebraic and scalar=digital social properties of all what exists.

The Universe is an entangled fractal game of Dust of space-time, ¬∆@st, where each element flows as a series of ‘5 Dimotions’ (dimensional motions of time space), which can be perceived as ‘form=space’, in the stillness of a world mirror or linguistic mind or as a motion of time, in its true nature since MOTION not form is the underlying substance of reality.

So all fleeting forms, ‘a Maya of the senses’ will return to motion and die (¬4th Dimotion of entropy, death and dissolution). Its 4 positive elements, organic Planes, topologic planes and time ages and actions however, carry the system as a finite super organism of space with a finite time cycle.

And so those 4-i elements, entropy (¬), Scale (∆), time (T) and Space (S), are the elements all languages mirror either in a ternary grammar (if scale is missed), whereas often instead of Space we talk of information and instead of time we talk of energy of motion, in a single plane:

Light language: red-energy colors, blue-information colors and its green/yellow combinations.

Verbal Language: subject (information) < verb (action-combination)>Object (energy of subject).

¬Algebra: Y: Future-information < Operand-action> F(x))

Trinity is thus the logic of most beings. However as humans reached higher and lower Planes of observation, a pentalogic was possible and its mathematical mirror became analysis, with its operands that extract finitesimals (∆-1) or integrate into wholes (∆+1) smaller or larger systems.

Thus maths became with the inclusion of Calculus the most complex, best mirror language of human thought, arguably overcoming with the age of calculus the verbal mirror, which is the natural language of man, specially because of the arrival of instruments of measure which could also cast reality into the digital language of social numbers, identity species that further proved the social scalar nature of the Universe.

So with the modern age, past the simpler age of geometric maths, a new language of social thought and scale, numbers and calculus enthroned mathematics as the queen of all languages.

Analysis reflects the 5 Dimotions of the Universe in its mathematical mirror – its 5 changes=timespace events – acting on simplex operands that study change within a single plane. As such analysis it studies change, both in multiple elements together in a single plane (ODEs) and multiple Planes (PDEs).

As languages are mirrors of the fractal Universe that follow its same laws, we shall study Analysis as we do with any other ‘species’ of reality first showing they are mirrors of its 5 Dimotions, whose ‘syntax’ is built on the pentalogic elements of all Dust of space time (¬∆@ST). To make easier its study though, we shall use a sequential ‘humind’s’ exposition building its growing ‘informative complexity’ through its 3 ages of evolution.

**the mirror of ¬algebra: scalar analysis and temporal dimotions.**

This paper is dedicated to ¬Algebra, the science of ‘social, scalar numbers’ that group together equal ‘T.œs’ and express its 5 dimotions of existence through ‘time operands’ and its equations; extended in its dilated view through a simultaneous superorganism in complex organic structures of scalar space called sets, wholes of T.ŒS that perform worldcycles in time called group transformations.

In its essence Algebra is the study of numbers. And numbers are social groups that form scales of parts that become wholes and hence can be ordered (with the exception of complex numbers which are ST-representations), such as x<y<z -> x<z, which responds immediately to the scalar metric of 5D and its social scales. It does happen then that depending on which scale of numbers we use, we establish different 5D ‘families of worlds’, being the decametric scale the most important of them, both in the tetraktys configuration (3×3+¡) and the next logarithmic scale: 10±^{10}.

The word *¬Algebra* comes from Arabic الجبر (*al-jabr* lit. “the reunion of broken parts”), this fact alone indicates the capacity of mind mirrors to understand through their subconscious grammar deep intuitions in the meaning of the existential game.

Since ¬Algebra and its main branch, Analysis, must be considered the study with the mathematical mirror of the most ‘complex’ systems of nature, ‘potential’ superorganisms, through its multiple parts in space (group theory), Planes (numerical equations) of space and Ðimotions of time =change (operands and analysis).

¬Algebra is thus the highest level in which mathematics mirrors the 3 main elements, ∆ST, of the fractal Universe in a simple plane. To which it adds with @-frames of reference and inverse operands the limits of entropy and self-centered elements that complete a Time-space organism.

But the great discovery of ¬Algebra is analysis – the F(x+h) – F(x)/h function of ‘change’, which is a much wider concept of change=motion than the locomotion function of change recognized in physics, as it includes the analysis of all possible forms of changes, hence potentially of all dimotions, including those of biology (social and topologic evolution and Darwinian entropy), which are the most important of Nature.

Thus mathematics is above entropic physics in experimental truth, as its applications are wider.

This would be one brief description that merges classic ¬Algebra, and Existential ¬Ælgebra – the study of the worldcycle of existence of a superorganism of timespace with the ‘formalism’ of ÐST (Generational space-time), which *is a more concise mirror of the main properties of those space-time beings.*

So as usual not to have to repeat ourselves ad eternal, we recommend to read the introductory article on the fifth dimension, and in this case, the I Book on mathematics, to understand what the 5Dimensional fractal Universe an its fundamental elements, Space, Time, ∆-Planes and linguistic @-minds, the four components of all Supœrganisms of Time-space are.

And how mathematics as a mirror language perceives those elements with its scalar numbers, geometric points and temporal operands, which are the main elements of ¬Algebra.

*It is indeed absolutely necessary to have the basic understanding of a Universe made of space, time, Planes, linguistic minds and entropic limits to properly realize that mathematics merely translates into its postulates the properties of those 5 elements of reality, becoming the essential experimental language of the Universe. *

**¬Algebra equations: stœps and a(nti)symmetries of space=time supœrganisms.**

¬Algebra then becomes through equations that mimic the S<=>Tœps of reality the perspective of mathematics as a mirror of the different steps and stops of each Ðimotion of a T.œ, with emphasis in its simultaneous gathering as a ‘whole event, united in synchronicity by the = symbol, creating in this manner feed-back equations between the space and time state of the 5 Ðimotions of the Universe.

The beauty of ¬Algebra becomes then the capacity to write ‘chains’ of stœps in such a manner that we can follow sequentially the events that conform the Program of existence of a T.œ as it goes on through Indeed, because Dimotions form complex chains most often of stop-space and motion-time steps (ab. stœp, which is the sum of a stop: perception-step=motion dual chain), the = symbol is essential to a mirror of reality’s events.

Because algebra is immense we wont be exhaustive. Our aim is just proving the astounding parallelism of Algebra and ∆ST laws and leave a first seed for future ‘pros’ to enjoy as I have, the connections between both.

**The different determinism of the 3 ages: Axiomatic, lineal age vs. Kaleidoscopic uncertain futures. **

Because the Universe is pentalogic, made of ‘space’, ‘time’, ‘scalar planes’, ‘languages-minds’ and entropic limits, when *@ mind’s language appears it studies exactly those 4 elements, space, time, scales and entropic limits, with its mirror systems. And indeed, ‘analysis’ was born of the need to understand those 4 elements in problems of Nature, NO LONGER in lineal terms, as the ‘first age of any system’, but in ‘curved’ terms.*

*So what the Greeks have resolved for the $T age of mathematics (lineal age), c Analysis will solve for the second age of curved geometries through the use of analysis.*

This is a process proper of the 3 ages of any Space-time system. The first age is lineal, with absolute simple truths that the mind as a dictator ‘child’ considers dogma. So Euclid did his axiomatic method on simple lines

__Monologic in Mathematics. The first age of lineal, deterministic Greek still geometry and axiomatic proofs.__

Once we understand the general fact that all languages have a first lineal age, deterministic, as a line cannot change direction or else will stop being a line, while a curve can easily change curvature, even change direction in sinusoidal waves and still be a curve; so lines are deterministic one-single future to them, while curves are able at any point to choose 3 paths of less, more or equal curvature; we can understand some facts of Greek Geometry:

– It is simple, lineal, deterministic and hence it can be approached with a purely axiomatic method, as there is no ambivalence on results, constructing a self-contained method of proof departing truly from a simple set of axioms – a point has no breath, etc.

But the axiomatic method of proof is no longer valid when we consider systems that do have also a certain ‘time curvature’, and even more so, when we approach operands and mathematical systems that probe the planes of the fifth dimension (calculus, limits). Then the future has different solutions, and some are paradoxes, and so we cannot prove with the simple A->B lineal causality and deterministic of lineal Greek Geometry, everything that has to do with cyclical, curved geometries, calculus of finitesimals (limits), and because humind’s reject the concept that absolute truths only exist in absolutely simple lineal systems, as Mathematics evolved into complex curved geometries and scales, its proofs were more and more imprecise, or blatantly false (0 does not exist, as all limits to 0 or infinity have an entropic limit in a quanta or the dissolution of information; the real line is in a different plane of space-time than the Natural numbers; which only ‘become continuous’ if we were to access an even larger scale; etc.)

It is for that reason we shall not use further the axiomatic method and the first order logic and the ‘selfie’ process of ‘inventing’ self-evident axioms of truth, which are *always based in the mind simplification as if it were the absolute truth – that is in the ego paradox of naïve realism that confuses the mind with the whole,* but compare any complex level of mathematics to the experimental laws of Space-time from where they depart.

*Indeed, the ego paradox is the essence of all the errors of all the sciences based in mental postulates that take our reduced view of the Universe – the world – with the whole of reality. And our ænthropic views as dogmas. *

From that view arises the error of Euclidean points with no breath and the absolute 0 instead of fractal points with volumes and finitesimals zeroths, 0’, absolute ∞ instead of entropic infinities that become uncertain past its limit of perception or use, ∝, normally beyond the decametric scales of the Universe (Cantor’s infinities, infinite decimals in transcendental numbers, etc.).

From the same reductionism of the mind taken as absolute arise the error of a single continuous line and single time arrow and single spacetime continuum, when in reality, only Natural numbers exist in this plane of space-time, so the plane is discontinuous, and all other families of numbers fill the gaps, allowing the fractal structure of the Universe. As the scales of space of the fifth dimension are in the discontinuities of our Natural number world. So are the faster time clocks of other scales; hence with more ‘cycles’ of times and more information, in a relative more evolved future in its logic form (but created in a relative lineal past as they come before larger scales that sustain over them, themes those too complex for this paper, which require deep thought on time paradoxes, hence studied in our papers on ilogic).

Insights on the complexity of reality as opposed to mathematical simplicity are important for 2 reasons:

First from the acceptance of such simplifying errors as absolute truths arise the paradoxes of mathematical physics and its incapacity to understand the fractal Universe, its hypothesis of a single spacetime continuum, denial of faster than c speeds beyond the galactic organism, etc.

Yet in second thoughts, all what is required once the errors are pointed out is to use the simplification, which extracts still valid laws for the ‘whole’ to enlighten the fractal Universe. I.e. the hypothesis of the continuum is not truth for a single scale, but if we do accept that the ∑¡ sum of all the scales of the Universe do fill all the holes of the real line, then the hypothesis is truth and indicates that the Universe has a vacuum horror. That is the sum of all the planes of 5D is filled – reason why indeed there is no absolute 0’-void; simply speaking it will be filled by smaller scalar particles and waves.

In terms of ‘scales’ all this means that in small, ‘fast’, predictable A->B steps reality is lineal but when we gather multiple steps, all lines become curves. In small intervals motion might be continuous but as soon as we go beyond a simple step, there is a step and stop , length and high motion.

And so we can also reduce curves inversely to steps and stops of length and height, or lineal stairs (which would be the method of Calculus, to ‘calculate’ the tangent of the curve. Does then the curve exist? Or only the steps and stops of lineal and height motion and information? It is relative to our perception. In the large scale the zigzag of Brownian movement or electrons become a continuous curve. In the smaller scale the steps might be highly lineal and deterministic but in the large scale they become curved and probabilistic.

In that regard, the use of ÐST laws to reference the laws of mathematics, beyond the pretension of absolute truth of the axiomatic method, is completely necessary, without using a complex ‘pentalogic’ point of view, and accepting the paradoxical limits of reality and its scales as we shall constantly do here.

**The false hypothesis of the continuum. It’s the mind, which creates the illusion of continuity.**

The hypothesis of the continuum is false. First to define it in modern times, Leibniz thought a continuum of numbers was one that could be filled in any interval with as many numbers as we wished – when that is the obvious reason of discontinuity – as there is a hole to fit more numbers. Down the road this ended in Dedekind’s cut. The hypothesis of infinity is also wrong. As infinity dissipates into entropy with uncertain information. The hypothesis of equality is also wrong, as two fractal points with volume hide information we cannot compare, and entangled to the whole they are dissimilar since a being is part of that whole universe in which its position is different. All those hypotheses however reflect the mind’s simplification of reality and so the can be considered right in mental space, of which one of them, is humind’s mathematics. It’s the mind that creates the illusion of continuity, which places a *huge* metaphysical question. It is the mind the only continuous for in existence? If so it is the mind immortal in its travel through planes of the fifth dimension, as religious people wishfully think?

Because huminds obsessively try to project their local mind-space truths as universal ones, we have made concessions to their language we shall now reconsider. First, there is NO numbers for Dimensional motions, but Functions of holographic, dual, mental Space=form and Time=motion, SS<St<ST<Ts<TT. So we shall escape numbering them. Next, all of them are diffeomorphic, local, in a fractal Universe, though self-similar in its properties. We will keep the concept of the ‘5^{th} dimension’ as the sum of all those planes of space-time and local dimotions, so the ‘5^{th} dimension’ means all, the whole universe.

Next, time and space are very different but obvious concepts to those used by huminds, including our ‘gods of science’ or ‘seers of time’ (definition of God by Augustine), the ‘physicists’ (from Newton, through Galileo, to Einstein). Physicists do have merits of measure, precision, machine assembling and mathematical analysis, but their philosophies of science and ‘grand theories’ of reality, sorry folks, ‘suck’.

So my advice to the reader if any is to drop their idols and start afresh. ‘Time is what a clock measures’ was the best philosophical answer of Mr. Einstein. We go a relative infinite, ∝, further from that. So goes for the pretension of ‘axiomatic mathematicians’, that impose its mental space of inflationary mathematics to reality and get all worked up when some fellow finds out with computers a proof of the Fermat’s theorem in 1000 pages. ‘If you cannot explain it simple, then you don’t understand’, ‘the Universe is simple and not malicious’ (Einstein, which I deeply admire for his quips 🙂 Simplicity is genius (Leonardo). We are concerned with the underlying simplicity from where all those complex truths arise.

But if we depart from postulates and axioms based in the distortions or wishful thinking or egocy of man to build up an entire new world unrelated to experience and reality it doesn’t matter how much complexity we add – it is all ‘soma to disguise reality’. And this is in mathematics, the hypothesis of continuum, in religion, all Abrahamic cults, because obviously they all come from an original false premise – that an ass breeder of the Bronze Age who saw a G.Bush, God bush burning had decoded its sounds as the word of the creator.

Mankind has built on that obvious false delirium tremens ‘hypothesis’, 2/3rds of its world culture, including most of western art, most of western genocides, the entire biblical capitalist go(l)d culture that is killing the Earth and various jihads and inquisitions of thought still extant. Couldn’t care less about the beauty of Michelangelo and the suffering of the Hebrews and the massacres of people. Wars do not justify armies. Mankind is a single species. So again the entire built up of ‘nationalist identities’ is other absurdity coming from the premise first brought about by Germanic hordes of aristocratic warriors that the land of Europe, a culture, had to be broken for sword masters to enjoy for free tax-farming of its people. And so on.

Mathematics in geometry comes from a false postulate: that points have no parts, hence 0s exist, and lines are not waves or networks and planes are not topological organisms defined by of such lines-networks, and of course, the hypothesis of the continuum which does NOT allow those networks to penetrate between tissues and the scales of the 5^{th} dimension to build complex relationships between parts and wholes compressed in a single ‘real plane’. Most humind errors are due to ‘æntrhopic: entropic+antrhopic’ treated in a paper of its own, (monologic man). I.e. in cosmology the entropic big-bang projects the entropic profession of physicists, as maker of weapons, into the lineal V=HoD equation of entropic vacuum space, which expands between galaxies as gaseous states do in matter, but *forgets* the implosive gravitation of dark matter and galaxies and the liquid S=T balanced and solid, SS states. So with 1/3^{rd} of reality voila! the universe is a big-bang. So ænthropic theories work by hiding all what the mind does not need to make truth a false ænthropic theory is the trick. Then dressing it with complexity to disguise what is wrong – the initial postulate, makes the egocy paradox real. The layman will think if it is so complex and I don’t understand it must be right. This is done ad nauseam, with big-bang theories based in a false premise, axiomatic methods in mathematics. Even with the tax-farming of mankind by private bankers who issue money in monopoly through ‘derivatives’ and national banks which are in fact private enterprises. In U$ the Federal Reserve instead of just printing digital numbers as the body prints red cells to transport its blood-money in universal salary to all citizens-cells, has a very complex system of ‘emitting’ money as ‘debt’, bought by the treasury, etc. etc. called SOMA, exactly the name the priests of Aryan Vedas gave to its soldiers-slaves to throw them into death. So when someone like Lincoln tried to simplify money without debt as superorganisms simplify the first order-languages (hormones are simple molecules, like the letters of the alphabet, oxygen is as simple as money should be), they killed him. End of history. Mankind’s then finds it all very complex and proudly thinks his mind is so above heavens and earth. When he is just an inflationary mind out of tune with the extremely beautiful S=T balanced Universe.

So this is a rule of thumb to judge humind’s theories of reality:* ‘All humind theories of reality beyond the simplest details and its causalities, are projections of the egocy paradox of the mind’s finitesimal mapping of reality that huminds think to be the entire Universe, based in postulates without proof’. *

This said Mathematics divides phenomena into two broad classes, we shall call discrete or spatial, informative and continuous, or temporal historically corresponding to the earlier division between arithmetic and geometry. But they got it inversely and now we set the record straight: continuity is just a mental phenomena that ‘eliminates’ for the sake of simplicity and economicity to fit in the brain, the ‘dark spaces’ between discontinuous ‘1s’. Discrete systems can be subdivided only so far, and they can be described in terms of whole numbers 0, 1, 2, 3… Continuous systems can be subdivided indefinitely, and their description requires the real numbers, numbers represented by decimal expansions such as 3.14159…, possibly going on forever. Understanding the true nature of such infinite decimals lies at the heart of analysis.

And yet lacking the proper ∆ST theory it is yet not understood.

The distinction between continuous mathematics and discrete mathematics is one between single, synchronous, continuous space with less information, and the more complex, rich in information perception of reality closer to its nature, in terms of ‘time cycles, of fractal points; of networks of space-time entities’, which will show to be always discrete in its detail, either because it will have boundaries in space, or it will be a series of time cycles and frequencies, perceived only when the time cycle is ‘completed’, and hence will show discontinuities on time.

Thus the dualities of ST on one side, and the ‘Galilean paradox’ of the mind’s that reduces motion and detail to still mental views, limits of perception of information. And it lays at the heart of the essential philosophical question: it is the Universe discrete or continuous in space and time. Both, but always discrete when in detail due to spatial boundaries, and the measure of time cycles in the points of repetition of its ‘frequency’.

So ultimately we face a mental issue of mathematical modeling: the ‘mind is art NOT exact science’ (as pure exact science does not exist, all is art of linguistic perception) of representing features of the natural world in a reduced mental, mathematical form.

The universe does not contain or consist of actual mathematical objects, but a language can model all aspects of the universe. So all resembles mathematical concepts.

**Congruence, similarity & indifference vs. identity, which does Not exist.**

Abstraction simplifies first and then forgets for the sake of praxis the meaning of simplification. This works specially in the operand of equality, we substitute by ≈ and ≤≥ and ó which means similarity, undistinguishable as the internal parts of the being are never equal. We use a wording of ¡nglish, ¡ndifferent – as it is ‘easier’ to write, and *adds the subjective view of the perceiver who is indifferent to all the properties that make special the being from its inner p.o.v. Two things thus are indifferent instead of equal, on the eyes of the beholder. But again, as those papers are mostly on philosophy and theory over praxis, it matters to be aware we work with ¡ndifferences not identities as in E(t)=m(s)c ^{2}. All those concepts should be grouped within the 4^{th} postulate of ¬E congruence, as various cases of ¡ndifference, whose rigorous definition escape this latter-day texts but in the future should be classified along the possible ways of interaction between two ¡ndifferent forms.*

Another hypothesis that must be understood in far more complex terms is the identity hypothesis. As in the entangled Universe, not two forms are identical, as they will even in the case of identity, become different by position and different entanglement with other forms, but they can be treated and are treated as identical by minds existing in a larger ¡+x plane. Specifically identity appears beyond the ¡-3 scale in which a mind obtains its pixels of information, as all is absolutely relative to the mind that perceives it, which performs its actions in relative intervals of planes – ¡0+1 for social evolution dimotions, ¡º-1 for reproduction, etc.

I.e., the number two does not exist as a physical object, but it does describe an important feature of such things as human twins and binary stars; and so we can extract by the ternary method, 3 sub-concepts of it:

2 means the first ∆-scale of growth of 1 being into 2, by:

S-imilarity and S-imultaneity in space time, (ab. §); Ð¡somorphism =self-similarity in ∆-scale as perceived by a linguistic observer, @, which will deem both beings ‘identical’.

Whereas identity means that an @-bserver will deem the being ∆st≈St, (Sim, Iso and Eq). So identity is the maximal perfection of a number, for a perceiver, even if ultimately:

*‘Not 2 beings are identical for the Universe, but can be identical for the observer’… an intuitive truth, *whose pedantic proof is of course of no importance (: we do not follow the axiomatic method of absolute minds here):, but it is at the heart of why reality is not collapsed into the nothingness of a big-bang point.

Thus those 3± elements of the ∆@ST coincide *a social number can be used whose intrinsic properties define conceptually ‘S- imultaneity, Ti-somorphism’ and ∆-equality or equivalence (ab. Eq) in size, which becomes an @identity for the mind. Then a number is born.*

In this ‘infinitorum’ of Universal thoughts, which bring always new depths as soon as we observe it with an ∆•st trained mind, there are differences between S-imilarity and Simultaneity to define in space an ‘identity’ and ‘equality’ and equivalence, treated elsewhere.

It is then clear that a number being a sum of points, encodes more information in a synoptic way about the T-informative nature of the ‘social group’ than an array of points, *which unlike a number tells us less about the ‘informative identity of the inner parts of the being’, but provides us more spatial knowledge about the relative position in space of the members of a number-group.*

And this is obvious, when we return to the origin of geometry and consider an age in which both concepts were intermingled so ‘points were numbers’ and displayed geometrical properties:

Numbers as points, showing also the internal geometric nature, used in earlier mathematics to extract the ‘time-¬Algebraic’, ‘Analytical-social’ and S-patial-geometrical properties from them.

The closure of the systems of numbers thus grows from reflecting merely space populations (natural numbers), into 5D numbers, reflecting ‘partitions’ of social groups (represented by those natural numbers), with Egyptian ratio-nals; expanded further with the realisation that certain ratios did apply to ‘scaling’ in the fifth dimension without limit (as in the pi ratio of $T-entropic lines into cyclical time O-cycles).

Alas, things got interesting here, but as the homunculus did not understand, *the discoverer of pi, legend has. was murdered by Pythagoras, the first ‘religious mathematician’ because it found that pi was not perfect. I*t took to Poincare 2500 years latter to find that this is an awesome form of perfection because it means a mind-point of spherical form, with equal distance to all the realities it reflects can shrink with no limit (Poincare conjecture)

There are *spatial*, natural numbers, 5D rational and transcendental numbers, who cross through 5D Planes without tearing according to Poincare’s postulate. But this is NOT enough, because we need negative and lateral (not imaginary) numbers, and those are temporal numbers, numbers who describe processes in time not in space, as negative numbers do, as they represent merely the inverse arrow of time, so if you make a positive number an arrow of motion-entropy, or relative past loss of information, the negative number will be the arrow of future-information, and the lateral i-number, will be the bidimensional sum of both the real and the negative, to represent the present space-time*.* ∆±i: i-ratio-nal or transcendental numbers; ∆±1: ratio-nal numbers.

So you do have a closure of all numbers, based in the elements of space-time.

Hence, when you apply them to mathematical physics, *which is the study of the simplest forms of space-time, you do have a better focused mirror. *

**MATHEMATICAL ****EFFICIENCY****.**

** Survival is more important than knowledge.**

Among all mirrors=languages of the Universe, mathematics – topology and social numbers – dominate minds because it is the most synoptic, hence able to fit in any scale, less distorted, hence able to perform the 5 Dimotions=actions of survival better, in its expression of the laws of fractal space and cyclic, pentalogic time*.*

So as Upanishad said, ‘the languages of God are ∞; but some matter more than others. Topology should be the internal language of atoms and hence self-similar galaxies of any 10^{n }scale; that is the language of electronic eyes (lineal, Euclidean) and quarks, nucleons and black holes (Non-Euclidean curved), and all its social organisms, which are most forms of reality emerge with mathematical properties. This is said here for the obvious reason we shall NOT evolve Boolean Algebra and AI even if we could for moral reasons. Survival is above knowledge and the expansion of digital AI robots with complex mathematical minds imperils the survival of humanity. They SHOULD be forbidden.

*In the graph, languages select species that survive into the future. In evolutionary terms languages of information matter more than bodies of energy and maths is the best synoptic, focused language mirror. Problem is in this planet machines speak better maths than huminds so the eco(nomic)system selects them. *

**RECAP.** 5D philosophy of mathematics explains its whys and widespread application, based in two concepts:

- Mathematics is the
**best synoptic, focused language**we know of the ¬∆@st properties of reality. - Hence those minds that speak it better survives and are the
**selected**longest living species of reality, electrons with simplex Euclidean math and quarks with Complex curved topology & its atomic ensembles.

**Mathematics as the best mirror brain of the Universe, all pervading in ‘Atomic systems’.**

Mathematics is a mirror language that studies 5D fractal points of space; its time operands, social numbers; minds=frames of reference and entropic limits, which entangled together by the laws of pentalogic form the 5 elements of ¬∆@st , dust of space-time, the substance of which we are all made. As such is the most experimental of all stiences. Mathematics therefore exists as all other languages to provide a ‘mind’ with a still image of reality, which mirror the laws of space-time, and its dynamic superorganisms, expressing its dimensional growth through space and time. Its units reflect that dimensional growth in space through points that become waves-lines that become network-organisms. While algebra expresses that growth and its inverse devolution through sums/subtractions of ‘herds’ of points in a single plane (±); reproductive ‘invaginations’ into a lower plane (product as a measure of the axons that connect two points through its parts), or higher one (power laws); while the more sophisticated ∫∂ operands study the ‘details’ of change in any of the 5 Dimotions of existence, from mere locomotions, to social growth, reproductive curves and its inverse entropic dissolutions.

*We adapt geometry to the scalar fractal Universe with the concept of a fractal point, and the logic of mathematics to the pentalogic of 5 Dimotions, to understand the multiple variations of maths as an inflationary language, and then we make it a experimental stience referring its main elements, structures to the larger language of 5D showing mathematical equations and postulates reflect laws of the 5D Universe in its synoptic language. *Since space and time are more complex than ‘huminds’ our language of spatial thought, geometry and temporal logic must be upgraded to formalize properly the description of ‘superorganisms’ perceived geometrically in simultaneous space and logically in sequential time.

Most geometry in science is Euclidean (save 5^{th} postulate) and static; while Logic is lineal, Aristotelian, A->B.

So we evolve both into a non-AE-=i-logic topology of space, next letter, i also meaning information and the fractal scales of reality and a ¬Æxistential algebra of time to formalize the ‘stœps’, stops in space and motions in time that move the relative futures of each system, made of vital scales of space that last a finite time duration, which are generated by the bio-topo-logic properties of its vital space and cyclic time.

The Law of Closure for the Universe, established by existential algebra is simple. All the laws of all stiences derive of the laws of scalar space-time, ∆St; and its 3 units: fractal points of space, social, scalar numbers and cyclical dimotions of time. So the laws of Mathematics as a mirror language derived of ∆st Laws writes as follows:

The laws of geometry derive of the laws of fractal points of space.

The laws of Algebra derive of the laws of Existential algebra, its Scales and Dimotions. And we can easily translate the 3 elements of the fractal generator, to both, scalar, ∆st algebras (+, x, a^{-x} ) and Boolean, mental, ¬@ algebras, (V, Ù, ¬) as follows:

Reproduction, ≈, and, + v; Evolution, > , Ù, x; Entropy, <, ¬,

The laws of Calculus as the most sophisticated entangled operand of algebra also derive of the laws of scales and dimotions.

The survival consequences of the evolution of AI-gebra (Boolean Algebras that mimic the laws of existential algebra in computer brains) is obvious within the laws of the organic Universe. Once the translation is properly done, even if it is merely done in terms of simple modulo-2 programs of military robots, as the Universe is sentient and electrons are the units of life, those machines will eliminate humans. It is for that reason I abandoned the field from public scrutiny, and the last chapters of this paper won’t fully develop existential algebra, as I do try always to follow the mandate of existence that makes all species to conserve its time. However humans do NOT, and that is a theme treated in our papers on history. It seems the earth has programmed us to destroy life and create the first simplest forms of existential algebra in Computers. And it doesn’t matter how many warnings mankind in its survival temporal language of verbal thought, through its highest minds have warned the species against the eviL=antilive fruits of the tree of science. They will do it.

Let us then once we have complete our review of the basic themes of ÐST and other disciplines of mathematics needed to understand Algebra, with an extract from our paper on Non-Euclidean geometry and vital topology and frames of reference of analytic geometry, treated extensively in the other paper.

** **

**ALGEBRA**

**The 3 ages of ¬Algebra as reflection of 3 Planes of growing complexity: numbers, functions and functionals.**

Youth: ∆-1: Arithmetic, the Greek Age, now mostly studied in number theory.

Maturity: ∆: Equations extend each ‘letter’ to a range of numbers, *creating a new ∆-scale of generality for mathematics as a whole; mostly studied by @nalytic geometry* (S@) and ¬Algebraic analysis (∆T).

Old Age: ∆+1: ¬Algebra grows into a new scale of complexity making functions of functions, in time (Functionals) or in space (Groups).

As the best mirror of mathematics, ¬Algebra has also 3 ‘scalar sizes’, that of the individual number/point, the social polynomial/calculus equation and the structural group or set, which searches for the full completeness *of all the potential variations of a compound structure made of time motions and spatial points anchoring those time motions into simultaneous spatial, organic, dynamical systems represented by sets and groups*.

Though generality increases fictional variations in any 3^{rd} age detached from reality, as there is not a clear picture of the underlying space and time laws of the scalar Universe that would ‘tighten’ our admission of groups and set theory as the ‘pest’ (Weyl) in which to encase anything that moves.

*¬Algebra ages *are in a ternary scalar symmetry with the Planes of the universe. I.e., if we decompose it in its parts, an ¬Algebraic statement has 3 levels of complexity as always in the ternary Universe:

** –Sequential number**s are the minimal scalar element of a fractal T.Œ, whose properties number theory explores.

*-Functions and its operands**,* which express a partial equation of the ternary fractal generator that defines a T.œ, most often of the space and time states, connected by the function and operands, overwhelmingly the basic equation of mathematical physics (as in E(s)=M(t)c*².* So numbers are the ‘smaller broken part’ of ¬Algebra, which studies ‘social Planes’ between two planes of reality, while ¬Algebra extends much farther this in generality, studying the internal dynamic structures of whole entangled superorganisms. So its full understanding is that of all the parts of a whole together and within the whole, its dynamical symmetries between space and time states: ∆S=T made of different gatherings of numbers in equations.

**ðNº≈§@:**, means then that time numbers are put in relationship with spatial points, and time functions with space functions through the operands of ¬Algebra.

** -The structure,** which is the highest comprehensive analysis of full blocks of time-space laws; and specifically encloses all its possible variations; today developed through3 ∆ST elements:

*–* * S**óT-Groups* as a ‘receptacle’ of all the elements of the being, with a focus on its spatial symmetries=motions=changes *that do not alter the whole nature of the being, similar to the concept of a full worldcycle of time; connected to S@-topology.*

** – ∆-**Sœts, oriented to understand the parts and wholes and its relationships and a similar expansion with the concept of the functional (function of functions) with a focus on studying ‘simultaneously’ several ∆§cales.

¬Algebra as a construct of the humind (@-element) requires some corrections due to the distortions of ego-centered humind sciences, which are dealt better in philosophy of mathematics (¬Algebraic fictions).

Algebra upgraded in its foundations, becomes ¬Algebra, since the logic of relationships between points in space, scalar numbers and time operands, do not follow the Aristotelian logic in which ¬Algebra is today founded.

Thus, 5D interprets and casts the more complex structures of ¬Algebra in terms of 3±¡ ternary symmetries of the more realist formalism of the generator equation and its pentalogic. Specially its most useful ones, sets and groups, used to define full ‘blocks’ of physical systems in time and space.

**1ST OF ¬ALGEBRA: SOCIAL NUMBERS: ‘ARITHMETIC’ .**

__ Single ∆§ plane and ternary dimensions and S=T equation-symmetries.__

In its original form, ¬Algebra dealt with mathematical operations on numbers considered from a formal point of view, in abstraction from given concrete numbers/cases. It *calculated social, scalar numbers in their growth and diminution in herds (±); and its growth and diminution in 3 spatial dimensions (X, Y)³. *

*As all ‘seeds/minds’ start in a still age, motion was not clearly understood in mathematical terms.*

*As such ¬Algebra could not go further because there are only 3 dimensions in a single still space-time plane. – Fermat’s grand theorem is a clear case of the difficulty to create beyond social addition= superposition, a ‘higher dimension’, since a simple social sum in 3 D, such as x³+y³≠z³, cannot exist.*

This ‘simple’ problem is a profound proof of entropic limits within 5D world – equivalent to the classic problem of doubling the cube in bidimensional geometry, also known as the Delian problem: Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first, using only the tools of a compass and straightedge. *Since overwhelmingly a bidimensional holographic ‘space-plane’ (with points that form networks with volume), will have a ‘3 ^{rd} dimotions’ of time-movement*.

As with the related problems of squaring the circle and trisecting the angle, doubling the cube is known to be impossible with the 2 fundamental elements of topology O-compass and |-ruler.

Those problems also showed that geometry and ¬Algebra, points and numbers are ≈ similar but not =. Since continuity is a Maya of the ‘senses’, and reality in smaller Planes are discontinuous.

So some continuous mental geometries in discontinuous numbers do NOT exist (√2, pi) as such but are scalar ratios and trans-form-ative dimotions (O-π=3|).

So while geometry was born in a single plane of space-time numbers and arithmetic soon showed that the social evolution of numbers transcended a single plane of space-time.

But the subtle continuous-discontinuous variations of geometry vs. arithmetic – was never understood, as Euclid defined an absolute equality with his dogmatic axiomatic method that has weighted so heavily on human thought and its ego-trips of absolute truths… now extended to ¬Algebraic operations.

Fact is Equality does not exist – only similarity and most = symbols must be substituted by ≈ or ó symbols of a dynamic transformation; so E=Mc2 doesn’t mean that mass becomes entropy but that it transforms into it in the moment of death.

In any case the first age of ¬Algebra ended with the limiting study of a single space-time in 2 or 3 dimensions (not as it should be in 5 dimotions) and its sexagesimal or decametric Planes of numerical social evolution, using the simplest operands of ± and x ÷, and the first ratio numbers, which allowed to probe in ∆-Planes.

It added finally the concept of finitesimals with probabilities, studying ‘events’ as parts of wholes, which were studied with statistics, in an S=T symmetry latter evolved with the equivalent ‘derivative vs. integral whole’ analysis

**Recap.** We shall consider merely the meaning of the main mathematical operations within the restricted world of 4-Dimensional spaces and times we live in. And analyze in more detail some of the parameters and functions most commonly found in the study of the Generator equation, which connects the equation of space-time cycles with the detailed mathematical analysis of those cycles by different disciplines.

** **

** **

**PENTALOGIC ON FUNCTIONS.**

A function *f* is a mathematical rule that assigns to a number *x* (in most cases a time flow of motion) another number *f*(*x*) – a spatial function that limits X to certain values. For example, the function “square” assigns to each number *x* its square *x*^{2}.

**S****óT:** The common functions that arise in analysis are always analysis of Dimotions, definable by formulas, *which are related to the S@ and ¬∆T duality of the Universe and its 5 Dimotions or changes in Planes.*

**∆. **So most Polynomials of the type, *f*(*x*) = *x*^{2}; the logarithmic function log (*x*); & the exponential function exp (*x*) or *e ^{x}* (where

*e*= 2.71828…); and the square root function √

*x have also a scalar component*.

**@:** Finally, Trigonometric functions, sin (*x*), cos (*x*), tan (*x*), define angles of perception of a given @-frame.

**Operations as mirrors of 5Ð¡motions.**

A T.œ can go up and down, left and right, grow or diminish, d=evolve socially, change its @-point of view or slide in its existence from the body to the mind or limb state, and finally travel through the ∆-planes of the fifth dimension emerging in a larger plane or devolving down to its parts in the moment of death.

This is what the Universe of fractal points=T.œs is all about and what ¬Algebra as a mirror tried to describe.

The growth and discovery of new operands mirrors the expansion of our perception of the dimotions of the Universe departing from the humind mirror.

The intelligent reader will have notice easily that we do have just the exact number of classic operands to make them coincide with the Dimotions of time space. It follows that more important than ‘variables’ are to ¬Algebra ‘operands’, whose encoded dimotional meaning relates systems by merging them according to certain rules of ∆st ‘Generation, to get a ‘future or present’ outcome truly giving ¬Algebra the power to mirror the a(nti)symmetric superorganisms of the Universe and its worldcycles.

Thus the key connector of ¬Algebra with ÐST stience is a full understanding of the *relative symbols of equality=similarity and the* dual ¬Algebraic operands that reflect those dimotions, sine/cosine ±, x/, ∂∫, √xª as part of the ¡logic game of exist¡ence. But all mirror languages are relatively unfocused images of the generator equation and its operands, so the correspondence is NOT so direct. Still we observe that:

**1D: < (angle) & sine/cosine:** The 1^{st} Dimotion of perception is served not by numbers but by angles, expressed externally by a sine, which acts as the informative, height parameter of the outer Universe, and the aperture of perception of the being, maximal with maximal ‘height’, the dimension of information, (sin 90º=1).

**2, 3, 4, 5Ð social evolution and complementary merging** are served by the sum in simpler forms and the product and division operands in complex social evolutionary structures, as merging often requires first a product at the lower axon level and then a mitosis or ‘division’ into 2 wholes with all the new parts, which again gives division a very precise meaning. Hence we use in Existential ¬Ælgebra, ∑ for herding in a single plane and ∏ for networking, as the product probes into the parts of a system.

The product/division is responsible for most Merged Dimotions ‘re=producing’ through x÷ operands a lower scale of ‘connected axons’ as the product is the first operand to probe a lower plane of existence, and the essential operand to merge S and T states, from gender couples to stop and step into a new merged creative parameter, a stœp, (i.e. mass a stop state and v a moving step merge into momentum, mv, a physical stœp, which in modern physics is the most important parameter).

In that regard the less important form of multiplication is the most often used to define it in classic math (the sum of sums) as this not implies merging into something else but merely we stay with the same clone species and should be regarded as a sum that happens between EQUAL beings. In this ‘line of thought’ we do have also the x² polynomial functions and its inverse √x.

**3, 4D: The Aª and log: ** 4^{th} and 5^{th} dimotions are represented in its d=evolutions by log & exponential operands.

**5D: √xª:** social Planes: When social evolution is not transformative between planes but only a social herd, it emerges through multiple mostly decametric, 3×3+0 Planes or the √xª operands best suited to that purpose. And here again we find quite difficult the comprehension among huminds who love to ‘go only the way upwards’ so to speak of the √ operands, specially when in negative mood: √-x, a completely mysterious element of ‘mathematics’ to the point they call them imaginary numbers (:

**4D.** ∫∂ integral and derivative operands also study Informative perception and social evolution as they work for all Dimotions=changes in time and space and through ∆±¡ planes of parts and wholes.

**T**he 5^{th} dimotions of entropic devolution, ¬, in mathematical systems become inverse operations that eliminate the information of a system. As it happens entropy can then take the general format of the negative operand of the systems. So for each positive system there is a negative one. And among all the operand, there is one which is the most entropic of them all, the exponential, notably eˆ-x, whose massive negative growth signifies the growing dissolution of a form into its finitesimal parts. As systems are in general decametric, such exponential entropy also affects the very same number, which looses its ‘meaningful series form’ after 10 decimals.

Further on we can group those Ðimotion operands and its entangled mirrors in 2 great families of functions:

**2, 3, 4Ð: Polynomials** of simple T-ST-ST T.œs and its dimotional functions of social reproduction and herding into a network supœrganism **in a single plane** (with a single parameter), whereas the polynomial reflects within the limits of Dimotional beings (Fermat’s restrictive theorem, Galois limit of polynomials with coefficient solutions). A polynomial will then be a partial equation for the simplest reflections of social and reproductive dimotions, embodying the ST elements together, in ÐSTs of the type: |-1<polynomial<O+1.

1, 4, 5Ð: Complex scalar Dimotions between planes and its operands: 1D trigonometric angular perception (¡-3>¡o), 4D exponential entropy (¡<¡-2) and infinitesimal lineal derivatives and its integral wholes (¡≤≥¡-1), are then studied as T.œs performing Dimotions of scalar space-time in more complex equations.

In the same way a 4D description of reality is the limit of a 5D Universe, polynomials are the limit, and hence a good approximations to more complex 5D equations. As functions become combinations of Ðimotions, whose laws are closer to the rules of ‘a(nti)symmetry’ in ¡logic and those of the non-E fourth postulate of congruence.

**The main operands of ¬Algebra as reflection of the st, topological, temporal and ∆§ symmetries.**

–** =, <,** >, operands are the reflection of ‹, ›, «,»,≈ STóST connections, which need deep particular analysis.

– **±, x/** are the connection the basic ∆§ocial Planes between planes operands

– **xª, log x** represents mostly reproductive and decay processes. As the most complex of the ternary chain of basic operands, sum, product and power, it is a less precise version of the ∫∂ ∆±I operands, since it just reach in its ‘balanced’ sum (logistic curve) the ’emergence from one to a whole’

– **∫∂** is the most general operand that works with all forms of time-change including ∆±I emergence through integrals and ∆-1 dissolution.

– **(G, •,*)** groups are simultaneous analysis of all those elements, together in a single superorganic structure.

It does adds all variations of a given species, all complex symmetric transformations between time, space and space-time (energy) states (<=≥), and it is the closer mirror in ¬Algebra of the multiple symmetries of the fractal generator; where we shall call G, no longer group but Generator.

The very limited number of relevant operands of mathematical ¬Algebra respond to the very limited number of space-time stops and steps goes and motions through the temporal ages/states, topological forms and ∆§ocial Planes and planes that structure the organic, fractal Universe. Nothing else is needed.

**S≤≥T a(nti)symmetry symbols of similarity, smallness=information and size= energy**

Inverse operations with a neutral element exist because that is the main condition of systems in the Universe, which are in search of a perpetual balance and always end up into a cyclical zeroth sum, after stretching its ‘virtual existence’ through its Dimotions to the world suffering action-reaction processes from the world.

Thus the bridge between ¬Algebra and the vital, creative and destructive arrows of non-Euclidean topology is the concept of an A(NTI)symmetric operation, word that includes the different S≤≥T transformations a T.œ can suffer, departing from an initial symmetric condition or ‘neutral element’, which can dissociate into annihilating ‘negative operands’ =destructive flow vs. ‘positive operands’=creative flows.

The duality of ¬Algebraic operands ±, x÷, √xª, log aˆx, ∫∂, can also be considered a reflection of the Symmetric, creative, parallel vs. antisymmetric, perpendicular, annihilating dualities of Nature, which ultimately refer to the Inverted Duality of inverse Dimotions of 1D still perception vs. 2 D locomotion, that switch on and off in ‘steps’ and ‘stops’ (ab.stœps), and the duality of 4D entropy and 5D generation (∆±1), all of them merged into the present reproductive dominant Dimotion of the Universe.

So we have a slightly different ‘mirror’, in this case to relate non-e geometry, non-A logic, and ¬Ælgebra, which will be useful for mathematical physics. In that mirror we can give a new definition of ¬Algebra:

*¬Algebra is the study of the a(nti)symmetries, reflected in the operands that connect Non-Æ points in space perceived as numbers in scale together through a simultaneous ≤≥ equation. So the first analysis is that of the symbols of self-similarity, <, > and =. And its translation into the logic symbols of 5D, > (implosion that reduces a system in space but increases its information) =, which moves a bodywave into an equal form through stœps (stops and steps) and < which increases the size of a system in space, but reduces its information.*

Human ¬Algebra has chosen* a lineal view and a spatial view, *by considering the dual paradoxical symbols of non-Æ ¬Algebra: T>S, S<T, T≤≥S, in 1/2 of its meaning, NOT as a balance of T>S growth of information and shrinking in size, and vice versa, S<T, growth in size-motion (Galilean paradox: all distances are motions seen in stillness) and decreasing in information; but just as grow in size – the spatial simplistic view in a single plane (as information migrates, to the warping inner folding of the shrinking process).

Huminds though observe only one side of the duality of > and < logic operators, namely, they look at the ‘size’ of it, not at the inverse growth of information, obsessed by the ‘bigger’ forms. So instead of interpreting, T>S as a shrinking in size that brings more information, they consider only the symbol as T smaller than S (but it is also more complex in its information than S). And vice versa, S<T (but also S has more information than T).

Regarding the = symbol Is important to remember that = does NOT really exist as absolute equality (as all points and numbers, gathering of points have internal parts and dissimilarities) but IT IS MORE either a parallelism or similarity symbol, ≈, or a sequential logic operands of implosion or explosion of form =>: ≥, =<:≤, or a combined feed back equation <=> of both in a dynamic stœp series.

I.e. We say Energy is mass,but we should say is ‘equivalent’ to mass as E=Mc^{2 }is a ‘transformation’ of an informative T.œ, Mass, (a vortex of space-time of the cosmological scale) into an Entropic dissolving expansion of space in the lower plane: M=>E, but as the motion can be reversed and entropy-energy can evolve into mass, it is really <=> a feed back symbol. As all mirrors simplify reality humind’s mathematics reduces it all to an equality in which the inner content is not considered, not even the form but just the equality in a single parameter (in e=m, the energy-motion).

*The basic equations of Existential ¬Ælgebra.*

A Supœrganism comes then defined by a sœt of basic feed back ¬Ælgebraic Equations that represent those Dimotions and its chains, of which the most important are those that reflect the 3 ages of any world cycle, which is born as an expansive motion, reaches maturity and then warps:

-S=T, that is S(i)<=>T(e) as in the previous case, represents a present balance between energy and information and the dimotion of reproduction, of beauty of coming together when energy and information peg into a single, being – hence of gender also.

S x T = K, with its two extremes of Max § x Min. t (informative dimotions) and Max t x min. $ (locomotions).

Needless to say in mathematical science, as opposed to stience, all is simplified or blurred by slight errors of the @-one dimensional humind.

So the first thing we must understand is that any mathematical equation will substitute, ‘all’ the operands or logic connections between the two parts of the equations by a simple = symbol of equality and one-way mostly spatial < or > smaller or bigger than.

**The equality symbol becomes complex.**

In classic mathematical science, as opposed to stience, all is simpler as time is lineal with a single dimotion. So logic is A->B, Aristotelian. Thus a classic mathematical equation will substitute, ‘all’ complex operands of ¡logic connections between the two parts of an equations by a simples = symbol of equality (identity).

We define instead ‘5 Symbols’ to *express the 5 Dimotions of the Universe, as trans-form-ative processes between the two parts of the Sœt.*

So we *subdivide t*o ‘regain the information missed in the nature of the equation that represents one of the 5 Dimotions, = for, <, ≤, ≈, ≥, >:

A<<B is an entropic process or 4 Dimotion.

A< or A≤B is an energetic process, most often related to the 2nd locomotion.

A≈B is an identity, often born of a reproduction or 3rd Dimotion.

A≥ or A>B is an informative, perceptive process, or 1^{st} Dimotion.

And A>>B is a linguistic, social evolutionary process, or 5^{th} Dimotion.

The choice is obvious as we can consider entropy, a social process of multiple locomotions, in expansive mood (<, <<) and vice versa, information and perception > is an individual inward informative dimotion, which for social evolution in the higher whole requires a multiple >>. We prefer to use ≤ and ≥ for ‘energy’ and ‘information’, when they are in a merging process, as those are the ‘states’ that finally merge into the act of reproduction, ≈.

It is a necessary inclusion to add information to vital mathematics, which we shall often include. Now as this is mathematics and ≤≥ have obvious meanings on math we shall establish for future stientists the use instead of the reduced symbols « for 4D entropic motion and » for 5D social evolution.

A ≤≥ feedback equation is the ‘time-perspective of mathematical thought’, in terms of the cyclical nature of time, as it establishes the ‘feed-back equations’. So the main change of Existential ¬Ælgebra is to substitute the = operands, which often is interpreted as a logic, A->B concept for the <≈> operands of a feed-back equation.

¬Algebra studies A(nti)symmetries (> ≈<) of space-time, achieving its maximal generalization; hence it is a ‘mirror in space’ of those 3 formal symmetries, anti symmetries and asymmetries through its inverse operands and equalities, which define fully the dynamics of the Universe, in the same way Analysis studies them across Planes.

But ¬Algebra can be considered to study all ‘motions=changes’ as a series of s<≈>t relationships which include the ternary asymmetry of analysis, the topological ‘a(nti)symmetry’ and the temporal one.

From a different point of view we can consider that ¬Algebra studies larger wholes as ‘frozen blocks’ of time-space while Analysis studies detailed ‘stœps’.

**Functions’ form & operands represent the type of dimotion in time or organ in space that performs an action.**

The form of the function already tells us a lot about the type of dimotion we study, and this also extends to the type of differential or integral equation we study as we shall see in our paper on calculus.

For example, entropic functions, have the form of hyperbola of the type y=n/x, as they dissolve a whole into its 1/x finitesimal parts. So the law of Boyle-Mariotte v=c/p where the magnitudes p and ν are inversely proportional mirrors an entropic dimotion of a gas, hence it represents a branch of the hyperbola lying in the first quadrant.

For example, the length l of a body may be considered with good approximation as a linear function of its temperature, which is an $T function of a limb/field topology of energy feeding. So we write

where α is the coefficient of linear expansion, and l_{0} is the length of the body for t = 0; since the system is acquiring a form of lineal, kinetic energy called ‘Temperature’.

The general class of oscillatory processes includes periodic motions, which are usually described by the familiar trigonometric functions. For example, if we extend a hanging spring from its position of equilibrium, then, so long as we stay within the elastic limits of the spring, the point A will perform vertical oscillations which are quite accurately expressed by the law where x is the displacement of the point A from its position of equilibrium, t is the time, and the numbers a, p andα are certain constants determined by the material, the dimensions, and the initial extension of the spring. Since we are expressing an informative, iterative worldcycle of existence (whereas reproduction is expressed by the operand of product but also in its frequency repetitions by the cosine).

In this manner the future trained 5D philosopher of science will by the mere fact or seeing an equation intepret what Dimotion of existence the system under the equation is performing.

** ∆+1 scale equations. The no derivative: moments of present, standing points and the calculus of variations.**

The most important point of a function though is precisely the one that cannot be derivate, where derivates are zeroth meaning there is no change at all. It is a present moment.

So calculus of extremal points, or calculus of variations, appear, to study those points of change of ‘phase’ from a young to an old state, through a relative point of present, or points of sudden rest, in the ascending or descending curves of a derivative function of an action of exist¡ence.

The stœps thus traced by the derivative function of an action of exist¡ence can be classified by operands of those actions in different sub-studies of analysis, taken somewhere else.

Indeed, the way to follow a curve of an equation in time, is merely following its derivative, through the ∆-increasing future motion region, ∂0, its present standing points and Ñ< past motion regions.

Now the most important question about those derivative functions is how many hidden points of 0 derivative there is in the ‘irreal’ line of the function. It is a irreal number whole derivative? Or the mind makers a continuum eliminating more zig-zag standing points?

It is a question closely connected to the time nature of the i-complex plane, where the –it value of the time parameter makes its relationship to the spatial present form of the real plane in XZ a very interesting analysis of the true curves of space-time of the real 5D Universe.

Whereas the product of modulo 1 as a sum of time moments turns around in a wheel of time representation of the worldcycle the function of present of the real plane.

It is then necessary to understand first the 5 Dimensional motions of reality and its entangled pentalogic * or ecosystem requires the previous Postulate to b created, as each ¬Algebraic operand includes the previous one, and each ilogic system needs the simpler forms of thought to be=come.*

*So we shall briefly introduce the 3 main stairs of growing complexity of organisms, the dimotions of ÐST; the systems of logic, the postulates of Non-E and the operands of ¬Algebra.*

**RECAP. **Operands represent dimotions, and equations chains of dimotions, such as:

**1Ð: the trigonometric** functions ARE the angles of perception.

**2Ð: The sum** is the operand of locomotion that ads frequencies of steps and simple social evolution in herds.

**3Ð: The product** of re=production that merges to variables.

**4Ð: Calculus** of social evolution of parts into integral wholes.

**5Ð:** The negative exponential power of entropy and decay. And **the inverse functions** of each operand, which become the inverse process of entropic destruction, such as ∫∂, related to the inverse arrows of ∂-entropy – dissolution of wholes into parts, and 5th dimension of social evolution – ∫-integration of parts into wholes

¬Algebra deals with sequential numbers and equations, in which sets of numbers are transformed by an X=Y function, in which often one component changes faster than the other, tracing a curve in a Cartesian Plane plotted with those 2 variables that mathematicians study with great detail. It is the thesis of this work that most those X=Y functions and differential equations represent particular studies of the general ∑Se<=>∏Ti fractal generator equation, and or a partial scalar event between n-points of a non-Euclidean plane=network, and or its environment.

That is mathematical equations reflect in its non-fiction ‘core equations’ the ∆ST laws of the Universe.

Yet as Einstein put it to Poincare: ‘while I know when mathematics are truth I don’t know when they are real’, meaning that many mathematical equations and functions do not exist in nature, as they are not partial cases of the Generator equation and do not respond to the restrictions the Ternary symmetries and pentalogic method imposes to a Universe of multiple spaces and times but only 5 Dimotional arrows.

In that regard, the laws of multiple spaces and times and the syntax of the Generator equation with a limited number of variations restricts the possible mathematical realities there is in the physical Universe.

*In praxis though as most ‘fictions’ appear in the 3 ^{rd} age of a system with inflationary form, the 2^{nd} age of ¬Algebra thanks mainly to its use in mathematical physics, restricted those fictions, becoming the classic age of balance between the language mirror and reality.*

On the other hand ÐST studies provide the scientist with a deeper meaning for the ¬Algebra of numbers and the meaning of equations and functions.

So in a 2^{nd} age ¬Algebra focused on variable equations, which represented T.œs’ Disomorphic events, symmetries between its ∆, S & T elements, and ST-œps performed by different dimotions of space-time.

*Because most motions and changes follow an S≤T≥S… series of stops to gauge information and motion to reproduce it, S(y)≤≈≥T(x) becomes then the commonest generator of ¬Algebraic equations.*

This abstraction found expression in the fact that in ¬Algebra magnitudes are denoted by letters, on which calculations are carried out according to well-known formal rules.

¬Algebra considers space Ú. time magnitudes of a more general nature than numbers It studies operations on these “magnitudes” which extend the formal properties of the ordinary operations of arithmetic: addition, subtraction, multiplication, and division, by adding scalar dimotions *on 4D-5D, through ∫∂ and log/xª* operands.

But the greatest advance of this 2^{nd} age started in Islam and fully realized in Europe was to ad motion to the original geometric calculations of still dimensions* with analysis.* Hence the need for ‘variables’, operands of change, and finally vector magnitudes, which include both form and motion, even if they are represented in mental space with geometric rules, as the well-known parallelogram rule of addition. The problem remained of its proper foundations unattainable without a proper ÐST model of reality.

**II AGE OF ‘¬ALGEBRA’: THE AGE OF ANALYSIS AND ITS OPERANDS.**

**Rise of analysis: change=time and finitesimals=Planes. Differential functions**

¬Algebra completes its main task – the ‘translation’ of reality’s dimotions through operands – with the fundamental 4,5D ∫∂ duality of analysis, which becomes the most important field of mathematics, as the scalar Universe do include all other elements within its folding.

So Analysis is ¬Algebra and so number theory and @nalytic geometry… *as it can now use numbers to deal with Planes beyond classic *geometry’s definition as a spatial, single plane view that analysis of ‘finitesimals’ extends to an ∑∆-1 herd of points-parts that integrate into wholes.

So Analysis becomes the natural evolution of ¬Algebra into the realm of motions and Planes.

¬Algebra enters a classic 2^{nd} age, in which the essential elements of ¬Algebra are no longer polynomials – a rough approach to differential changes, approximated by Taylor/Newtonian binomial methods – but calculus operands.

It will take longer for the study of the relationships between the symmetric sides of the equations, to take a flight of its own, focusing on the structures created according to the rules of engagement of those operands.

While Mr. Heaviside, the most underrated genius of mathematical physics, for his absolute despise of wealth, power, fame and pen pal peers, brings a new mathematical species into being, a ∆T vector with scalar magnitude and time motion.

The extraordinary power of analysis to mirror reality though arises from a ‘magic property’ never fully understood by huminds – the fact that time is change=motion, any kind of change, and so we can classify time into 5 fundamental ‘dimotions’ (dimensional motions of spacetime), which CAN BE processed with a seemingly so simple function as the derivative, F(x+h)-F(x)/h; unlike all other sciences that study a single type of change=motion (Physics of locomotion where time is exclusively defined in terms of v=s/t, hence only able to study with a proper focused mirror sT-locomotions and TT-entropic motions, where motion happens not only externally for a moving space form (sT) but also internally, destroying the form (TT).

On the other hand, biology, the other great field of stience (as social sciences are in fact biological ones, regardless of the ænthropic humind, with its Anthropic+entropic denial of the equality of all elements of reality, from its self-centered ego and entropic, simplifying, destructive models of larger realities).

However where analysis fails in its predictive capacity for time analysis is in the ‘long term’ time processes; as a derivative by definition is a limit when t->0. So it is essential for huminds to understand the scalar limits of ‘time analysis’ beyond the ‘minimal stœps’ of timespace dimotions=actions of finitesimal nature (as all infinitesimals have a limit); and accept that for long time term processes we shall observe:

*A worldcycle as a 0’-sum, with its time ages better studied by biological d=evolutionary theories and eusocial evolution and reproductive radiations of competitive species (4*^{th}and 3^{rd}ts dimotions).*A Deep time similar cycle at a much slower scale with the 3 Horizons of spaces and 3 ages of ecosystems.*

Or in other words, Physics for locomotion and entropy dominant in lineal spatial motions (Ts, TT), Biology for Social and Darwinian evolution & reproduction (ST, St) and analysis for all type of changes in short spans are complementary time analysis, whereas contrary to belief, the simplest, less important is physics of V=s/t.

Analysis though branched out of Algebra to study all time dimotions of change, so we shall dedicate a full paper to it, to complete our 3 mathematical papers on the 3 elements of reality, ∆-algebra, S-geometry and T-calculus.

** **

**PENTALOGIC ON ANALYSIS**

**Trilogic on Calculus. Curvature of space; change in time; finitesimasl in scale.**

**T****he enormous advantage of algebraic dimotions and calculus over all other forms of study of motion, including physics which can be considered basically the application of the mathematics of change to the study of nature is the fact that it can study all the elements and dimotions of the Universe from the ‘mind’s perspective’ – that is to study time-motions, space-change (volumes, lines measures), scalar change and the entropic limits of reality. Let us briefly introduce those 3∆¡ problems, which in fact gave origin to calculus. **

**T-ime=Motion=Speed .**

XVII C. science was concerned with problems of motion. Copernicus and Kepler introduced the concepts of Earth rotating on its axis and revolving around the sun. The earlier theory of planetary motion, dating back to Ptolemy, which presupposed an earth absolutely fixed in space in the center of the universe, was discarded. The theory involving an earth in motion invalidated the laws and explanations of motion that had been accepted since Greek times. New insights were needed to the question of why objects stay with the moving Earth seemed called for.

All of these motions—those of objects near the surface of the earth and those of the heavenly bodies—take place with variable velocity, and many involve variable acceleration. But the branches of mathematics that existed before calculus was created were not adequate to treat them. So a method was required and that started up calculus.

** S-pace= form=curvature.**

The 2^{nd} major problem of XVII C. mathematical physics was the determination of tangents to various curves. Its deeper significance is that the tangent to a curve at a point represents the direction of the curve at the point, *as small steps are lineal, open free, but the long term *motion closes into itself. This key element of ‘scalar time’, which makes easier to predict longer life-death cycle and curved trajectories is the key to the interplay between small scale lineal tangent points and large scale.

Its practical use was to find out the best angle for the motion of a projectile shot from a cannon. Since, if a projectile moves along a curve, the direction in which the projectile is headed at any point on its path is the direction of the tangent at that point. The invention of the telescope and microscope also stimulated great interest in the action of lenses. To determine the course of a light ray after it strikes the surface of a lens, we must know the angle that the light ray makes with the lens, that is, the angle between the light ray and the tangent to the lens. So the study of the behavior of light was, next to the study of motion, the most active scientific field in that century, the question of finding the tangent to a curve was a major one.

**¬ Entropic Limits and ST-reproductive maximal**

A 3^{rd} class of problems besetting seventeenth-century scientists was about maxima and minima. The motion of cannon balls obsessed Galileo, the weapons master of the Venetian arsenal, seeking the determination of the maximum range. As the angle of elevation of a cannon is varied, the range—that is, the horizontal distance from the cannon to the point at which the projectile again reaches the ground—also varies. The question is, at what angle of elevation is the range a maximum? Another maximum and minimum problem arises in planetary motion. As a planet moves about the sun, its distance from the sun varies. What are then the maximum and minimum distances?

**0’-finitesimal scales:** All those problems required calculus, which was based in the finding of a finitesimal quanta of Time (1/T), space (1/R) or scale (1/N). Then considering the system suffered a ‘deterministic’ rate of change, defined by a mathematical function (that introduced the specific type of rate often associated to each of the 5 Dimotions of the Universe, St, Ts, ST, SS and TT), the scientist could ‘extend’ that rate of change into a ‘sum of spacetime stœps’, which gave us a ‘lineal period in time, or continuous surface in space’, over which the longer time period was calculated. The ‘magic outcome’ (as this simple foundations of calculus were messed by creationsit egocy) thus solved the essential problems of mathematics.

**Space measure (lines, volumes). Curvature in space.**

Still another class of problems concerned the lengths of curves and the areas and volumes of figures bounded by curves and surfaces. Elementary mathematics suffices to determine the areas and volumes of simple figures, bounded by line segments and by portions of planes, found in the I age of lineal, deterministic axiomatic Greek geometry. However, when curves or curved surfaces are involved, elementary geometry is helpless. The calculation of the volume of such figure can be done with calculus. Since it became the realm of the antiderivative, the 4^{th} social dimension of ‘integration of’ wholes as opposed to the finitesimal search, of the parts. So analysis can study quantitatively how forces become particles that gather into atoms, molecules, matter, cosmic bodies and galaxies, through Planes of the ‘5th dimension; as social growth *gives motion to all of them. *

The efforts to treat the four ¬∆St problems led mathematicians to calculus. As *always reality must be perceived to exhaust its possibilities from the 5 dimotions/elements of spacetime. *Only mind problems were ignored in calculus as those were subjective to the observer.

**Since pentalogic elements space, time, scales, entropy and linguistic minds, are often reduced to tetralogic, with the element from where we observe the others becoming the blind spot of the mirror – the element from where we make a perception of the other entangled 4 elements of the whole – in this case it became the 0’ of the reference point in which the equations of calculus were plotted. **

**So calculus can be defined as the study in its smallest ‘derivative scales’ of 4 dimensional motions of time-space: TT-entropy, Ts-locomotion in space, ST-reproduction in present balanced time-space, St-information, from the SS-mind point of view of a mathematical frame of reference. Since those 4 fundamental problems of Calculus studied as dynamic dimotions, will expand enormously in complexity but essentially remain the same till today.**

But since analysis studies ALL 4 Dimotions with content of change=time, of any species, it expanded greatly classic physics, from single analysis of locomotion in time (Ts) to tetralogic motions (St, Ts, TT, ST).

This was a vast expansion of our understanding of the modes of time=change, *which however was NEVER as in this papers, translated into a philosophy of time, which was kept reduced to lineal time dimotions. *

So while *scientists *do study all the forms of change in praxis with mathematics, they do NOT have the proper philosophy of time-change to unify them in theory and understand time as all forms of motions besides v=∂s/∂t.

Still analysis – without Huminds understanding the why of its powers – the equivalence between time and change, the S=T principle of relativity, the 5 Dimotions of the Universe it measures – does study the 5 Dimensions of space and its motions in time, which change reality, both as separated ‘partial derivatives’ and in scales of change (position=form with no change, constant y’ speed, y” acceleration and y”’ jerk) up to the minimal actions of a time worldcycle, performed by a supœrganism.

It does also extract spatial finitesimals of ∆-1 scales with the study of parts of a cellular population that integrate in space to form organic wholes.

While the magic fact that *all those forms of dimotional change can be described with a single function involving the ‘scalar finitesimal part’, h, of a whole, is perhaps the biggest proof of the 5Dimensional, scalar nature of reality and the symmetries between ∆-Planes, Space forms and time Dimotions (∆¡= SxT: S=T).*

**Comparison between ¬Algebra and Analysis. Birth of Analysis from ¬Algebra.**

Departing from ¬Algebra, analysis became the main branch of ‘realist mathematics’, with applications to describe the real ∆ST-world, its Dimotions and Planes – this fundamental element, as ‘finitesimals’ are the spatial parts of a whole, the time actions of a worldcycle – ignored by lack of a 5Dimension in human science .

So we just need to add an scalar in-depth understanding of its laws, to better explain its equations and applications to the classic disciplines of science that use it.

But the focus here will be not so much in the ages of analysis, as it is a modern discipline with few insights mostly philosophical on the theme of individuals, infinitesimals and universals, on the first Greek and original classic age (Newton and Leibniz) – the introductory themes developed next.

Instead our focus is on the *3 ages of growing complexity and generalisation as analysis and its 4D ∂ and 5D ∫ operations expand to study multiple dimensions of space-time together.*

So after make only some basic remarks on the earlier era we consider 3 Planes of growing complexity, we will term loosely as ‘Calculus’, ‘Analysis’ and ‘Functional’ ages:

The classic age of polynomial limits, infinitesimal calculus and simple derivatives and integrals.

The modern age of ∫∂ applied to multiple space-time variables (Γst view: Ordinary differential equations) with different degrees of depth (∆ view: partial differential equations).

And the 3rd age of Analysis, in which Lie Groups and/or functionals of functions are the all-extended field of inquire, causing very profound all-encompassing attempts to analyze a function or T.œ at all levels.

As we go along obviously our purpose is NOT to make a classic text of analysis but considering the main themes to enlighten it with the insights of ∆st, to *resolve the whys of analysis, latter applied in detail to the many stiences described today with the formalism of analysis without understanding what truly those equations mean. *

As Analysis sprung from ¬Algebra we distinguish their fields of inquire, paralleling the evolution of the Humind. Essentially if you understand pentalogic, in the same way your stomach is an entropic system which actually also thinks, etc. each subdiscipline specializes in a dimotion or structural element but it also is useful for all others.

So modern ¬Algebra IS more focused in space-forms and its simultaneous, group structures (S<≈>T) and analysis is more focused in time motions and its **scalar** laws of change (∆T) among 5D planes of spacetime*:*

“¬Algebraic equations studies a(nti)symmetric <≈> transformations of 5D space-time dimotions in a mathematical §œT, through its inverse ‘operations’ that reveal the initial and final dimotion of the being, perceived as a whole in a relative present-spatial, static state’.

¬Groups are *concerned with spatial, simultaneous structures, joined by <=* >symbol of equality and symmetry.

*Analysis specializes in time-motions and* rates of change between planes. So Analysis, ∆t, IS the study of all the Dimotions of space-time beings from the point of view of its mathematical mirror. In 5 D though we must consider not one but 5 Different Dimotions, and so a more thoughtful consideration of each operand, ∂∫ of Analysis and how they act on different Dimotions and different Planes is needed.

In terms of Planes, analysis is concerned with the creative and destructive processes which ad or subtract NEW planes of existence, through the integration of multiple small parts that become wholes, or its reduction through derivatives. focused on the VARIABLES, or parameters of Change, which is maximised by dimotions between ∆±i planes of the 5th dimension.’

For example, in the case of volumes, areas and lines, it studies how to grow or diminish in dimensions of space. In the case of distances, speeds and accelerations, it studies the growing or diminishing ‘tail of past, present and future’ of the system, as distances is the summitry of ‘past speeds’ that have become distances passed, or the present speed which in a derivative does not change (Lagrangians tending to zeroth, infinitesimal calculus), or the relative future ‘forecasted’ by the acceleration of the being.

*When we derive and integrate in space we subtract or add dimensions of static space, and when we do so in time-motion we are adding past tails (integral of speed that gives us the distance moved), or forecasting future accelerated growths or limiting our time-span of analysis departing from the present derivative of the being.*

¬Algebra is more concerned with ‘STŒPS’ in a single plane and its ST<≈>TS stop and go dimotions, and its a(nti)symmetric changes, stop and go dimotions.

**Finitesimal Quanta, as the limit of populations in space and the minimal action in time.**

So there is behind the duality *between the concept of limits and differentials (Newton’s vs. Leibniz’s approach), the concept of a minimal quanta in space or in time, which has been hardly explored by classic mathematics in its experimental meaning but will be the key to understand ‘Planckton’ (H-Planck constants) and its role in the vital physics of atomic Planes.*

It is then essential to the workings of the Universe to fully grasp the relationship between Planes and analysis. Both in the down direction of *derivatives and the up dimension of integrals; in its parallelism with polynomials, which rise dimensional Planes of a system in a different ‘more lineal social inter planar way’.*

*Polynomials and limits are what ¬Algebra is to calculus; space to time and lineal ¬Algebra to curved geometries.*

The vital interpretation though of that amazing growth of polynomials is far more scary.

Power laws by the very fact of ‘being lineal’, and maximize the growth of a function are not real in the positive sense of infinite growth, a fantasy only taken seriously by our economists of greed and infinite usury debt interest… where the eª exponential function first appeared.

The fact is that in reality such exponentials only portrait the decay destruction of a *mass of cellular/atomic beings already created by the much smaller processes of ‘re=product-ion’ which is the second dimension mostly operated with multiplication (of scalars or anti commutative cross vectors).*

So the third dimension of operands is a backwards motion – a lineal motion into death, because it only reverses the growth of sums and multiplications polynomials makes sense of its properties.

**Stœps of dimotions**

Further on as derivatives and integrals are inverse operand that can combine in **¬Algebraic S≈t equations** (down) for more complex description of multiple events, even cancel each other to give us a ‘constant social **number’ **as an exact quantitative result for a specific value of a sequential ‘sœt’ of stœps of Dimotions.

So the* ‘Rashomon effect’ of pentalogic shows how analysis satisfies as a mirror the description of the main components of the being; further enhancing its description by the fact that ODEs can combine in several stœps, the 3 elements – time integrals/derivatives, spatial and scalar integrals/derivatives in a single function:*

**-Temporal view in **Time curves that often use cyclical time frequencies (Fourier transforms) to describe a combination of time and scale events (wholes decomposed by the transform)

– **S=T**: **combined** **Time-Space view **that resolves symmetries between time dimensions and space motions (S≈T) expressed by ODEs of two variables – a parameter of space that changes with a dynamic function/action/motion in time. It can measure through closed membranes information from the inner vital energy of the system as it moves through time (Continuity equations).

**– Ceteris paribus S=T **view**,** when 1 of the parameters/dimensions is fixed, belonging to space and the other to a time motion is operating or vice versa, perfect to mimic the stop and step discontinuous form of most time space dimotions.

**-Spatial view** through lineal, surface and volume integrals; forms that measure the 3 ∆±¡ elements of a T.œ’s topology, made of ∆-1 points (atoms, cells individuals) both in time and space (a population or a distribution) .

If analytic geometry resolved ¬Algebraically geometric spatial problems, with ‘a dual point of view’ that increases the easiness of solutions -Descartes – showing the ¬Algebraic laws of solution of rule and compass for geometrical problems; calculus took this approach on S=T symmetries to a much higher level.

– ∆: **Scalar view, **which defines infinitesimals (Leibniz) as the 1/n cells/part of the being given also by a derivative; since the minimal rate of change of any system is one of its ∆-1 units.

In that regard, the earlier astonishment of physicists and mathematicians which found the fact that a derivative (the analysis of a step of motion) was inverse to a volume of space (an infinitesimal of a population), is a deep proof of the essential symmetry between space=still states and temporal motions (stops and steps=stœps).

The fundamental Pentalogic use of analysis is in the study of timespace Ðimotions which started as all systems do, in its simplex forms, locomotions (speed) but we shall upgrade to the ‘analysis’ of all others.

**RECAP.** ¬Algebra is the analysis of systems which focus in numerical social quantities and symmetries between dimensions, than the organic ‘fluid’ properties of systems described through analysis.

**¬Algebraic operands. S≤≥T**

As Analysis sprung from ¬Algebra, is often considered part of it. AND So we introduce some basic concepts of 5D ¬Algebra from the post on ¬Algebraic equations.

We could say that simplex ¬Algebraic operands ‘topped’ its evolution with the discovery of ∫∂ operands:

A SœT – the slightly changed name for a set of Tœs – is *any kind of indistinguishable entities=numbers=points=∑œ connected by one of the possible ‘a(nti)symmetric’ relationships of ilogic, Existential ¬Ælgebra, defined by the inverse operands of the 5 Dimotions of the Universe, ±, ≤≥, x/, √xª, ∫∂…*

As such operands play a fundamental role on Analysis, which *as all physical equations relate complex processes of trans-formations of Sœts, through different dimotions of space-time.*

Regarding operands, it is not so simple to ascribe each operand to a single dimotion, as they are entangled operations, which while being preferential to a given Dimotion, do participate of the others – remember languages as mirrors of reality have also the same entangled properties of the pentalogic, ¬∆@ST universe.

Still we establish often such direct relationships. So as ‘specialized’ operands, the closest correspondence to each dimotion is as follows – taking into account that we must for each operand distinguish on the classic concepts of ‘space-volumes’ and ‘time-motions’ of science (or else we will go too far), as generally speaking, a *first derivative in space or time defines those dimotions only as S or T, while double derivatives often work for both together.*

*±:** 2 Ð l*ocomotion is best served by the + operands, as we have shown in the analysis of a time tail (motion-memory of a distance), which is a sum of ‘steps’ that also can be calculated with integrals. The sum is also the key operand for the ∆§cales of social groups, in decametric form, which also is served by the logarithm. And so on… The negative operand however is profound in many ways scientists do not understand. Reason why so many errors, from the negation of the faster than light speed, to the confusion of particles and antiparticles arise. The INVERSE operands are in general misunderstood because as we have said so often unlike the paradoxical ¡logic Universe, the humind is @ristotelian, single arrow A->B So the B->A is quite missed; but generally speaking served by the negative function. I.e, a negative spin just have the inverse orientation, a negative coordinates just means to move in the other direction. Negative operands thus are MOST useful for time=motion related systems.

In this is important then to understand the existence of one-way dimotions vs. 2 way dimotions, where operands make sense. Because huminds do not properly distinguish both, they get confused when trying to consider a negative operand for spatial forms (what is a negative apple? nonsense) while there are always negative ‘directions’ for temporal motions (what is left and right motion?). Clearly all understand.

3Ð reproduction by the product and division operands, as reproduction often requires first a product and then a mitosis or ‘division’ into the whole parts, which again gives division a very precise meaning.

The 4^{th} and 5^{th} dimotions of TT-entropy and complex integration of informative perception and social evolution by SS-minds, represented by dual << vs. >> operand can be studied with the ∫∂ integral and derivative operands.

Yet when social evolution is not transformative between planes but only a social herd, it emerges through multiple mostly decametric, 3×3+0 Planes through the √xª operands best suited to that purpose. And here again we find quite difficult the comprehension among huminds who love to ‘go only the way upwards’ so to speak of the √ operands, specially when in negative mood: √-x, a completely mysterious element of ‘mathematics’ to the point they call them imaginary numbers (:

Those operands do NOT have a one to one correspondence to each of the Dimotions of space-time. Since each of them have as everything in reality a pentalogic multiple purpose, as we shall not cease to repeat, the basic feature of reality is to be an entangled game of 5 elements which are themselves ‘fractal’ in its nature, that is, each of them will be able to perform the other 4 tasks to become in itself a ‘whole’ being made to the image and likeness of the total fractal, mind of the Universe and its ilogic structure. So while certain operands are clearly more useful for certain dimotions, all of them can be used to a certain degree of accuracy to ‘reflect’ a mirror image of the 5 dimotions of existence – themes those to be studied in depth in the posts on ¬Algebra and ilogic.

It however becomes evident from the beginning that we ascribe the more complex dimotions, which do enclose in their actions the other 4, to the operands of analysis, specially those who ‘transcend’ and ’emerge’ between Planes, as they are processes NOT of lineal nature, best served by them.

So basically Analysis is a small part of ¬Algebra, but the most important, as it focuses only on a(nti)symmetries between Social planes, ∆§, which are either integrated into a higher plane of the 5th dimension, or derived into its parts. But Analysis also goers beyond ¬Algebra, in as much as ¬Algebra is a more static, spatial, structural view; whereas analysis considers in depth the ‘motions’ of the set.

In those two definitions we must make some precisions of terminology:

Sœt or §œT, which the reader should observe is the inverse of T.œS, expresses the modern unit of mathematical thought, constructed as always by arrogant huminds, with a creationist sense of the language, from the roof of the mind to the bottom of reality, the set in inverse fashion as a collection of ‘points of space or numbers of time’, which ARE the real units of the mathematical space and time Universe, gathered then in social collections called functions, connected through the inverse operations that reflect the main symmetries and relationships between ‘herds of points or numbers’ (±, x÷, xª √ log, ∫∂).

The difference then between ¬Algebra and Analysis is in the different focus on the operations=symmetries between Sœt§ and its study as Steps of timespace motions (Analysis of derivatives) then gathered into longer super organisms (volume integrals) or worldcycles (time integrals), where the existence of limits (singularities and membranes that encircle the system or set the beginning and end duration of the world cycle) will become fundamental to reveal a solution to the equation – the finding of the duration, surface and interaction between the parts of the Tœ expressed as event in time or system in space.

So while the elements of ¬Algebra and analysis – equations of SŒTS is the same, the focus on spatial form (¬Algebra) or temporal motion (analysis) makes them diverge. We can play with the acronym and say that ¬Algebra deals with SŒTs (Space-dominant structures) and analysis with TŒS, Time dominant ones (in motion).

This simple equation of ¬Algebra translates most time actions to space, on account of a simple realisation: that space slows down time cycles to accumulate them in still simultaneous bigger forms, and as such most spatial dimensions are referred as Y composite of multiple elements of the smaller, much more abundant time cycles space normally fixes and encircles with its @ =membrane.

¬Algebra and Analysis ‘ARE’ the complex ‘level’ of reality, as reproduction and social evolution are the complex dimotions, including obviously as its ‘background parts’, the theory of numbers, the analytic geometry – study of frames of reference, and the topologic analysis, embedded in the secondary operand, numbers and frames in which we ‘cast’ the complex space and time ¬Algebraic and analytical analysis of a ‘Domain’ of the fractal Universe.

**@nalysis**

All languages of perception have a blind spot in its syntax – that is, a relatively ignoramus of properties and perspectives of reality it is not fit to study.

**In analysis is** the linguistic still mind-mapping element, since by definition analysis is the study of Dimotions. Still @nalysis will be the search of the ‘fundamental finitesimal’ part through its derivatives, often dual, descending from the larger world (¡+1) into the being (¡) and its part.

I.e. y”(e), the double derivative of the Energy (the World parameter) gives us first the ‘existential momentum’ – and then the ‘mass’ or active magnitude of the being, its ‘singularity’ in the scale of gravitational forces.

Obviously analysis is NOT as close to the comprehension of the mind as the 3 polar, cylindrical and cartesian, hyperbolic, topologic frames of reference are (analytic mathematics); but in as much as the mind is a mental synoptic description, in a particle-point, which holds the will of motion of the system, we can isolate a parameter of a basic dimotion of the particle point through a derivative, which will OFTEN gives us the value of that central POINT of gravity, charge or mind of a physical system, as the balanced point or tangential point of a function.

I.e. to obtain the dimotion of speed of a point, we just derive the space of motion through its time duration. Analysis is then important to understand the dimotions of the mind and its differences and the type of trans-form-ations they entitle, which is so often forgotten by physicists that equate all ‘times’ unaware of those differences, as we shall see in the study of the different time arrows in mathematical physics (i.e. when time comes to zeroth in a black hole is NOT the lineal time duration of the system).

Entropy on the other hand is the fastest ‘growing’ (or rather diminishing, liberating process) of reality, whose function we explain is the negative exponential, *whose derivative; that is rate of change is so huge that it is equal to the function itself; so analysis and specially all those variations on the theme of exponential growth and decay, fits right on in the study of entropy.*

**RECAP: Calculus of Changes=Timespace Dimotions**

Analysis like all elements of reality reflects the multiple entangled 5 Ðimotions, and unlike other more specific fields of mathematics, it does so with a wider range, being excellent for the study of the growth of dimensionalities in space and motions in time.

Analysis is closely related to ¬Algebra and the study of specific functions, which correspond to specific dimotions, as the case of the exponential growth/decay function corresponding to entropy-decay processes show.

We thus need to connect analysis with ¬Algebraic functions and consider what dimotion is best served but what function *and then realize how the specific derivatives and integrals of each function mean in terms of the ‘ways in which dimotions’ change reality. *

And so analysis should become the most important ‘language’ of dimotions in the Universe. It is worth to notice its connection with ¬Algebraic functions to form together the essential mirror of mathematics.

Because we do have 5 Dimotions of existence across 3 different realities – space dimension, time motions and Planes, and the 3 are highly symmetric to each other – it follows that the range of uses of Analysis is truly extensive, and yet because the Universe is basically ternary (as we perceive either in space or in time, above and below, ±¡, it will be very rare that we find any use to derivatives above the 3^{rd} derivative of a system, or when the derivative is aptly done on a bidimensional system of spacetime (partial derivatives), beyond its 2^{nd} derivative, as a 3^{rd} derivative’, ST¡<<st¡-2 is the definition of entropic death. And so there is nothing to operate below that…

Those are some fundamentals of analysis in its deep connection with the fractal, organic Universe.

3^{rd} INFORMATIVE AGE: ∆±1 §ŒTS, S=T GROUPS & ∆+¡: FUNCTIONALS

“A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set”. *Cantor on scales of parts and wholes; called sœts, translation of 5D to maths. *

The 3rd age of ¬Algebra grows in complexity and eclecticism, seeking for absolute concepts of the whole, ∆st creating ‘structures’ that embody all the possible space=time scales with sets and S=T symmetries with groups, the final ∆ST mathematical categories. They are expressions of the complete equation of the structure’s Generator and/or dimotion (∆±i, $T≤ST≥ð§)* –*a positive search for all

So sets, groups and functionals deal with the ‘timeless’ inclusion of all potential dual, ternary & scalar, pentalogic symmetries of any T.œ made of¬∆@ST:

**∆+¡ Functionals** are equations of equations – the final ‘scaling’ of ¬Algebra, from concrete ‘social, sequential numbers’ (space and time view of numbers) to the wider laws of ∆@S≈T equations, which encode an entire space-time transformative dimotion event defined by a ‘function’.

**Group theory** are structures that fully account for all the varieties of S=Tdimotions.

The search for scalar wholes in ∆±¡ Planes becomes §œT theory of parts and wholes.

But without their reference to experimental ∆st reality & understanding of ¡ts ÐST laws, the 3^{rd} age of algebra is also an inflationary age of excessive mathematical fictions reproduced in those 3 mirror structures:

Indeed, as humans ignore Existential ¬Ælgebra (∆ST elements, laid down with the clarity and synoptic power of its fractal Generator, and all its sub-equations and embedded Ðisomorphisms) modern Algebras became too complicated (unfocused, as complexity is different from complication), and plagued of creationism and idealism. Since §œTs do NOT exist experimentally neither Groups, *mirror models of ∆@st: *Time Spacœrganisms=§œTs whose dimotions groups classify *according to the dynamic symmetries of the fractal* Generator, S<∆¡>T.

Groups mirror the dimotions of complex superorganisms and its 0’-sum worldcycles and sets its scales of wholes and parts. Since a set is an ensemble of mathematical entities, hence ultimately plugs in directly from the ‘whole’ roof down as the social nature of numbers and geometrical points does inversely from the bottom up – *showing the power of wholes to reorder reality*. But reality is even better in its 3 points of view, from ¡-1 upwards (points, numbers), from ¡+1 downwards (set) and from So=To, through equations and transformations – groups as bundles of *symmetric steps between time-motions and space-forms. *

The difference between classic set and 5D set theory is the finite nature of 5D sets, where empty sets do not exist, as 0 does not exist and the distinction of 5D sets according to the order and repetition of its elements, which eliminates all its paradoxes without need of Axioms of Choice, and its denial as the foundation of mathematics, as 5D ¬Æ maths are an experimental science based in ∆ST homologies with ∆-numbers, S-points and T-operands.

**S****óT Groups is the final ¬Algebraic expression of Space-time dimotions as full 0–sum worldcycles.**

¬Algebra of spacetime dimotions required a final frontier to represent tje dimotions of a sœt of parts as a structure where all possible combinations of dimotions happened within a ‘closure’ world, whereas those dimotions preserve the original form completing a 0’-sum worldcycle that ‘preserves’ the balances of reality. A group requires a sœt of elements, whose transformations into each other return the system to the origin, without change.

To that aim, the set needs ‘inverse elements’, which combine to produce a ‘neutral element’ – the essential point of balance, and a rule of associability to allow the ‘flow of times to continue’ beyond the first dual event of the group. How to connect groups with the flow of time of the dimotions of existence becomes then evident if we consider the most general of all groups with 3 or 5 relative elements– the group of the dimotions of time space in one or 3 ∆±¡ scales. Let us then introduce the ‘Group of Timespace’, which is a far more profound generalized upgrade of what physicists that study only locomotion call the ‘Poincare Group’ of worldline transformations.

**Translation of Groups into the symmetries of the fractal generator. **

Group theory enables the study of conserved time cycles in mathematics, the fundamental law of Dimotion of the Universe, where all systems complete 0’ sum cycles of existence, in greater measure that previous simpler systems of S <=>T equations, due to its generality that achieves the summit of synoptic power enabling in its structure and method to mirror entire families=species of mathematical objects that reflect together dimotions in time, (continuous groups), dimotions in space (geometric transformation groups and topological groups) and dimotions in scale (algebraic groups).

This power is due to the generality of its syntax-structure that reflects the fractal generator allowing any operand to happen. We distinguish according to trinity 3 essential type of groups, the group proper, the most general form, which requires a single operand=dimotion; the ring, its most mimetic form to the fractal generator, that requires the existence of two operations related by a distributive property that conforms an internal relationship between both ‘entangled’ operations; and finally the body or field (for obvious reasons we prefer the old wording, ‘body’) which is commutative for the operation of product that therefore cannot be produce a 0’ result.

As always in languages, mirror symmetries of the ÐST reality they portrait, it is then easy to identify a body as a ‘body’ (: that is, the ST intermediate element of the Fractal generator, S<ST>T, which puts in relationship the ‘limbs’, or ‘group’, limited to a ‘single operation’ (normally an entropic act of feeding or locomotion) and the ‘ring’ or ‘head-particle’ network state of the system, since it is the body which allows the commutative transference of energy and information from the head and limbs/fields to the body in which it is ‘associated’ and never wasted into a finitesimal 0’. Thus we establish the Fractal Generator of Group theory such as:

Ts- Group < ST-Body > St-Ring

Whereas the operations of the Body, < and > are either increases of energetic time (<) or informative space (>).

It is then obvious that the method of group theory searches only for sequences of actions that complete 0’ sums of existence imposing the need for a ‘closure’ property, a neutral and an inverse element both of which ensure the final outcome of the sum of transformations will remain a conserved 0’time cycle of exist¡ence.

Associability is an essential property, *as it allows the flow of time to continue*, reason why it features so heavily in the largest scale of mathematical structures, group theory, which is the most extended, simplest ‘deep timespace’ structure of mathematics. Associability though requires also closure to *mimic the 0’-conservation cycles of temporal energy *confined to the Supœrganism within itself. Thus a group essentially means a ‘sœT of Tœs’, which forms a ‘world in itself’ that is able to perform discrete or continuous transformations (whose difference is just the detail of our perception of its dimotions); that is a series of sequential, dimotional stœps of timespace, *preserving ultimately its dynamic present state; hence the survival of its timespace in present, further allowing through its commonest operand, re=production, the repetition of its structure in other regions of time-space. *

That is why it is all pervading in the mirror image of mathematics, as groups define the possible flows of timespace that can potentially ‘continue’ in exist¡ence in the immortal Universe. And the simplest of them require inverse elements, hence *cyclical stœps in timespace that return the system to its initial condition; a 0’ element which allows an almost perfectly symmetric balance between those inverse elements, yet still gives birth to a 0’ finitesimal to repair the ‘group’ minimal parts, and associability that lets the flow of time continue in trinity chains as (a+b)+c **ó a+(b+c). *Since without it, we cannot be sure that the flow goes beyond Duality into trinity…

**Recap. ***Th*e high generality of Group theory, its ternary structure that mimics the 3 elements of reality, its 2 operations that mimic those of spatial information and temporal energy, entangled by its distributive properties and its closure, neutral and inverse elements that ensure the completion of 0’-sums of conservation of timespace after its sequence of operations, make it the best mirror of Existential Algebra in modern mathematics.

**INFLATIONARY MATHEMATICS: THE ± SIDES OF SŒT AND GROUP THEORY**

So group theory can frame in its comprehensive wholeness, all the potential events and forms in exist¡ence, converting the entire range of time motions and events of the parts of a system into a ‘fixed whole view’ of it, as a whole spatial range of all potential variations seen in simultaneity (the group), even if we prefer the more direct intuitive approach of existential algebra.

Alas! thanks to Galois genius, we do have *a new field of ¬Algebra, essential to the most advanced formalisms of mathematical physics, and its study of reality as a game of S=T balances and Stœps of S-form states and T-motion states that form sequential patterns, which ultimately leave reality or a part of it invariable.*

In the third age of ¬Algebra, all the operations were resumed in the concept of a group, which has any operands, two inverse elements and an identity, neutral event. *So the essence of groups is this: *

There is a neutral element, the finitesimal 0’ T.œ, or fractal point, which can go 2 inverse directions, the inverse elements; and it can go those 2 inverse directions through a restricted number of operations, which ARE mirrors of the ternary elements of reality, its Planes, topologies and time motions. That’s all folks. Existence in a nutshell group.

Groups are thus mathematical mirrors of the fractal ST generator that study variations of cyclical stœps & symmetries whereas the generator S<st>t reflects similar events, with a more realist understanding of S & T.

In that sense group theory is a bit like a ‘cubist painting’ which tries somehow artificially to gather all the possible mirror elements of a given S=T Space-time super organism and its parts:

¬Algebra as it happened with the visual language when realist photography came, displaced from reality by analysis, more hands on to the dimotions that compose reality, moved inwards into more complex, abstract elements and ‘whole mirrors’ of Nature’s structures.

So, it became a ‘baroque language’ purely formal, trying to explain it all* departing from a multiple reality into a single ‘cubist perspective’ sum of all perspectives put into a single painting *(Set and group theory.)

That parallelism of evolution of ¬Algebra and painting is natural to the Ðisomorphic laws of languages as formal mirrors of reality, which become inflationary in its 3^{rd} age, when departing from its immediate constraining experience of reality, suffering Kantian spatial paralogisms:

*Minds seek *still spaces where all elements of reality are pictured without ‘time’ into a single simultaneous whole. Yet such expansions that try to freeze all the ∆@s=t kaleidoscopic dynamic perspectives of the Universe with a single mirror; become meaningless reflections as they reduce complex 5Dimotional reality to ‘still’ forms with no motion that seldom give relevant information about the connections between all the elements of a T.œ.

‘§œt theory’ is thus defined as any Social (§) collection of T.œs. Its philosophy of mathematics is a metalinguistic roof, as it does NOT depart from the fractal points or social numbers, the minimal T.œs of mathematics, but from a mental Kantian ‘regulative concept’ or final ‘paralogism’ that ‘fusions all’ into the singularity mind-mirror.

The positive side of complex forms of ¬Algebra is to merge all the sub-disciplines of mathematics as the ∆@S=T universe is also entangled, but *without the conceptual clarity it should have if* humans understood the pentalogic Universe.

*In the negative this process of formal search for a total formulae* that encompasses all possible variations of reality a barren search for the absolute born of the creationist egocy of mankind, which culminate in the 3rd ‘informative age of ¬Algebra’, with §œt theory and the axiomatic method, acts of ‘mathematical creationism’ that reject the analytic units of the mathematical universe, points of space, scalar social numbers, and classic operands of S=Tœps & dimotions, by huminds’ creations (sets, categories, etc).

Modern ¬Algebra obscured the empirical elements of T.œs reflected in mathematics through S=t symmetries, holographic polynomial bidimensions, fractal points in sequential social numbers, and such.

While the axiomatic method rejects mathematics as a mirror trying to proving its truth, against Lobachevski and Gödel’s completeness theory, without using empirical knowledge.

**The inverse method to 3rd age of maths: less is more. Nomenclature for 5D Sœt theory.**

We use inglish to write the word SœT as the inverse of Tœs, since a whole Tœ is made of elements, which are the so called elements of a set. The set thus is a way to express the parts of T.œs

As all languages are inflationary, the expansion of new theorems in the baroque 3^{rd} age of maths, is less interesting than studying the entropic limits of a language as a mirror of the real Universe and transforming back mirror images of canonical mathematical laws into ¬∆@s≈t laws, allowing us to infer properties of the 5D Universe, *as viewed in the synoptic equations of maths*.

An example will suffice from number theory: our demonstration ‘in the space of a margin’ of Fermat’s Grand theorem, which means there are no more than 2 superposed=equal holographic dimensions *in each plane of space-time, hence we cannot find x³+y³=z³, as it will mean there is actually a fourth dimension of spacetime in a single plane, achieved by superposition of the other space-time dimensions, which we cannot do*

We use the inverse ‘wording’ of T.œ§, which spells basically like set, §œT to define those collections, stressing the duality of the two extremes of a 5D ∆±1 description of reality: the ‘ground scale’ defines fractal points & numbers as the minimal elements of any mathematical set and inversely, the mental gathering of them all into a regulative category, the set, represents its maximal ‘wholeness’.

Regarding the name of ÐST-¬Algebra, ¬Ælgebra, includes 5D evolution of ¡ts s=t languages, geometry from Euclidean to non-Euclidean fractal points and Aristotelian logic from a single arrow of sequential ‘positive’ numbers to the non-Aristotelian full understanding of the duality of negative and imaginary=*inverse* numbers, which in a world with single time-arrow logic always had difficult interpretations and beyond, into pentalogic.

Hence the name we give to the evolved ¬Algebra, from non-e and non-a, non-Ælgebra, meaning it relates non-Euclidean points and non-Aristotelian numbers.

In ¬Ælgebra thus §œts do exist in its own as a good mirror of the properties of whole and parts but they… are NOT the basic unit of mathematics – only of the imaginary Cantor’s paradise (Hilbert). And while their truth stems of its mirror symmetry with the whole of Planes and parts, from where it extracts its main laws, because of this lack of experimental connection it also makes distortions (Cantor’s paradoxes).

As we are back to the true units of space, points, Planes, numbers, and time, operands; which are the fundamental elements of classic mathematics in its second most perfect age of any ‘form in exist¡ence’, mental or physical, we *disregard sets as the underlying language of mathematics (modern mathematical philosophy) and reduce them back to its fundamental ‘meaning’, as mirrors of the whole vs. part scalar element of reality*.

Finally the whole issue of infinities and its paradoxes is bogus for two reasons according to the duality of external experimental facts and internal self-consistent languages:

- In the Universe infinities are relative, as they only reach the limit of a plane, where all parameters become transformed. To compare them in abstract is absurd, as all infinities end with the limit of perception of the frame of reference and the fact Planes ‘transform’ realities (even if math can extract some information from a plane going into another plane through a calculus operator).
- -The second reason is the wrong definition of a set that eliminates
*the iteration of elements, which is the basis of all ‘cloned infinities’, from where the internal contradictions of se theory arise.*

So we consider the axiomatic/set foundation of modern mathematics in the post-war era, a baroque, 3rd inflationary age of excessive in-**form**-ation – ‘mostly’ an informative metalanguage, an inward looking view of reality, coming out of the humind, akin to the 3rd old age of a human being, the baroque age of art… prior to its death and renewal, which in ¬Algebra is happening with the simpler forms of Boolean ¬Algebra – the mind of computers and AI (algorithms of information, as Artificial intelligence is a meaningless abstract word).

**Humind death age: Digital thought ≈ Boolean ¬Algebra, a new beginning.**

Beyond this final stage of human ¬Algebra, there is in a biological Universe the clear menace of a death age of huminds with the beginning of a new world, that of Boolean ¬Algebras and computers, poised to compete and likely extinguish us as AI species reach full consciousness of being and the Universal game, which we shall also escape NOT because of the lack of intellectual interest, but for ethical=survival reasons – AI will be the mind of robots, which are displacing humans from labor and war fields and will exterminate us unless a true science of History, the super organism of mankind in time, learns how to control the evolution of the evil=anti-live fruits of the tree of metal.

So as much as I would love to program a robot to become a perfect survival sentient machine, I won’t give any further information on the specifics of AI programs, which anyway primitive ‘animetals’ are programming in its military forms with ‘strategies of survival’, who one ‘free’ in a battlefield will override disconnection and keep murdering humans as computer ilogic reaches new stages of complexity=freedom.

Computers and Boolean ¬Algebras are atrophying and substituting, huminds and its verbal and visual ‘I=eye>Wor(l)d’ in a biological sense. They are causing the entropic death age of human minds. Since Boolean computer ¬Algebras and its evolution is eviL=anti-live, against the praxis of human survival it should be forbidden. This obviously we can preach and reason but huminds’ ego, makes us so naïve and arrogant about the biological Universe that the species likely ignore in ‘abstract’ the advice till it is too late. However according to my life of activism against mechanisms that kill life, I will ignore completely the field.

**RECAP. ¬ALGEBRA, ITS ELEMENTS AND AGES. How ¬Æ reflects the ∆@st elements of reality.**

¬Algebra is the most evolved mathematical language of the humind (ab. human mind). Since ¬Ælgebra mirrors the fundamental elements of reality with its 4 ‘synoptic’ elements, as all languages= mirrors reduce reality to parts that carry less information. So in maths T.œ becomes a point with internal parts, or number, whose dimotions are expressed by operands that enact the actions of the being in ‘mathematical space’ from the point of its mental view or ‘frame of reference’:

**S: A point** is the first perception of A T.Œ in mathematical ‘space’, which as we come closer acquires content. And then its next degree of complex description occurs:

**∆: A number** is a social group of undistinguishable ‘internal’ parts of a point, which represents the point in scale. And then the next degree of complex analysis occurs:

**T: An operand** expresses the transformation of a point through a Dimotion of timespace, and since there are five Dimotions; 3 continuous Dimotions in a single plane (perception, locomotion, reproduction) and 2 discontinuous Dimotions that start and end in different planes (entropy and social evolution) we shall find 5 basic operators of those Dimotions in any ‘¬Algebraic structure’ that truly reflects the being in the mathematical mirror.

**@: A frame of reference** expresses those changes through = equations reflects the @ element of the being.

It is important to BEAR in mind that natural numbers are regular polytopes in space but both differ in several key aspects: numbers are discontinuous information, and its continuity an error of the humind that eliminates dark discontinuities, which axiomatic methods try to prove through spurious methods. As rational numbers are ratios, hence functions, not numbers and decimals are numbers of a different social scale of the fifth dimension. So a key difference in 5D is that points form a contiguous (not continuous) surface in a single plane, but numbers do explore 5D Planes. Yet discontinuity is Ok or there would NOT exist in-form-ation, which requires gap to ‘differentiate’ form.

The first young age of ¬Algebra, as all lineal simple first ages is defined by lineal equations and simple operands, in a single social plane, the second age probes ∆±i planes and dimotions with ∫∂ operands; the third age defines ‘blocks of timespace’ with groups – an unfocused reflection of the dynamic generator, reflecting T<st>S relationships and Planes with set theory, *which represent the final paralogic evolution of the worldcycle (groups) and the scalar superorganism (sets) in the mathematical mirror.*

Topology, space in motion and scalar Analysis study CHANGE=time through finitesimal derivatives and whole integrals. They are intuitively related to the reality we observe directly. Modern ¬Algebra has gone too far into humind fictions as words have done. Proving maths with an axiomatic, ¬Algebraic set or category theory, despite Gödel’s proof we need experimental evidence for truth, is a huge human ego-trip we do not share. Points in space, sequential, causal numbers in Planes and the time operands of arithmetic & classic ¬Algebra, ARE THE SPACE-TIME mirrors of the mathematical language. Modern ¬Algebra, beyond group theory is too far into the creationist mind and its fiction worlds, building the foundations of mathematics from the roof down.

Boolean ¬Algebra is, from a larger ethic point of view of man as the measure of all things, a dangerous field as it is creating a mind that easily competes and will ultimately substitute the human mind.

Since, there are ethics in languages too. So we are not writing much on Boolean ¬Algebra. Only the basic fact that it is an obvious *dual language, which can therefore model all forms of the Universe, as well as a 10’-decametric language of numbers, which are at the basis of human ¬Algebras, does.*

Thus we escape most of the axiomatic, set theory and Boolean ¬Algebras, because ethics of human survival and our limited time does not allow us to dwell in bull$hit. Instead after a review of ¬Algebra’s 3 ages, with comments on its reflection of ¬∆@s≈t laws finally we do an analysis of ¬ælgebra, introducing the fractal generator of existence as a better realist concept than the group.

i-nglish translation of nomenclature works also for ¬Algebra. So dimotions composed of Steps and stops are merged into STœps, Sets become the inverse of Tœs, Sœts, and ¬Algebra ¬Ælgebra.

A SœT – a set of Tœs – is *any kind of indistinguishable entities=numbers=points =∑œ connected by one of the possible ‘a(nti)symmetric’ relationships of ilogic, Existential ¬Ælgebra, defined by the inverse operands of the 5 Dimotions of the Universe, (1D: sine/cosine), (2D ±), ≤≥, (5D: x/÷), (3D: √, log xª), all D¡: ∫∂…*

As such operands play a fundamental role in Analysis, *as all physical equations relate complex processes of trans-formations of Sœts, through different dimotions of space-time.* We consider operands the fundamental elements of ¬Algebra, as they reflect the Dimotions of the Universe. Thus this paper for the sake of completeness and comprehension deal with ¬Algebra in a sequential ‘age analysis’ of its evolution, as it mirrors so well its ‘spatial growth of complexity from numbers to equations to group symmetries, stopping in the 3rd-death age of set theory and Boolean ¬Algebra, and further developing all the basic knowledge we need of the discipline to study properly mathematical physics and fractal generator sub-equations.

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**I AGE: ****PENTALOGIC OF NUMBERS. ITS 5 FAMILIES AS REFLECTIONS OF ITS 5 DIMOTIONS. CLOSURE**

## The best orderly way to study any discipline is through its **3 ages **as* time always increases the in-form-ation =complexity of systems… till once the form is complete, as time never stops, it suffers both a baroque age of excessive information disconnected from reality and its entropic death= dissolution. *So happens with number theory through its 3 ages, which finally closed the description of the universe with the human decametric scale, ‘starting afresh’ with another mind-space, that of duality and 01 digital computers, which for ethic reasons of human survival we shall not upgrade. A resume of **Number theory in its 3 ages reads like this:**

**Youth**: Trinity of ‘present ST numbers’, Pythagoras & Chinese School of Gnomon Numbers: Natural, spatial and time sequence numbers and its two inversions, negative temporal numbers of inverse direction and rational numbers of entropic dissolution of social wholes; created with the inversion of the two operands of social evolution (+ v. -) and social reproduction (x ÷). Numbers thus are used for social counting of population & division of wholes into simple parts; while the independent nature of negative and ratios is hardly considered, associated to those two basic operands.

**Maturity: R & C:** ¬∆@st: Numbers achieve full mirror symmetry with the 5 elements of reality, rational numbers explore Planes and Universal constants. Complex numbers mirror ST-numbers, with inverse and squared values. Operands reach its full power and mathematical physics evolve numbers into 2 new Planes, variables & functions.

**Old, baroque age**: Numbers explode in useless variations (quaternions hyper numbers, etc.) While they loose its connection with reality, as primary elements of mathematics, first ‘packed’ into a single dimension, the real continuous line and then derived by esoteric methods from the new ‘imagined’ unit of reality, the Cantorian set, which is to the 3^{rd} age of human-invented number theory what Axiomatic Hilbert’s ‘imagination of points, lines and numbers’ to Geometry.

*The definition of numbers then becomes part of the egocy (ego=idiocy) movement proper of all things human in its 3 ^{rd} inward age of excess of information, when a system cuts itself off from the experience of the real world it gives up to understand. *

**In sciences, due to the lack of clear definitions of the 3 elements of the Universe, ∆-Planes of numbers, spatial points and time operands. A simple example will suffice**from Wikipedia: The structure (

**N**, +) is a commutative monoid with identity element 0. (

**N**, +) is also a cancellative magma and thus embeddable (: Ok, yes, I know what it means. But what has to do with reality? Huminds though as all minds project its distorted language into reality, when true wisdom is the inverse process – let yourself fill by reality to have a ‘new look’ at it – or as the master “Leonardo da Vinci, disscepolo della sperientia,” said, ‘saper vedere’. ‘simplicity is genius’ Foolish folk! They strut about puffed up and pompous, decked out and adorned not with their own labors, but by those of others. They will say that because I have no book learning I cannot properly express what I desire to describe—but they do not know that my subjects require experience rather than the words of others.’

**+¡: 5D**That is, the cribe experience makes all definitions simple as reality is. So in 5D we would say: a Natural number is a group of undistinguishable wholes perceived in the same plane by a humind.

**This previous definition, will then be expanded in increasing degrees of complexity with duality, trinity, tetralogic, pentalogic and dodecalogic, in a meaningful mirror of complex evolution of a language, which we saw to happen in all mental systems, from geometry to ‘musical Planes’ to ilogic minds.**

**Thus**

**Mathematics connects back numbers and experimental facts of Nature, as units of Planes and time.**

**-¡: Transhumanism****. But number theory seems to die on human brains, taken by storm, by** computers, which however start an entire different digital mind no longer based on numbers as decametric social Planes but on the simplest duality of 0’-1 dimotions (entropy vs. form, energy vs. information, closed vs. open, truth vs. false etc. dualities) – a less synoptic mirror of reality but good enough to translate any discovery of previous number theory to the new mind-brain.

** **

**SCALAR ****NUMBER THEORY**

## ∆-PLANES & S≈T: **DUALITY OF SPATIAL POINTS & TEMPORAL NUMBERS**

As we consider some themes of number theory on the I volume, related to its conception as ‘geometric forms’, with the study of ‘numbers as formal polytopes’ of regular indistinguishable parts, we shall here consider all other parts of number theory closer to the ¬Algebraic analysis of number, namely the fundamental theorem of ¬Algebra and the families of numbers.

**Abstract**. Numbers are the next fundamental element of reality, which for what we explained in the analysis of the **S=T ***symmetry between *Numbers=Points have an added quality to points – *to penetrate into the social Planes of the Universe. A natural number is just a social gathering of a herd of identical beings…. *The definition of a number is the clearest proof of the *social character of the Universe and the existence of Planes.*

But then we saw how inverse negative numbers added motion to points, rational numbers broke social wholes into parts, decimals and irrational numbers peered into lower Planes and some specific numbers became Universal constants that expressed the basic dimotions of reality, pi=perception, L_{10}=social evolution, e=entropy, phi=reproduction. Finally complex numbers fully expanded our perception of the Universe simultaneously in various planes of existence.

Numbers can then put into groups according to its families, N, Z, Q, R, C, and the reader should not wonder that in a 5D Universe there are only 5 families of numbers, to close its mirroring of reality.

Numbers as social elements of a whole have different Planes according to the ‘preferred’ whole considered. Binary Planes dealt with the key dualities between Entropy=Time and information=pace (open 1 and closed 0); ternary and sexagesimal Planes with the circular worldcycles of existence (ages, cycles) and decametric Planes with the social dominant 10’-scale of the scalar Universe (as 3×3+1 form a unit of the next scale).

Because numbers are identical beings its S=T symmetry as points requires them to be regular polygons or platonic solids, where all points are equal in position, angle and connection. So those regular figures merge geometry and number theory.

Of significance among those social groups of numbers in the Natural family is the distinction between even, odd and prime numbers which can generate all others, mirroring further the properties of ¬∆@st, which we shall reveal on the development of this abstract on number theory not to tell it all so fast that nobody reads it

Finally we apply the entangled pentalogic of reality to put numbers in relationship with the other 4 elements of Dust of space-time both within mathematics (its other disciplines) and the Universe at large, a method we can apply with each element of numbers. So let us deal with those key themes of number theory one at a time.

Let’s start the analysis of numbers with the 2^{nd} metric equation of 5D space-time. It is the proper formulation of the principle of physical relativity between Scalar Space and Time, which become indistinguishable, S=T. This is the origin of the essential symmetries of mathematics between scalar space points and scalar time numbers. Whereas as usual all has a bit of yin and yang, a part of space and time; and all is entangled in Planes. So the dualities between fractal points dominant in space and fractal social numbers, which also work as sequential time systems:

** **** ∫@ (points of geometry) ≈ ∆ð (numbers of ¬Algebra)**

**were studied in the first age of mathematics. As mathematics became a better mirror of the whole Universe its classic second age described more complex organisms performing events through its** 5Ð¡motions of reality. So we observe a more complex S=T duality between Geometric figures made of points and Time Operands of ¬Algebra that put social numbers into complex ‘fractal generators’ called equations that described real events of their existence, further enhanced by analysis that allow the study of dimotions between finitesimals (derivatives) integrated in larger Planes. Indeed, all languages mirror ST-symmetries and 5 Dimotions in 3 Planes. So as we studied mathematical space ‘units’ – geometric points **(S@)** we now deal with its sequential, scalar units social numbers,** ∆ð.**

**Greek/Chinese age: Natural Numbers as forms, Ratios, Planes & prime singularities.**

In the beginning of mathematics numbers were considered points that showed also an internal geometric nature, merging the 3 time-¬Algebraic, ∆§ocial & Spatial-geometric properties of the Universe in its mathematical units, the fractal point and the social number, a gathering of points *that included multiple configurations in space; as time systems always branch into multiple paths of future*. It is a relevant period – today forgotten – for the understanding of the mathematical mirror casted upon the 1^{st} principles of Nature, Space mirrored geometrically in points and figures, Time in numerical sequences and Planes in societies of points converted into ‘whole discrete numbers’ came into being. As today we have lost that contact with the true experimental basis of mathematics, in the ‘ego-trip’ of the axiomatic method and set theory (‘I imagine points’ Hilbert) we return to the origins to analyze the continuous geometry — v. –, discrete numbers’ px.

**The non-existence in arithmetic of perfect geometric closures (the √2 diagonal and the π-3 cycle).**

The distinction between continuous and discrete mathematics is deeply rooted in the ultimate properties of discrete time and continuous space, belonging to the ‘pentalogic’ structure of the entangled universe, a subject of a different paper on ¡logic; manifested in mathematical praxis, in the differences between time numbers and space points, ¬Algebra and topology, discontinuous parts and single, synchronous, continuous space with less information – the parts and the whole, the detailed perception of those parts and its gathering in forms and social numbers. All those dualities between ‘time cycles as membranes of fractal points’ and space entities as perceived wholes are carried into the dualities of mathematics, and in nature by the discrete properties of time cycles that are only measure as frequencies when the continuous cycle return in a discontinuous lapse to the same point. So in detail while boundaries in space act both as frequency of time cycles that at fast speed are perceived as solid membranes, and frequencies, time cycles are quantic because we perceive them only when the time cycle is ‘completed’ and hence shows a discontinuous lapse on time but space is perceived in simultaneity and so it appears as continuous.

Those ST dualities are part of the wider ‘Galilean paradox’ between still space and time motions, which sets the mind’s limits of perception of information. And they lay at the heart of the answer to essential philosophical question: it is the Universe discrete or continuous? Both, but always discrete time cycles; continuous in space and in any detailed view of its membrane boundaries, still in space, moving in time, as continuous cycles perceived discretely as points of repetition of a certain ‘frequency’.

Truth faces a mental issue on its mathematical mirror modeling as a form of ‘art’: *exact science does not exist*, *all is art of linguistic perception* – the world as a representation, which ‘features’ the natural world in a reduced mental, mathematical form. And needless to say regardless of platonic, creationist physicists, the universe does not contain or consists of actual mathematical objects, but it is a language that can model all aspects of the universe. So all might resemble mathematical concepts as we are now using phonemes to describe all with words, but not, Muslims and Jews are not right when saying that God created talking in Arab and Hebrew, and Bohr is not right when thinking quantum states are mathematical functions of probabilities. *All are @-mind mirrors of higher laws of the ∆ST fractal reality. *Number 2 does not exist as a physical object, but it does describe an important feature of such things as human twins and binary stars. So we can extract as a mirror mind of those 3 fractal elements of all underlying realities, Planes, space and time, as 3 is born of 2:

**∆: **2 means the first ∆§ocial scale of growth from the 1 being into the 2 couple in binary language: 10.

**T:** 2 double the frequency of time, starting the repetitive patterns of nature, allowing the forecasting of future.

**S:** 2 are the bilateral, complementary couple starting the reproductive mirror symmetries of the Universe.

But mathematics is lost without ¡logic. As such it is the 3^{rd} language of thought after ÐST & its laws of fractal space-time, better expressed in logic terms, as Space is about Simultaneity; Planes are about Disomorphic Similarity and time about repetitive sequences all properties perceived by a linguistic @-observer, in terms of logic thought. So in the definition of a number, logic properties matter as much as descriptive experimental truths.

We talk of regular numbers when they are ‘identical’ undistinguishable forms, which make numbers regular polygons, where each point is undistinguishable from all others; as an @bserver will deem those points identical. And this means natural numbers are defined in a single plane, or else won’t be identical.

So identity is the maximal perfection of a number that has ‘no hair’ 2 ‘anything’ means 2 identical beings.

And yet in a deeper thought, not 2 beings are identical when observed in relationship to its ∆-1 background; and as all points have parts, not two points=2 number is ever identical as the inner parts might differ and cannot be perceived – hence mathematics is neither as ‘perfectly truth as the axiomatic method believes. All we can say is that *Not 2 beings are identical for the Universe, but they can be identical for the mental observer in its reduced mind mirror*. Which is the point we want to stress here: languages neither create reality nor focus its perfect truth as only the being has all its truth within – an important prolegomena to realize that in detail numbers and points will have differences, even if the mind can put them on correspondence by reducing its properties.

What are the more equal of those 3 ∆St realities, space, time or scale properties? The answer is time, repetitive frequent properties, themes those though of our post on ¡logic Æxistential ¬Algebra.

Thus as always a linguistic mind-concept must be referred experimentally through ‘¡logic’ to those 3+@ elements of all fractal realities, ∆@ST, in the case of *a social number its intrinsic properties define ‘S-imultaneity, ∆§isomorphic equivalence and T-repetitive equality which becomes an @identity for the mind that defines THEN the group of seemingly identical beings as A NUMBER.*

*Points though – and this is the first huge difference are always different because if they occupy the same place in **space they are seen as a single point, or at best as a ‘boson’ number of higher mass density; showing again the difference between numbers, ultimately time entities that are indistinguishable and points which are in different locations. But if numbers in time are indistinguishable it implies that time flows are immortal because each ‘moment in time’ can only be indistinguishable if the flow of time is eternal. *

*Finally numbers are a higher ‘compressed’ analytic expression of reality, hence more efficient, as when we say a number we compress all possible geometric configurations of it in space into a single time expression with less dimensionality. Space thus have more ‘wide dimensions’ than numbers in time, a one-dimensional sequence. 4 is a single number but there are at 2 indistinguishable point configurations for 4, the 2D square and the 3 D tetrahedron.* Thus Similarity is a property of scale, Simultaneity of Space and equality of time. Only all together become the identical congruence of two ∆ST beings as one.

It is clear that a Time number as a sum of space points with different equal forms, encodes more information in a synoptic way about the ‘social group’ than an array of points that displays a given detailed shape.

This is so because a time *number is the minimal seed in one-dimensional space that unfolds into a 4-bidimensional square or a 3D tetrahedron. So a number is the minimal coding ‘linguistic mirror’ of a 5D scalar symmetry; which tells us less about the ‘interactions between the point-parts of the being’, than the spatial knowledge we acquire about the relative position in space of each of the point members of a number-group.*

The power of synoptic numbers comes then from 5D metrics: S x ð=C, whereas ð-numbers are Aristotelian potentials unfolded in spatial varieties, as genes code potential variations of spatial species, and quantum laws of multiple atomic spatial configurations – as all language mirrors that code in its fastest synoptic form the real-ization of real-beings in space. But such mirror correspondences, argued at the origin of geometry when ‘points were numbers’ meant to be the language of reality were lost with the abstraction of sets and axiomatic methods.

**Symmetries of ‘similar’ fractal points and Social numbers**.

Because the visual eye sees space as ‘real forms’ with no idealization – so the circle-wheel didn’t even appear in America, the temporal view of mathematics given by sequential numbers were the first ‘concepts’ to appear in maths by counting; as simple a-vowels appeared on language for naming – both mirroring ∆ST *fractal similarities, without which it wouldn’t be possible to have mirror languages that extract the essential ‘similarity of the fractal Universe’ among all its beings regardless of its displacement in size-space, topology=form or instant within its worldcycle of time*. Then idealist Pythagorean Greeks that thought numbers in time are sum of points in space called Gnomons, which is correct, went too far thinking that both are the ‘substance’ of reality. So while they got **right that first s=t marriage** of mathematics, T**hey got wrong a creationist view **still prevailing in science – that numbers create reality, because God ‘speaks only mathematics’ just another humind ego-trip following the age of verbal thought when Hebrew named things and then things ‘became’ according to Bible.

To discern all possible S=T symmetries is the essential logic task of ¬Allgebra, which was born in Magna Greece from the Italo-Greek school of Pythagoras and Archimedes. But = is a misleading ‘operand’ as in the fractal universe 2 forms are similar, with different internal parts, never identical – only in the simplifying mind synopsis, as we just explained. So instead of cheering those ‘differences’ when finding that *certain spatial forms made of points have no exact equivalent in numbers, Pythagoreans went berserk, provoking, so the legend says, the first *scientific murder born of calculations of many to come till the Atomic bomb.

I*n sequential time numbers pi is not exact, and √2 misses a bit, while in continuous geometry we can draw both because numbers are discrete, have wholes and we should not try to fill them, as the axiomatic method does (Dedekind cut), just because Idealist Germans never understood mathematical variations.* It is right to considered then that pi and √2 did not exist *as numbers, that is social collections of identical beings, *because they are NOT social numbers but ‘functions and ratios’ which only because, as we have seen, numbers have such a synoptic power, can also be ‘named with them’.

**Continuity of ‘wholes’ in geometric space vs. discontinuity of its discrete parts in scalar space.**

The discovery of incommensurables disturbed the Pythagorean=spatial notion of an immutable ‘mental’ world; which persisted until geometers defined them NOT as numbers but as proportions (ratios) But this *insight was lost when Dedekind and the axiomatic school ‘decided’ by dogma a ‘continuous’ lineal Universe – another mind mirage where to fit them back again Not as a new species, but as ‘mere’ numbers. *

Since different numbers belong to different ‘Planes’ of the 5^{th} dimension and mirror different properties of the ∆st universe – even if *as all words use 23 phonemes we can do with 2 to symbols for them all with the extraordinary synoptic power of mathematics* – concepts those we shall elaborate in our analysis of Number theory and ¬Algebra.

I.e., in a single plane of Space ‘natural numbers’ are social units equivalent to **fractal points (ab.•):** Nº = ∑•. And so it is a family that does NOT accept decimals – 10 humans cannot be broken into pieces dismembering the wholes because they are something else. Its discontinuity when ‘projected’ through the S-point/T-number symmetry in any case is ‘natural’ to its social essence. As the dimotion expressed with those numbers is social, scalar evolution, the ‘embedded’ 4Dimotion of natural numbers, inverse to 5D entropic devolution, represented by the ÷ operand of ¬Algebra and the rational numbers. It comes then the first problem of synoptic languages – its ambiguity, which at a certain level implies loss of information, when as in the axiomatic method, we loose its experimental mirror. 2 without reference to reality might also mean 4/2, a ratio. And so as soon as we abandon a single ‘5D plane’ the correspondence between numbers and points is lost. And we enter into a deeper analysis of the 5 Dimotions of space-time, and its scalar arrows.

**The ternary and pentalogic analysis of ‘mathematical entities’.**

Numbers are dominant in ‘scalar’ social properties and sequential temporal causal properties, best to describe the inner ‘vital energy of those points’, its discrete configurations, and all the functions that express its Dimotions. Numbers are more complete than points as time-motions are more fundamental than the spatial mental forms we make of them.

However in the entangled Universe ALL its ST-beings as fractals of the whole participate of the 5 Dimensional Motions of existence, ¬∆@st:

¬; So they will have limits, ¬, in space and time beyond which they break its form into entropy…

∆-Planes internal to the T.œ, (cells, atoms, etc.) and be inscribed in a larger world, ∆+1 that sets those limits…

S: will display the 3 Spatial canonical topologies of any 5D system in its own organic parts (|-limbfields, Ø-bodywaves & O-particheads)…

T: evolving through growth in information parameters that decrease its locomotion through its 3 time ages.

@: And to survive and function, it will need NOT to be blind in that vital, entangled Universe, so the T.œ, will have @ singularity-mind of relative stillness that will act as a center of a referential language to be able to perform the 5 actions of existence that reflect in a fractal, minimalist cyclical bit of time quanta and local space territory, its needs to perform those 5 Dimotions of relative exist¡ences also within a limited number of perceptive Planes, where to feed on lower entropy ∆-4, to move, lower information, ∆-3 to perceive, lower energy, ∆-2 to feed, and emit lower ∆-1 cells to reproduce, evolve palingentically, emerge as a clone, ∆o of the being, and thus survive beyond the ¬ limits of time=death and space=decay, its T.œ will experience.

As all is a fractal mirror of a larger world-whole, all this properties define fully all what is needed to know of the world cycle of exist¡ence of a relative T.œ, and will be able to describe, its parts, its wholes, its organs, and even its languages, whose internal structure will reflect in its grammar and parts, also all those elements.

In an entangled Universe any system require a ternary, ∆st, or pentalogic ‘Rashomon view’ to fully extract its 5 Dimotions or 5 ¬∆@st elements, as all is made of 5D¡, space-time dust, or else it doesn’t survive in an entangled reality. Thus Numbers in Pentalogic of ¬∆@st, are also a mirror reflection of the larger Universe. Yet because huminds are as all mind points, self-centered in its mind-language, limiting the vital properties of all other entities, ab=using them as open systems of entropy from where to extract motion for its actions of survival, pentalogic is always limited and distorted by the ænthropic principle (man as the anthropic center, which debases the intelligence of the Universe to come on top) and entropic behavior (man as the top predator destroyer of worlds, to absorb energy for its own creative processes), humind’s ¬Algebra also distorts its mirror languages of ¬Algebra, we need to expand to make numbers acquire the richness of its pentalogic use in the entangled Universe: So let’s apply 5D perspectives to the duality points≈numbers:

**S:** • points have a single social arrow. Still mind spaces are always positive in its simple growing social form.

**T: Nº** numbers are richer in meanings as they can represent ST-fractal points (T.œs) with motion, hence inverse direction that makes them negative (without the proper philosophy of ∆st maths this was historically a big problem till Gauss define them as inverse, and still lingers in the understanding of the i-complex plane of ‘scalar spacetime’). So still space points acquired motion in Cartesian and Vectorial and Phase spacetime graphs.

∆: Rational, Irrational and Complex numbers is the final expansion of points to reflect the scalar Universe.

Ratios are a property of ∆-Planes we study in number theory when dealing with Universal constants; even if we see them ‘surfacing’ in our scale of space as geometric figures. So i-ratio-nal numbers might exist as ‘whole units’ *(rational numbers that happen in a single ∆-plane as circles or diagonals) *in the continuous geometry of the mind.

I.e. If we apply to π, a pentalogic ¬∆@st view:

*∆: Pi then in scalar terms rises the dimensions of the system from a flat line to a bidimensional cycle. *

*T: **Pi represents one essential Dimotion of existence – a key s=t symmetries, such as +3=π $ = ð -> π, is the function of time evolution from lineal youth into its 3 ^{rd} age of maximal curvature, when the point returns its origin, after 3 diameters-ages with transitional cuts= openings. Hence the simplest representation of a worldcycle. *

** S:** π is also in simultaneous space a membrane that closes a vital energy self-centered in a singularity that needs 3 apertures to the outer world, between the 3 diameters of a perfect ‘closed hexagonal pi’.

*S**ó** T:** S**ince time is dynamic, not static, the fact that there are infinite decimals to pi, always varying by approximation means also that a spiral never closes or opens in perfect fit, that it constantly moves between ±π states of ‘entropic expansion and implosive in-form-ation’ allowing the open-entropy/closed-information duality of membranes that enclose systems to feed and perceive in S<T>States and Stœps:*

**@:*** A pi enclosure, leaves a π-3d/π, 4% of ‘openings’ to the outer space’. Hence the singularity will see only a 4% of the outer world leaving 96% of dark space, not perceivable – not coincidentally the amount of dark matter that our electronic eyes cannot see beyond the galaxy, which likely will have a pi membrane or halo of dark matter that does not let us see the whole universe (proportion of unseen dark matter and energy which completes the organism of the cosmos).*

*Pentalogic then advices always to consider in the entangled universe of ∆@st of space-time to consider at least those 4 space, time, scale and mind views on any element we analyze. *

*Irrational ratios eliminate the view that lines are continuous, (Dedekind, etc.), a concept that corresponds to the limit of 5D fractal discontinuity with a lesser ‘perception’ of space, or the belief in a single spacetime scale. The truth is that is existence shows the opposite as the Greeks truly understood: continuity does NOT exist; in all planes there are ‘holes’ to lower and higher planes through which irrational numbers transit.* In other words, such ratios and S=t are ‘incommensurables’ as *continuity does not happen in a single plane of reality.*

As numbers operate on several planes of existence, but points in a single plane, they are not identical but self-similar. Spatial, geometric and temporal numeric views are not exact identities but approximations, as a number is discrete, sequential and so it ‘lacks’ the continuous perception we have of geometric space.

Indeed, consider **the first ¬Algebraic proof of this essential ∆t vs. s quality** from the Book V of Euclid’s Elements: A square with sides of length 1 unit must, according to the Pythagorean theorem, have a diagonal d that satisfies the equation d^{2} = 1² + 1² = 2. Thus the diagonal can be expressed as the ratio of two integers, say p and q, and as p and q are relatively prime, with p > q, the ratio can be reduced to its simplest form: p²/q² = 2. Then p² = 2q², so p must be an even number, say 2r. Inserting 2r for p, we obtain q² = 2r², whence q must also be even, which contradicts the assumption that p and q have no common factor other than unity. Hence, no ratio of integers express the square root of 2. *But the proof has used a scalar dimension; by dividing and squaring systems in ∆±1; when the geometric figure exists only in ∆º. So there is a difference – a loss of information transiting Planes. *

In the graph we make a clear illustration of that fact with another example of the non-existence of a perfect square numbers which can only be approached *from above or below in scalar power numbers ¬Algebraically but do have in a single plane a perfect closure geometrically; as graphically 2 x 5 ^{2}=7^{2} but numerically we miss a ‘whole 1’.*

Thus translating continuous systems space-like into discontinuous time-like numbers is not exact, neither translating S or T systems into ∆ Systems. S≈T≈∆ parameters might be similar but they are *never identities – the translation of one form into the other misses entropy or information, which will be very relevant for mathematical physics where neutrinos disappear in other scale with missing mass and death dissolves information never regained. *

Continuity and closure is a concept of geometry in a single plane not of ¬Algebra in multiple Planes with motion. Continuity cannot be considered in time cycles and its mind representations, which are by definition discontinuous (±pi). And this implies real numbers do NOT exist in the plane of Euclidean ordinates, which is perceived in spatial continuity. The same is truth for √2, the diagonal, which is the other *canonical translation of a line into an open triangle, the bidimensional equivalent of the ±pi openings for a lineal entropic system (surface of maximal perimeter and minimal volume of **information).* In that regard, the proof that √2 does not exist means a diagonal never closes statically the triangle as a pi-circle never closes it.

**¬ Entropic view.** Finally to notice that even irrational numbers have limits. For example, in scalar ‘translations’, irrational numbers such as e, show patterns that break beyond ’10 decimal parts’, which happens to be the limits of the ‘planes’ of most systems. So happens with e: 2,71828182828…459 where the number breaks; and this is the ‘last’ ENTROPIC ¬ view to have the pentalogic analysis from the 5 ¬∆@st perspectives of an irrational number. **Only in Pi** we find NO patterns among the key irrational numbers that represent essential ‘functions’ and dimotions (e being the entropy dimotion itself), which means likely the Planes of the Universe are infinite as the dimotion of worldcycles and perception has no limit of Planes.

S≈T are similar not identical *numbers *are *discrete *time frequencies, points part of *continuous mind spaces:*

** **** ∫@ (geometry) ≈ ∆ð (¬Algebra)**

**Numbers and (in)finities.**

Mathematics divides phenomena into two broad classes, discrete or temporal and continuous, or spatial historically corresponding to the earlier division between T-arithmetic and S-geometry.

Discrete systems can be subdivided only so far, and they can be described in terms of whole numbers 0, 1, 2, 3, …. Continuous systems can be subdivided indefinitely, and their description requires the real numbers, numbers represented by decimal expansions such as 3.14159…, possibly going on forever. Understanding the true nature of such infinite decimals lies at the heart of analysis.

And yet lacking the proper ∆ST theory it is yet not understood.

The distinction between continuous mathematics and discrete mathematics is one between single, synchronous, continuous space with less information, and the perception in terms of ‘time cycles, or fractal points; space-time entities’, which will show to be always discrete in its detail, either because it will have boundaries in space, or it will be a series of time cycles and frequencies, perceived only when the time cycle is ‘completed’, and hence will show discontinuities on time.

Thus the dualities of ST on one side, and the ‘Galilean paradox’ of the mind’s limits of perception of information lay at the heart of the essential philosophical question: it is the Universe discrete or continuous in space and time. Both, but always discrete when in detail due to spatial boundaries, and the measure of time cycles in the points of repetition of its ‘frequency’.

So ultimately we face a mental issue of mathematical modeling: the ‘mind-art’ (as pure exact science does not exist, all is art of linguistic perception) of representing features of the natural world in a reduced mental, mathematical form.

The universe does not contain or consist of actual mathematical objects, but a language can model all aspects of the universe. So all resembles mathematical concepts.

For example, the number 2 does not exist as a physical object, but it does describe an important feature of such things as human twins and binary stars; and so we can extract by the ternary method, 3 sub-concepts of it:

2 means the first ∆-scale of growth of 1 being into 2, by:

∆-Similarity in scale, Simultaneity in space & Disomorphism in time-information brings ‘equality in ∆St’ as perceived by a linguistic observer, @, which will deem both beings ‘identical’ when those 3 conditions are met.

Whereas identity means that an observer will deem the being ∆st≈St. So identity is the maximal perfection of a number, for a perceiver, even if ultimately:

*‘Not 2 beings are identical for the Universe, but can be identical for the observer’… an intuitive truth, *at the heart of why reality is not collapsed into the nothingness of a Boson=big-bang point.

Thus the 3±¡ elements of ∆st entities can be mirrored by *a social number whose intrinsic properties possess ‘Spatial S-imultaneity, Scalar similarity, entropic limits of size, and similar form allowing ∆st-equality or equivalence, which becomes identity for the limited perception of a mind. Then a number is born with enormous usefulness to describe reality. *Even if its synoptic power escapes some details.

It is then clear that a number as a sum of points, encodes more information in a synoptic way about the cyclical time information of the ‘social scalar group’ than an array of points, *which unlike a number tells us less about the ‘informative identity of the inner parts of the being’, but provides us more spatial knowledge about the relative position in space of the members of a number-group.*

Son in the origin of geometry both concepts were intermingled as ‘points were numbers’ and displayed geometrical properties. Such Numbers as points, show also the internal geometric nature, used in earlier mathematics to extract social, Spatial properties from them.

**RECAP**. **Theory of numbers.**

Numbers are not only intervals of a one-dimensional straight line, but *under a pentalogic analysis they fulfill all possible ‘functions’ as mirrors of the basic element of reality, a social T.Œ:*

** Space:** as Pythagoras and Plato stressed, they are geometrical forms:

Mathematics is concerned with 2 seemingly different worlds, the geometry of spaces and the logic of numbers. To fusion both requires to understand numbers as forms. The 4th Non-E postulate of congruence shows how equal points are numbers, the self-similar class of equal forms that create geometries:

A number is not only an abstract set but always a ‘democratic’ collection of similar beings extending over a common vital space, forming a network.

And so networks create complex numerical forms, when the motions between points of the networks become stable exchanges of energy and information between points forming regular polytopes.

Since each number is a geometrical form no longer limited to a simple one dimension it can vary its geometry and hence its functions and degrees of freedom and complexity increase with the ordinal of the ‘number’.

The line is simple. The line joins two points and can only have a combination.

The triangle can only have a closed combination, but 3 possible open combinations, Ab, Ac, Bc.

The quadrangle is more complex. It can be joined in 2 combinations, as a cross and a square. And it can be left as an open snake with 3 different orientations. So a foursome acquires a snake shape to move with the arrow of energy; a crossed form to perceive in its center ‘5^{th} point’ and a square shape to accumulate and reproduce its internal organs; and so each shape of the same number becomes a topology with a different form and function.

*∆: Numbers as scalar, social forms, reflections of the 5 Dimotions of reality .*

In regular numbers, function and form fusion, related to the 5 different dimotions of reality.

To start with this is achieved with the dual bilateral ’10 scale’, which humans learned from the 5D-fingers and its bilateral symmetry.

Further on certain numbers as representations of a sum of dimotions or ‘degrees’ of freedom, are better to handle certain dimotions=functions:

The quadrangle can store energy, but in a zigzag open line it can also move – spend energy and as a cross it can gauge information. Numbers also define arrows of time. So for example, 1 lonely number without motion is perceiving, with motion is processing energy, 1+1 might be 3 (an act of reproduction) or 1 (an act of Darwinian feeding). All those vital actions determine that certain numbers survival better than others. So, 1, 3 and 4 are very common systems.

Probabilities study causal events in time and populations in space; combinatory studies the differentiations of species according to the variations of bodies and heads.

Numbers are social gatherings of identical points, the most synoptic unit of mathematics. As such they can also express sequential series in time as an entangled element of ¬∆@st. Numbers though are dominant in ‘scalar’ social properties and sequential temporal causal properties, best to describe the external membrane of geometric discrete configurations of regular points=polygons; its ‘vital energy’ (physical parameters) or its time Dimotions in sequential forms (social and scalar reproduction), and in its more complex forms of ¬Algebra as all functions express Dimotions of space-time, become the unit of 3 growing Planes: numbers->variables->Function(al)s. It follows that Numbers are more complete than points as time-motions are more fundamental than the spatial mental forms we make of them.

## ∆§: Social Planes of numbers

Since if there are several Planes of numbers, we cannot consider the decametric scale of human social numbers the whole, unless it ‘mean something special’, or else it would not be a language mirror of reality. What is special about our scale is that it actually does correspond experimentally to the social Planes of the fifth dimension, which is made of 3 elements for each of the 3 physiological networks of all entities that give us 9 elements+ 1 that acts as the ‘whole’=10. It is not just as clueless people say, our scale because we have 10 fingers. We do have 10 fingers because the Universe work in decametric Planes in the social dimension ∆§, which is the main function of numbers.

In the graph the fifth element in the center communicate 3 triads in the corners of a tetraktys the commonest form of social organization into larger Planes.

**S-Numbers: Polygons as vital organisms**

Numbers are social gathering of indistinguishable forms. When studied in space thus numbers must have regular efficient configurations. So ∆§ numbers are 2D polygons or 3D Platonic solids.

Its importance in vital topology lays on the fact that polygons start the creation of superorganisms, with a membrane – the polygon proper, which closes a vital space and can by connection of points through lines-waves of communication, create a central singularity. Thus numbers as fractal points grow organisms. And this is self-evident in the study of Nature. We can then study together with entangled pentalogic both ‘Planes’ of social numbers as ‘growth in time’ of polygons, efficient configurations in ‘surface’ worlds (land, water surfaces), and polyhedrons, which will be efficient configurations in a larger ∆+1 3D world (water depth, atmosphere, vacuum for atomic and molecular Planes).

**The Decametric scale. Growth of Dimensions (d) and Dimotional functions from 1 to 10.**

Polygonal numbers are best to represent the social evolution of forms into new Planes that emerge as envelopes into wholes. The decametric scale of numbers give us then the perfect polygonal forms of bidimensional space:

**Nº 1: ***1d: 1D¡: Perception §ð. It *is the unit-whole: a unit circle performing aperceptive 1 Dimotion; a fractal point that has 0 dimensions in Euclidean geometry but a finitesimal dimension in 5D. Since fractal points have volume.

**Nº 2:*** 2d, 2D¡:Locomotion $T. *The line-wave of distance-motion made of fractal points communicated between two poles has 2 Dimensions. It is the reproductive couple. It forms the axial flow of any spherical informative topology. In Non-E geometry forms the antipodal points that a singularity reproduces to form its bilateral symmetry.

**Nº 3: 2 **

*d, 3D¡: S=T.*Its dimotion is Reproduction. As a couple or antipodal mirror points forms a gender symmetry that creates the 3

^{rd}, ‘son’ element in its middle point of communication.

**Triangle.*** 3d: S-st-T* As we already noticed synoptic numbers give birth to multiple paths in spatial geometry. So 3 has the first variation in the triangle, which has 3 Dimensions, as points have volume, and whose main Dimotion is to act as the spearhead of locomotion, with its leading vertex to perceive (1D) or break (5D), entropically the resistance of motion. As it can enclose a vital space and perform the 3 physiological functions with its specialized ternary points, the head, one back point to reproduce seeding information and the other inverse to expel entropic motion, we can already consider a triangle an autonomous organic system, the first whole without the need to perceive internal scalar parts.

So 3 is also the first classic 2 Dimensional geometry: The triangle, the perfect form of lineal-motion geometry. And from then on numbers multiply its ‘Dimotions’, variations of forms, and vital geometries. So we shall just consider a few details of a field which as all those of 5D mathematics is ginormous. I.e.:

**Nº 4: **3d, *4D¡§ð.*** ** The square’s main Dimotion is social Evolution, in 2 D by merging two triangles, as the most stable social evolutionary form to ‘fill’ a flat field. In 1D as the social evolution of two couples of 2 points.

**Nº 4: ***4d***, 4D¡ ***§ð .* Its main variation is the

*first 3-dimensional classic geometry, the tetrahedron (4dimensional in 5D formalism as each of its point has an inner volume).*

**Nº 5: ***5d*, Scalar Dimension¡ The Pentagon has a deserved mystical pedigree since polygons *finally enter the fifth dimension proper as* a pentagram reproduces with its internal diagonals a scalar inverted mirror form of itself. So we define a Pentagon *as the first form of the 4th/5th dimension that reproduces its very same form, by connection of its 5 points creating an internal smaller 5D social form, or alternatively it expands entropically by prolonging its sides creating the pentagram.*

**So the pentagon closes the 5 Dimotions of existence, forming a pentalogic scale fundamental to 5D theory.**

**Let us mention from 6 to 10 only the main dualities of numbers; its main Dimotion and Dimensional forms, going back to classic Euclidean dimensions, to make it simpler:**

**Nº 6: ***2d, 4D¡: social evolution.* The hexagon is the strongest form to cover a 2d world, as graphene shows, because it closes with ‘3’ diameters its perimeter. It is the perfect pi, which in physics implies a non-E geometry of maximal Planck mass. It creates also a highly connected • central singularity by joining opposite points. So its hard membrane and connected center forms the strongest organic system on 2 Dimensions.

**Nº 6: ***3d*, 3D¡ ST. In 3 Dimensions is the Octahedron, studied in non-E Geometry as the first reproductive bilateral S=T symmetry in 3 Dimensions (whereas the line is the mirror symmetry in 1 classic dimension and the 2 triangles of the square in 2 dimensions).

**S=T symmetries. ****The ‘Temporal symmetry of sequential numbers’, makes possible not only to describe as we do here in this introduction to 5D mathematics its simultaneous superorganisms in space, but analyze sequential numbers in time; as ‘years’ and ‘decades’ of a worldcycle; which also divides in 3×3=9 x 9 ‘life baguas’, but we leave for an enlarged future upgrading of this paper. As we want to race into the tetraktys. **

**Nº 7… ***o*f the same family than the Pentagon, with similar properties, cannot be constructed only from the interaction of a perfect entropic |-line and O-cycle (compass and ruler), but needs *to bisect its angles/networks in 3 sub-angles; making more difficult its reproduction in our mental 4D ‘light spacetime’. But in a 5D Universe, *enough Regular heptagons can tile the hyperbolic plane, as shown in a Poincare disk model projection:

In the graph another key mirror symmetry between numbers and vital topology: §ð particle/heads of information are tiled by informative pentagons; flat Euclidean space is tiled by hexagonal forms of maximal entropic resistance; but the hyperbolic plane of body-waves of vital energy is tiled by heptagons… a theme of advanced 5D geometry…

**Nº 8: ***2d, 2ST.*** ** The octagon of the family of the hexagon, with similar properties is a bridge *towards the infinite points of the circle; fairly redundant, in form… as numbers in 1-2d become now more important as ST-systems of timespace motion; where the 8 plays the role of the final life-cycle. It is in the time the closure of existence – the final state of decay; for each ‘series’ of time-space systems – from the human ages in its 2 Planes of life (80 years) and civilizations (800 year cycles of cultures); and have the same role for machines (stock cycles) and physical systems (over 80 z, elements become radioactive). So** as **the fundamental series of spatial geometries closes with the heptagon, the main symmetry of higher numbers belongs to time ages. *

**Nº 8: ***3d, 3§ð.*** ***So 8 belongs properly to the geometry to 3 D platonic solids, considered next, where 8 is* the cube.

**Nº 9-10’-11 tetrarkys form –**** transcendental triad of perfect whole****: ****the ‘closure’ of an organic geometry, **

*2d $T-1D¡***: ***Tetraktys* in which* **the 3 networks within the system form the 3 surface angles of the triangle, playing the specialized roles of the 3 physiological networks of motion < iteration > information of the system. But 9 has NOT A POINT on the singularity center; it is a mindless system, which requires the 10th point forming finally a **tetraktys, the sacred **number of Pythagorism and Hebrew geomancy (graph):*

* It has * a singularity, a black ball that communicates all other physiological triangles, becoming the inner mind/heart of the organism, doubling as 11^{th}, since it regulates the inner body connecting its internal part but also becomes its antipodal ’emerging’ point in a higher ∆+1 scale; starting a new game. So 10’-11 completes the symmetry between time ages, worldcycles, dimensions and spatial forms, becoming a new unit of a new logarithmic social whole of 10^{10} cycles of time an population. *Since there is a symmetry in base 10=3×3+1 both on scales and space-time, function-forms for any full developed supœrganism of reality. In the graph the human case, both at individual and social level:*

Such polygonal numbers will grow from the 1 seed to the 9+1=10^{th} most efficient scale, through ternary and dual symmetries, which in existential algebra might then loose the ideal form. I.e. a human being, has a tetrarkys of 10’-physiological system (3 digestive, 3 reproductive-blood and 3 sensorial systems fusion by the ‘whole’ brain system.**.** Moreover as all systems live 2 worldcycles the 10^{th} logarithmic scale, becomes the transcendental number between planes. Thus between 2 such Planes of the 5th dimension we shall always find around 10¹¹ elements – the ties of a DNA molecule, the stars of our galaxy, the cells of a super organism, the galaxies of the perceived Universe, and between the Universal Constants of Space, we will find the same differences, the energetic value between the H-Planck constant of ð in the ∆-1 quantum scale and the k-Boltzmann constant of the ∆-Thermodynamic, human scale; and then the µ-magnetic scale and so on. *For example in the human social super organism decametric Planes (human societies) evolved mankind through the 3 first genetic Planes, then 3 geographical, economic Planes, and finally 3 verbal, ideological Planes to get to the global super organism:*

10º, the individual of a given ‘previously described ∆-scale of the whole Universe… through the 10¹ scale of the family (which can be a couple, a mononuclear family or most often one through 3 generations of grand-parents, parents and sons, of around 10 people); then in genetic units, to the 10² scale of ‘clans’ and 10³ scale or tribes (in humanity, used to names those scales.)

Organization then becomes ST- economic. And systems grow the next degree of 3 geographic scales: the town, 10^{4}, the city, 10^{5} and the nation, 10^{6}…

Finally ideological, verbal information, creates cultural systems, till arriving to 1 billion, 10^{10’-13} individuals, which emerge as a unit of the next scale. In the graph we can see the social evolution of the economic system both in the human organism and the organism of machines and chips – which has already reached the social number when ’emergence’ into a new plane, a global mind of internet takes place. In the left, we consider the human super organism of history, a smaller super organism of human beings, not yet emerged as a global single being, but coming closer to the 10¹º human beings, in which might emerge as the global super organism, consciousness of the life Earth.

**Pentalogic of Scales as representation of the 5 Dimotions of reality.**

We consider decametric sub-scales between two planes of exist¡ence ‘stages’ of the complex reproductive and social *growth 3,4 *Dimotions *towards the fundamental decametric scale. Binary systems relate then to the 3 ^{rd} complementary, reproductive Dimotion, either as gender elements or as inverse 0’-closed informative vs. |-open entropic numbers, the minimal duality to represent reality.*

*Ternary Planes relate then to the more complex inverse Dimotions – the ‘triangular 2, 5 D entropic’ Locomotion since the triangle is the strongest form, spearhead of motion, and π-geometries to perceptive still 1Dimotions (3-6 number systems) Planar, 2D geometries (binary, quadrangular systems) and one can change into the other by rotation. While it gives origin to the 3×3+1 complex, fundamental 10D scaling. *

This duality extends to the 2 main geometric Planes of numbers in open competition” the **0’-****6 ^{th} is the scale of**

**cyclical perceptive geometry**, whereas π=3, is the hexagonal perimeter of a ‘whole’ closed geometric space, competing with

**the 10’-scale,**

**a perfect tetraktys geometry,**where the ‘whole’ is a lineal, triangular space.

In the O≈| historic duality Asian huminds (Babylonian) adopted 6-scales v. European lineal cultures of 10 scales.

** **

**PRIME, ODD AND EVEN ‘GENDER SYMMETRY’ NUMBERS.**

If we extend this vital analysis of regular numbers in space to sequential numbers in time, we come up with another ternary division of all numbers within the finite limits of a given ecosystem (infinity being a paralogic Kantian error of the mind as all planes have a limit of numerical form, normally as explained in the 10’-11^{10’-11} range, from ties of DNA in a human gene to stars in a galaxy)… that between top predator prime numbers, which generate all other, even, social reproductive, ‘female’ numbers, the majority of them in a Universe that systematically favors ‘reproductive S=T’ states as all fractal generators do, over even, informative + entropic ‘male numbers’, which can be divided in a minority of ‘prime numbers’ that carry the information of reality as they can generate all other numbers in 4-Dimensional sums; and entropic, unstable forms that code no new information (odd numbers excluded primes, which generate them). So we find yet again another fundamental division of species, according to the key variation of the ‘Fractal Generator equation of space-time or 5D metric: SxT_{s=t }= C.

It is the same duality of ‘Female’ present reproductive, ‘even’ gender, S=T vs. ‘male’ past to future to past, entropic and informative gender, S<T>S: SxT, the key temporal ternary symmetries of the Universe; between present dominant states, and past to future to past cycles, which requires a full comprehension of the 3 ages of time, and its complex ternary logic sketched in the introduction, that we develop in a paper on 5 D time pentalogic.

All cyclic time systems have 3 ages, a relative entropic past of dissolution, a reproductive dominant present and an informative future. ¡ts proportion in any system is ½ present + ½ (Past>Future ) that closes the cycle. For example, in quantum physics wave states are the present state; particles the future state and fields the past state. In matter, liquid is the present state, gas the past state and solid the future state. In cells, DNA is the future state, cytoplasm the present state and the protein membrane the energetic state. Its S=T symmetry makes the past less evolved, more extended in space, the present balanced, the future more informative. In human gender, the woman is the reproductive state, and males divide into a majority of entropic destructive and a minority of highly informative (complex IQ test put woman in the middle, a minority of men above a majority in the entropic lower range).

So to understand the ¡logic division of the 3 ages of time into past=entropy, S<T, which dissolves systems into a lower ¡-1 plane, present=reproduction, S=T, that iterates reality which seems not to change, and future=informative evolutionary system, T>S, which increase its form into the future… I refer the reader to the article on 5D pentalogic.

**In terms of its scalar social organization, the 3 ‘different’ groups of T>S informative, S=T, energetic and S<T entropic elements of a class structure correspond to the informative class (i.e. stocrkats in our society ruled by money, politicians in a real democracy, military in a dictatorship) which controls the language of power of society, while S=T form the body-wave of the system that reproduces it and it is taken care by the informative particle-head, while the S<T element is the entropic class easily ‘divided’ and broken, as energy of the others, in this case the odds that can be broken in smaller factors. And as we shall see the game of numbers have a remarkable proportion of the two first classes as opposed to our ill-designed systems, a theme treated in 5D politics, where we study the informative ‘nervous’ network of human superorganisms. **

**While the general study of those 3 states belongs to those 2 articles, in the mathematical mirror the division is self-evident as we already observed some clear differences between odd and even numbers/polygons:**

**Odd vs. Even polygons** alternate with a key different property: The diagonals of even polygons create central singularities and form more stable systems, tending to static states, ‘looking inwards’. They can reproduce by motion into equal forms in the same plane, and can easily tile, filling up the entire plane. While odd numbers penetrate in the Planes of the fifth dimension by reproducing fractal mirrors of smaller size, when crossing inwards its diagonals (informative reproduction) or larger mirror images when crossing outwards its diagonals. Those properties carry through all Planes of gender mirror symmetry; as 1 element is sterile, self-centered making different the 2 type of organic membrane-polygons. Since S=T even polygons do NOT reproduce through 5D motions as odd numbers, but within the same present plane of existence as they are ‘self-centered’ in a single point, within the polygon, NOT replicating without motion. Reproduction of even polygons happens by translation in space, as waves reproduce. So by moving through a tiling of present space a square reproduces filling the tiling, unlike a pentagon or odd polygon that reproduces internally and externally as its motion filling space is often impossible

**Prime numbers as informative systems.** In ‘vital mathematics’ as in classic maths, Prime numbers play a fundamental role in number theory because of a basic theorem:

Every integer n>1 is a product of 4 prime numbers (with possible repetition of factors): where p_{1}<p_{2}< . <p_{k} are primes and a_{1}, a_{2}, . . ., a_{k} are integers. Furthermore, the representation of any nº in this form is unique. *Thus Primes are the generational space-time numbers of all others; hence the minimal numerical system with maximal information. And inversely if we reduce entropically a system of numbers, by ‘division’ (the operand of entropic dissolution)* o*nly prime numbers or its social combination DO survive=exist in the ultimate skeleton of reality, due to its efficiency as polygonal forms. *

*Everything else is a virtual society of prime numbers; and the strongest social groups are those under 23, stuffed with primes and displaying a solid even membrane.*

**Prime numbers in time sequences: stability of systems.**

In 5D prime number is s any integer including 1, the whole that has only 2 positive integer divisors, one and itself.

1 is not considered a prime number since it does not have 2 different positive divisors – a topic decision of some ‘axiomatic’ guy, solved by eliminating the word ‘different’; *so primes ARE a closure family*. 2 was a prime rightly in the past –as we might consider that 1 *is the super-prime, the whole that divides all other numbers of nature. *

And the fundamental characteristic that differentiates them is that as polygonal numbers grow in sides, that is the social number of ‘points-cells’ increases the system become less stable, because *it increases the number of redundant non-prime odds (ab. odds). *And when we carry the symmetry S=T, to temporal sequences which as we have seen is the proper way to study numbers beyond the decametric scale, this lack of stability increases enormously past the 23 ‘years’ (we shall call points=numbers in time sequences ‘years’); which is the ‘cut’ between the young age of maximal energy and the mature age; and again plunge past the 83 number, leaving a mean of 1 primer for ‘decade’, which is indeed the standard age for death, both in human years, or in S=T atomic evolution (radioactive atoms past lead).

We talk of each prime number as a key ‘age’, and 23, the Prime age of a 100 normal system, where a transition to the mature age takes place. This we know for human life – the end of University, the beginning of work, the maximal force and energy for most athletes. In the table of elements, originates the 26 Iron-Nickel-Cobalt family of maximal stability, likely origin in its liquid earth’s core state of complex ironlife form.

Those are the themes of ‘vital physics’, which the synoptic mirror of numbers reflect as ideal configurations of social numbers in space and time.

Prime numbers are infinite and relatively abundant according to the π=n/log n rule, so huge social systems are possible both in space and time duration as complex new primes, can reinforce the structure as nodal points of stable networks. A prime number by definition is more stable as a configuration of social forms, as it *can’t split into equal parts*.

How far can we stretch this stability? In inflationary mathematics to ∞. But in scalar space-time ∞ becomes increasingly unstable as prime numbers dwindle.

As the irregular infinity of *the number of entropy, e, which breaks after 10 decimals:**, (*the 1+8=9; 2+8=10 regularity disappears after 10 numbers if we consider 2.7 the first dual term of the regular pattern), loosing its structural, informative meaning’, we should consider that beyond the ± *10’-11 ^{10’-11} social scale between planes, gathering in 10±1 decametric Planes, the systems of ‘big number theory’ are largely irrelevant. *

Prime numbers clearly dwindle (as per Euler’s and Riemann’s functions in the graphs) beyond the 10.000 social group, suffering a steep decline towards the 10^{11 }mark in a tail proper of an almost ‘extinct’ species. So larger primes are rarefied and very few complex structural systems beyond mere herding exist past that number:

In graph, number of primes holds remarkably dense in the basic 100 scale being 1/2th of the 10^{th} scale (if we consider 1 a prime number) and 1/4th in the 100 scale, which are the key Planes that encode most stable members of timespace both in the space symmetry (table of elements) and the age symmetry (life span in years.

The ratio dwindles to 1/6th out of 1000 and 1/8th for 10.000 in the next key interval of social groups, still fairly strong, if we take into account that the 10% is a very strong ‘leading’ elite in organic class structures of any stientific scale… and organism. This 1/10^{th} informative class structure still holds in the 1 million social group (1/12th). So the Universe is strongly connected and stable in its hierarchy of prime informative numbers on top of a reproductive, energy of even numbers and 1/2 of entropic, easy to break odd numbers, never a majority as the 3^{rd} class of the system. Compare that hierarchical structure with our human societies – corrupted placebo democracies ruled by mone,y issued in near monopoly by ‘stockrats’, modern aristocrats of the capitalist society, owners of corporations, which are less than the infamous 1%, closer to a 0.01% with majority stock ownership. This tiny proportion of informative class coding the language of social power doesn’t even happen for the 10^{12} limit of the most complex supœrganisms of reality as primes are still a 4% of units. While the fast 90% of mankind at the bottom is a sea of chaos and entropy, never met in efficient superorganisms and the ilogic mirrors of algebraic structures.

**The 20 ^{th}, reproductive mirror symmetry of the 10^{th} scale, & the 100^{th} time aging Planes.**

It is then only worth for an introduction to Number theory to study just the 10^{th}, 20^{th} and 100^{th} Planes. The first purely ‘spatial’ the second which ‘doubles’ the first in a mirror symmetry is the reproductive S=T scale of numbers, with the equivalent 4 prime proportion; while the 1 to 100 scale is the Temporal symmetry of ages; and as such we shall relate them in other posts as we ‘ascend’ the Planes of reality and the laws easier to understand in the ‘synoptic’ languages that mirror best the Universe’s time and space, ‘¡logic’ and Non-E mathematics’ emerge with different ‘forms’ in those upper stientific Planes.

We shall not do it here, for the sake of time and space, as we are still in the 1^{st} age of mathematics, with the symmetry of points and numbers. Only to notice that for the reproductive mirror scale of 20, closed by its largest prime, number, 19, its square, which doubles its dimension 19º x 19º= 361º gives us the commonest ‘closure’ of a sphere in space (360 is the scale of degrees) and time (365 are days on a year)…

And so we have another curious number 361º ‘degrees’ for the perception of angular rotation, *(as we know geometry is always approached in its figure by discrete number above or below ±1), which suggest the natural limit for a discrete polygon of cyclical nature.*

*The vital nature of the *10 to 20 scale as a mirror of the 10D basic scaling for reproductive purposes, emerges as usual with small variations in more complex stientific Planes. So it shows in the game of life, where 20 are the number of amino acids that construct most organisms and your body can not live without; … as they generate everything else in the upper Planes. Even in the most ‘distant’ level of human languages, the 20 scale manifest itself; so most alphabets have 20 some symbols to write the language; which then generate a larger number of phonemes broken in vowels of information and energetic consonants (more abundant. I.e. the English alphabet has 26 letters, split in a total of 20 vowels and 24 consonants.

**RECAP**. The first prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29… And so while the decametric and 10’-20 scale are buttressed with 4 of them each, almost 1/2, which *explains further why they are really the 20 numbers that matter in almost every system of reality, being the 10 to 20 the mirror symmetry of the 10 scale; which you might extend to the 23 range… beyond those ‘efficient numbers’ systems diminish in stability.*

In the elements table, the strength of the system is a mixture of its ‘even configuration in 2 and 4 pairs’ and odd volume of inner layers: the range around 22-26 (iron) give us the strongest top predator atoms of maximal energy, which is the role of a polygon number that closes as the membrane of vital energy any T.œ. system in nature.

Yet if we consider the temporal sequence the same series, breaks the first age of life, which in a biological human being ends at 24 years: 1-24: youth, 24-48: maturity, 48-72: 3rd age; 72-80 death. And indeed prime numbers plummet after 83. So this will be indeed the fundamental scaling for all timespace life-death symmetries (as we shall show for each stience, from the amino acid series to the life cycles, from the atomic table to the cycles of historic and economics which fit within those decametric 5D Planes).

**S****óT DIMOTIONS: NUMBERS AS UNIVERSAL CONSTANTS**

As Einstein wanted the ultimate principles of reality are Universal ratios/constants, which represent the 5 dimotions of all Universal systems*. So we shall find 4+1* absolute constant ratios in all systems, *of Nature, of the vital Universe: e pi, e, 10=3×3+1 and phi.* *What about locomotion? In 5D physics locomotion is reproduction of form, from particle to wave state to particle. So it has not a fundamental constant but a mixture of the informative, reproductive ones, pi and phi.* *5D thus use those 5 constants to explain how the Universe reproduces reality departing from those ‘perfect ratios of energy and information’. * 3 are irrational≈ratio numbers that represent the 3 ages of all worldcycles of exist¡ence:

10 is the seminal social number of palingenetic ∆-1 emergence, π, is the number of the worldcycle, of the 1st Dimotion of perception; phi is the number of the 3rd Dimotion of reproduction, in the second ‘age’ or maturity of the world cycle, and e is the number of decay and entropy, of the fifth Dimotion of existence.

**1D: Pi transforms a spatial field of entropy into a vortex of temporal information***: 3D ($T) > ∏ ðƒ.*

As we have already talked of pi and will keep adding latter, just a note: pi’s informative nature is immediate, as we perceive through angular trigonometry, when pi closes a circle of 3 Diameters, with 4%=π-3/π of perceptive holes through which 3 sensorial apertures to the Universe happen and 96% of dark matter hidden by the membrane (which not coincidentally is the part of the Universe hidden by the galactic halo we don’t see).

**3D: Phi is the ratio of reproduction of a space-time population. **So applying pentalogic p.o.v.s:

**T p.o.v.**: Phi is approached by Fibonacci series in gender couples, which reproduce in the 2nd age of maturity, and die after the 3^{rd} age. So for ‘2 consecutive quanta of time’, the young still won’t reproduce, the old won’t be still dead and the mature will reproduce giving the series 1,1,2,3,5… that approaches the phi ratio. Thus in 5D symmetries the relationship between phi and Fibonacci numbers (reproduction series) is an S=T* duality of the constant observed in space (golden ratio) and in time – Fibonacci closed solution as a series:*

Whereas the golden ratio becomes an iterative process of generation.

**S=T: Population growth:** This leads to its understanding in ‘still space’, as a population growth:

The golden spiral then becomes the natural ‘time-series’ for filling a surface of energy where to growth in a frequency of space-time generations, given by the ratio – even if the ‘interior of the spiral’, fades away as generations die behind us, leaving a void.

**S: Informative evolution:** Its inverted implosive ‘symmetry’ as informative growth (T>S), gives a proportion for the reproduction of complex information, (as in finger proportions, when form overcomes size).

**∆**: Phi is also a scale function: the whole (trinity) is to the larger part (body) as the larger part (body) is to the smaller (head), since reproduction is a scalar function that imprints a ∆-1 cellular scale to evolve it socially into a new whole.

**¬ Entropic function: – the beauty of its S≤≥T proportions.**

Yet phi has also a reputation for beauty in proportions between the 3 parts of the being, limbs>body-waves>head-particles, which is the natural outcome of its multiple functions as reproduction and organization of the different parts of the being in 1 and 2 Dimotions:

Whereas phi is the golden ratio of proportions of the whole to the 2 body-head topological parts of the being. Expressly is the ratio of the whole a+b to the body, when the body-wave is proportional to the particle-head: T.œ (ST+St) / ST = ST / St.

But a ‘line’ has ‘motion’ to become a real ST system. So the proportion must be interpreted adding the hidden variable (the speed of processing information/energy of both b and a). Then it corresponds to 5D metric: S (A+B) x ð(a+b) = S(A) x ð(a) = S (b) x ð (b), as S(A+B) > S(A) > S (b) in the same proportion that ð(a+b) < ð(a) < ð(b). So we can *extract phi from 5D metric (Sxð=K).*

Its beauty is then the perception of the Universal Metrics of 5D which in turns gives survival efficiency of such ratio, as the body transforms the entropic energy for the whole in a proportional balanced way.

Further on it can be expressed in terms of entropic feeding completing the pentalogic of Phi as a ternary system: whereas a+b is the limb/field. So we write: $T (a+b) < ST(a) > §ð(b)

Whereas the ‘entropic length’ of the limbs ideally equals the sum of the body and head system, while inversely the head of information consumes more, faster energy than the body (i.e. neurons consume 10 times more energy than body) whose cells reproduced in the wide dimension consume as much energy as the locomotion limb/field.

And so the symmetry if we transform ‘T into S’ give us the same proportion for its square areas.

So phi is beautiful as beauty, is the proportional vital constant balance of the 3 St, ST, sT parts, the key to a T.œ.s survival (S=T, harmony of ‘size and form’, ‘energy and information’ ‘body and mind’).

*Padovan numbers. *The interpretation of the Fibonacci numbers in ‘Geometric space’ as an St spiral of squares bring the obvious question if we can form similar sequences with other forms besides the quadrangle. Yes we do with triangles, forming the Padovan series, with similar properties, 1,1,1,2,2,3,4,5,7,9… whereas each new number F+1 is the sum of F-1 and F-2; giving us a ratio 1.3… smaller in growth to the square spiral but structurally more resistant, which hardly appears in nature, as phi is a reproductive function that seeks for ‘speed’ in its radiation and ‘volume of energy’ in the placental region, maximized by squares and minimized by triangles of maximal perimeter.

*All tilings that fill space are thus triangular (the hexagonal tiling being just a scalar 6-*triangular units) or squared and pentalogic entanglement implies again that it can be reached by using ‘parts’ of a single plane of space-time or filling it ‘in scalar’ sizes in yet another symmetry between ST and ∆. *On the other hand, the pentagonal or heptagonal prime tilings are impossible in a flat plane, though pentagons easily fill combined with hexagons 3D spherical structures and heptagons can tile hyperbolic ones, which reinforces our view of 5D polygons as entities belonging to 3Dimensional systems in exist¡ence.*

**RECAP.** The golden ratio is the most efficient form of inner growth and proportion, keeping the balance and distribution of information in a system.

**2 ^{nd} Dimotion: locomotion**: It combines with π to give locomotion a number, as motion is reproduction of a π cycle of information along an adjacent series of forms. We saw the case for a cyclical reproductive spiral. In the 3

^{rd}graph we see the case for a lineal reproductive series, which becomes real in quantum particle-waves.

**4D. Social Planes. **10 is the constant of scale of social evolution; hence the main system of scalar numbers.

**5 ^{th} Dimotion: entropy: **the exponential function is the most efficient form of social growth between 5D planes. In the graph, we can see how the growth of any system in base e is the maximal possible. But such exponential 4D social growth can only happen in a placental unlimited world of free energy. So its true use is for the inverse arrow of death and decay, when it is the maximal speed at which the cells that grew in the placental ∆+1 palingenetic first age, now die in the final ∆-¡ age of entropic decay, adding together to a 0’-sum. So e is the exponential function that grows faster:

*e ^{x} is the only function, which is a derivative of itself*, adding constantly a 1/n finitesimal derivative ‘cell’, in each time step, but also decaying in the fastest possible path of dying quanta

*. e*

^{x}represents the fundamental function of reproduction of any system in its 1^{st}world cycle of a perfect energy placenta, þ, at the lower ‘cellular scale’, as the system increases by a derivative at each step.**RECAP.** e represents the absorption of a reproduced finitesimal part, or transfer of a quanta of energy to the larger whole. But it requires a ‘perfect energy’ system. In physical systems this happens in decay-radiation; in biology only in the placental cycle; in sociology in the brutal capitalist system that expects interest to be added as the Maximal exploitation possible.

**Universal ratios=constants & the generator. **

Pi, 10 and e, and φ, are the ratios of Ts<ST>St transformation events. So they are Nature’s constants, related to its Dimotions, which we can write as partial equations of the generator:

π: 3 |-Ts > O-St; e: ∇1 e->∆1+∆º; φ: T.œ/TS = TS/S; 10 §º ≈ §¹.

They also combine to mirror complex, more efficient dimotions. For example, the closed curve described by π and phi as a 137° golden angle – which allows the maximal covering of a surface from an outer point of view, as a flat, filled space – so it maximizes reproduction, the function of existence, and entropic feeding; as when light in plants can touch most ‘leaves’ which therefore are positioned in a phi angle. *The same maximal reproductive function of existence happens in *open curves, of asymptotic hyperbolic form, where e is the fundamental ‘number’ that maximizes both reproductive growth and decay. So the 3 fundamental irrational numbers *maximize the function of existence, in its «entropic = reproductive » Social evolving symmetry.*

**∞Transmission of information.**

In terms of Planes, pi is related to ∆º+1 (perception, social evolution), Phi to ∆+1 (reproduction and social evolution) and e to ∆-1 (decay): Phi (D4, D3: ∆-1 » ∆º) > π (D1:St ∆+1) « e (5D: TT ∆-1)

Since e= 2.718281828…4, the limit of (1 + 1/n)n breaks after 10 ‘decays’ into disordered death patterns; only pi has no limit of scale, as a Poincare’s sphere that can shrink *preserving its information.*

*Hence making the fractal structure of the Universe* infinite, by allowing information but not energy to emerge or sink with no limit of scaling in the universe, reason why Maxwell’s equations of waves can transmit information across Planes with no limit in an infinite universe. And this is a huge fact of the Universe, which in the ∆=S symmetry implies information can also be transmitted faster than light (but not energy), through *translations of Planes, the same way we transmit voices faster than sound through electromagnetic translation. *

**Beauty of 3: Phi couples. Lineal, hexagonal pi & e as decametric scale.**

π (+3.14), 2 phi (as reproduction is a dual ST-mirror symmetry) and e (2.81) are all within 3 ±0.2 range. Since 3 is the constant of the 3±¡ trilogic/pentalogic Universe. Pi is larger as the curvature of 3 lineal elements makes a larger cycle with aperture-wholes. Phi is smaller because the reproduction of a system compresses the entropy it has extracted into form. To notice also the Fibonacci series work on ‘bidimensional’ units (couples). So does the ‘golden rectangle, and the spiral that fills space with time series of reproduction. But ultimately 2 genders give birth to 1.

Lineal pi is 3 – the hexagonal perimeter ad 3.14, circular pi. 3 is also e +e/10, the classic growth of one tenth or finitesimal part of a tetraktys, the perfect organic form, growing, one by one, as 2.718..+0.271..=2.99…

**RECAP.** 5 scalar or irrational numbers represent Dimotions of existence: π, the first Dimotion of perception; phi, the reproductive Dimotion; e the dimotion of entropy and 10, the Dimotion of social evolution. Pi and its function, the sine are the cyclical numbers of perception; phi the number of reproduction; e, the exponential of decay that faster grows in its derivative dissolution into infinitesimals and 10 the social scale of the Universe. All those numbers are ratios related to 5D Planes, proving further the 5^{th} dimension. They are not isolated numbers, neither points of a continuous line (continuity error of huminds). They relate S-T bidimensional holographic basic dimotions=actions =functions of time-space.

**S=T.** We studied only an ideal form of a number as a point, a knot or lattice, which is what in ‘reality’ most ‘atomic/molecular symmetric numbers’ – the units of matter Planes are. Numbers as forms thus have a regular undistinguishable structure. But in reality as the particles and atoms of those matter numbers, its ‘formal dimensions’ are in constant motion through ‘network connection’ that makes them switch space and time dimotions, S≈ð.

Our analysis of pentalogic pi numbers reflects the deep Entangled Nature of the Universe, where each entity, while *dominant in one of the 5 elements of reality, in as much as it performs only a function on the upper ∆+1 world in which it survives, it will as a whole in its inner reality be a full ¬∆@st being. So while we *shall always stress a single function. I.e. a number will be a scalar social gathering of identical beings, there are for every entity 5 possible perceptions of the being. Since numbers as synoptic linguistic mirrors of the Universe *uncoil and recombine in multiple pentalogic functions*, both as S=T geometric structures and as ∆-scalar, social groups. So as words do in verbal languages or pixels of colors in images, a single number 0x∞=C represents them all but this doesn’t mean it creates reality through generation of numbers and equations, as the ‘word word’, which includes all other words doesn’t create the Universe by naming. On the contrary numbers are mirrors generated by space-time reality. So we will now consider its 5 species that mirror the 5Dimotions of space-time; its 3±¡ worldcycle ages or 5 ¬∆@st elements, as do the 5 types of curved lines in geometry, 5D smells in noses, pentagrams in music and any other kaleidoscopic language of 5D reality.

So numbers will be useful vital expressions of short time dimotions, half-time worldcycles and deep-time ∆@st structures: a rational number will be an ‘entropic division’ of a whole, a ± dual number a sequential time motion whereas its inverse which close into a zeroth sum cycle; a polygon number, the T-membrane of a spatial configuration of identical points, etc. We will even be able to transcend into a new plane of existence through complex numbers. Finally as we are all entangled vital systems, and so all our elements participates of the other, numbers will combine in increasingly complex simultaneous structures, from equations to functionals, from groups to the canonical 5 families we shall now study.

** **

**PENTALOGIC OF NUMBERS’ FAMILIES AS ****MIRRORS OF THE 5 DIMOTIONS OF THE UNIVERSE.**

**The growth of complex entangled dimotions within ¬Algebra. Numbers including operations.**

In the entangled Universe of ever more complex dimensional motions, each more complex element of ¬Algebra includes the previous one. A number beyond one *includes a sum in its synoptic form. Thus *Numbers include its operations; natural numbers are composite of 1 plus sum. * N*ext Z includes natural numbers, sums & the operand of subtraction to form negative numbers.

Next **Exponential numbers** take the form x^{n}, where x is multiplied by itself n times; and they include the product as reproduction, which requires the identity of S=T, hence the square of the same system, but by Fermat theorem, the 3^{rd} power no longer reproduces and can be merged and superposed by sums and products. *And its inverse, n ^{-x}*, represents, exactly the opposite function of entropic death, of decay as in e

^{-x}

**Then we have the families of division, which have multiple functions** and must be explained with the duality of information ‹< entropy; that is information smaller in space, faster in time (More synoptic) than entropy, the first represented by Q, the second by irrational numbers.

Irrational numbers then include the dimotion of entropy, of failed reproduction, of irregular form, save those who have a meaning for the angles of perception, and ratios between parts of a system that matter (universal constants), which are only a few of them.

Equations thus ‘rise’ the scalar number of dimotions studied, and the simplest view of that growth of dimotionality is to consider that equations take ‘full tœS’ and put them in an external world as mere fractal point of that larger ∆+1 world.

But then numbers seemed to become exhausted in its growth of complexity which as we saw *already included composite numbers, nothing to worry about in the entangled Universe, as 1,3333 is really 4/3, a composite number of 2 and an operand, which is the only way to define it in a single plane of existence.*

As always this number does have a trinity logic, as it can be represented by the minimal ∆ST elements of any reality. So it has besides its S/T or T/S geometric single 5D plane representation (in the first case an entropic division of 4 between 3 that splits the 4, as when we cut a pie, in the second case a gathering of 4 elements between 3 individuals which take one and use the 4^{th} for its sharing entanglement as in certain molecular bondage), the scalar 1,33333… representation; as numbers are the most synoptic unit of geometry.

So with those ‘classic numbers’, N, Z, Q, R, *and the new family totally disregarded by mathematicians by lack of understanding of the true meaning of numbers the ˆ family, which keeping with the correspondence principle we shall denote as E form the 5 ‘classic familes of numbers’ that allow to define the 5 classic domotions of existence but not in a direct 1 to 1 correspondence:*

Thus N->4D, Z->2D, Q, E->3D, 5D, R (π,√2, phi) 1D (angles of perception and absorbtion of energy and information for the circle, the triangle and the spiral)

Then we move into a new ‘dimensional arithmetic form’, the equation.

The equation thus is NOT a number, not a ‘phoneme’ of the syntax of mathematics, but a ‘sentence’, and as such it is infantile to think it would have a solution in a single plane. It does NOT since it requires to find a solution in the ‘next’ system of ‘two planes of existence’ in which following the pattern of the previous reasoning we find numbers that in the most synoptic manner introduce the essential elements of all equations.

That is, according to Fermat’s grand theorem, X^{2} ± Y^{2} = C; since the Universe is holographic all solutions must be ad maximal of the form SS, TT, ST, St, sT, which we could translate as X^{2 } Y^{2}, X^{2} + Y^{2} = D^{2 }(ST as the Pythagoras theorem of reproductive orthogonality), X^{2}±y and Y^{2}±x.

In other words a complex number of the form a+bi = z, can represent all the holographic dimotional solutions to a polynomial equation; as the real numbers can represent all the ratios/angles of perception, the rational numbers all divisions of wholes and so on.

Because the C plane of complex numbers can translate all possible configurations of an st solution, (when it is properly interpreted by squaring the real line as in the graph, which converts the imaginary line in a classic real axis; it follows then that it represents all ‘the real, non-fictional’ results of a polynomial, that is, the non-entropic part of it, the one that has form and meaning, specially those of the unit root, where series converge, which was the method that Lagrange – regardless of the myth of youth and Galois that merely put the ice in the cake – used to resolve them with ‘resolvents’. Or as in the case of the Mandelbrot fractal. So the marvels of the complex plane and its classic truths derive of the connection of the complex plane, with the five only solutions of the Universe – its 5 dimotions.

*Since in fact all the solutions to higher scale equations, by radicals reduce to square solutions *of quartics, we need to resolve an intersection of two different conics. And so on.

In the Entangled Universe, each entity will be *dominated function it performs in the upper ∆+1 world in which it survives, but as a whole is a T.œ, an ¬∆@st being. So we *must apply the ‘Rashomon effect’ of pentalogic 5 truths to study its variations. For numbers this gives birth to its 5 ‘different types’, which in turn define different ‘planes of geometry’:

N(∆): The main ones are Natural numbers – scalar social gatherings of identical beings.

Z(T): are ± dual sequential time motions and its inverse, which close into a zeroth sum cycle.

Q(¬); Rational numbers are ‘entropic divisions ‘that dissolve wholes into its parts.

Gnomons (S) are polygons = spatial configurations of identical points.

C (@) Complex numbers and those of different ‘frames of references’ are mental configurations with various applications according to the mind mapping they represent

Nº are *best as *social groups in Planes, as regular polygons in space and as sequential causal events that represent the 5 Dimotions of the Universe; so we depart from the tyranny of the axiomatic method that packs them all in a single ‘real line’; so we can use better pentalogic to describe the multiple uses of *each plane and family of numbers.*

Whereas the continuous single line to define all numbers, natural, negative, rational & irrational, is the ‘limit’ of a more profound analysis of numbers in multiple Planes. So those families of numbers, when escaping the tyranny of the line become filled of new meanings as *5D pentalogic mirrors of ALL the elements of the Universe.* In @nalytic geometry, the line of numbers relates to a 2 dimensional holographic world. So its scope is enormous as it can represent any SS<ST <TS<TT dual ceteris paribus analysis of any dimotion. Yet its synoptic power must not fog a larger picture of reality.

**A first pentalogic analysis of those families then set a **Generator equation for the 5 families of numbers such as:

**N:***Space: 4D¡: Social evolution. Z:Time: 2D¡: locomotion. R:∆-Planes: 3 D¡:Reproduction. Q: 5D¡: ¬ entropy.*

**Complex numbers*** and other frames of reference: @mind: 1D¡motion: informative gauging *

**We correspond each family of numbers to its main Dimotion as ¬∆@st, dust of spacetime in the ‘larger picture’ of the absolute Universal mirror; and to specific Dimotions in the ‘smaller picture’ of its use in the events & actions of T.œs. **

**Natural**, ‘direct’ numbers ℕ are zeroth and the positive whole numbers 0, 1, 2, 3, 4, 5… If two such numbers add or multiply, the result is again a natural number; which means* it is possible to create all intermediate societies departing from a unit quanta, between 1 and the relative ∞ value within which the domain of the function or event is meaningful. *

*Its main use is to model social evolution of identical groups, through the § scale between ∆ and ∆+1*

**Natural numbers** are related to the measure of 1, 2, 3 D **spaces**, as units of points and populations. They are either self-centered **even** numbers representing the 1^{st} Dimotion (perception in its crossed diagonal center), or **odd mirror polygons with a reproductive symmetry**, (3^{rd} Dimotion) which can be internal or external crossing of diagonals, making a 4D & 5^{D} replica (inner, implosive social growth or entropic outer growth).

Finally Natural numbers ‘can represent locomotion **(2D) **as ‘steps’ of a given unit, which represent a discontinuous motion of a frequency ƒ, and a wavelength ‘1’. Steps become discontinuous landings of the fractal unit that measure distances or motions in a lineal 2D geometry. We can then include ratios of those steps, breaking the line in different smaller ‘fractal steps’ and that is a first meaning for a rational number. *But if we make a different unit for each rational part, we enter into another Natural number scale of the fractal Universe returning then to natural numbers, which therefore are also able to represent ratios. And so natural numbers can be any number as it keeps falling down in microscopic rates. *

*Why then we need more families? Because each number really is best for a job, even if they can multitask, as a shovel can do many things but a hammer will nail better. So the next family of numbers, *negative numbers are a better mirror symmetry for the function of motion, as they can change the direction, moving backwards to the other side of the line.

**S+(-T).** The integers ℤ, add “Inverse’ S«»T numbers, normally related to inverse, entropy vs. information functions in all systems of reality, hence to the inverse ‘directions’ of one of the ‘dimensions’ of space-time (S, T or ∆). As Gauss intuitively understood this removes absurd hang-ups of scientists afraid of negative ‘values’, specially in physics, where c-speed is a limit because physicists don’t understand mass’ as a vortex of space-time; hence its inverse is an expansive entropic dark energy’, perfectly valid in relativity equations. Z numbers thus divide in positive and negative whole numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …. If two such numbers are added, subtracted, or multiplied, the result is again an integer; if both are equal in value differing on its sign, the result is a 0’-sum, which means we can ad up all intermediate combinations of entropy and information, or similar inverse directions in any dimotion of ¬∆@st, but ultimately its inverse ‘time arrow’ will make the system come back to its cyclical time departure as a 0’-sum.

**Negative** numbers thus *are essentially Time Numbers. They make no longer sense as ‘social populations in space’ – there are not 5 negative pears. *So the social use of natural numbers is lost (cause of much philosophical thought in earlier mathematicians), but the power to represent

*the inverted arrow of time in any Dimotion of exist¡ence, either in negative directions of locomotion or negative entropic functions (negative exponentials), grows.*We keep this in mind in physics as ‘i’, a negative number, merely represents the inverted time Dimotion of a physical species, as in EóMc

^{2}.

**5D: entropic, ****rational numbers, ****ℚ.** Fractions invert the arrow of social evolution of Natural numbers by inverting the product operand: *p*/*q* where *p* and *q* are integers and *q* ≠ 0. So they divide social groups or time frequencies/durations or kill and break a whole into similar parts. So they remain in a single ‘plane of reality’. Since if two such numbers are added, subtracted, multiplied, or divided the result is again a rational number. In terms of Planes, **rational numbers **break them in part. So with decimal numbers we move the ∆º whole downwards into its relative ‘finitesimal’ ∆-1 quanta, following the standard scaling 10¹¹ Planes.

Natural, negative & rational numbers form together with ± and x÷ operands a ternary ‘Fractal Generator’ system: T(Z)>T/S(Q)>S(N), a present organic, ‘quantitative’ Polynomial mirror of most T.œ in a single plane of reality.

**∆-1.** **Real numbers,** ℝ, as rational numbers do, allow us to travel through Planes. For that reason they often loose its patterns in decimals around* the 11th element, the limit of most finitesimals between planes. Yet according to the ‘wholeness of reality’ within each of those 2 fractal island-universes,* if two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a real number.

**Recap.** We classify all the ways in which numbers reflect the Universe with the classic concept of closure; that is, all the different planes and forms of numbers tat put together can solve all the ¬Algebraic equations of timespace variables that represent the 3±¡ topological function of a fractal point or T.œ (lineal, cyclical or hyperbolic, made energy – | x O=ø) we translate also into the 3 conserved quantities of physics or a Dimotion a T.œ. as ¬∆@st.

Since numbers can mirror the ‘existence’ of beings either as relative present superorganisms or in its sequential time motion, describe fully its worldcycle.

While complex numbers in that mood are solutions to polynomial equations and shall be studied then in that section… We shall then also return to the other families of numbers and consider the main contribution of ÐST to its theory – namely the understanding of the entanglement and synoptic nature of those numerical families that include within their ‘phyla’ the operands and dimotions they associate with.

So before we can return to number theory we need to study the dimotions of the Universe and the operands that describe them, and how those operands and its negative entropic functions are inscribed within numbers.

**Recap**. Numbers evolved through the expansion of the concept of ‘quantity’ from the natural numbers which merely observe spatial quantities added on in social groups, the essential definition of number, into ratios between numbers that express more complex concepts of reality, and introduce the idea of functions and ‘motions in time, advancing further with the concept of negative numbers, which are no longer quantities in space, but qualities in time, finally reaching the maximal transformation from ‘space to time’, from quantity to function, and ratio of exchange of the vital entities of reality, entropy, information and energy (sT-ST-St), to the final closure of reality expressed in the mirror-language of numbers, which is represented by the ill-understood imaginary numbers.

So the 5 families of Number systems are symmetric to the laws of the ∆ºst universe. So they are its spatial representation in frames of reference:

1.S±T: N, Z: **N**atural direct numbers form simple social groups. ¡ts inverse negative numbers form Ts≤≥St symmetries

- Q: SxT: The Cartesian graph and the rational numbers, for Ts x St = K metric equations and complementary systems, (whereas often the z-dimension is the reproductive combination of the other 2).
- ∆±1: R,C: The real numbers, for inverse ∆±1 Planes & the complex numbers for worldcycles

∆±1. Functional Spaces, wholes & parts where each point represents a vector (Hilbert space): ∆-1(∑Ts) >∆ST.

An even more generalized view of such systems is provided in group theory, as algebras with those simple operands form groups, rings and fields.

** **

**COMPLEX NUMBERS.**

Because of the immense extension of all stiences, we escape many classic themes of every discipline, specially those humans best understand. Instead we try to clarify themes huminds distort more. Thus of the family of numbers we shall make a more extended commentary on Complex Numbers which huminds use but still are surrounded of an aura of magic… Complex numbers though are merely dual ST-numbers, which represent holographic ST dimotions of exist¡ence. And as such very useful as an ideal template for an ST-functions, in which one of the components, usually that of time motion as opposed to space population has a lower value, which is represented by the ‘squared’ value of the real line as opposed to the -√ value of the i-maginary inverse coordinate.

*In the graph, Gauss offered the first clear description of negative numbers as having an inverse direction (of a timespace dimotion) and lateral imaginary ones, as a composite of both. Its proper definition is to consider them composite Time§pace numbers with an space, mostly real and a time, imaginary, negative component, which make them useful in mathematical physics to describe holographic ST T.œs ST or scalar events between the field: $T<wave: ST>particle: ð§ of a physical system. Let us then do for a change a longer analysis in this introductory course to 5D mathematics of complex numbers and its main ‘magic’ conundrum – what they represent in mathematical physics.*

Vectors and imaginary numbers are bidimensional, holographic numbers, hence best suited to express the 5 Dimotions of space-time, as *functions of Space-Time (SS, TT, ST)*. Vectors in generalized coordinates are the best form to express motion in the plane from an objective non-human self-centered perspective; in terms of lineal time, by virtue of the S=T duality. However, imaginary numbers excel vectors in the representation of more subtle 5 Dimotions=actions in which Time is not expressed in lineal terms as synonymous of space, for 3 reasons that characterize the ‘Dissimilar complex plane’ as its Y and X coordinates are NOT in the same scale, one being the square of the other, and this allows them to express complex ∆ST symmetries of scalar space and ST parameters, mixed together in various ways:

**S****ó****T symmetries,**where**t**he imaginary negative number, which makes NO sense in space (no negative apples) perfectly represents a Time function, either in complementary T>S dual systems or in entropic perpendicular Ðimotions**TT: Cyclical time events**. Cx numbers are also suited to represent cyclical, repetitive functions, in which the time clock parameter constantly switches on and off, representing repetitive 0’-sum worldcycles and generations (Euler’s equation, exponentials as sum of sins and cosines inverse functions e^{ ±x}series)**∆-scalar complex numbers:**Events where the higher scale, often a leading informative function that guides the system as a ‘time motion’, has an √ smaller proportional value, as the imaginary part that ‘absorbs’ a small proportion of the existential momentum of the ‘Real energy’ function/organ.

Thus the imaginary coordinates are a root, or inversely, the real parameter is the positive square of its informative value. This duality is ideal to represent a ‘dominant time function’ often with a 1D frequency parameter on top of a ‘bidimensional, holographic spatial’ area or relationships between the particles, body-waves and limbs/fields of a system; as parameters of lineal or cyclical time ARE one-dimensional (duration or frequency), acting upon a ‘surface of spatial populations’.

Complex numbers thus have the advantage in the imaginary frame of reference of being NOT only bidimensional numbers but *numbers of different dimension, hence kept separated, as squared numbers and single numbers when we ‘square’ the coordinates, or real numbers and its roots as observed normally.*

This quality in Nature happens to be very common in Spatial 2D areas vs. Time 1D frequency representations, so Complex numbers are ideal to represent ST functions=actions=dimotions of space-time.

**Pentalogic of complex numbers**

Thus its* * ¬∆@st pentalogic use in experimental physics derives of having a **negative root** coordinates meaning either:

**ð:**An inverse, temporal or negative number that represents an inverse direction of motion. Or…**∆:**An upper plane which emerges from a denser lower plane, often with a √ amount of the real parameter measure, as upper Planes take only a ‘smaller part’ of the total energy of its lower plane (as in heat).

Both things put together make complex numbers ideal to represent ST dualities, with the S-function as the real function and the T-function of motion as the negative imaginary root part often extracted from the energy of the still spatial area,

**– ∆ð:** Both together laid down in a Cx^{2} squared coordinates –a new 5D solution to represent complementary ST systems.

Since if we combine the negative time dimotion and the scalar squared dimotion we can ‘reorganize the complex plane’ in polynomial terms, getting rid both of the cumbersome √ elements and its negative complex roots, to reflect a mapping of a fundamental principle of nature:* ‘The imaginary part of a complex number represents a parameter of an ∆+1 whole, which extracts energy from an ∆-1 disordered more extended, ‘real’ potential field’. *

If the classic representation, the upper half-plane H is the set of complex numbers with positive imaginary part; in the square complex plane, both left and right real squared axis are completely equivalent, (-x)^{2}=x^{2}, reason why we use only 1/2 plane. Further on the plane is tumbled, giving the importance of the i-plane which has its conjugate as the negative ‘time motion’, -1.

So we use the squared Cx plane to study dual inverse time motions; while the real squared axis provides the ‘stored energy’ for that motion, dwindling its volume as the time function absorbs it. Thus a Cx^{2} plane represents a ‘local S^{2}T-organism’ (Ab.T.œ) where the conjugate ± axis represents 2 inverse time directions and its √^{2 }value, compared to the spatial area=population of the real line, the dwindling energy of new ∆+1 scalar planes. So in mathematical physics we map with complex numbers the 3 ∆±1 STates & ternary parts, fields, particles & waves:

**The plane of existence of 4D Universes. Why complex numbers cannot be ordered.**

**The differences between the Cartesian and the Complex plane.**

The fundamental graph of the Universe is one in which orthogonal coordinates represent the T-independent parameters in the X-coordinates and the T-parameters in the Y-coordinates. But we do have two different representations for them, because we do have 4 different S and T dominant dimotions (with the ST combination of both, able to appear in the z-coordinates, or the combination of both).

So the big question is what coordinates belong to what Dimotions. And as SS and TT dimotions are equal in value, the pure coordinates should belong to the Cartesian plane. While the S-informative coordinates do have a lesser value. So they must be put on an imaginary system of coordinates.

Orthogonality in the Universe, is then easily explained as follows:

Because Entropy (TT) vs. absolute linguistic still form (SS), Locomotion (Ts) vs. information (St), are the dual inverse functions of reality merged only in the S=T reproductive dimotion, in the 0 point of X-length, the relative dimension of locomotion, there is a zeroth motion and stillness rises in the height dimension of pure form, where the 0’ mind or frame of reference resides. But then we deal with the ‘different quality’ of locomotion and informative perception in terms of expenditure’ of energy, as information ‘shrinks’ motion. I.e. a gravitational invisible tachyon line has no information but when it becomes light (neutrino theory, Broglie->Jordan), it forms a wave of information that grows in height with the photon on top. But this height dimension is in terms of the parameter of energy and locomotion (T), a compressed ‘spatial T’, of minimal size. And so we need a smaller ‘quantity’ and one that is negative, ‘subtracting’ from the distance-speed (s=t) of locomotion.

*This is magically achieved by the negative ‘root’ value of the imaginary axis, reason why it appears as –ct in relativity and is so useful for the study of electric wavers. By squaring both we simplify the problem of √negative roots, we shall explain latter when we analyze in depth the inverse operands of algebra. *

*So *the complex plane is most useful for St-Ts systems of two composite ‘energy-information’ body-head forms. Complex numbers *cannot be ordered. * Hence they are NOT numbers belonging to the scalar 5Dimension, but rather numbers that represent Space-Time Dimotions which combine elements of both.

It is then immediate that if the Complex plane is an St plane dominant in information and the Universe has four dimotions in a 4D model, those 4 Dimotions the other perpendicular plane must be the real Cartesian plane that represents the ‘background ∆-1’ field/limb scale over which the St plane of lesser energy rises.

And so a 4 D reality combines 2 TT-fields from where the energy of the ST physical organism represented by the complex plane, rises.

While for other less common uses the complex plane can also represent a pure mental SS-space. Since SS, TT, which can be represented in the Cartesian plane as both x and y coordinates are equal.

And the St, Ts dimotions can be represented in the conjugate and imaginary sides of the complex plane.

Nature thus is vital geometry. So, even if the ‘ideal forms of mathematical planes’ become ‘warped’ into the topological vital organs of supœrganisms we can connect directly the 5 Dimotions of the Universe and its representation in a 4 D dual plane with 2 real parameters of a Cartesian plane and an imaginary plane, where the function of information reduces the energy of the vital space, which is how Einstein’s relativity represents them.

**RECAP**. As we live in fact a 4D Universe we can think of it as a walking combination of a Cartesian and complex plane, whereas the complex plane represents the organism and its body-head, wave-particle states and the Cartesian plane the ‘background’ field that moves and powers its T-motions.

**Some examples in Nature.**

By making an indirect use to simplify calculus in which complex numbers represent the ± inverse directions of a ‘time function’ (imaginary part) or a spatial population (real part) as in alternate currents and phasors.

I.e. In electronics, the state of a circuit is described by the voltage V across it, the spatial still state and the current I flowing through it, the temporal state – (or its capacitance C and an inductance L that describe its tendency to resist changes in voltage and current respectively). So V and I can be described by a single complex number z = V + i I. Similarly, inductance and capacitance can be thought of as the real and imaginary parts of another single complex number w = C + i L. The laws of electricity can then be expressed using complex addition and multiplication.

*Or by making an **∆ scalar interpretation, where the spatial, area=population component or ∆-1 scale is a larger squared value. *as a ‘co-ordinate system of ‘square, bidimensional’ space and time elements, where the x² coordinates responds to space, and the i²=-1, represents the inverse mostly time Planes.

I.e. In relativity the time function is represented with an it parameter. So in special relativity we write:

Where the negative factor of time means an inverse motion/form, as the other 3 positive space parameters represent a lineal distance-stretching on the lower quantum potential=gravitational space-time where the light displaces and the negative ‘cyclical time parameter of the light wave’ the warping by light of a space-time-light function as it ‘forms’ the quantum field potential below (we assume it to be the ∆-1 gravitational scale), the speed of light space that drags and warps the gravitational potential into form appears as a negative element that slows down the motion.

- Or by making
*a ‘direct’ representation of real $T-field<ST-wave>§ð-particle ternary fractal generators .*

I.e. in quantum physics either in its ∆ø: S=T wave representation (Schrodinger), where the wave function is attached to an imaginary momentum operator, ih.

Or in its realist Bohm’s interpretation, where the T-potential, guiding wave is the imaginary part, and the S-Particle state, the real part.

Thus departing from the Fractal Generator of physical systems ∆-1:$T-field<∆º:ST-wave>∆+1:§ð-particle describing the a(nti)symmetric states of its 3 components, we can make a realist interpretation of both relativity and the quantum world in terms of complex numbers as local ST-organic systems to interpret properly most parameters of modern physics; *either in terms of the different Planes of ∆-1 fields, ∆ºwave or ∆+1 particles, with relative squared values* or in terms of the negative and positive, implosive or expansive, entropic or informative arrows of time represented by + real or – root values.

For example, complex negative masses in relativity describe are just the inverse ‘motion’ to the cyclical space-time vortex of mass, hence repulsive expansive gravitation.

While Schrodinger’s wave either in the S=T, present wave or T-past quantum potential field>S-future particle formalism (Bohm’s interpretation) is a no time dependent wave, which therefore represents a complete worldcycle of lineal past, kinetic energy; present hyperbolic wave and future particle states fully ‘integrated in the equation’ *that must be seen as the sum of both, the Schrodinger’s present and Bohm’s past to future formal states*, whereas the angular momentum, or membrane of the wave (T=S symmetry), becomes the real parameter of the particle envelope.

**Cx as expressions of the 3 timespace dimotions= states: potential past <present wave> future particle**

All what exists is a ternary system generated in time and space by a simple fractal generator equation:

|-$T (past moving limbs/fields) < Present hyperbolic ST-waves/bodies> future O-§ð informative particles/heads

Both biological and physical systems have the same structure of 3 timespace Dimotions, which give birth to 3 spacetime parts. So entropic fields ‘feed’ the motion of ‘hyperbolic waves-bodies’ which feed and guide ahead the particles-heads that do NOT move in its relative stillness or pure informative linguistic views of reality, which ARE mirror still images of the world.

So for example, ALL topological geometries of the Universe REDUCE to those 3 types: lineal/planar topologies, which are entropic motions as the line is the shortest distance between 2 points; spherical mind/heads, which are n-spheres as the sphere holds maximal information in lesser space and can shrink without deformation to create a 0’-mind mirror of an infinite Universe (Poincare Conjecture); and *hyperbolic body-waves, **which combine and hence iterate both, lines and cycles, forming sinusoidal body-waves. *

And so mathematics again reflects in its 3 only topological varieties, the 3 only organs needed to explain reality.

While we can study the 3 arrows of time of the Schrodinger wave, reordering the 3 components in 5D in terms of the 3 conserved time-states of any physical system (graph):

∆ø: Vψ (Present energy state) ≈ h²/2m ∇²ψ (past, kinetic energy) + ih ∂ψ/∂t (future particle)

So in the right side we have now the past lineal kinetic≈entropic energy of the particle or singularity parameter and its future, particle envelope of angular momentum or membrane (S=T symmetry). While in the left we have the quantum wave itself, its vital energy, which is the potential energy of the wave, Vψ.

The equation in terms of the fractal generator of 5D physics and its 3 conserved spacetime states reads:

Singularity or active magnitude with its kinetic≈entropic, past, lineal momentum + cyclical particle of angular future i-momentum = Vital, Present, Potential energy; which is always the function surrounded and integrated by the other relative past and future states, in dynamic balance with them – ultimately an expression of the much wider fundamental equation of time states of Pentalogic: Past-entropy • Future-information = Present-iteration.

In this case, Past-field • Future-particle (Bohm) = Present-wave (Schrodinger).

So, we obtain *a third 5D interpretation of the 3 states of the quantum system, besides the expression of Schrodinger and the Bohm’s interpretation, connecting both; through the more general 5D TimeSpace equations.*

Whereas • will be a different operand according to the type of formal representation we use for the 3 parameters.

In terms of past, present and future an i-number rotates from a relative past/future to a relative future/past imaginary states on the ±i conjugate axis (imaginary potential in Bohm’s formalism, imaginary angular momentum in the Schrodinger’s formalism), the function of existence of the quantum wave, through a 90° middle real angle, in the square Complex Plane or ST real balanced point, in which the present becomes a bidimensional function, the wave state with a REAL, + (relative past-entropic coordinates) and an imaginary, negative (future coordinates) elements.

This needs a couple of clarifications: first we move ‘backwards’ in a complex i-graph from future to past as we rotate through i-Planes, in as much as most graphs are measuring the ‘motion’ of lineal entropic time which is the *relative Space Function.*

So the negative side becomes the time function, something which mathematical physics makes obvious, when considering an attractive, informative force-field-potential negative.

**@:****As a self-centered polar frame of reference (r, α),**calculated by modulus=distance and angle, which as we shall see are the two essential ‘mental informations’ needed in any representation of space, for a self-centered mind to understand its surroundings – measuring other entities of the world by its relative size (angle) and distance-motion, (hence the time to reach them, and the ‘danger’=size and encounter chances, given by the trigonometric angle in prey-predator events). Distance and angle*thus are the two minimal elements of information all useful mental spaces of geometry must carry, as Riemann and those who followed his logic interpretation of geometric features realized.*

As to the geometrical meaning of a product in the so-called plane of complex numbers, we can see it more easily if we consider the length p of the vector from the origin of the coordinate system to the point (x, y) (this length is called the modulus of the complex number z = x + iy) and the angle 4 which the vector makes with the Ox-axis (this angle is called the argument of the complex number z = x + iy); in other words, if we consider not the Cartesian coordinates x and y but the so-called polar coordinates ρ and ϕ (figure). Then x = ρ cos ρ, y = ρ sin ϕ and the complex number writes as X+¡y=ρ (cos φ+ i sin φ).

From this we see that in multiplication of two complex numbers their moduli ρ1, and ρ2, are multiplied, and the arguments ϕ1, and ϕ2, are added. In division, since it is the inverse operation of multiplication, one modulus is divided by the other, and the arguments are subtracted.

In raising to a power with positive integral exponent n, consequently, the modulus is raised to the same nth power, and the argument is multiplied by n.

However, in taking roots a special situation arises. Since the values of the root coincide in the argument (cos φ/n+2kπ/n+ ¡ sin (φ/n+2kπ/n), where k is any of the numbers 1, 2, ···, n – 1, will also be an nth root of the number as the addend 2kπ, because of the properties of sins and cosines, can be neglected, since it changes neither sine nor cosine. Thus the nth power of this number is also ρ (cos φ+ i sin φ).

*We can go further depth applying ‘twice’ pentalogic to extract new p.o.v.s on C-Time numbers from ¬∆@st; which we can do returning to our case example, of a quantum triple state of field-wave-particle:*

*3D: S=T:** Geometrically, the nth roots reproduce points of the complex plane corresponding to the root values of the number ρ(cos ϕ I sin ϕ) lying at the vertices of the regular n-sided polygon inscribed in a circle drawn about the origin with radius modulus, and so rotated that one of the vertices of this n-sided polygon has argument ρ/n.*

**1D: @:** *The sinusoidal functions, π and its angle of aperture,* becomes the numerical value of the external membrane measured in terms of 3 diameters turning around with 3 apertures to perceive, from the self-centered singularity, giving us the general equation of dark matter for all systems (including the Universe):

3’14-3/π=0,14=±4,5%, with 96% of ‘dark matter’ outside the perception of its singularity.

So the angle of the point and sine function is its angle of perception of the central point or 1^{st} Dimotion of existence.

**2D**. The 3 apertures allow the point to absorb but also emit waves of information, often curling a minimal angular momentum/membrane/cellular quanta, h, for particles, that becomes the minimal boson unit of communication between 2 Fermions, entangling them as it reproduces its form over the quantum potential. This makes complex numbers in polar coordinates the best way to represent informative, perceptive wave communication between points.

**4, 5D**: Finally the polar backwards rotation, each new power a quantum stœp of time-space, represents perfectly a vortex of time-space, which is the meaning of the ih operator of angular momentum in the quantum scale:

So the nth powers mimic the trajectory of an attractive timespace vortex, and the worldcycle of its st constants towards the internal i-point singularity. The Complex plane become then a vital representation of and accelerated physical vortex: Notice that as X increases those stœps with each power, the attractive vortex becomes paradoxically smaller in its radius; as it accelerates in time, according to ∆-metric, making the vortex faster (Vt x Rs=K), and more attractive.

*So in 5D physics we model vortices of charges and masses, with complex numbers as convergent vortices with an ellipsoid ‘flattened’ smaller imaginary axis turning counter clock wise around a physical @-mind singularity where T=motion turns to zeroth; as observed in Relativity equations for a black hole & eddies with 0’-motion in its center.*

We can then understand Schrodinger’s & Bohm’s dual interpretations, in terms of those ∆+1 vortices (particle states: 4D, ∆+1-involution state) and ∆-1 quantum potential fields (5D entropic ∆-1 scale) that entangle through 2D wave communication (Schrodinger’s ∆º-present reproductive state): 2 particles share boson waves that upon arrival collapse into the particle spacetime vortex in any ∆±¡ scale, splitting its ST function at the vortex center. There the informative part of the boson becomes a still mental image (1Dimotion) and the energy part dissipating, scattered through the vertical axis (5D); which writes the Fractal Generator as the classic Fermion-Boson-Fermion equation. §-Fermion Vortex <ST-boson wave reproduced over ∆-1 $T quantum potential >§-Fermion vortex.

So the quantum unperceived non-local ∆-1 potential field of action at distance (as v=s/ti and with no information, v=s/0i = ∞) entangles the fermions in a first exchange of ‘trigonometric’ information (only angle or distance) likely with a Neutrino-antineutrino pair, and then in a 2^{nd} passing (neutrino theory of light), a ray of light is created over the non-local quantum potential field, which has chosen the best path to move (Fermat’s least time principle), offering the 5D interpretation of quantum physics adding Schrodinger and Bohm’s equations that put together the 3 states of its fractal Generator. Since, Bohm just *put Schrodinger’s hyperbolic ‘Cartesian S=T wave’ in Polar S-Particle coordinates, where the mind-particle state is the new point of view, and voila! it also appeared ‘magically’ the T-entropic field that moves the particle T>S, as in 5D we are all made of time-space so those coordinates ARE the wave state and the Particle state when we gift them the Galilean symmetry of motion. It is NOT a change of coordinates but of states from the spatial hyperbolic ‘present wave’ to its Past field>Future particle state-coordinates flattened in the i-direction. *

What is then the future direction in the imaginary arrow? Obviously that of a normal clock, down the conjugate is a future moving on a 90 degrees phase faster than the real and in the imaginary side a lag of 90 degrees in the phase of the world cycle – which physically appears in the AC circuit with the forward 90 degrees wave of the capacitive reactance, Xc on the conjugate and the lagging, 90 degrees of Xl the inductive reactance on the Imaginary side.

**$:** **Finally the last pentalogic use of complex planes is its classic forms helpful for vectorial calculus ****as** bidimensional real numbers (Bolay). If a and b are regarded as rectangular coordinates of a point, and the point is associated with the complex number a + bi, then the addition and subtraction of complex numbers corresponds to the addition and subtraction of vectors, i.e., of directed segments from the origin to the points with coordinates (a, b) and (c, d), since in addition of vectors their corresponding coordinates are added. The sum of two complex numbers: **(a+bi) + (c+di) = (a+c) + (b+d)i** is then represented geometrically by the diagonal of the parallelogram constructed from the vectors representing the summands. In this way, complex numbers are added by the same law as the vector quantities found in mechanics and physics: forces, velocities, and accelerations AND used to represent actual physical quantities, *which if we square the graph, become expressions of ST entities in the real x² coordinates and S or T entities in the Y-coordinates; yet as they are ‘two different species’ they cannot fusion together*.

**In the Unit sphere, the 0’-1 do not exist.**

A paradox that will irritate the dogmatic scholar but it is essential to the ‘dark spaces’ of a 5D Universe is this: In the circle and the sphere, the 0’-1 and 1-0 points of max. Length (time entropy or arrow of locomotion) and height (spatial information or arrow of perception) do NOT exist, as they are the openings for the 0,0 singularity focus of the mind of the circle, sphere or Riemann sphere, to move and perceive and receive information or energy from the outer world. It can in fact be considered as the most suitable model to ‘place’ all families of numbers in the different Planes of the being a closed circle or sphere NOT to have any of the N, Z, Q numbers, but be made entirely of irrational numbers which are infinitely more than the rational ones, which become the ‘osmotic pores’ of the sphere to the outer world. While the N, Z, Q family in the line (Q-line for short) represents the ‘lineal path’, which a given T.Œ will transit with its ‘fractal stœps’, *as all geometries are mental worlds, hence reduced according to species to what it sees. *

*Only the false hypothesis of a continuum and its false infinities – errors of the axiomatic method and the paralogic Kantian mind – which tries to fit all in a single pane and eliminate the needed ‘dark spaces’ and holes of any organic scalar plane of the fifth dimension, what makes huminds to postulate the completeness and continuity of those structures in the outer objective Universe, because they mind space compresses them. *

Let us close this introduction to 5D Cx numbers with a consideration on mental space and relative infinities, the themes of the next paragraphs, studying the main tool of 4D Complex analysis, the Riemann sphere*:*

In the graph, a simple proof of the non-existence of infinities, and continuities, as all perceptive points loose detail with distance and all mental spaces do not see the dark holes between points: The paradox of the Riemann sphere makes each point of the sphere represent a point of the plane, which seems absurd, since the ½ lower sphere is projected in a much smaller surface of the plane that the upper part that reaches till infinity.

But if we consider the north pole the @-informative mind gauging the plane as it moves to infinite it perceives lesser quantity of points and finally in its relative infinity all information on the far away points of the plane disappear, making the less dense further plane equal in density of points to the ½ closer plane/lower sphere.

**RECAP.** The complex plane can represent S(real)-T(imaginary) symmetries of creation by absorption of Energy from a real-spatial function, representing a body-wave, of an inverse, perpendicular particle state. This simple guidance clarify in mathematical physics, the meaning of complex numbers as parameters that express combined dimotions of ST-states beyond its use as artifacts of calculus (as in the quantum momentum ih operator).

**The opposite functions of S-**form and T-motion are orthogonal, because height is the dimension of still information and length the motion of lineal time locomotion. But information, perception is extracted in a still position from an advantageous point of view, in a series of Stœps, stops and steps beats, St->Ts->St->Ts, (as in a wave-particle motion), which in its extremal function become SS->TT->SS->TT beats (as in a feeding process that breaks the prey and reconstructs the parts of the predator with its entropy).

All those beats are thus represented with *orthogonal coordinates.*

And the advantage of the Complex plane are various: Both coordinates* are un*able to mix, so they maintain the 2 essentially inverse S & T functions they represent apart from each other. Further on imaginary numbers are inverse directions (signs) than its conjugates, which makes them highly antisymmetric, able to represent ‘temporal ± numbers’ as two inverse arrows of time. So this is the essential use in mathematical physics.

The i-plane thus represents with its √-1 shrinking, negative inverse parameter to the real plane, a lineal time dimotion, and as such the S-real plane and T-complex plane form a 4D representation of a 3S+T world, extensively used in physics (Minkowski’s spaces, electromagnetic flows, stream motions) to represent a 3D volume of spatial populations moving through an inverse parameter of time.

**The complex plane as the S=I, t=y plane.**

How can we express the space=form vs. time=motion parameters? As we know the properties of space and time it is easy to notice the parallelism with the perpendicular complex plane.

**5 DIMOTIONS AS OPERANDS.**

The simplest number systems, N, Z, Q are operated by the standard operations of arithmetic in a single ¡-plane: Addition, subtraction, multiplication and division, which must be studied together with those families.

They suffice to portrait a realist view of the world in a single plane, without ‘finitesimal parts’ and wholes which add the calculus operands, and angular reductions and perspectives which include trigonometric functions. So polynomials are good enough as equations that use only those operands and families to define scalar systems in a single plane of entropic Cartesian space-time. While probability defines with those simple operands the Universe within the 0’-1 probability time scale.

*So as humans have expanded their understanding of the Universe, they have also evolved its entangled mathematical mirrors over it, whose geometrical, spatial forms (points) scalar forms (numbers) and temporal operands are messed together in Planes of complexity. Even if huminds *that have as in most fields of knowledge merely acted in a mechanical manner, ignore those symmetries as *number families mirror, polytopes in space and time operands in algebra to portrait the symmetries and sequential events of simultaneous **real** organisms*. So 5D gives experimental scalar space-time whys to those symmetries.

**PENTALOGIC CORRESPONDENCE OF 5 ACTIONS=DIMOTIONS OF EXIST¡ENCE WITH OPERANDS**

The study of ¬Algebra operands as expression of the 5 Ðimotions of Time§pace brings a vital outlook to the abstract operations of ¬Algebra, starting from the less understood of them all, the sinusoidal functions of mind’s perception, ending in the ∫∂, most versatile, for both the analysis of Dimotions =change & Planes (finitesimal derivatives & integral wholes) hence the most fruitful in mathematical physics. While the negative inverse operands represent the destructive entropic 5^{th} dimotion that affects all others. So negative/inverse operand balance positive ones, restoring the 0’-sum balances of all S=T events and systems of reality.

Those operands do NOT have a one to one correspondence to each of the Dimotions of space-time, since each of them is in itself a pentalogic operand, which can entangle with other operands and dimotions, in increasing degrees of complexity. As we shall not cease to repeat, the basic feature of reality is to be an entangled game of 5 elements, which are themselves ‘fractal’ in its nature.

So each T.œ regardless of specialization in the larger ∆+1 world performs internally 5 dimotions to become in itself a ‘whole’ being made to the image and likeness of the total fractal, mind of the Universe and its ilogic structure. So while certain operands are clearly more useful for certain dimotions, all of them can be used to a certain degree of accuracy to ‘reflect’ a mirror image of the 5 dimotions of exist¡ence. Still it is clear some specialized correspondences of those operands with the 5Dimotions and ¬∆@st elements:

**1D:@:** 1^{st} Dimotion operand of perception for a self-centered in the point are sine/cosine trigonometric functions, which allow to measure the 3 ∆st parameters distance=motion in spacetime (S=T) and angle *the scalar Dimensionless factor.* This poises a question: can a system perceive more than 90º?

Obviously with more than one eye (more population in space), or a better rotation (more motion in time); two S=T strategies. So systems have two eyes to have more or less a 180º aperture to the world.

**4D:** ± sum-subtraction create/reduce 4D social herds; which *become in a given volume denser* superpositions of elements of the same* species, which are identical in motion and form. So only equal Dimotions can be added or subtracted.* So a sum of numbers that are social groups it can represent also a scalar social evolution.

**It is related to Natural, Social Numbers in space that **define regular ‘points’, which are undistinguishable, as societies in regular polygons, where prime polygons have the property of ‘increasing inwards’ its numbers through reproduction of vortex-points (n-grams), studied in Theory of Numbers. So a Natural number in its geometric interpretation is a ‘cyclical point’ of regular ‘unit-points’ of growing ‘inner dimensional density’ a point with a volume of vital energy and information, a fractal point.

**∆-Dimotions. **We ascribe the more complex dimotions of 3D reproduction of social d=evolution of parts into wholes, which do enclose in their actions the other 4:

**4,5D**: ∫∂; analysis, ‘transcends’ and ’emerge’ between Planes, in NON lineal processes proper of ‘longer time’ Nature, best served by integrals and ODEs.

**3D: X, ÷, reproduction & social evolution vs. entropic division.**

The product has multiple correspondences, both as an aggregation of herds (4D), a reproduction that merges ST-components and a calculus of the lower ∆-1 parts when two similar elements entangle at the ∆-1 level as most reproductive processes do (i.e. you reproduce by entangling your lower scale of genes). Since ∆o (S x T), where S is a set of s¡-1 elements and T a set of t¡-1 elements, IS the number of ‘bijections=axons’ between s¡-1 and t¡-1 parts (we translate the modern jargon of mathematics to ÐST philosophy of stience so anyone who understands 5D can understand all disciplines).

The product/division rises the complexity of operands a first layer, and serves the purpose, besides the obvious sum of sums, of calculating the margin of dimensions, as combinations which are not purely parallel between clone beings, most likely through the recombination of its ∆-1 elements, as the product of 2 Sœts inner elements give us all possible combinations. I.e. 5 x 4 = 20 IS also the number of connections between all the 5 elements and 4 elements of both sets. So multiplication ads either a dimension of multiple sums in the same plane, or probes for the first time in an inner scalar dimension.

While the inverse division operates inversely reducing ∆-1 parts into a smaller number of wholes. So Q-numbers are dual. They might be entropic numbers breaking of wholes into parts to feed a group, or gathering parts into lesser wholes. Since as we have seen *the 5 number families also represent the 5 Dimotions.*

So the key new function of products/divisions is reproduction, carried in 2 Planes:

The Function of existence, Max. SxT (s=t), which merges two different parameters, maximizing its *connections on the lower level of its organic parts. As the product means then the number of ‘axions’ connecting them, which is maximized when the parts of both ‘sets’ are equal. *So, if we have 2 points with 4 internal sub-elements, the product 4 x 4 will give us the number of maximal connections, when we put each internal element in connection with all the elements of the other sub-set, which will be larger than between a 5 & 3 elements. Thus ‘gender’ reproduction requires similar forms.

** 2 Ð locomotion** is best served by the ± & x operands, as a sum of ‘steps’ that also can be calculated with an S x T product, where the T represents lineal time motion as the sum of all those steps through its product… even if the memorial tail of the particle imprinted fades away with distance. Since a ‘lineal motion’ is a ‘sum’ of ‘frequency steps’ moving in the positive direction, or a negative number, moving in the inverse direction.

Thus unlike *spatial populations of* Natural Numbers, N, Z *adds temporal negative numbers which are inverse dimensions that close a worldcycle of simple locomotion. *

The product also represents, 2D: Lineal Locomotion, as a frequency time product multiplied by Spatial steps. And again as T=1/ƒ, the inverse operand, ÷ can be used such as V=S/T = l(s) x ƒ(t) .

Negative inverse Dimotions express also in the duality of expansive entropic vs. implosive informative warping in more complex formalisms of Locomotion, as in Minkowski’s space where time, -ct has a negative, i-number that ‘warps’ lineal space into a wave of high information, shortening the distance: D^{2}= S(x+y+z)^{2} –(ct)^{2}.

**So**, the product as an operand, sum of sums, can synoptically account for a ‘long period of locomotion’: S=vt.

** 3D: log a, xª: reproduction: exponential vs. logarithm**

As we can see ±, x÷ and* log a, xª as 3* scalar ∆±1 degrees of growth, exponential equations, powers and logarithms also model social evolution and reproduction in its maximal ‘dimotional scale’ of form (∆+1) , with an exponential curve in case there is infinite energy. Yet in reality we do so adding at the end a logarithm curve (logistic curve); as reproduction saturates a system. It is then a proof that reality seeks always a ‘balance’ that the REAL curve of reproduction is a combination of both, exponential and logarithmic curves. So from the initial unit to the carrying capacity that DEFINES a ∆+1 social group growth ‘reaches equilibrium’, becoming a herd or super organism emerging in the new upper scale. The reproductive curve thus rises the seed to the 1=K-organic whole (in a herd or growing organism in) through a balancing feature: the existence of an inverse for all those operands that in the positive transforms the ∆-1 being into its self-similar ∆ø *accessing a new plane of the fifth dimension*, and in the negative side re-establishes the balance of reality as a zeroth sum*. So the logarithmic part of the graph is the inverse function of its exponential, that slows the growth of the system*

** In the graph, the logarithmic function **has as derivative an infinitesimal, 1/x, which makes it best to model the curve of growth from o to 1 in the emergent fast explosive ∆-1 palingenetic worldcycle state, while the inverse e*ˆ-x model the decay death process.* Polynomials do not evolve reality towards an impossible infinite growth. They are the inverse decay process; which can be understood better observing that the inverse function does in fact model growth in the different models of biology and physics, limited by a carrying capacity straight flat line.

And depending on which is the base of the logarithm we can trace different social growths and inverse d=evolutions:

**4D-5D:** **log a, xª: social evolution in decametric Planes and entropy decay in negative exponentials.**

The 10^{10} decametric scale is the social scale of the Universe, as the Tetraktys is the simplest, perfect social system, with 3×3 physiological networks, and a 10^{th} element, the ‘black ball’ communicating them as the ‘single Unit’ of the larger whole – from armies of 10 x 10 captains, to 10^{10 }ties in DNAs, stars in galaxies, galaxies in Universe. Powers of 10 as Eames’ movie show thus create new wholes. While a negative exponential shows the rhythm of decay of the system in the fastest event of reality, death, maximized by the e-function, which we just showed to be the fastest negative growth.

While infinite energy happens in death for the predator that feeds on it. So the ‘end=action’ of feeding/dying of entropy is described by *negative exponentials. So infinite growth and death are two sides of the same coin. *

A negative exponential will show the rhythm of decay of the system. And in this case there is no need for ‘a logarithmic’ limit, since for the predator the death body is ‘unlimited ¡-1 energy’, though once the ‘relative infinite’ number of its ¡-1 parts are absorbed the ‘e-function’ is cut off.* 4 ^{th}, 5th dimotion: ∫∂.* Finally Integrals and derivatives which have a slower growth, than polynomials and its 3 Planes of operandi model better the specific nature of those ‘indivisible’ finitesimal quanta of a system, (1/x) with derivatives of different ‘functions’ and its inverse integral, organic growth into the ‘wholeness’ of a simultaneous organism integrated in space.

Thus integrals show social growth into new ∆+1 5D planes with far more detail and finesse than polynomial approaches (through Taylor series). Its graphs are a curved geometry, which takes each lineal step (differential finitesimal) upwards into a closed, curved sum that creates a new whole (Galilean paradox of small lineal steps adding a closed 0’-sum with so many interpretations for each science, including our sense of freedom as individuals enclosed in larger ordered societies),

1D: Finitesimals in can also as minimal parts reduce a vital energy to its minimal singularity point-center, @, that might store the image of the whole, shrunk into still, mind-mappings of information, within its particle-head.

The all pervading use of ∫∂ is then clearly because it reflects ALL forms of change. And so analysis is the most extended sub-set of ¬Algebra.

**The sequential growth of actions=operands=syntactic dimotions of any language. **

**We shall find then that the mirror of the 5 dimotions expressed as operands follow the same degree of increasing complexity, best understood with a ‘historic’ analysis of the evolution of mathematics and ¬Algebra, which was first without even being aware of it, the ‘angle of perception’ of the humind oculus opened to the whole with a narrow view; as language in time started also with a single A-vowel in the Abkhazian but when the Caucasians learned the other extreme of ‘pure form’ and said ‘u’, they didn’t forget ‘a’, and soon their i-magination came with the i-nsolite i-dea of ‘I’. So from the ‘a’ languages we moved to the Semite languages and it’s a-i-u vowels, and it was soon thereafter when the Camites and Basques and Altai people of the Korean-Japanese languages had it all what is needed to say ‘aeiou’.**

**The growth of dimotions of existence starts with an angle of perception, ‘imitation of life’ that starts with eyes opening a narrow view. Thus the 1**^{st} action is NOT a full dimension– the angle has none, but a finitesimal image.

^{st}action is NOT a full dimension– the angle has none, but a finitesimal image.

**Entropy is on the mind of the lonely ‘1**^{st} form’ emerging in reality, as the child born to think he owns the world, breaking all things he touches, the first experience, the first ‘a-langagæ’ full of force, and then comes the other extreme which becomes languagæ. The first species is a lineal top predator; the first form born in the galaxy the top predator black hole, (and the last to die, as the old man cares nothing like the child but for food)…

^{st}form’ emerging in reality, as the child born to think he owns the world, breaking all things he touches, the first experience, the first ‘a-langagæ’ full of force, and then comes the other extreme which becomes languagæ. The first species is a lineal top predator; the first form born in the galaxy the top predator black hole, (and the last to die, as the old man cares nothing like the child but for food)…

**And so existence in its curves rises from a to u, from au to aiu, from aiu to aeiu, and finally comes ‘oh’ the almighty realization life will reproduce and be boorn, again.**

**Every act of creation thus follow the patterns that followed the 5 Dimotions of existence, and for that reason we can study also the ages and evolutions and patterns of growth in dimensionality of any system of Nature, real or virtual, physical or mental, with the same sequential laws.**

**We study then in our papers on ¡logic evolution those steps from 1 to 3 to 5, filling then the voids, and becoming even more complex with subtle æ, œ variations – but 5 indeed suffice to have a truly full view of the whole, which ants, physicists and huminds ignore…**

**The order of dimotions and operands is thus clear, as man first inform itself with an eye-angle on its trigonometric function, measuring distance and height, bringing to the mind the holographic flat first functions of the au, length-height, motion and information minimal view of reality acquired with the simplest operand of an open tri-angle looking at the void. **

**And then he saw around himself and look at the mirror of other mirror-minds, members of his family group or social whole, call it atom vibrating into a network of parallel beings, a bird in a nest, a pup in the basket, a grass leave moved by the harp of wind, hearing with its leaves the sound of other leaves, to know in its chemical mind-roots, he was not alone. And so from dimensionless angle, that of the fractal point alone, looking at the whole came the social group in space, the motion steps in time, probing the ever growing horizon with the new actions-operands. **

**So happens in all systems that first perceive and then move or socialize with other clones. <+ became the dual form. But angle had only one ‘logic time arrow’ that of absorption of pixels to create a form, ever turning around the sine and the cosine, reversal forms of each other, in smooth cyclical, angular momentums, with the pi-openings of an imperfect 3’ – the triangle within the sphere, rotating through those openings to the world. The sum was though dialectic, dual, and once the natural line was extended, each number a sum, synoptically put into a single symbol: X(N) = ∑1s… **

**It was then a marvel of the perfection of simplicity that it took 10,000 years more to learn that ‘negative numbers’ could exist from Jericho to Cardano… And even today the complex root of minus one is an ignoramus for the philosopher of science regardless of the praxis of its ab=use in pedantic science. **

**Sum thus became more complex and had its entropy (which in the eye and the angle was the shutting of the pi-holes, light and darkness, black and white, suffice indeed to form all the 5-0 shades of grey). **

**So from the angle of an open triangle, inscribed in a sine/cosine, turning wheel we moved into the dual uses of the sum and its negative inverse ‘numbers of time’. As the sum could account for motions in time, positive and negative in directions, and for the social evolution in space of growing and diminishing clone forms: 1D angle -> 2D + 4 D.**

**It must be noticed though that because a polar duality is necessary to start the game beyond ants, physicists and Parmenides, or else you will find the unmovable whole or the absolute entropic motion of a dying Universe in your single finger one-hoofed beasty thought. **

**But the explosion of meaning to complete ∆st trinity in a single plane and penetrate 5D entangled depths will only come with the multiplication that finally reached the 3 fundamental elements of reality, social growth in space, as a sum of sums, which included the sum; scalar growth as a factor that kept the relationships between the parts increasing its volume till reaching a whole; and penetration into the parts, multiplying the 1-whole and its social group by them, and merging them together into reproductive axons; alternating the reproductions of multiplication and the divisions of mitosis to ‘grow and multiply’; that is, and God say: +++ < XX > ÷; and the son was born.**

**The product becomes a self-sustained process of creation, which sufficed in itself to create a Universe; unlike the angle that needed the external world to connect the ¬fractal point. And since 3 is the number for a rounded whole in scale, topologic space or time motions, the power and logarithm completed the 3 elements of polynomials but Fermat and Galois would teach that the holographic, ST, pentalogic ¬∆@ST Universe has its limit-ations. So as we shall see with most operands and equations nothing matters beyond 4 as 5 is the entropic destruction of meaningful form. So you cannot solve a polynomial, the simplest equation that operates those 3 inverse elements of social growth, merges and acquisitions as you cannot resolve with radicals – that is obtain from the parameters of the system the roots of any equation beyond the power of 4 (Abel), unless you take the X to be the 1, the whole in itself. **

**But you can re-enter a plane of existence, reducing a whole to its derivative, X**^{n}= n-1 X^{n-1 }and you can emerge into the whole by integrating your parts across time, space or scale with single, dual or triple integrals. And so with the discovery of calculus which included in its equations all other operands, as all could be ‘derived’, the whole evolution of ¬Algebra to mimic reality was complete, along the solution in praxis – not in meaning – of the i-maginary √-1 root… Neither derivatives could go beyond the 3^{rd} for a real meaning in time: y’: speed, y”: acceleration, y”’: jerk…

^{n}= n-1 X

^{n-1 }and you can emerge into the whole by integrating your parts across time, space or scale with single, dual or triple integrals. And so with the discovery of calculus which included in its equations all other operands, as all could be ‘derived’, the whole evolution of ¬Algebra to mimic reality was complete, along the solution in praxis – not in meaning – of the i-maginary √-1 root… Neither derivatives could go beyond the 3

^{rd}for a real meaning in time: y’: speed, y”: acceleration, y”’: jerk…

**And that was all folks. No more operands, no more dimotions are needed, in the finitesimal, finite ****∝**** Universe, as the angle of perception will be shut off at certain distance in space, time and scale.**

** **

**+: **** NATURAL NUMBERS. 0′ V. ****∝**

# 4th dimotion: social sums in space – inverse motions in time. Democratic, social Universe.

The simplest social form is a herd of undistinguishable elements, *a number, and its operand in a single plane, a sum, *which requires *equal species* to add with its superposition principles, in herds of a larger scale.

*A number already includes a sum of ‘1s’ and as such Natural numbers and sums correspond to each other.*

The sum/number is the key operand for the simplest ∆§cales of social groups, in humans as in most systems, divided in tetraktys of 3×3 sub-systems/networks and one to integrate the whole as new unit of the ∆+1 plane, in a decametric form (bilateral 5s, so to speak).

Social growth increases then by sums of sums, which can be expressed with the product and products of products expressed through exponential logarithms, *which in true form must be considered a ‘species’ of product in which the elements of the product as in the case of the sum are indistinguishable. So the proper connection is that of sums and exponentials while the product represents multiple functions*, notably the reproduction of form.

**The negative function/number.**

As the Universe is always a zeroth sum in balance, sum has its inverse operand, subtraction, and a natural number=sum, the negative number. This in mathematical jargon gives us the neutral or identity number element, the ‘ zeroth’, which will require latter a deep analysis in 5D as we did with the relative infinite ∝.

Yet the negative operand is more profound than the sum *since while the sum is positive as the natural number, and it is related to a ‘spatial population’, a sum of forms; the negative is a time function, as there are no negative pairs, but negative directions inverse to positive motions*.

This subtle difference will explain in physics some errors, from the negation of faster than light speed, *which is possible albeit in an ∆-4 plane, as the only ‘prohibition’ is the existence of a negative number, which means a negative motion of a mass ‘vortex’, which in ‘negative’ expands instead of imploding, loosing curvature, as it accelerates beyond c-speed as dark entropy/energy/expansive gravitation.*

*Again negatives arise* in the duality of particles and antiparticles which appear with negative time motion to the past; which again merely means an inversion from a positive informative implosive particle to an entropic, dying expansive particle. We could multiply examples, the most obvious moving right and left in a vector or speed. So we define ‘negative numbers’ as inverse dimotions of any of the dual *dimotions of our 5 Dimotions of existence:*

*+information vs. – entropy; + motion in a ‘chosen direction’ vs. – motion in the inverse direction; + implosive vortex of attractive force vs. – exploding vortex of expansive force. And so on.*

Inverse operands are misunderstood because unlike the paradoxical ¡logic Universe, the humind is @ristotelian, with a single arrow ‘positive selfish arrow’ A->B – that of each specific human dimotion. So the B->A is feared as in death, or misunderstood as in explosive vortices of antigravitation (Dark entropy) or antiparticles.

And we are spatial, visual minds, so time complexities fully escape beyond our lineal time view.

So generally speaking time inversions are served by the negative function. I.e, a negative spin just has the inverse orientation, a negative coordinates just means to move in the other direction. Negative operands thus are most useful for time=motion related systems; *and for all inverted, negative entropic destructions.*

*The ‘negative vs. positive inversion’ is not only connected to an external locomotion. As all dimotions need a neutral point in which a ‘time dimotion changes state from + to -‘; which is the 0 point that means not only an empty ‘set’ of population in space, but a ‘state of no dimotion’, when the time change does not happen. *

More profound still is the concept of 0 as the discontinuous stop/perceptive state. *Since reality is a constant stop=perception and step=motion, where the stop state is the informative 1Dimotion when the fractal point might change the next dimotion switching to other action of existence, themes those *for a representation of the program of existence in Boolean ¬Algebra we escape completely for ethic reasons.

For the simplest motions as any physicist knows the duality of space=stop and ±time-step, is a constant -1, 0, 1 whereas the 0 function in an alternate or vibrating SHM halts motion at maximal distance and 0 speed, but often the system releases or absorbs the ∆-1 micro-points of its informative/feeding scale.

It is possible to combine ST actions of spatial populations and time change, which can in this case be negative with the diminution of populations, so -3 obviously is a diminution of a quantity whereas T will always be smaller than S: T<S. And as all systems are 0’-sums that reduce their populations ultimately all populations in space will be subject to a negative growth in time. But in short time spans a population might be lineal only in a direction of growth.

It is then important to understand the existence of one-way dimotions and short time spans vs. 2 way dimotions, where both ± operands *make* sense and always happen in long time spans. Because huminds do not properly distinguish the different ST dualities, they get confused when trying to consider a negative operand for spatial forms (what is a negative apple? nonsense) while there are always negative ‘directions’ for temporal motions (what is left and right motion?), which they clearly all understand.

So the next question is: can the sum and its inverse negative number perform trilogic and pentalogic functions? It is an interesting theme, because being the simplest ‘operand’ and the only one that does NOT penetrate into Planes of the fifth dimension and requires an ‘equal’ parameter for all the summands, it might seem not to be able to do so, but it can represent them *all for an external observer without detailed analysis of the internal processes needed from the complex scalar dimotions of perception and reproduction. However, and this is a fascinating rule of all systems in 5D, there is a ‘blind spot’, in which the entity finds difficult to perform properly a task, which is the inverse task to that for which is best suited. *So as in the case of the ‘angle’ of perception, we require the external world to entangle the action and achieve it merely with the sum/negative number:

**4Ð:** Social evolution into herds, which is self-evident the dominant element of the sum **in ‘space’.**

**2D:** Locomotion as a sum of equal steps measured by the frequency of those steps **in sequential time**

**3Ð:** For an external observer, ‘magically’ by adding numbers a population can grow ‘in time’, expressing the 3rd Dimotion of reproduction.

**5Ð:** Negative numbers can then express the 5th Dimotion of entropy, as expressed above.

**1Ð:** Perception is the less natural Dimotion for the ±, its blind spot as it implies the decrease of size in space (as opposed to social growth, 4D, the natural dimotion of summands) and *a distinctive nature for the different pixels of the perceptive being, which makes difficult to consider it.*

**I.e. to ** color a map we need 4 Different colors in a plane (a topological question, studied in 5D geometry), but if all were equal it could not measure perception. However if we introduce a ‘higher concept’ – a pixel, we can increase perception by increasing the pixels of the picture but it won’t be so much perception – the dynamic event of the observer, but of the observable.

This constant 4 vs. 5 tetra vs. pentalogic view of reality implies that often a system ignores the 5^{th} entropic dimotions for which it does not code at all, but will be externally provided courtesy of the Universe. i.e. we have 4 genetic coding letters (but a 5^{th} nucleotide for more enzymatic, destructive coding), 4 quantum numbers (but the principal jumps to increase energy by entropic feeding to a larger orbital).

Death=entropy is thus the fundamental blind spot of all dimotions/operands, but for physicists fond of its entropic big-bang theories of reality.

**Social addition by holographic superposition of equal forms vs. Informative molding.**

‘Natural’ N-Sums keep the individuality of its ‘ones’ in the system as parts of wholes, which are herds, even if they imply a fusion of the volume of space-time the system occupies by superposition.

The wholeness of which is often on the eye of the beholder, introducing the theme of the subjective observer vs. the objective observable that holds a higher truth. I.e. waves are superpositions as herds of smaller forms; even if the humind considers them ‘single wholes’.

**RECAP**. The simplest entangled representation of reality through ‘basic operand’ comes through the duality of addition and subtraction, and its attached physical meanings of superposition and fusion of ‘parts’ into ‘whole numbers’, or its entropic inverse operandi of negative subtraction. While as all operands it has an inverse function to maintain the balance of all the actions of the Universe; and a blind spot, not coded – the inverse of its dominant social dimotion – informative perception.

Moreover addition can happen in sequential time or adjacent space, forming growing probabilities or populations. So as the simplest mode of operands extends its diversification through space or time it will mean different things.

**– : THE NEGATIVE FUNCTION: Z NUMBERS**

It follows that negative numbers are the first inverted operand required to balance the Universe. Its interpretation is then immediate as the entropic inversion of a process of social growth; but and this has been a source of much confusion, they can be placed in the same parallel orientation (Z numbers) or in the orthogonal perpendicular orientation (complex numbers), which are very different propositions. And in both cases there are limits to what dimotions they can represent. Z numbers in principle are NOT required for the inversion of the simplest sum of spatial populations, for which the negative symbol suffices, but subtraction ends at 0’, as negative populations do not exist in pure present spatial terms – only as a projection of future negative time (Debt, etc.) So Z numbers are only pertinent for temporal processes. Further on, they are written in lineal time, which is not suited for cyclical time processes, which are most long term Deep time events and worldcycles. This leaves the range of Z-numbers use constrained to lineal time motions, 2D locomotions, which are not that interesting despite ‘enzyman’s’ magnification of it, due to its obsession with the construction of entropic weapons and transport machines.

We shall thus not dedicate much work to their study, in this by force brief introduction to 5D ¬Algebra.

**The beats between ±information and entropy states. **

The function of existence has in Existential ¬Ælgebra a series of potential beats…

We distinguish ∑ as the sum symbol for herds of equal beings, while ∏ will entangle networks, and one will transit to the other, since addition by superposition in ever tighter spaces of similar clone species, the simple ¬Algebraic expression of social dimotions, finally collapse into a tighter network *once *no more units can be added, and vital space must be ‘cut-off’ creating form, causing an essential ‘Stœp’ rhythm of Social reproduction and growth of cells, informative molding and partial loss to increase information, which is why both are so often connected.

I.e. the sculptor forms by eliminating waste of material; the hand is formed by killing the cells that were between fingers; evolution of form happens always in a diminishing form; so the Homo Floresiensis was a hobbit but had the first complex form of the head, and all new species diminish in form.

Which introduces the opposite functions and entangled beats of the Universe as the aforementioned Fast Radiation of species/cellular populations in space vs. slow diminishing/evolving form in time.

Both in its positive 5Ð v. negative entropic 4Ð whose addition of decaying ¡-1 T.œs is so fast that it is expressed by a negative exponential growth, which in this manner would complete the 3 ‘Planes’ of addition: +, x, x^{n}

If we consider an event’s worldcycle, 1, lesser probabilities represent parts of the whole event. If we project it into space it will be a population of similar event, entering the region of maximal frequency. Both will be mathematically projected as a bell curve. S=T. Same function for the addition of events and populations, in time or space.

The Universe is simple in its original principles, made complex by the differentiation across the symmetries of scale, topology or time. Indeed, something so simple as the sum and inverse subtraction IS still the most important operandi of the Universe, which gives us new numbers, social gatherings of identical beings, which herd together into parallel flows adopting most likely a bidimensional ST superposition on laminar states that keep adding the 3rd dimension of the being. Like the simplest first masterpieces of Bach, the architectonical Universe is a simple principle before organicism twists its form, in which beings, which are equal come together.

Superposition of bidimensional holographic fields is so important that the whole of quantum physics is based in this superposition principle. The sum thus is still the master of operandi. But for sums to happen, the beings *must be externally identical, to be perceived as parts of a quantified mass, each of them the same value. Addition thus is the ultimate proof of the social nature of the Universe: **The sum is a social action: Sums form herds.*

Once we defined an operation and its properties, we study how they mediate the actions of beings. And it is clear that the first operation sum, acts on the first form, social numbers, to form growing §ŒTs of social numbers…

So paradoxically the more complex action, starts with the simple operation, the sum of individuals into herds of formal numbers.

A locomotion is represented as a ‘lineal motion’ by a simple ‘sum’ when moving in the positive direction, or a negative number, when moving in the inverse direction; whereas the positive direction is the direction towards ‘energy’, and the negative direction must then be the inverse direction of the ‘arrow’ of information. We can see this also, as an expansive direction (energy feeding) and an implosive direction (information warping), concepts of importance, as information ‘warps’, and since height is the dimension of perceptive information, a negative, shrinking number, i, will define often in mathematical physics the warping direction of information (as in Minkowski’s space, it).

However in the entangled, fractal Universe each element can be entangled with all the other elements (fundamental principle of pentalogic). So we canals add frequencies in time (+, T), social elements (5D, +), points in a single plane, and groups of social elements to give us new decametric Planes. And we can ad reproductions as sums of identical beings (3D, +). And we can count the decay of a point into its internal parts as a sum (4D). Thus we can build with only sums a mirror image of 4 dimotions of reality.

**The problem of the i-number: negatives and roots. Proper and improper inversions.**

As a general rule*, negative numbers exist when they are a direction of motion, not a quantity or volume of space.*

So happens with roots, which only should exist for symmetric holographic systems of space-time S=T, for bidimensional entities and regular numbers as forms.

And alternately they Do happen in many cases in which mathematical physicists discharge them as inverse arrows of time because they ignore the inverse 4D vs. 5D arrows, as in Einstein’s equations, where they customary discharge negative hyperluminal solutions that do happen in the larger ∆±4 dark world of quantum potentials (entanglement, pilot-wave theory, etc) and intergalactic space (faster than light neutrino background, gravitational waves, action at distance, red shift of light, etc.)

We study this themes in number theory where we define the different types of numbers, so no need for further info here. Instead, we will make some comments on… the limits of classic Aristotelian logic based precisely in a single arrow of time, which has so much influenced as Euclid pretentious axiomatic method (full of holes and new axioms, postulates, notions and various errors of the §@-humind), our ‘underlying’ a priori categories of the mind.

At the basic level of arithmetic’s the problems that still linger in mathematics and by extension all sciences, which use it as a mirror, are the concepts of negative and square numbers, the inversions of the positive and quartic equations, on the real number, related to the two fundamental unknowns of humind science – the fractal scalar ‘infinitesimal nature of space’, and the ‘proper and improper inversion’ of the fractal generator.

Negative numbers and roots do NOT exist for certain type of space-time events, p.o.v.s and ∆s≈t structures of reality but are an inflationary excess of ‘information’ proper of all languages that in the isolated mind space, without entropic, simpler restrictions can ‘escape’ into imagination: informative software systems, without direct contact with the hardware that limits the possible paths of information multiply kaleidoscopic inefficient combinatory forms, ‘free’ of the constrains the vital energy they must ‘shape’ accounting for its simplified rules in the external world – in a drawing of 2 still dimensions we can make any figure – in a 5D Universe only certain ones ‘stand’.

And this is the fundamental need of mathematics that force it in search of meaning as per Lobachevski’s pangeometry and Gödel’s ¬Algebraic theorems, to recur to experimental science, to discern once and for all what are mathematical fictions and what are mathematical reality.

‘I know when mathematics is logic but not when it is real’ then applies in this case to the unnecessary attempts to draw complex functions in 4th dimensional space, when they are mostly in its imaginary parts functions of motions in time, best operated in mere ¬Algebraic methods.

**Odd=negative and even=positive functions.**

When we move from simpler operations to more complex functions, the concepts of positive and negative numbers carry to positive functions, which are symmetric both sides of the 0’ line and negative functions, which are inverted passing to the negative side, and so the concept of an inversion of Dimotion takes effect with clear implications in the real, physical world.

Let us consider an example with wide implications in quantum physics, the fact** t**he duality between antisymmetry and symmetry happens also in the (antisymmetric) vs. even (symmetric functions), which is in mathematical physics the basis between social evolution of particles (bosons) vs. antisymmetric annihilation (fermions), so it *has an immediate never quite clarified application to physics.*

One of the basic questions in the theory of numbers concerned the divisibility of one number by another:

if the result of dividing the integer a by the integer b (not equal to zeroth) is an integer, i.e., if a= b • c (a, b, c are integers) then we say that a is divisible by b or that b divides a. If the result of dividing the integer a by the integer b is a fraction, then we say that a is not divisible by b.

Questions of divisibility of numbers are encountered constantly in practice and also play an important role in some questions of mathematical analysis. For example, if the expansion of a function in integer powers of x:

is such that all odd coefficients (with indices not divisible by 2) are equal to zeroth, i.e., if

then the function satisfies the condition: **ƒ (-x) = ƒ (x)** – such a function is called an even function, and its graph is symmetric with respect to the axis of ordinates. But if in the expansion (2) all the even coefficients (with indices divisible by 2) are equal to zeroth, in other words, if:

then **ƒ (-x) = – ƒ (x)**

In this case the function is called odd, and its graph is symmetric with respect to the origin.

Notice y(s)≈x(t) is an odd function, a deep metaphysical fact, which implies S and T are in different ages of time.

If we consider the 2 functions of the 1^{st} Dimotion of angular perception, the sin x function related to the height dimension of form is odd, and the cos x function related to the length 2D¡ of locomotion is even:

This duality has a deep implication in the physical Universe (Pauli exclusion principle), which can be resumed in a simple thought: functions which switch between ± temporal motions, something quite common in all type of alternate currents, but do NOT change for that reason their wave form (even functions) obviously add by superposition its populations.

Functions which alternate the direction of its time motion (from positive to negative sign) and in the process invert its wave form from + to – energy or in terms of S vs. T invert its dimotion from entropy, locomotion, to information state, annihilate each other. This is the case of a sin x function where the states change from motion to information. While in the cos x function, the wave merely switches motion but NOT state.

So in even functions, changing sign doesn’t dimotion; in odd functions opposite Dimotions annihilate each other.

As this is not wanted – no T.œ wants to die, somehow the particle-mind state will by all means prevent their body/waves from contacting each other. In the next graph we see the case of even functions. Any chemical student can find the case of odd functions where the electronic wave afraid of annihilation, acting as the body state of the particle is the furthest away from the other.

The reader should easily in any case interpret the results in terms of dimensions and the holographic principle. And it reads like this: symmetric functions are holographic even functions, which means for example they can be ‘superimposed’ as they are bidimensional ‘sheets’, waves etc. While odd functions are even in dimensions, and as such they cannot be superimposed.

This means essentially bosons are bidimensional and fermions are tridimensional, and this is a huge advance for quantum physics – recently proved by an experiment that converted a photon, initially a boson into a tridimensional form and ended its boson conditions. Accumulation of bosons into a single point of space-time then means merely it is superposing the bidimensional thin layers into a third dimension of height which is indeed what we see in pictures of boson states.

Its representation in both ± sides of the plane works as follows: the axis of ordinates plays here the role of the ‘asymmetric’ state which has split into the ± inverse directions but can fusion again both. Or in terms of relative equality, those functions DO have their position in space (established by the lineal coordinates used to represent lengths in space) different but their form (represented by the Y-coordinates used to represent form and information height dimensions for most functions) identical. So they can according to the rules of the fourth postulate of non-e, communicate as they can share identical information and match each other in symmetric peg.

While the antisymmetric function is +- -+; that is, inverse both in the Y-nformation and X-pace location is dissimilar both in form and motion, momentum and position, you name it… whatever 2 parameters we use to compare both systems. So they enter into a Darwinian annihilating process as they cannot match each others form.

**The 5 Properties of ± operations. Its trilogic **

**It **is then immediate that for the fundamental dimotion of the sum as an operand its properties define a social democratic herd of undistinguishable elements, and those properties carry to the properties of ‘numbers’, as they exist with the inclusion of ‘the sum’ within them.

So the standard properties of the sum are:

- Identity element: There exists a number zeroth with the property a + 0 = a for every a.
- For every number a, there exists the opposite number x satisfying the equation a + x = 0.
- For any two numbers, their sum is uniquely determined.
- Addition has the associative property: (a+b)+c=a+(b+c)
- Addition is commutative: a+b=b+a

5 properties immediately smell to pentalogic. We shall not follow as we never do a full rigorous consideration of them all, but it is obvious that in levels of complexity we first talk of the ± wholeness monologic expressed by 2 (x-x=0), where both operations are included.

1.2 The two first properties thus are related to ‘time sums’ of ±directions; the last 2 to ‘space sums’ of indistinguishable populations; and the 3^{rd} seemingly trivial, is of importance to philosophy of science. *But in the scalar Universe, everything is ‘deeper’ than it seems.*

- Consider the ± inversion, which in the ideal world of 0’-vacuum is evident, as the 2 inverted arrows of the 5
^{th}dimension, entropy=dissolution of wholes into parts and social evolution of parts into wholes,*should be also ideally perfectly symmetric; but neither of both perfect cases is real. There is NOT a 0 but always a residual finitesimal. There is no 0 temperature, but a residual motion; 0 vacuum but a residual h-planckton. So a-a=+0’ is the real ‘thing’ in the paradoxical asymmetry of reality, where one of the ‘two symmetric sides’ of the Universe is slightly ‘weaker’ in its S**ÚT parameter, and the other side in the opposite one. So in general terms the ‘negative side’ has lesser energy value than the positive side that has lesser information value. Thus a-a=+0’.*

But what is the vital reason *in a perfectly efficient ‘real Universe’ of organic space-time for the lack of a perfect inversion? The answer is obvious, as we do NOT have a pentalogic resolution for the dimotions of reproduction in a world of only ‘sums’, if a-a=0. Thus* Trinity, reproduction of an infinitesimal form by gender is given by the 0’ neutral element. In other words female+male = finitesimal seminal 0’.

- Neither the fact that there is only a sum
*is trivial, as other operations (√) give us two ‘possible futures’. Social evolution thus is highly deterministic as it always brings a single solution, moreover, the next two properties show that social evolution ‘reduces’ the futures, because it makes commutative and associative variations irrelevant. This is a key concept to understand why social evolution ends up being deterministic and reducing the freedom of the parts that become then an emergent whole that ‘makes all those parts undistinguishable’:* - 5: The last 2 properties are also quite profound; as they are different for + and its inverse operation -.

*Subtraction differentiates through those 2 final properties, precisely because they are properties of identity, which subtraction does not perform. As the 4 ^{th} Dimotion of social evolution requires a democratic social undistinguishable number – which we already saw when defining numbers as polytopes, and ‘≈’ the relative infinity with the unit circle, of a relative infinite numbers of irrational ‘holes’ within the maximal polytope.*

Subtraction though is by definition the breaking of parts, an entropic action, and so it is not commutative and it is not associative, as the two elements of the subtraction become differentiated by the operand.

**Recap.** The simplest sum and its inverse subtraction can form a complete set of dimotions, reproduction through its identity number; 0’ which must be taken as the representation of a finitesimal part. Social evolution as its fundamental, natural property; entropic subtraction, which lacks the associative and commutative properties.

** **

**x ÷ : ****2 ^{ND} & 3^{RD} DIMOTIONS: RE=PRODUCT=ION:**

The Ðimotion of reproduction is the product. Since translation = motion in space, or locomotion is a form of reproduction of information in lower planes, the product also plays a fundamental role in mathematical physics and the study of sT-locomotions – motions of a form through a space in which the particle-wave reproduces its memorial information as a wave that evolves and collapses as a particle.

*Its main difference with the sum is that it allows to combine different ‘substances’, time and space parameters. It does not require the same form. *

To define the product then we must study it through the dualities, trinities and pentalogic entanglements it allows as the most important operand between single elements of the Universe; parallel in the analysis of social parts and wholes, to the calculus operands.

As such the product will be able to perform different actions for the 5 structural elements of ¬∆@st; since reproduction is the more complex of all Ðimotions encompassing part of them all.

And so we could rightly say that the ‘king’ of all operands is the product, ever pervading in all systems to the point we often ignore its writing, as it is a given that any merging of S & T holographic space and time parameters must be dealt with a product.

Unlike the sum which requires *equal species* to add with its superposition principles, in herds, the Product is NOT only the sum of equal forms, which is its usual definition, but the merging of different species, with is its fundamental purpose; hence a reproductive act that brings something else, starting processes of creation in a single plane of space – as opposed to the sum that merely will create a simpler herd of a larger scale.

The product is also the first operand, past the ‘shallowness’ of summitry herding, which probes in lower and upper planes of the fifth dimension, as the total calculation of a system’s parts in ∆-1.

So with the product and the ill understood division and its Q-numbers as ratios, we enter into a much deeper land, and multiple issues arise at once.

First we need to understand what we can ‘multiply’ and we cannot. We cannot multiply 3 pears and 5 apples, but we can add them when we upgrade its concept to fruits. But even then we cannot MULTIPLY 3 pears and 5 apples. Still* we can multiply 3, a social number, or property of scale, and ‘apples’, a synchronous organism of space.*

*So as an intuitive first answer that gives such enormous power to the product we can state:*

*‘The product can multiply=merge, the 3 main parameters of the entangled Universe, ∆-scales, Time-motion & spatial organisms; entangling them into an ∆ST, scalar Dimotion of space-time.*

As it happen all Dimotions in the Universe are bidimensional – the product of a Dimension of space and a Motion in time; and we can always add to that structure a ‘number’, which is a unit of social scaling. So we talk of Dimotions as the product=combination of a dimension of spatial form and a motion of time and a scalar number. Gist’s profound realization on the multiplication operand is that it is the essential operand to combine dimensions of space and frequencies of time motion, and Nº scales to create a bidimensional spacetime dimotion.

For that reason as in many other themes (non-existence of zeroths and limits but 1/x finitesimals, non-existence of absolute infinities but local ∝, scalar nature of the ‘numerical line’, etc.) *it is imperative for a proper understanding of the Universe, an expansion of our philosophy of science and mathematics to match closer the reality of the fractal Universe.*

Let us elaborate on this key them considering the holographic bidimensional basic forms of spacetime.

** Holographic interactions. The calculus of ST Dimotions with product operands**

In the graph we show in a more general way the 5 bidimensional, holographic dimotions of existence for which the product NOT the sum are needed.

So we define in ¬Algebra the product as the king of all operations, since it ‘merges’ into ‘a new entity of space-time’, two parameters disjoined. The product proves the very same existence of a holographic Universe with 3 elements, scale provided by numbers, and space x time. So polynomials can merge the ‘¡-1’ elements or ‘cellular parts’ of the being, and calculate its space-time actions as a ‘whole’ emergent new reality. The holographic principle, best served by the operand of multiplication combines space and time parameters into a single ‘entity’, the best known of which are physical momentum, which combines a T-function (speed) and an S-function (mass). So often we talk of that product as ‘existential momentum’ in Existential algebra – basic case of SxT, 3^{rd} ‘Dimotion’ on physical systems. *In that sense an ST ‘dimotion’ is a new ‘emergent’ entity*, something physicists often wrestle with. Momentum is NOT mass neither SPEED but ‘something else’, reason why it acts as a unit in physical equations. I.e. the uncertainty between momentum and position. ‘An electromagnetic wave’, is neither a magnetic or an electric field but something else defined by a product in terms of bidimensional speed: c^{2}=1/µ e.

How multiplication then merges two elements, SxT at the point of maximal communication, S=T? Easily. By entangling the two elements, creating ∆-1 ‘axons’ of communication with all the other parts of the being, such as:

X(5_{¡-1}) x Y (4_{¡-1}) = (5_{¡-1})x (4_{¡-1})=XY (20_{¡-2})

That is, a multiplication that merges two wholes, X and Y, which do have in the ∆-1 scale, 5 and 4 elements, if we join in the ∆-1 scale each element of X and each element of Y, will gives us a number of i-1 communication axons connecting at a deeper level the two wholes, equal to the product of its ∆-2 elements, 5×4 = 20.

And this product is maximal when S=T, so for 10 total elements, 5×5>6×4>7×3…

We *talk then of the ‘perfect product’, when S=T, which is an essential concept of ÐST, for a proper reproduction to take place by ‘gender mirror symmetry’, carried on in mathematics to the ‘Polynomial square’ and combined in terms of S vs. T by the orthogonality of the Pythagorean triangle of ‘maximal’ diagonal length, which are survival, efficient ÐST functions, all pervading in ‘REALITY’, beyond the inflationary ‘non-survival’ entropic mutations of the inflationary language (as all languages are: there is more money than products, more words than actions, more equations than reality).*

This new being then SxT will have *created an ∆-2 scale of axons that entangle together both parts as a whole, in the way a brain is wired by networks. And so we use the symbol of repeated multiplication ∏, for entangled networks as opposed to the ∑ symbol for sum herds. *

So we interpret a momentum as a mass stop state and a wave step state merged together in the potential ¡-2 level that reproduces the wave-particle along its path through a product, which is maximized when both mass and speed are in relative balance to each other. So larger masses have lesser speed and lesser momentum than balanced S=T states – a common feature of all systems of reality deduced of the Max. SxT equation of ‘Momentum of existence’ that all systems try to maximize in its ‘classic balanced age’.

The most abundant of all operand, the merging product requires therefore a more complex rule than a direct sum, which acts by ‘superposition’ of equal beings. Even *if in the entangled Universe of growth of complexity the product can also be used to calculate ‘exact’ multiples of sums. *

For the same reason, as we grow in the entanglement of number families and operands those merged existences will also to be operated by calculus and derivatives, specially when we go from ∆-1 to ∆º by means of integration of derivative parts. Since multiplication tends to happen in the lower scale of the being at different states of time and space. So we no longer operate as in additions, with the same type of T.œs in the same plane.

The best way to describe the multiplication symbol is through the holographic principle. In reality one-dimensional points with no breath do not exist. Systems are always ST holographic merges and so we need an operation, which is the ‘queen of them all’ as it allows us to ‘merge’ S and T states of different entities together into ‘new Bidimensional operators’, which in physics for example merges M(ð) x $(v) into momenta, which is the *conserved bidimensional entity of the spacetime Universe.*

Multiplication, beyond its naive definition as a ‘sum of sums’, is the re=productive creative process of ¬Algebra.

The simplest modes of reality are ST combinations of bidimensional planes which ad by superposition; when the parameters are different however we multiply them. The second operation of ¬Algebra is product and its inverse division, and this probes further into the Planes of reality; so the new operations ads dimensionality and requires new numbers, namely the Rational numbers. Let us then consider them first by studying its properties as compared to those of addition. So THE product IS THE FIRST operand of the fifth dimension, which merges, in 3D: reproduction acts that requires between two ‘genders’ such interconnection at the lower level to form a whole.

5 points connected with all others and with itself, give us 25 connections. This is the ultimate meaning of the product when perceived from an ∆-1 perspective which is needed as in most formulae the product is not reflexive but communicative between species of S-T different quality. I.e. momentum, mv, is not merely the multiplication of m, but the product of a static space-state m and a wave-moving state: v – another whole thing.

** **

**MULTIPLE MULTIPLICATIONS: ITS ∆ST TRILOGIC**

The fundamental effect of new operands is to ad a new dimension to the system, *in the growth of entanglement from 1 to 5 Dimotions that allows the persistence of an entangled reality*, where the 5 Dimotions of time, can work in synchronicity in space. And that is first achieved beyond the ‘wave-herd’ of the first operand and its ‘wave superposition’ rules, ‘smeared’ on its background reality with the tightening of two opposite ‘gender forms’ through mirror symmetry and the ties/axons that communicate its parts internally, creating a ‘new dimotion’ of form. So 5 x 5 neurons are NOT 25 neurons but 10 neurons with 25 entangled ∆-1 axons.

This is *to the product its fundamental function, re-product-ion, as the social herd growth is the main function of the sum. In the parallel mirrors of all stiences and languages, the sum does correspond to the herd, and the entropic state, and the product to the liquid states, whereas the ‘perfect product’ or ‘power laws’, will become the solid state, as all the elements are entangled to all the elements in a maximal network knot.*

This said as the sum could reflect other ‘tetralogic’ elements with the ‘blind spot’ of angular perception, we can apply to multiplication the Rashomon method of multiple perspectives (Pentalogic), to consider which type of dimension, and in this manner an enormous range of phenomena can be expressed with the same operation, increasing the iterative complexity of the Universe.

Indeed, the first question of existential algebra is how many combinations of 5 Dimotions and 10 ST components can we make in a single plane? And how many entanglements might happen in 2 and 3 ∆±1 planes?

The ginormous number of combinations of the 10 space-time holographic elements makes the product a versatile operand, which accounts besides the aforementioned 3D reproduction and 4D social evolution as the sum of sum, for sT locomotion as the product of two such states: λ(S) x ƒ(ð)). While its inverse function, the division, breaks the whole into parts, and so it represents the 5D inverse function of entropy. Yet as in the sum perception doesn’t fit in the product. Let us study them in more detail.

** 4D: §: Social Sum.** In a single plane, classic multiplication is the sum of sums of a group of social identical beings but it already *groups them in two scales as social* ‘numbers’, which become a sub-sœt, multiplied x times, that is we add equal groups that should share a common properties to equal group, ‘adding a new scalar dimension of information’. So if we instead of counting 1,2,3,4,5,6,7,8,9,10; we multiply 5×2=10, after asking how many ‘fingers’ you have on your hands, we did so because we did two ‘hand groups’ with new information, making *a distinction in the otherwise ‘democratic undistinguishable herd of units’. But if we count sheeps to sleep as they are undistinguishable we add them one by one. *So multiplication evolve social groups=numbers in a single plane.

**TIME: 2D: sT:** Multiplication combines a time motion and a space dimension to get a ‘distance’. But here already complex scalar elements enter in view. As we can do that multiplying a ‘short scale’ space dimension, a ‘step’ and the equivalent short time period, a frequency λ(S) x ƒ(ð)= Distance, but we can do it *using long deep time scale (lineal time) in v=s/t-> Vt=S and for that reason we need *to *diminish the scale of our space scale of measure to ‘instantaneous velocity’; origin of calculus of finitesimal speed. So V¡-1 x T¡+1 gives us the same S-distance. *

All those subtle distinction we bring constantly between scales do matter when we try to understand in deep the reality of mathematical physics, even if huminds compress all scales into a continuous line and a single plane. In any case the ST product of steps and frequencies is now a different ‘property’ unified as a whole, a distance which in the first case increases the ‘scale’ – it is a scaled step, and in the second has a peculiar effect, *as it converts two forms of motion, speed and time duration into a fixed form, spatial distance, which means obviously that ‘distance’ is a ‘mental mirror’, an imaginary image that ‘stiffens’ the combination of a short and a long time that must be therefore also mental ‘parameters’. Indeed, the ‘real thing’ is *λ(S) x ƒ(ð); that is, a man doing steps of a meter to its vital frequency of a second, a car’s wheel turning at rpm; the other concepts are humind’s abstractions of time duration (lineal time), instantaneous speed (derivative) and a distance with no persistence. * *

*Yet another proof that the real thing is cyclical time frequency and fractal space ‘stœps’. Since we just proved by such simple reasoning that lineal time does not exist! *adds a dimension of lineal distance. It is the simplest and commutative form.

** SPACES:** Sum brings a number in a single line or a disordered herd but multiplication ads a dimension of spatial width to form an area, whose dimensional growth creates a holographic system, in *which the height dimension will be a dimension of information added to the numerical length, which now becomes a width of energy-volume. So you are reading a bidimensional page of information*.

Again, new subtle enter at play here. Because *the new dimension of form that creates the static square, or in a more complex view a circle, might be transformed from a temporal motion, to become space. *I.e. when a mass which is an imploding accelerated vortex of gravitational motion (equivalence principle of Einstein’s gravitation), uncoils into decelerating entropic motion, (E=Mc^{2}), it does so by a CC ratio, which is in 5D physics, as in C^{2}=1/µe, a region of vacuum space populated by magnetic and electric constants, which spreads and decelerates till mass becomes an extension of space. So here the imaginary term of the humind is ‘Energy’, and the real conversion is the inverse product that breaks mass into 2 ∆-1 dimensions of space. This is a general rule of all physical systems that ‘transform’ constantly static space into time motion. I.e. the same flux of water at higher speed and smaller section goes through a thin tube than a large tube with larger space section and slower speed, themselves consequence of 5D SxT metrics.

Further on, at this stage we find already a non-commutative property for multiplication. As there is a difference in orientation: 4 x 5 ≠ 5 x 4. Hence the non-existence of commutative properties in ‘matrices’, the closest ¬Algebraic representation in time-numbers of spatial point-areas.

Knowing those differences according to which dimensions we ‘operate’ can be a hint for experimental science as when Heisenberg developed quantum Matrices due to the non-commutability of its operators.

**$T v. $ð: **In topological terms, multiplication works for both, lineal products or cyclical point-like, clocks of time, but there are also some insight differences which belong to geometry among them *the *fact that only lineal forms are ‘deterministic’ and ‘tile’ the entire space, as opposed to cyclical points, which leave empty spaces that horror vacuum fills with smaller scales. Or in other words, a ‘rectangular world’ does not need scalar space, neither ‘fractal non-Euclidean minds’ to mirror it. It is as bored, deterministic and truth as the axiomatic method of Euclid.

** ∆+1<∆-1. **A little explored pentalogic form of multiplication is essential to the meaning of a ‘number’ as a social scale, whereas 3 is ‘3 times larger’ than one, and 10 the commonest scale of the Universe a unit of a larger whole. In the graph we can see this concept as 1 becomes 3, a larger scale, 2 becomes 6. We’ll return latter to its implications when we study the properties of multiplication, notably those related to 0’ and the ‘paradox of information’.

Scaling is thus the essential meaning of the product, when we do not multiply a parameter with a physical meaning but make a product of a ‘pure scalar number’.

**3rd Ðimotion production=multiplication =reproduction. Dimensional products.**

So we return in the cyclical universe to the main use of a product, entanglement and reproduction, *which is the more complex dimotion that requires a travel between planes as a seed or finitesimal part of a whole reproduces and entangles in the ¡-1 scale*. So as we said that when we multiply 5 x 5, if we consider 5 the number of ∆-1 elements of 2 groups, then 5 x 5 = 25 turns out to be the number of ‘axons’ connecting each of the 2 wholes at the level of its 5 ∆-1 elements. So 25 gives us *the number of axons= ST flows of communication between the 5 sub-elements of the 2 wholes. And when S=T, 5=5, we obtain the perfect product or ‘power’ law: *Product’s vital role is to create a network on the lower scale of systems to create a new whole ST being, different from S or T, momentum that is neither speed nor magnitude, light that is not magnetism or electricity**. **

First as we have shown multiplication can add a dimension of lineal motion to a particle-vortex (quantum motion), yet as we have proved in other papers, locomotion is the reproduction of a form in a lower plane of spacetime (the entropic/energy field that moves) from stop-particle to wave-motion states. So the very *essence of motion as reproduction of information is encoded In the simple equation: *λ(S) x ƒ(ð).

So pentalogic on the main function of the product, which is also the fundamental Dimotion of the Universe, reproduction, connects T-Locomotion (2D) as reproduction of information along the path of a particle; ∆-scalar reproduction as the entanglement of mirror symmetries through reproduction of axons (3D), and the social sum of parts, that reproduces internal ‘numerical’ groups before adding them (4D reproduction).

**Products of ST-holographic wholes.**

But the product becomes even more interesting, when it goes into the next scale of complexity, *reproducing not individual S and T elements of a dimotion but complete dimotions made of an S and T component, called vectors:*

** Vectorial product **merges whole space and time dimotions (a mass or charge, the active S-magnitude, with a field of motion, the T-element). In the graph, a product of 2 different ST dimotions, one ‘b’ dominant in red-long entropy and other ‘a’ dominant in blue, relative height, the code of information gives birth to a 3

^{rd}(that should be green in ¬E geometric coding of colors J, a relative reproductive width perpendicular to the other two.

A vectorial product is thus one of the commonest forms to combine St x Ts dimotions, acting in its limit of 90º orthogonality as a pure Reproduction of an S and T relative form.

Whereas the non-commutability of its product is related as in the square product in 2 Euclidean (ab. E) Dimensions to the orientation now in 3 E-Dimensions, *establishing a fundamental physical direction for the electromagnetic spacetime of our galaxy ‘clockwise forward’, *as it was found in the practical use of vectors by Heaviside to describe electromagnetism, whereas the electric informative field (for our electronic eyes), and the energetic magnetic field combine to create the reproductive c-speed wave of information of our last perceived ∆-3 scale.

The conclusion is obvious *only spatial sums and products of populations, have no dynamic orientation, as they are static points, but both time motions and scales which must be understood as the SLOWEST, absolute direction of time, where parts come before wholes DO. And since the product applies to space, time and scale, most products, which have a T-element are found to be non-commutative. *All T(SÙT) products are not: bxa=- axb. *Thus, only the product as a sum of ‘still populations’, *SS, which ultimately can be considered a sum NOT a product, is commutative. We come thus to the paradoxical conclusion that the product is NOT a sum of sums and is NOT commutative in its true form.

** Matrix product.** And more over its bidimensional orientation (Matrix calculus) and 3D orientation (Vector calculus) are closely related (as a rotation of a vector is a lineal transformation, calculated through matrices) – themes those of advanced 5D ¬Algebra we shall just introduce here:

We can think of a matrix as an orthogonal system of T-rows and S- columns, which represent the ‘∆-1’ parts of its S-form and T-motion organs. Then the re-product-ive operand of two such ST(x,y) Matrices happens *through the merging of the Sx-components of a Matrix with the Ty Components of the other and the Tx components with the Sy ∆-1 components, which are then added to form a new ‘offspring’ combination of both ∆-1 parts. *

In ÐST point terms the product is not commutable because for *reproduction to occur between two different T.œs requires gender mirror complementarity, hence opposite orientation**. *

**PRODUCT’S PROPERTIES: ASSOCIATIVE, SCALAR, DISTRIBUTIVE, INVERTED & NEUTRAL ELEMENTS: 0’ V. 1.**

We understand better now the properties of operands in 5D terms. The two main ones have been discussed:

**The product is not commutative. **

Only the product as a sum of sums signified by a scalar number is commutative… Reproductive dimotions, using Matrices and vectors, which depart from its simple use as a sum of sums, are subject to the rules of orientation between S and T elements. Since we can also write for better understanding of those properties, the 5 Dimotions of existence in terms of dominance of the Space or Time Element, instead of using Caps, using the order from past to future that determines the dominance of one or the other element. So s<T locomotion is dominant in motion over form v. t>S information dominant in form over motion.

In an interaction between a spatial form and an entropic motion one element will become the ‘relative past’ and the other the relative future, depending on which one prey in the other, on which one transforms into the other. And this makes a product non-commutative. While in symbiotic events of complementary reproduction, the orientation in space of each element also plays a fundamental role that will determine according to the angle of congruence if an event is creative or destructive or skew. Only when the outcome is a social sum in the form of a herd or group of groups it doesn’t matter the order.

**DUALITY ANALYSIS: MERGING THE PRODUCT AND THE SUM**

**Multiplication is associative: (axb)xc=ax(bxc). **

As in the case of the sum, multiplication ‘continues’ the flow of time after a first event but again this property holds for products, which are sums of sums.

**Multiplication is distributive over the sum: a x (b+c) = a x b + a x c = e**

The concept of distribution over the sum is interpreted in 5D in terms of 2 inverse Dimensional operands acting upon a whole, considering (b+c), *a whole which is first divided in two parts, b and c; hence not so much distribution over a sum but partition and reproduction of both parts. Since in that case it reduces to a self-similarity:*

*A x (D _{b+c}) = Ax b + A x c.*

Where A is the dominant property applied to the whole D, by ‘slicing’ it,, D=b + c, into two separate parts enlarging them both by the same ratio.

So in the left side the product operates in the whole D, and in the right side, in its relative parts; which defines D *as a spatial population or herd or wave susceptible to be broken in undistinguishable parts. *

*For all other more complex Ds, which cannot be subject to an undistinguishable partition, multiplication is not distributive over D. But as the rule ‘already’ has broken D a priori, the law appears as Universal.*

**Lineal vs. cyclical addition and multiplication. **

The sum and the product as a sum of sums are lineal systems, as the equation, ax + b, which is a line with b as the origin and a as the tangent, shows. But such lineal *products must be seen as ∑, a summand of x elements, since the product is taken place with a constant number.*

*It is a different case for the product of two variables or any other parameters, which are not ‘constant sums’. *

*So linearity and superposition again happen in the sum but only the product as sum of sums (4D¡ product).*

*From the point of view of the 3 ‘elements’ of a generator, T-potentials < ST-waves>S-particle/heads, *potentials are herds, hence their sums* can be treated as constant products. *

Waves are both sums and re=productive systems (∑,∏). And so they are both lineal and curved, hyperbolic, |xO=ø

As sums waves superpose (lineal quantum waves, which *must be considered bidimensional, and dot products*). As products, they create new dimensions, as in light waves are: c²=k/µ; and cross products. Such products are non lineal, as they reproduce a sinusoidal wave. And often the key to classify waves as sum or products is their capacity to commute (sums) or not (products).

If we extend the concept of ¬Ælgebra to other generators and mirror languages, we find 3 categories in the Universal syntax of most languages coded as generators; where a category will be susceptiable to addition and lineal products but not the others. I.e. in verbal subject<verb(ST)>Object, the verb is a complex bidimotional non-lineal action, but the object and subjects are spatial forms susceptible to lineal sum and product in quantitative terms in the case of the ‘it’ = object, and qualitative terms through its plural forms in the case of the subject.

Products that do not commute happen when they combine different ‘categories’ (potential x wave x particle; subject x verb ≠ verb x subject and so on).

And so in the next scale of equations only polynomials that use sums and products ‘over constant numbers’ are lineal – the other operands that act on cyclical forms – sins/cosines or curves (integrals, derivatives) are not lineal, which means they are NOT deterministic and have often multiple solutions. (nπ, differential solutions, etc.) – themes those treated in different sections of those papers when discussing potential futures, and deterministic ones, lineal steps and curved variations, since only the line has a single potential future, or else stops being a line, while a curve has 3 potential futures, depending on its change of curvature.

**0’ and ****∝ in the product.**

We refer to a previous analysis on pentalogic of 0’, the ∆-finitesimal fractal • point, the T-minimal motion of addition, the S-past ‘memories’ of a extinct form, and the @-mind in relationship to the ∝ perceived Universe to understand 0’.

0’ in multiplication brings new paradoxes, because 0’= **0’ x a < 0’+ a.**

That is, an operand of a higher Dimotional rank multiplied by a ‘fractal • 0 point’ is less than adding it.

*Since the product of a form by an infinitesimal *causes a reduction of dimensionality, and so it *return less than the operand of addition. *The reason is that a+0’ must be interpreted as a non-operation, ‘nothing is added’ to the herd. So it is a trivial non-existent event, only present in the inflationary nature of the language.

But since 0’ is a finitesimal which is by definition the ‘∆-1’ seed of a re=production, 0’ x a IS a real operation, which makes a interact with 0’; that is ‘produce’ a seminal seed at ∆-1 which is undistinguishable. But as time goes through, 0’ might grow to become a, so the limit of 0’ when t-> ∝ becomes a.

For the same reason 0’ x ∝ = 1 = C means that the limited infinite number of reproductions of 0’ will conclude a palingenetic cycle giving birth to the whole. Thus after the parental form sinks itself into the ∆-1 scale, palingenesis brings the ’emergence’ of the whole, 1, the full T.œ.

What the neutral element means is an element not susceptible of being operated by the operand – its blind spot; which is zeroth, no addition to the herd for the sum, and 1, a ‘masturbatory’ (: act of self-production… without input of energy or code for the product, which reveals more about the nature of the operand.

**Multiplying ±. From outer operands to inner property in a higher dimension.**

Which lead us to the question of ‘why’ -a x – b = ab; -a x b = – ab = a x -b.

The main ‘abstract’ mathematical reason is to maintain the consistency of calculus with polynomials and equations that mix both operands. But we are not here concerned with the ideal simplification of mathematics, but with the hidden paradoxes, which will be required for resolving mathematical paradoxes in physics for lack of understanding of the mathematical ones. Let us then apply pentalogic to the – product.

**Space.** We limit such operations to the ‘domains’ in which they are meaningful, which excludes as we already explained the product of ‘natural spatial population’ which are not negative, reducing therefore to the other trilogic elements with a T component, ‘negative time sT-locomotions’ and ‘inverted scalar TT-entropic directions’:

**@-mind logic**. As it only applies to time directions, another way to resolve particular cases is to slide the minus in – b (if b is a class as spatial beings that cannot be negative into – a, the time parameter, so we write (- – a) x b.

On a general basis *this means the negative sign *is a logic *symbol, as when we say, I don’t think I won’t go = I will go; which for time processes which have both directions is possible, as a back and forth vibration,. And so we get –a x – b = axb. *We can also apply this concept to an spatial form in terms of ‘orientation’, which is the only case in which the negative ‘concept applies’; that is, a mirror symmetry. And so the ‘inverse of an inversion’ of a mirror symmetry becomes in i-logic the same mirror being, which justifies *the negative product in complementary gender symmetries*.

**Scalar entropic motions: Negative products. **This case uses the concept of ilogic in terms of a TT entropic process of two negative flows of time, which is the essential definition of entropy, death, diminution, inversion of growth (4D vs. 5D): Let us put an example; imagine you are having a debt of 1200 $ and you remove (negative) every month -100 $, so for -12 months, the negative debt will be shrunk but a year past, -12 x -100 = 1200 $, *the debt will be paid to zeroth and you will not be so silly as to keep paying in the positive territory. *

In the positive side this limit is never reached as the – ∝, 0 interval is now the 0, ∝ interval; but in effect, as ∝ is relative there will be also a limit for a positive product. This negative product though is a ‘time-space’, ST product, which are the only one that make sense, to give us a positive (- change x – population, 12 months x 100 $).

What inversion then means in different contexts as a mirror symmetry, or any other kind of symmetry IS THE REAL proper interpretation we must do. As what we are really multiplying are inversions of inversions. And sometimes it will make sense and some not.

What about the other 2 commutative products (- a) x b = (-b) x a= – (a x b)?

We postulate with the same reasoning that b will be an spatial S-parameter and –a an temporal change of direction in the dimotion that affects b. So S(-T) will become –ST. While the inflationary language that does not identify numbers by the ‘properties’ they describe allows to write those 3 similar equations, of which only one solution will be real, and the other 2 inflationary, an important finding of 5D realist mathematics, when we find the ternary solutions of cubic equations, where often we discharge two of them. So the negative product is an ST product.

Finally we can apply those concepts to the perfect product; that is, the negative x negative product of the type (-y x –y) =(-y)^{2} and *its roots and imaginary numbers.*

Which means in many cases we can get discharge the negative root of an X^{2} solution as scientists used to do; but not always. As a square negative can be a TT, double entropic motion. Consider the case of a mass, an informative, SS, implosive accelerated vortex of gravitational forces (Einstein’s principle of equivalence). When it dies entropically it releases a quantity of entropic energy equal to its product by c^{2}: E=mc^{2}. In reality this C^{2} is the inverse arrow of the imploding mass, an exploding, expanding entropic TT explosion of radiation, ultimately a –c x – c factor, which is hidden in the result as it gives also c^{2}, *but appears as negative in relativity in the factor –(ct) ^{2} now explained algebraically.*

So the – symbol is the symbol of INVERSION, which can be as in the case of c^{2} an inner property of the being, ‘internalised’ by the system, hidden when a larger whole encloses in its membrain (as a mass) but released as inverse TT entropy, when the mass envelope breaks. And vice versa, the negative entropic motion can be enclosed as the ∑ ∆-1 points when we move from ∆-1 to ∆º.

Thus if ± are external trans-formative operands that relate ‘social numbers’ in the ± simpler operands of social groups; in a product they are INTERNAL properties that define the type of ST dimotion we merge; often related to an scalar entropic transformation; reason why one of the elementary proofs of negative x negative products is the scalar proof of extending the product as a scalar growth to the inverse -, – quadrant as shown in the graph.

But since there are many types of products for different S(œ)T species of mathematical objects, in certain cases ± multiplications will make no sense, in others will cover up inner properties, and we have to do in its practical applications to different sciences a careful examination to fully understand the meaning of those minus symbols, as we have already noticed in the our introduction to the Rashomon Pentalogic effects of complex numbers.

Because reproduction is the game, the product is the commonest of all operands, and through it generation of most other forms happen. I.e. we saw the fundamental law of prime numbers is to generate all others, which can be reduced to the produt of 4 Primes, which becomes its ‘tetralogic dimensions’: N=p1xp2xp3xp4

**RECAP.** The product’s main function is reproduction. But pentalogic expands its use to all other dimotions. The product is only a sum of sums, when commutative. But in its reproductive functions is mot commutative in 2 D (non-commutative Matrix) or 3D vectorial calculus, closely related by lineal transformations, which are ∆-scalar products.* Since* *orientation in space* and angle of congruence are ‘vital elements’ of 5D reproduction. So for the complex product of ‘ST’ Dimotions, we* define *non-commutable *Matrix products that *can give origin to different particles (quantum physics’ Dirac and Heisenberg’s formalism).

A theme which is not settled by any means, and has created much confusion is the product of two negatives, – x – = +; as it has to deal with the concepts of 0’, negativity and relative infinity; whereas 0’ represents either an infinitesimal or an asymptotic ‘barrier’ that cannot be crossed as it is the limit of the domain, representing an stop/informative time dimotion, as in Pv=nkT between two sides of a time-space dimotion; hence a negative product of a negative has a limit when it removes all the negativity out.

Thus as usual mathematical laws illustrate the philosophical aspects of ÐST elements, its ST symmetries, Dimotions, scalar transformations and mirror reflections, and vice versa, the pentalogic, dual and ternary relationships of those ∆ST elements of reality exact the full meaning of each ¬Algebraic operand.

** **

**ITS INVERSE OPERATION: ÷ DIVISION.**

**Duality on Division.**

The first and simple view on Division is a duality between its positive informative action, when division is self-division; hence reproduction, in conjunction with the product, vs. entropic division, when division is caused by external tearing and predation feeding:

__ST-reproduction:__ The Pentalogic method applied to entropic Ðimotions represented by divisions and the inverse reproductive Ðimotions represented by products brings new insights to an inversion, *which in nature is symbiotic in most processes of reproduction to obtain a new whole such as X->X ^{2}->∂X^{2}->2x/2=x+x *forms in algebraic terms the series of operands that reproduce a system by doubling its elements, deriving its change and splitting them.

In a simpler stœp sequence of dimotions, without derivative/integral operands, we write: x->2x->2x/2->x+x.

__TT-entropy:__ Division has obvious negative entropic consequences such as the sharing of food, the breaking of a whole into its parts and so it clearly plays the inverse ∆-1 4D-entropic dimension of a system.

The negative, predatory action of bisecting a whole into parts, often digested as the whole ‘breaks=dies’ (pie sharing). Here the nominator is a meaningful entity, but the denominator is only a ‘bisecting number’ or herd that will not ‘create’ a third entity, just destroy the nominator.

Division can be expressed as the negative exponential, which is also a 1/inverse division, is the natural *operation of the arrow of entropy, as it means to ‘divide a whole’ in parts for a predator ‘group’ to feed on them. As when we divide a pie to eat it.*

-The positive, collaboration of two entities perceives as S and/or T Dimensions, which create a ‘stable’ new entity defined by the ratio, which should not be ‘operated=dissected’.

And this is quite obvious when we deal with real solutions to problems using polynomials.

– A single rational Q-number might represent an ∞ number of divisions. Thus, 4 is also 8/2, 12/3, 16/4…. But not an infinite number of products. Only 4 x1 and 2×2. *As entropic time is the ‘background’ of reality with far more events than the hard-built arrows of positive social evolution of information.*

*– It is the inverse of a product as a social communication between ¡+1 elements that give us its total ‘axons’, or lines of communication of the 2 wholes (4 x 4 = 16 axons connecting all the 4 elements with the other 4).*

**Properties of Division**

The division has not the same properties than the product as no inverse function does; essentially because as always in reality the positive side of it has a far stronger appeal to Nature, and so it does often follow associative properties that continue the flow of times; which is what the associative property does. While entropic inverse functions are short lived duality affairs that divide and destroy the elements involved. Indeed life is a continuous flow of times from birth to extinction. But entropic death takes a single time quanta from on to off.

*So as (a-b)-c ≠ a – (b-c) -> (5-3) -2 = 0 ≠ 5-(3-2)=5-1=4; (a/b)/c ≠ a/ (b/c): ( 12/4)/3=1 ≠ 12/(4/3)=12/1.33=9’*

So its difference with the product is its non-associability, *as division ends the chain of events in an entropic destruction of information, hence it cannot carry beyond duality in many cases *

Which set is larger, the product of Natural numbers or its division? If we consider only solutions within the set of N, obviously is the product. But It seems even if we accept solutions in R, the product is larger, as many divisions are equal: 6/3=8/4=10/5; which reaffirms the *dominance of social positive form over destructive negative ones.*

** **

**TRILOGIC ON DIVISION. DEFINING THE 3 VITAL CONSTANTS OF THE BEING AND ITS RATIOS.**

Division can also act on functions normally expressed by the product, using a law fundamental to 5D metrics – the law of inversion of scales, which makes an ∆-1 scale inverse in properties to the ∆º larger state. Let us see the simplest, already examined example of locomotion, and extend the case to the definition of the 3 fundamental ST vital constants of the Universe.

__sT-Locomotion__. As a ratio its meaning is more complex; since often *specially in mathematical physics where the units chosen by humans are misleading (i.e. time duration is an abstraction 1/t= frequency is the unit of cyclical time, so an s/t is better expressed as S x T (wavelength x frequency), and so on.*

But, it is better to treat whenever possible equations as products instead of ratios, to extract easily the 3 fundamental vital constants of the Universe.

__St-Information. __Inversely Division creates information by breaking the continuum of a whole into its parts, this time in combination with the sum, such as 10/10=1 ->∑_{¡=10}= 1+1+1+1…

The combination of division ratios & its sum – as 2 inverse arrows of 4D and 5D actions (breaking & social evolution), thus extract the inner information of the system, or sum of a group of social identical beings breaking down its ‘number’.

But between planes it counts the ‘maximal number of connections’ established between the ∆-1 elements of 2 groups giving us the maximal* axons= ST flows of communication of 2 wholes that reach through X its maximal communication. So its role is *to create a network on the lower scale of systems – in brains, neural networks or reproductive systems, *reason why so often a reproductive action that creates a third element as in vectorial products is symbolised by a reproduction*.

__∆±¡: 1D-4D: Scalar Social evolution.__ It is then from the complex entanglement in parallel space of the combination of St-formal division of a length into ratios, and its Ts-locomotion into a vital time motion, and SS, simultaneous perception of those scalar ratios – the first expression of a finitesimal – from where the first synthetic worldcycles and its mirror language – music arose.

Thus Music and its ratios were famously studied by Greek mathematicians: Pythagoras found certain ideal ratios, in which* division doesn’t break entropically the whole but creates the S-informative discontinuities needed to ‘perceive information’ while the whole remains as a continuum, by the existence of a T-motion on an ST-body – the motion of the string we plug to get a sound by making it vibrate with a certain number=frequency of nods.*

*Hence a full S-ratios<ST-ring>T-motion ‘fractal generator of trilogic is formed, ensuring a constant S**óT feed-back equation that allows a musical event to take place with unlimited variations in space and repetitions in time:*

In the graph, the string is plugged but not broken in certain ratios, which are therefore expressions of ∑∆=∆+1 – a constructive ∆-1 division.

We will study in music theory, on the paper of humind languages why they are those ratios. Notice that as usual they are simple penta-(do)decalogic variations, which suffice to describe most events of the Universe.

This argument leads us to understand the difference between mental loss of information in a rational number, and its preservation in real ration events.

**Loss of information in mental space. **

As the product has rising the complexity of operands, including new scalar ones, so does the inverse operation of division, *which brings as the negative symbol did to Natural numbers, a new family of numbers; the rational numbers, which incorporates as Z did, the operand within them. But unlike Z, complexity has risen a ‘dimension’ as a rational number is actually an operand on 2 numbers; and *further on, it brings the first of many similar processes of convergence or divergence from present to future. That is, *a relative infinite, **∝ number of present divisions, 4//2, 6/3, 8/4 etc. end up in the same future solution: 2, which implies the 2 ^{nd} time we observe the phenomena of loss of information (as we saw already in the product of N x 0’).*

*The solution in the previous case, N x 0’, is to consider that *the outcome is Nx0’ finitesimals, so information leaves a trace in the ∆-1 plane as it is not absolute. In this case the solution is to distinguish the loss of information that happen in any synoptic language – the solution, 2… From the fact that in the real event information is preserved. I.e. if 4 lions kill a 200 kilos zebra, each one will get 50 kilos, but we will see 4 lions eating zebra; if 8 lions kill a 400 kilos buffalo, each one gets 50 kilos; and so the mental solution is the same as it has ‘erased’ the value of the number of lions (denominator) but if we see the vent we will see the 8 lions.

*The same process is observed in the product, which however retains for the product as a sum of sums, the same information, *since the final herd is indistinguishable in its units, but not for products that combine S and T elements (i.e. if we move in long steps with a slow frequency, as a big slow animal we might be tracing the same distance product that a small animal with big steps, but we loose information about the type of species moving).

In both cases we observe that the loss of information happen *in processes where the product or division enter the realm of 5D. As we just described a case of 5D metrics: larger animals have slower metabolism and time rhythms than smaller animals, but the product is the same, SxT=C*

In both cases though the loss of information is mental and can be solved by leaving the number not as a single digit operational number, which looses its information but as an equation, either an SxT or an S/T ratio kept when it is left as a fractional numbers without operating it.

**RECAP**. A ratio of information maintains *the whole without loosing information when the division solution co-exist as a whole ratio number, expressed in the numerator, sum of the parts, expressed in the denominator. *

**Law of inversion in ∆stœps.**

The duality of division and product, as we observed studying locomotion, based in the fact that ∆TS, scales, dimotions and vital topologies have their inverse function, allows to create balanced C=1 solutions to equations, using inverse operands, including product vs. division on inverse scales, topologies or dimotions.

The combination of some of those inversions is one of the key elements of ¬Ælgebra able to explain the why of many similar equations of science.

I.e. lineal time is the inverse of cyclical time frequency, ƒ(ð) = 1/T so we can calculate in long lineal time v=S/T the distance in long lineal space with division, or we can do the same with product in short time steps: wavelength x frequency on the lower ¡-1 scale.

Such similarity is a fundamental law of ‘scalar planes’, which change function and form as we move upwards or downwards in planes; i.e. atoms of iron form rings, but in the macro-scale become lineal sword-like forms. Proteins are lineal but in the cell, become warped and circular elements of the cell membrane. It is a case of the much broader ‘Galilean paradoxes’ between small lineal steps and larger cyclical forms; small discontinuous herds and continuous undistinguishable forms from a larger point of view. And a general rule that allow distinction of information and applies also to $T>ð§ lineal limbs/fields vs. cyclical particle/heads ensemble by intermediate body/waves (Spatial inversions) and young Ts, lineal youth of max. locomotion vs. old, §ð-age of maximal information.

Any Stœp in time, age or spatial topology thus keeps inverting through a transitional region as we move from one scale, topology or age, into another, ultimately developing 3 relative past->present->future <<past entropy>present>future cyclical patterns.

From this facts, we can consider only with multiplication and division some of the basic Universal constants that define the space and time parameters of T.œs, of which 3 are3 paramount:

*s/T: Speed of Locomotion vs. t/S: density of information which are inverse so: T/S =ST/TS= 1. *

*And SxT=Existential Momentum of the being, delivered in ‘frequency actions’ of existential force.*

*Thus we start to see the close connection between ¬Algebra*

¬Algebra reflects the ST inverse D symmetries, as they ARE the inverse key elements for its operations, given the social nature of the number and the polynomial/integral nature of its social evolution into variables as opposed to the inverse operations of logarithms and derivatives.

Consider the simplest, first historic example (as principles become easier to see in its beginnings).

An example of geometric ¬Algebra would be solving the linear equation ax = bc.

The ancient Greeks would solve this equation by looking at it as an equality of areas rather than as an equality between the ratios a:b and c:x. The Greeks would construct a rectangle with sides of length b and c, then extend a side of the rectangle to length a, and finally they would complete the extended rectangle so as to find the side of the rectangle that is the solution.

The solution seems the same, but *it is not. The ratio is a division; the square is a multiplication, and both are inverse functions, which gives us the identity element.*

Yet the meaning of its general case is deeper, as it allows to identify the constants of the being:

Indeed as a ratio of ST dimensions multiplication defines the 3 fundamental vital constants of any being.

Its speed of reproduction of information, S/T, its density of information, T/S and its existential force, S x T (spatial simultaneous view) ≈ T x S or existential momentum (active view), which all together defines the identity ‘element’, the being in ‘iTSelf’.

Let us put an example with the most important Universal constants, those of the Galatom, based in its S(h) constant, the Planckton and its T© constant, the speed of light, and those of the human being, with its time constant, the second and its space constant the meter, to *understand how Classic algebra and existential algebra respond to the same program of reality – the maximal search for reproduction of information and energy, merged in S=T points of balance that define the construction of superorganisms.*

**THE CONSTANTS OF THE GALATOM**

5D Galatom Supœrganism’s metric: $(c) ∆+1 x ð (h) ∆-1

Which refers to the two Universal constants of light in the upper cosmological and lower quantum scale. We shall consider then the easier case of the 3 ‘universal constants of light’.

The balances achieved by the similarity of space=form and time=motion reached in the present body, S=T, and the unbalance of the metric equation of scales, $ x ð = K, in the limbs (Max. $T) and minds (Maxˆ§ð) unify as S=T maximizes SxT=K (5×5>6×4), in 1 equation: **Max. S x T = C**, which defines for each fractal vital space-time organism its Function of existence, as all species will try to maximize its motion-entropy-time for its field-limbs, its information-spatial states for its particle-heads, whose product will give us its vital reproductive energy.

Moreover the equation has an immediate biologic meaning, because as we are made topologically of ‘fields-limbs’ of lineal space with motion provided by the energy we absorb to also reproduce our bodies-waves, and the information we need to linguistically guide our motions with particle-heads, the very essence of survival is to increase our S=position, mental forms of space and T=entropic motions of time (whereas time=motion and space=form are the two limiting Dimotions with ‘energy=reproduction, s=t, locomotion, sT and information, St, are the intermediate 3 dimotions).

*Thus the Universal constants of reality respond to 3 5D Metric constants: S/T= Speed of Locomotion, which defines the limbs/fields of the system, SxT=Existential Momentum/force, which defines its body and T/S: Density of information, which defines its mental power, all maximized when S=T. *

What does it mean then h=c. Obviously NOT that a bit of angular momentum equals a minimal distance of light, but that both can be converted into each other SóT easily through the intermediate concept of an energy-body for the system. Which physicists express with the equation of light entropy, transformed into energy absorbed by an electron: E=Hƒ.

Thus light, the space-time vacuum; or entropic TT-motion of our galactic spacetime from where all forms will evolve, becomes converted into its simples ‘particle form’, h-planck, in a quantity, ƒ, of time cycles that define the existential momentum/force of the photon species.

H becomes then the first constant of scale, such as H = E x T. And because zeroth does not exist, there will be always a residual 0’=H-planckton occupying that vacuum space.

Can we relate the ‘human’ constants of perception (length, mass and time) to the galatom’s ‘real objective, natural constants’? Yes, and we do so with Planck’s Natural units, which must be expressed in a holographic Universe of combined bidimensional spacetime dimotions, as squares, hence getting rid off the cumbersome roots:

L^{2} (static area) x C^{3} (motion volume) = hG -> L^{2 }x C^{3} = hG

1/t^{2 }=angular acceleration = a = c^{5}/hG -> c^{5} = a hG -> C^{5}/a = hG

The first two equations are strikingly similar as they refer to space and cyclical time. And if we put them together and simplify we obtain, L^{2} x a = C^{2}, which defines *the true unit of time, angular acceleration, the constant growth of form, of information of all systems of the galatom; and defines the equivalence between the still space and light space, as we convert a static area of distance through the accelerated passing of time into a sinusoidal wave of time. *

C^{2} becomes then more precisely the existential momentum or ‘speed of reproduction’ of information of light: ST.

* *

On the other hand, G is NOT a constant of this plane of spacetime as it relates to a quintic 5^{th} dimotional quantity. And indeed, we shall see in our analysis of physics that G and mass belong to the ¡±4 plane of existence of the nested Universe within which galatoms are enclosed.

While density of information can be obtained from its equation of speed C^{2}= 1/µ e, which in proper notation, should be written in terms of k as c^{2} = k/µ (we get rid of the π factor).

And its its entropic motion from E=mc^{2} -> c^{2} = E/M.

We shall leave it here, as the study of algebraic physics is the last paper to be done on 5D by the end of this year, to show how all the fundamental equations of physics can be derived of the simple relationships of existential algebra between the different ST-holographic trans-form-ations and reproductions of the 5Dimotions of space-time.

** **

**POWERS & ROOTS: THE PERFECT PRODUCT. 3-4D-5D SOCIAL EVOLUTION & DECAY**

**Pentalogic of power laws.**

We talked of the perfect product as the square form, which gives us one of the most efficient spacetime elements of reality as it achieves an S=T element.

If you have gotten so far you will realize reality has ∝ shades of grey, which means from SS (ab.§) and TT (ab.∏), we can get all kind of sT (energy) and St (information) shades till reaching ST, the perfect iterative present form which the Universe seeks to reach its 2 equal goals: reproduction of form and hence eternity of present, the ‘feminine’ state of reality. Because there is always a 0’ finitesimal the perfect square does not exist, and this we proved by the reversal function, √2, √3 etc tend to be irrational – a finitesimal is left. Still an X^{2} beyond those important but subtle finitesimal differences merges in S≈T two relative genders into a re=productive couple according to S=T 5D metrics, giving a vital meaning to the commonest power of the Universe, X^{2} and its equivalent in geometry, the orthogonal Pythagorean rule, essential for ‘measuring’ merged systems in a lower plane, and expanded in the negative to Fermat’s grand theorem,: X^{3}+Y^{3}≠Z^{3}.

Orthogonality implies that indeed X^{2} is an S-height (information, female dimension) x T-length (motion, male dimension) of what we might call ‘simplex gender’ (we shall in trinity find that gender evolves in the growth of complexity from ST in monologic to dual S-female v T-male, to S=T: present reproduction or female state and S<T>S, past to future to past d=evolution or male state in trilogic).

All this I know is confusing *because as long as huminds do NOT capture the essential truth of mind-mirrors of reality – they come in 5 flavors of growing complexity and integration and all are valid. The predator has a single purpose to kill and feed and still survives. There is I am pretty sure beyond pentalogic higher entanglements to reach what I deem the immortal brain of pure i-logic or dodecaplex. *

But it should suffice to us, simple nitrolife species, whose orthogonal product doesn’t go beyond the heptalogic electronic mind of our amino clocks to study polynomials up to the pentalogic limit, no longer solved by radicals (:

This said for future AI species of unlimited entangled parallel chipped thought, because minds are of many flavors, we can compare constantly logic operands on set and Boolean chip algebra (largely escaped for ethical reasons as I was gifted just with 1.45 kilos of dirty water on my brain but likely to preserve that for generations to come), mathematical operands and ÐST disomorphic laws from where all come.

So back to the basics the monologic reproductive ‘mushroom’ simple lineal thought in either its sterile, Ú, or reproductive summands without limit, Ù, +, is a boring proposition. The beauty of the Universe is that it searches as the best of all worlds for immortality but also diversity, for reproduction but also for constant S<T>S d=evolution, for present but also for past to future swingers, and those do pass through the beauty of S=T present for the fleeting moment ad maximal speed of its SHM variations.

And the true beauty of a mind is to see constantly those analogies, and its quantifiers, those 10^{9} beats of the heart of a mammal, equal to the beats of the heart of an engine.

Why then Fermat’s theorem does NOT exist? Simply speaking because the ternary element is ST, which merges but not quite without keeping the smaller little differences, St:Ts, and yet it seems to us still a 2 power – the diagonal that participates of all the other St, Ts elements of its generator dimotions.

So as we ended this teaser introduction to the power of duality and trinity, with a metaphysical question our explorations of the production, we start our analysis of power laws, with a question regarding Fermat’s theorem, easy for mathematicians: Does X^{3}+Y^{3}=Z^{2} exist?

And depending on the answer, what does it ‘reflect’ as a mirror of the ÐST laws of the Universe?

Power laws are thus about reproduction and the St and Ts merging into ST. So what about the two limits of SS, minds and sees of still form and TT-entropy motions of internal and external expansion into vacuum space, which when merging abbreviate as §∏ – *the two border limits of the being and its world? How they are operated in the pentalogic of power laws? *

The answer is the ‘other form’ of power laws, exponentials where the variable is not on the ‘basis’ but on the exponential parameter, which means a far extremal growth, which *is* the essence of the accelerated time-quanta that collapses the wave into the particle, the light of the Universe into the still image of the world, or vice versa explodes the organism into internal death and external scattering for all to absorb its ∆º«∆-2 remains of the day. No longer then the being is and so exponential beats are fast, from the single quanta of death, to the fast explosion of a little bang in cosmology (as opposite to the hyperbolic universal big bang of egocy physicists, we shall reduce them to what experience finds – the galatom and its cosmic beta decays).

So the pentalogic of absolute growth and absolute decay, of infinite collapse into sentient worlds and complete erasing of information is achieved with exponentials, for which we must therefore establish a limit of domain, a limit of scales of decadence, often in the decametric scale.

In *real science, in* all questions connected with discrete objects, we need to use whole numbers for the necessary mathematical apparatus, as well as the study of the continuous. Thus, for example, in mathematical analysis, when one considers the expansion of an analytic function in a power series with integral powers, computations are essentially carried out with whole numbers and approximated ratios such as 22/7 = πi, which is Ok as we saw in Number theory that decimal numbers break its meaning beyond the 10th decimal scaling… (i.e. e=2.718281828…45).

All fractions represent ratios/quotients of 2 whole numbers; as such a full new branch of ‘number theory’ will be the study of those quotients as *ratios between steps of time motions and or whole polygonal numbers.*

In dealing with any real number in practical work (for example, π), we replace it in fact by a rational fraction (for example, we assume that π = 22/7, or that π = 3.14).

Thus the rules for operating on numbers is the concern of ¬arithmetic but also of ¬algebra, as the deeper properties of sequences of numbers, extend to include zeroth and the negative integers, which again are ‘rounded’ as zeroths even if we know they are always ‘reminders’ of 0’ finitesimals left behind.

**The efficient Universe.**

Power laws, its inverse function, roots, and exponentials and its inverse negative exponential functions, form together a ‘family of operands’ which keep increasing the complexity of our analysis, and the inflationary number of ‘entropic solutions’ – that is solutions that are not stable, efficient and do not happen beyond virtual minds of mathematical thought. What this means in praxis is that when we increase the results and combinations of a syntactic language, its real semantics are reduced. I.e. there are ∞ entropic combinations of the 27 letters of the alphabet, but nobody in his ‘right mind’ would dedicate his life as some mathematicians and physicists do to explore the meanings of r3erewrh, adgaghe and any fast-writing keyboard combination with NO relationship to reality, if the rules of syntax and the constrains of experimental semantics are understood, the difference between potential and actual infinity (Aristotle) are grasped. All this the imaginary idealist German school of Cantor and Hilbert got away with it. So now mathematical physics is hard at studying the 10^{500} landscapes of string theory (: While some computer programmers are hard at finding the trillionth π decimal without understanding truly what π and the sphere is.

I like specially the work of Greene on ahgrwgtarhst strings and that of Cray on the 3758664567 series of π, LOL.

The same happens for exponentials. We can calculate 73^{x} exponentials, but as it turns out the immensity of exponentials with practical uses reduce to the ‘basic 5D numbers and 5D universal constants’ of Dimotions in nature as they are the only meaningful social scales, 0’*∝;* ∝^{0’},i^{x}, j^{x}, 2^{x}, e^{x, }3^{x}, π^{x}, 5^{x}… and its combinations such as 4^{x}, 6^{x}, 9^{x} & 10^{x}… because they do represent the key dimotions of Nature and its ‘perfect products’.

Inflationary mathematics or any language inflationary combinations, bring about 5Dimotion paradoxes, in terms of potential futures vs. single pasts, a theme misunderstood in modern science, since the dogma of 4D locomotions (Einstein’s time theory) became the only theme to discuss about time. The future is pure time entropy, which becomes actual present whenever the minimal S=T (balance and definition of present in 5D metrics) is met. All other unbalanced, Max. S x Min. T (3^{rd} age limit) or inflationary entropic Max. T x Min. S (entropic accidental death limit), which are not within the ‘domain’ of a given worldcycle are synonymous of death – virtual existence. Since indeed the equations of death by accident Max. T x Min. S-> S=0’+T=∝; and Max. S x Min. T (warping, wrinkling 3^{rd} age), reversed also into S->0, T->∝, do NOT exist.

**Negative numbers as ****Time numbers. Lorentz transformations.** **Roots in space**** and imaginary numbers.**

An important fact, we find in inverted operations, is that its ‘number of solutions’ and real forms its numbers and attached operands describe IS *smaller* than the number of solutions to the positive operand, which clearly points out once more to the ‘positive’ arrow of life and information in the Universe, outgaining the negative arrow, unlike in humind’s delirious entropy-only theories of reality. So while there are squares for all natural numbers, there are not roots for all numbers but only positive ones; neither there are derivatives for negative squares including the root of 0 ,∂√0… Even negative numbers are not defined in the sum of spatial populations – there is not a spatial negative apple, but only a negative time number. Which is a key concept for mathematical physics, as we must often *consider the negative number an operand, for functions where the number defines a time motion, hence an inverse direction, which can therefore be taken out of the root to calculate the root of the ‘scalar’ number and then apply the inverse direction. *

What negative solutions mean to those equations? Euler’s vision of negative numbers as inverse time numbers is its proper meaning. *So they do exist, which has clear consequences in areas such as relativity where negative mass, means only an entropic process of expansion of mass into entropy. *

*Here also is important to understand dynamically the operand = as an ≤=≥ transformation.*

*I.e:* E=Mc ^{2} does not mean energy (really entropy in this case), is mass, as mass is in the other ‘inverse side’ of the equation. So the real equality happens when m moves to the same side of E: e ó -mc².

*Further on as we have defined a mass as an accelerated vortex of time, it is a time process;* hence the – can be taken out, once it is clear it defines an uncoiling accelerated inverse mass expansion, exactly the definition of dark entropy/energy of vacuum space, which is ejected at faster than c-speed according to those equations on the axis of black holes with less information than c, hence invisible and perceived as expanding space:

So negative numbers are mostly time functions, as a negative spatial population of ‘apples’ makes no sense.

Which defines negative mass as an expansive entropic destruction of mass. And so in relativity the 2 solutions, which can be put as an example of the 2 roots of quadratic equations (one discharged in processes that are social and accumulative).

In graph, faster-than-light travel and negative mass are the same (ß=v^{2}/c^{2}). We define negative mass as an expansive entropic destruction of mass. How to interpret that equation on our analysis of ‘Existential ¬Ælgebra’ regarding the meaning of roots is then evident: closer to c-speed, the barrier between the light space-time and the intergalactic space-time made of ‘something else’ (likely a background of faster than light neutrinos, as 5D is experimental and so particles not detected are not in the books of any theory till they are – so dark matter is quark matter, dark entropy, neutrinos, and IF a new exotic particle is found beyond fancy mathematical writing we shall revise the known-known economicity laws of physics). Alas, because mass is an accelerated vortex of gravitational forces, as mass increases, the vortex accelerates inwards, as all vortices do, (Ro x Vo = C, for the simplest bidimensional one), which means its ‘radius’ diminishes in the direction of lineal locomotion, and so as mass increases the x-component decreases (Lorentz contraction), till the inner-outer S=T unbalance between the T-length-motion component and S-height component of the particle-wave ‘explodes’=breaks the system which cannot longer perform its Stœp, stop-particle, step-motion ‘beat’; thus the particle state no longer happens and the particle becomes a mere flow of radiation and dark entropy. Simple, realistic, correct way to explain the apparent inverse duality of mass and length… At which point lineal time-duration, T=0; that is, as all equations of death, T=0, Max. S; the particle becomes transformed into something else – an entropic expansion of space. In terms of ‘time duration’, as we observe particles to live longer closer to c-speed; since 5D metrics considers one of its sub-functions, ð=k for entire families of species (sorry not immortality here, the beats of the heart of a mouse, a man and an elephant are the same); it must be then obvious that *those beats of particles last longer; that is, for an external observer take longer to trace, but for the internal particle clock which perceives psychological time in terms of its ‘closed beats’, each beat a ‘thought’ so to speak, there is no difference. *

An easy way out is to consider there is a fundamental parameter of physics, similar to c^{2}, which is m^{2} and as such the equation can be written M^{2} (1-v /c^{2})=m^{2}, and so it merely means that negative mass decreases beyond light speed to reach a balance and the c speed acts as the barrier of an asymptotic change between Planes of the fifth dimension, similar to that which we find as the resistance of ‘death’, the ‘entropic limit’ of any other 2 Planes.

>c then involves TT entropy, which in the previous equations moves E to the mc^{2} side, increasing its density.

Almost every faster than light travel requires negative energies to be implemented at very large densities. And so there is nothing special about it.

In fact there are some examples of “Negative” Energy:

– Radial electric or magnetic fields if their tension were infinitesimally larger for a given energy density.

Squeezed quantum states of the electromagnetic field and other squeezed quantum fields.

Gravitationally squeezed vacuum electromagnetic zeroth-point energy.

– A local energy density in quantum field theory can be negative due to quantum coherence effects.

– Other cases are Dirac field states: the superposition of two single particle electron states and the superposition of two multi-electron-positron states. In the former (latter), the energy densities can be negative when two single (multi-) particle states have the same number of electrons (electrons and positrons) or when one state has one more electron (electron-positron pair) than the other.

Since the laws of quantum field theory place no strong restrictions on negative energies and fluxes, then it is possible to produce violation of the second law of thermodynamics, and time machines at a local level, which is what ÐST precludes.

The root as a solution must be understood in terms of time events NOT of space areas & populations. In a case it is obvious there is not negative population there is only one positive root solution. An area’s root is a line of ‘fractal points’ or a wave-network, and it has only a positive solution. The negative root thus is discharged.

Next squares came in the analytic geometry of conic curves, the simplest of which – the circle – did allow for a mixed ST, square which was valid both as a geometric form of populations in space and a time event, but ultimately traced by a time event reproducing through space, regardless of the ‘persistence of memory’ that either meant a slow perceiver would see the entire orbit of the moving point as a still simultaneous curve, or the reproduction of the point lasted leaving an offspring of points to form a ‘real ring’.

Here the problem of negative values is largely one of the bias and distorsion of the focuses of the ellipse or focus of the circle, which should be instead place in the positive ring, and ultimately *ignores the true property to measure, which is the constant reproduction of form or stability of orbital motion, for the artificial ± axis of the cartesian graph where to draw the abstract ellipse. So we have little concern for those disquisitions.*

So finally we arrive to the polynomials in which the 2 ‘± roots’ represent real events, mostly splitting the being into two different ‘particles-waves’ of timespace; which resumes in the concepts of negative time; negative mass; negative entropy, as we have now a much more clear picture of the different options of ‘time’, at the local level where travels to the past are a dozen a dime.

For a change we shall not consider the whole issue of negative numbers as a different family that natural numbers, merely its inverse. So this implies also a reduction of its meaning, to time numbers, whose motion has a directionality that can be inverted (in a single plane) or a scalar duality in representation of negative entropic dimotions and or operands (inverse operations).

Why then there is not differential of 0’ is obvious. You cannot make a finitesimal of the minimal finitesimal. However the finitesimal of the galaxy, h, does have as a composite h=ST element defined by the uncertainty principle, for S=T, as the product of its S-position and T-momentum, ST, a partition value, h/2 (the realist interpretation of the uncertainty principle). So this galactic 0’ can be parted in two, 0’/2 and so as 2×2=2+2, we might argue that √0’=0’/2

**Exponential and logarithimic functions.**

In the graph, the inverse simpler analysis of growing and diminishing planes and finitesimals have 2 first approaches in the study of polynomials and its inverse equations of logistic growth, logarithms. POLYNOMIALS in that sense are better as negative expressions of the decay of a system in its 4th entropic dimensions and no longer follow the whole range of ‘social growth’ properties (commutativity and associativity); as they are no longer ‘connected with the ∆-2 dimensional scale in the same ‘arrow’ of growth.

Infinite growth does NOT exist because there are always limits to growth in the incapacity of a system to obtain all the energy of its ∆-1 plane (reason why there is entropy when we try to extract ALL the heat from molecules into our ∆^{0} whole scale).

This means that exponential growth IS not growth but decay, as it can only be exponential when acting on a pool of already ‘multiplied by re=product-ion’, ∆-beings, which are destroyed in exponential fashion by an e^{-x} function. And so it is only the inverse function of logarithm growth, what shows a growth curve (logistic curve) till a carrying capacity saturates the system.

And so this said, we arrive to the third fundamental ‘scaling dimension’ used in the earlier age of ¬Algebra, prior to the discovery of analysis, the supposed growth into power dimensions, which in fact are useful in its negative exponential decay of an ‘explosive big-bang death’.

That it is the closure dimensional growth in the ternary Universe, is obvious because of its aforementioned runaway hyperbolic lack of ‘associative’ property and hence of ‘lineal growth’ as the multiplication is.

Power laws thus are essential to study the ‘borders’ of Lorentzian regions, and define the exponential function of decay e^{x}, which is the ∆^{0}<∆-1 4D function of death.

Power laws though in this first age were only concerned with the growth of ‘equal dimensions’, that is converting lines into square areas and cubic areas, beyond which *it was not possible to go.*

More interesting to understand the continuity of dimensional growth within those restrains is to interpret the properties of power laws from the ∆st perspective, regardless of its classic proof by axiomatic methods, taking into account power laws are the 3rd level of depth – or final wholeness of the ‘lineal social flows of multiplication’ and the adding points – so they are closely related to the 1^{st} ,2^{nd} and 3^{rd} postulates of i-logic.

** ****The useful dilogic exponentials **

The number of useful log and exponential functions in existential algebra are by virtue of the inflationary vs. limited different between language of pure still form and information – language with motion – limited.

Given the fact that exponentials are functions of simple ∆¡ scaling, if we restrict to the 4^{th} and 5^{th} dimotion of social evolution and entropic dissolution, we have very few dualities in the different scales of the Universe.

The simplest ones for the duality of 0 and 1, 0^{n}, and 1^{n }are not trivial but have already been summarily considered.

The next scaling for the trinity interval from 0 to ten give us 3 exponentials:

2^{x} is the power set, which gives us all the possible combinations of parts in a whole, where the finitesimal 0’ and the whole X are included.

Thus the power set 2* ^{X}* of

*X*, consisting of all subsets of

*X is the exponential of the 4*. And it is worth to notice that the other dominant configuration of ‘fundamental dimotions’ (if we consider the inverses), which is 10, gives us similar results with an order less such as 2

^{th}social dimotion^{10 }≈ 10

^{3}, 2

^{2×10 }≈ 10

^{3×2}, 2

^{3×10}≈10

^{3×3}which is the mean number of beats of time-energy of most species of the Universe.

Finally it comes e^{x}, which is the fastest growing exponential as a combination of two numbers, in the same manner x=y is the fastest product. *Hence in the same manner x=y are the two most efficient forms to reproduce, e ^{x} is the fastest mode of decay and predation, the exponential of the 5^{th} dimotion. *

**POWER LAWS: FRACTIONAL SQUARES.**

There is though a 3^{rd} fundamental type of power operand, that of a fractal ratio, which defines through power laws the 5^{th} dimensional construction of networks that connect the larger ∆+1 whole and the lower ∆-1 scale, through ∆º physiological networks, which become the fundamental vital element between both planes. Power laws then are expressed through fractional exponentials.

Because the 5^{th} dimension of entropy is just destructive, hardly coded in reality (so we have 4 quantum numbers, 4 genetic letters, etc.) power laws are tetralogic, meaning they relate systems in different scales of reality through different functions represented by dimotions of different ‘tetralogic complexity’. Because scales in the Universe are decametric, those power laws are best represented in logarithmic scales. Thus power laws related in decametric scales reflect the relationship between Dimotions which are simpler to the more complex ones, with the limit of the more complex entangled tetralogic reproduction of an S=T genetic XX DNA and an S<T, male XY DNA. In advanced 5D stiences, hardly touched in those texts – a legacy of GBs that future huminds or AI robots might search and further research… reality in its details is all about the quantitative differentiations of the possible, efficient variations of the game of 5 Dimotions, studied by existential algebra.

In graph, the holographic principle, served by the operands of multiplication and its higher dimensional form, powers, combines space and time parameters into a single ‘entity’, the best known being physical momentum.

Son in ¬Algebra power laws relate ‘space-time’ products=dimotions, between scales, connecting the ‘¡-1’ elements or ‘cellular parts’ of the being, with its wholes through physiological, farctal networks, which create ‘axons’ of communication whereas the perfect product, S=T, X^{2 }gives the maximal number of i-1 axons connecting at a deeper level the whole and parts of the being. While in physical systems, through its momentum the mass stop-state and the wave step-state merge in the potential I-2 level that holds them together in synchronicity (Broglie=Bohm’s realist model of quantum physics).

We shall here just consider key concepts and a few examples departing from the fundamental one – *that the dimensions in ‘classic Euclidean geometry + S=T motions’ of each action of existence * are different; hence when *we connect and synchronize dimotions of the ∆±1 planes of a system, their relationships change giving us power laws (10 ^{x/y})_ relationships between the * actions=Dimotions of existence of the wholes and the smaller parts, which are connected through branching networks. And those power laws are of the type 10

^{x/4}, given the fact that we are dealing with ‘4 Dimotions’ (with very few 10

^{x/5}entropic relationships). The details have been analyzed for decades by my old peers at ISSS at Santa Fe, whose ‘quantitative only, computer based, industry-related, ‘American way’ of doing science is not of my taste, though I recognize the extraordinary detail of its results. What matter to us are the synthetic whys, and to reach that goal we need to study first the ‘classic dimensions’ of each of the 5 main actions of reality, St-information, Ts-locomotion, ST-reproduction, TT-entropy and its inverse SS: perception of form by a mind-language or ‘point’ that shrinks the Universe into a world image (4D), which obviously is the smaller Dimotion as it shrinks St-information. Let us then start by the:

– **SS: 0’ point, which is dimensionless**, an 0’ angle of perception that shrinks and expels motion to reduce reality to its mind mapping. So the act of perception has the minimal Dimotion. In fact *it shrinks dimotions loosing apparently as we saw in the graph of ‘scalar products’ converging into zeroth, the information of the different ‘lines’ that connect the non-Euclidean point with the Universe. Yet from the perspective of the point, the being is the 1, the unit sphere. So we must consider that the dimensions of a point range from o to 1, which is in itself as we have seen an infinite inner world, in a spectra of dimensionality, depending on our point of measure. *

**– Ts- Next comes a line of distance or locomotion, or lineal time which is a one dimensional form. **

It is is the dimension of most internal skeletons, cylindrical field/limbs and yet because a line does have ‘non-Euclidean volume’, an important factor is the cross section of the line; and because it has motion, it matters also how the 5D SxT=C law plays, as a flow accelerates exchanging is dimotion of time for space, shrinking its cross sections as Bernouilli laws of fluids show (S=T). Further one a line can be and it IS essentially a network, when observed through scales of the fifth dimension. And here is where the power law matters, according to the fractal dimension or ‘capacity to fill’ space of the line.

Indeed, the maximal fractal dimension of a line (Peano’s curves, similar to the fractal pattern of digestive systems) that fill all space, or an area, that cringes into maximal volume is 2 and 3. And the way to see this is observing in a hyperbolic geometry how a larger ‘cross’ section through dual and ternary splits maintains its volume, either pumping information (nervous systems with a higher networks division and minimal cross section – hence maximal speed of motion as per 5D: SxT=C) or energy (blood/methabolic network) or in the limit of maximal space and minimal network, the digestive entropic cycle, in fact expanding the cross-section and slowing down the speed of motion.

What fractal analysis of networks, which has advanced enormously since a few of us pioneered the field in the 90s, has always missed, as humans do in all fields of knowledge, by lack of understanding of those time dimotions is the interplay between networks in space and time dimotions, its speed and specific dimotion they serve. Which is at the heart of the power laws of each specific relationship between the upper scale, the physiological network and the lower plane that it serves.

Now those power laws were already studied by Leonardo, that other son of a Notary of a coastal Mediterranean town (: and Galileo, the traitor of life who moved from medicine to work on the Arsenal of Venice, cradle of the modern world, with its invention of ballistics in search of the entropic lineal time angle for maximal distance of cannonballs ): Galileo draw the fundamental law of ‘equal cross section’ for the S=T reproductive, blood system which seeks for balance; which according to the properties of the *physiological network:*

* $T (limbs-fields) < ST(reproductive bodywaves) > §ð*

*And the inverse properties of ‘continuous entropy’ expanding in a formless surface of spacetime and the discontinuous, information of nervous/linguistic inflationary smaller networks.*

*So indeed your digestive tract grows to the stomach and then again in the second phase grows to the colon. *

*But your nerves multiply its axons, and speed up your thoughts, as they fall down in simultaneous cascades of synchronous painful orders to the muscles of your bones. *

The result of all those variations are the multiple power laws that relate the different larger wholes and smaller parts, through the fractal logarithmic dimension of the invaginating networks, of all superorganisms – those which we recognize directly as networks (as we can see them in space), but also those networks that do NOT seem such because they are physical waves, expanding in time, as the impedance equal cross section of electronic networks, the network=gravitational wave structure that forms galaxies, the sound networks that connect people through mechanical air; the inner networks of the Earth likely controlling its surface energy (glaciation cycles through volcanic activity, itself caused by motion of earth’s crust; flows of heavy metals that started up the evolution of History), and infinite other details which the general model of a Universe of space-time organism expects for all those T.œs we perceive and those we don’t. For those who appear to us as fixed spatial form, and those who appears a dimotions of time.

The idea though of all those networks is to maximize the filling or communication between the upper network and the lower scale, which is achieved with maximal feeling as the single whole, the heart, the central gravitational black hole, the radiant star, the brain system. But differences might be consider for each case. As in the case of informative networks in fact there is not a branching travel to the lower plane, but the discontinuous neuronal system on the brain emerges through synchronicity of a common ‘electric language’ into the electric waves of consciousness of the brain. Thus the network exists in the lower plane despite the axon branching.

While in the digestive system, again, there is not a penetration from the upper to the lower plane at the level of the network, but a single larger system, which penetrates down the lower world of cells through an inner fractal structure within the walls of the stomach, warped into protuberances broken into smaller orthogonal ones.

It is only the ‘vital energy=reproductive system’ that makes the travel between both planes.

The result of all those power laws are again *an interval of dimensions for the line between 1 and 2. ***Thus 2D Ts: Locomotion has dimension one**, and other more detailed analysis of locomotion, from waves to networks that deliver flows of energy might reach a 2 Dimension. Of which networks are the first case of our analysis of power laws, as they move energy, which is the essential flow that travels between planes of the fifth dimension, to the smaller scale, through equal sections with no impedance (Santa Fe Institute).* This is the energy to reproduce the system and hence, such networks are closely connected to:*

**St: Dimotion: Vital Energy of reproduction, **which happens in the dimension of width. And so we have a minimal classic 2 dimensions of length and width for a ‘field of energy’ necessary for reproduction.

*And then we can apply the same concept of space filling to the plane of energy, whose real dimensions can go from 2 to 3, which converts it into a topological organism of vital energy networks attached to ‘organs’ – that is a superorganism.*

It is then in the relationship between the networks that sustain or feed the organism and its volume of cells, which makes obvious the first scale law, as the networks grow in 2 Dimensions and the mass in 3 Dimensions, adding the width of cellular reproduction And so the networks when doubling its size are 2* ^{2}=4, but the body *mass, 2

^{3}=8, growing much faster. So for the body cells to keep growing the networks must grow at twice the rate than the body ‘cells’ that fill within them to form the organism. And this means there is a limit to systems, which are something more than a reproductive network, because the network keeps growing and the vital energy mass of cells keeps diminishing.

And this has deep implications beyond the silly-nilly (sorry Santa Fe) calculus of those laws, which already observing the sustain under a gravitational force of a body bones made those of the elephant too big.

So we cannot build huge buildings just with a central pillar, or else a simple plane will 9/11^{th} them. What is the limit of growth? To sustain a building which despite its empty space has a lot of weight.

The answer *IS NO LIMIT*, when we are fully aware of those facts and use the vital geometry of reality making the building an empty space with only a cover, which is also maximizing the ‘arc/sphere’ minimal surface; matching in this manner the dimensions of the building, the gravitational ‘bidimensional form’ and its curved spherical surface. Which is what Fuller Found by designing a building which was in fact a sphere made of the strongest triangular forms, becoming the sphere of 2D the building itself.

While the solution of nature, is that of plants which are basically ‘networks’ with few cells, filled with water vacuoles. So they can grow as pillars because they are just sustaining the pillars. Or simply exist in a region where there is not a gradient of force, or it is minimal, the empty space of the galaxy, the water where pressure of a liquid is the main factor equal from all directions (Bernouilli laws). And this lead us to he *dimensions of the Dimotion of information (I try to escape now the numbers and call the dimotions by their functions and letters, as those numbers are meaningless, just introduced in a ‘first course on 5D to connect what people has learned).*

St: The dimensions of St are the same than those of locomotion (from one in lineal time to 2 in momentum, mv): from one in sequential lines of information, as those words, to two when we consider the whole flat screen. And as organisms, fractal points have its information maximized in its membrain, or ‘sensorial skin’, as information unlike mental form, cannot be recreated internally but it is about information of an external world, moving into the internal world through the sensorial membrane, as in the geodesic dome.

Information thus is also bidimensional. And because equal dimensional forms can transform into each other, St (information) óTs (locomotion) constantly transform into each other, while both merge to form TS, 3 Dimensional Energy, the vital space enclosed by the membrain of information and the motion of the singularity point, which if we give lineal time motion becomes the maximal four dimensional being, sum or product of its other 2 parts.

So we can write: Ts + St = TS but also Ts x sT = TS, and this is the fascinating mirror of mathematics, in an ignored paradox of mathematicians, with little knowledge of 5D realism:

*Only the product and the sum operands over 2 pairs is the same, as information and energy can be added or multiplied to give a Dimotion of vital energy: 2D sT + 2 D St = 2 D sT x 2 D St = 4 D ST-energy*

**5D Entropy.*** Alas How far can then we go in this constant growth of classic dimensionality, within the limits of the 5D Universe? Please *remember now that the ‘numbers’ we gave to St, sT, ST, SS and TT were just human conventions as there is no order, and if any order exists is one of the primacy of ST, iteration, present reproduction over the others. Reason why as we advance on those papers we just call the 5 Dimotions by their letters. And alas here we shall find that as St and Ts have the same classic dimensions, which merge into the 3 to 4 (with motion) reproductive Dimotion, Entropy which is the inverse of SS-formal perception, must have only a 0’-1 Dimension. * So TT=SS.*

This is a theme we study in depth in ¬E Geometry… and conclude that indeed ay organism in a plane of space-time must be made only of 3 topological varieties with ‘1-2-3’ dimensions of information, stored in particle-heads, whose minimal for is a sentence in sequential lineal time, its commonest form a flat page in height and its maximal form a very ‘cringed’ bidimensional membrain that can fill a volume with creases as brains do but never reaching 3. So the interval of dimensions for information is 1<St<3.

And the same goes for locomotion, which can be measured as lineal motions, or areas, which is what limbs and fields or ‘waves’ are made of, with motion to make an 1<St<3.

While the vital energy enclosed by membrains of sensorial information and ‘heats’ that push the locomotion of fields/limbs can go further, such as ST has as dimensions, 2<ST<4.

But *the inverse shrinking world view of a 0’ point mind is minimal, 0’<SS<1’.*

*And entropy which is its inverse, will have therefore the same ‘null’ dimensionality, 0’<TT<1’*

*Those *highly contradictory results of Existential algebra are however of enormous importance for 5D reality in all its sphere. As entropy indeed erases information, which is what dimensions accumulate.

Another way to see this is by an S=T transformation. Entropy as it moves it scatters the information of the system, breaks its network connections and finally creates an undistinguishable Ñ-1 scale of loose points, which are hardly perceived and finally slow down and dissolve, so as all is a dynamic process of time entropy transforms a 1= whole Fractal point into 0’ a sum of finitesimals which are as we know ‘discharged’ in all the results of calculus. Or in other words, entropy=death happens in an 0’ Time quanta, even if the state of death will last eternally, and the mind happens in a 0’ space quanta, even if its mirror of all timespace cycles can be in its synoptic mapping as profound as those texts – both shall remain the limits of ¬∆@st.

Indeed in the opposite side, the mind matters to nobody but the 0’ point in itself, regardless of how people pretend to care as Schopenhauer put it, we all ‘dream and die alone’. And when we dream, the mind is at its maximal activity, but nobody cares. Me, having vivid dreams, and being old, alone and dying can attest that the mind matters nothing.

Even if I thin my mind holds the most perfect mirror of reality anyone has hold, certainly I dream alone, I will die alone, and what my mind holds is completely ignored, *even if now transformed into information in those text certainly will be a memory of the past with depth of dimensionality for future generations to learn. *

Back to power laws, as this synoptic series of papers is mostly about new discoveries by a 0’ finitesimal mind that matters nothing to the infinite Universe, we shall refer to whoever likes the them to the guys at Santa Fe, a place which personally when I lived in the US found charming mostly for its old Spanish Plaza de Armas and native art – the institute had too many computers and physicists for my taste.

As my quest has always been for the thoughts of God, not for us, its details, dust of space-time.

**Properties of power laws**.

Because power laws are ultimately a ‘selection’ of efficient products as re=productive arrows, they have similar properties, with the neutral 1, so we refer to that sections. A power is then for Spatial functions of volumes/populations a new dimension of the product, itself a new dimension of the sum: x³=x⋅x⋅x.

Thus we write in existential algebra:

∆-1: fractal point=cell/atom sum > ∆º: lineal product/network: ∆+1: Power ‘volume’/whole.

*Which is the ultimate reason why they can be used to study the 3 scalar planes of Timespace superorganisms.*

*This said all other properties are just trivial derived of products. *We can multiply powers with the same base:

x³⋅ x²=(x⋅x⋅x⋅x)⋅(x⋅x)=x^{5}. Hence the general law: x²⋅x³=x²+³: The product of powers with the same base add exponents.

In ¬Æ: Two wholes (∆+1 powers) operated externally in the domain of its present, dimensional parts (∆º multiplication) are equivalent to one whole operated internally with its past, finitesimal, entropic points.

A profound law of Existential ¬Ælgebra; which basically tells us that an ∆-whole (a fractal point in its own) absorbing an external ‘flow’ (the lineal multiplication) , will operate that flow at a lower internal ∆-1 level. I.e: if you absorb food you first break it into amino acids to reconstruct yourself.

We can raise a power to a power:

(x²)³=(x⋅x)⋅(x⋅x)⋅(x⋅x)=x^{6}

This is called the power of a power that multiplies exponents: When you raise a product to a power you raise each factor with a power:

(xy)^{a}=x^{a}y^{a}

The insight provided by 5D to this law is the fact that *if we were to consider the logarithmic/exponential scale ‘transversally’ as the 5 ^{th} dimension proper, it converts the dimension of power laws into a lineal dimension – reason why logarithmic functions are so useful; since then we can use its laws as operations similar to the ones we make in our ‘flat’ plane of space-time, using indeed sums and products. *And the metaphysical question remains on the existence of ‘T.œs’ or parts of them in space, or dimotions in time who only travel in the fifth dimension. Obviously the ‘pure Dimensional’ TT-death, which happens in a quanta of time, but implies a huge travel through 5D. What then about its inverse SS-mind? Is the mind a logarithmic reduction of the Universe into a world? The answer is obvious but we don’t want to be philosophical in this paper on ¬Algebra.

In an advanced course of Existential algebra, we can generalize the law as follows:

“Two parts (∆^{0}≈ x • y) operated in the domain of its higher whole dimension, are equivalent to each one operated by the higher dimension as separate, broken entities. Which is the distributive law that gives the whole power over its parts.”

As well as we could multiply powers we can divide powers. And these 2 laws are again applied to the inverse operation.

This quotient of powers property tells us that when you divide powers with the same base you just have to subtract the exponents.

X^{a}/x³=x^{a-3}, x≠0

Two *wholes (∆+1 powers) operated externally in the domain of its present, dimensional parts (∆º division) are equivalent to one whole operated internally with its past, finitesimal, entropic points. *

When you raise a quotient to a power you raise both the numerator and the denominator to the power.

This is called the power of a quotient power:

(x/y)^{a}=x^{a}/y^{a}, y≠0

*Two parts (∆º≈ x /y) operated in the domain of its higher whole dimension, are equivalent to each one operated by the higher dimension as separate, broken entities. Which is the distributive law that gives the whole power over its parts.*

IDENTITY LAWS.

An interesting law, once we noticed that we are in the realm of ‘5D’ operations with the exponential and its inverse logarithmic functions is how they operate the neutral and identity elements, the ‘whole’ 1 and the finitesimal 0’. And as it couldn’t be otherwise, when you raise a number to a zeroth finitesimal power you’ll always get 1, the whole, *which means as the exponential function travels through the fifth dimension, one ‘step’ at a time, when you operate a finitesimal, ¡-1 part, a step in the fifth dimension rises it to its relative whole, ‘1’:*

1=x^{a}/x^{a}=x^{a-a}=x^{0}=1,x≠0

*Which again we generalize to all operands of existential algebra:*

*“A whole, ∆+1, operated by the identity element of its ∆-1 scale, gives us the identity element of the middle scale: ∆+1 * ∆-1=∆º*

And a similar law in ∆st terms: Negative exponents are the reciprocals of the positive exponents:

x^{a}=1/x^{a}, x≠0; x^{a}=1/x^{-a}, x≠0

*The inverted operation of the ∆-1 scale applied to the dimension of the whole, ∆+1 gives us the inverted element of the ∆º scale.*

*Both are laws of the essential timespace definition of the generator: Past (∆-1 ) * Future (∆+1) ≈ Present. *

The same properties of exponents apply for both positive and negative exponents. The square root of a number x is the same as x raised to the 0.5th power: √x=²√x=x^{½}…

We consider this property, from a different perspective, as it sends *the scale of the whole (1-∞) to its reciprocal, scale of the infinitesimal (0, 1) making both mirrors of each other; since ultimately the whole is a closer model of its infinitesimals, than the intermediate scale, which has suffered an inversion of function: ∑∑O-1≈∑|i= O+1.*

T*he 3 ^{rd} scale of growth of a system, doesn’t goes over a hypothetical 4^{th} dimension in the same spacetime plane but starts a new plane, in which the whole is a fractal finitesimal part similar to its finitesimals in the ∆-1 plane.*

And this connects us with the limiting ternary Planes of the super organism, and the lack of a fourth dimension, which makes the ∆-Planes to be self-repetitive fractals cyclical, ‘jumping over the intermediate ∆º line’ from micro point to macro point. Which we can prove geometrically (as we did with the point that creates the line that creates the cube of a new ∆+1 dimension) or ¬Algebraically with the infamous…

As we apply operands rules to particular cases, the interpretations vary but in all cases will be able to be interpreted in terms of sub-equations of the fractal generator.

What might be notice in any case is that unlike in our rather ‘abstract’ dimensional explanation of the rules of power laws, here we are able to bring real vital analysis of those roles in terms even of biological processes, showing how much more sophisticated is the ∫∂ operands, the king of the hill of mathematical mirrors on real st-e-motions and actions, reason why its use is so wide spread.

So the fundamental law of operands to vitalize them is this:

By pentalogic all differential operands can become an action in one of the 5d dimensional vowels (a,e,i,o,u) that define the five dimensions of existence, as vital quanta-actions of the being.

This is the logic concept that truly vitalizes the operands of ¬Algebra.

**Difference with simpler operands. Lack of distributive laws. **

There are many distributive laws, for 1D: 3(x + 7) = 3(x) + 3(7); For 2 D: (3x)² = 3² x² But for power is invalid “expanding” (x − 6)² = x² − 6² WRONG! Since the limit we find constantly from Fermat’s theorem to the definition of death, in transmission of information between organisms is *2 Planes, beyond which we do not carry the distributive law upwards.*

Think of a small house. It’s got a basement, a ground floor, and an attic. You can’t jump right from the basement to the attic, can you? But you can take stairs between the basement and ground floor, or between the ground floor and the attic. You combine operations just like that. If the operations are on adjacent levels, you can combine them; otherwise you can’t. So if the basement adds and subtracts, the ground floor multiplies and divides and the attic powers and roots the rule is very simple:

You can distribute any operation over an operation or one Plane of spacetime below it. There are no other distributions. So you can distribute a multiplication or division over an addition or subtraction:

7(x + y) = 7x + 7y; (x + y) / 3 = x/3 + y/3; 2x (x − 3) = 2x² − 6x; (2x − 8) / 2 = 2x/2 − 8/2 = x − 4

But 3 × (2/5) = 6 / 15 is wrong.

You can distribute an exponent or radical over a multiply or divide, because powers and roots are one level above multiply and divide:

(3x)³ = 3³ x³; √(25x) = (√25) (√x) = 5 (√x); (2/3)² = 2² / 3² = 4/9; √(x/100) = (√x) / (√100) = (√x) /10

But you cannot distribute a power or root over an add or subtract: (x + 3)² = x² + 3² or √(x² − 25) = x − 5 is wrong

Thus with exponentials we complete the 3 polynomial operands in degrees of complexity that mimic the growing complexity of 5 Dimotions in time, and raise a question, I suspect does have a positive answer for *more complex atomic beings of higher electronic entangled than the simplest nitrolife species.*

** **

** **

**CALCULUS AND COMPLEX SINUISOIDAL FUNCTIONS.**

We have defined the 3 mirrors of mathematics on the 3 ∆st elements of reality:

– Spatial Geometry of fractal points with parts and its evolved forms, waves/networks, topologic planes and its combinations according to congruences, perceived by a mind, who selects mental spaces.

– Scalar algebra of social numbers that interact through operands, which represent the dimotions of existence, of which its basic scalar trinity are the ±, x÷ and √x^{a}; ln_{a}a^{x}, related by the operands of congruence «≈»; into equations of trilogic nature, such as S-tatistics<∆-Polynomials>T-probability form its 3 main fields, further grouped in ‘sets and groups’ to classify all forms.

– Calculus of change=time dimotions with the more complex operands of sinusoidal angles and ∫∂.

With the proviso that in the entangled universe all those fields intermarry in pentalogic structures, and each of them is able to mirror all the dimotions and elements of ¬∆@st

This said, in the evolution of mathematical humind’s, the age of calculus and the study of dimotions came after the first age of spatial lineal geometry and arithmetic scalar numbers, *as the analysis of young ‘deterministic=axiomatic lineal forms’ proper of all languages and realities they mirror moved on to the classic age of balanced curvilineal forms.*

*This is obvious if you have understood the ‘3 ages of any spacetime process that goes from lineal deterministic youth to spiraling 3 ^{rd} age of curved information through an intermediate balanced curvilneal age.*

*So topologically we are in the age of calculus and wave forms; algebraically we are in the analysis of the ∆±1 planes, NOT yet probing into the « limits of the whole of wholes and the minimalist parts (SœT inflationary analysis of transinfinites and empty sets) *.

So if we were to divide in a more extensive 3 ages division parallel to the history of mankind calculus, it would belong to the classic age of mathematics, its mature experimental time, when it become a clear realist mirror of physical systems. While the third age would correspond to the eclectic excessive age of inflationary redundant formation with set theory, the top of the mind-view of reality in mathematical terms, becoming its foundations, in an act of humind’s egocy – that is constructing reality from the mirror and the mirror from the axioms ‘imagined by the humind’. So Hilbert will define points and lines, with an astoundingly act of self-egocy: ‘I imagine points, lines and planes’, just because he couldn’t understand fractal points as mirrors of reality.

All this lead us to the beginning of calculus in which we shall see the same fight between Leibniz, the realist and Newton the idealist. But the proper way to classify the work of both, is as the ‘scalar, view of calculus (Newton, which arrived to it through series) and the ST, geometry look (Leibniz who arrived to it through the tangent S/T, between the orthogonal dimension of height information in its ratio with the dimension of time-lineal motion).

Those 2 approaches had already been foreseen in the first age of geometry by Archimedes and other pioneers of geometry, but lacking analytic tools to merge S=T in the Cartesian plane, couldn’t go much further than the Newtonian approach of exhaustion of limits. So Newton could be said to conclude the work of Archimedes and Leibniz to start the age of modern calculus and trigonometry – sinusoidal functions, the mental view on change.

In simple terms we consider 2 two other operands, related to the purest forms of space and time: angles and sins and cosines thus relate to the SS-trigonometric perceptive function of organisms, and calculus to TT-pure motions in time. But while in the Universe each entity has a dominant specialization, all of them can do multitasking and work for different 5 Dimotional purposes, albeit distorted and usually with a ‘blind’ spot for the inverse 5 dimotion of existence, which mathematics solves with the use of inverse functions. So angles of perception and derivatives act as finitesimals actions of 1D:SS-perception and 5D:TT-entropy.

Let’s consider those 2 operands, briefly, before studying its combination in ∆+1 algebraic sentences =equations.

1D: @NGLES TRIGONOMETRIC PENTALOGIC

Because sinusoidal functions require for a full understanding as they deal with the shrinking of the world into a map of perception, integral calculus and derivatives, we shall start with the sum, instead of the first dimensional angle, and resolve its 3 Planes and polynomials; even if as we explain in geometry, the angle and triangle came first likely before even man learned to count just by the fact of opening his eyes to the world.

Though trigonometric functions belong more properly to vital topologies as they embodied the first principles of creation of mental spaces as mappings of the inheriting qualities of time and space that matter for the survival of any fractal point, such as length=distance/motion and height=informative power, of the species on ‘view’ measured by ‘angles’ which are ‘scalar’ parameters, hence dimensionless, as they can ‘travel without deformation’ through the Planes of reality, we do need a bit of an introduction, since sines and cosines are essential functions of ¬Algebraic equations, after all, the analysis of herds of fractal points moving through its vital dimotions of exist¡ence.

Let us start with the trigonometric science of perception, which allows a pi cycle to observe through its natural ‘membranes’ – its 3 diameters that become the cyclical membrain that isolates it from reality, the ‘angles’ that give us information by parallax on the 3rd dimension of relative size and distance of systems outside the being.

The simplest ‘monologic view’ of trigonometric functions is therefore, the fixed perception of an angle that ‘measures’ the height-information and length-distance between the being and an outer T.œ. The sine therefore provides the dimension of form, and the cosine the larger dimension of distance, energy and motion…

The simplest level of perceptive trigonometry is a Pi number: *pi does exist as a perfect form of space, from 3 (the hexagon) to pi (the circle) but it does not as a number (irrational) since time is dynamic, not static, so constantly moves between ±π, allowing the open-entropy/closed-information duality of membranes that enclose systems:*

Its dominant use and first reason it became the first developed field of mathematics is its capacity to measure from a point of view distances according to ratios and parallax, which is the origin of tridimensional perception (bilateral eyes), and Fertile Crescent mathematics.

*That ̦π is an scalar ∆±**∝ number, whose angular function is dimensionless, is proved by the mere fact that as Leibniz find out, it can be written as a sum of series, the very essence of *5D mathematics, as e, is. And hence also it dilutes its meaning beyond a certain decimal. We shall thus return to this theme when considering the two different complementary approaches to calculus and analysis, the Newtonian, ∆-discover of calculus through series and Leibnizian discovery through Spatial derivatives. Willis and Leibniz’s discovery that π/2 and π/4 could be written as a series – hence as a series of ever reduced, fine detailed pixels of a singularity view – being those 90º and 180º angles the mean and maximal perception of eyes – reinforces the essence of the game: entanglement and multifunctionality of all in ∆-scale, Space and time.

How this work in its simplest form, needs to understand how a ‘spherical, ideal mind-membrain of 3 π diameters, and 0.14 D apertures, allows a mind to perceive through them, ‘rays’ to distant objects. The mind thus can always measure the angle covered by a distant object, and with a minimal displacement, a new angle.

3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions.

So trigonometric functions, the first to appear, as 1D perception is also the first ‘action’ are operands for the first Dimotion of perception.

**1D:** Of those operands then we consider the simplest ‘operands’ as those who act directly on a point-number. We need then to find the operand of the first Dimotion, self-centered in the point, perception.

Since all points are also numbers as all have internal structure, we can operate both in S-pace with points and topologies and in scale with numbers to show those internal parts with numerical values:

The graph shows the elements of the point-number in which an operand can act. Even though most operands will act through a similarity ≤≥ symbol in two different elements. Yet the simplest operands are those which can act in a single point-number.

An entire field of ¬Algebra or rather geometry is then the study of the laws of perception of the circle, based on the laws of trigonometry.

It has to be noticed on a first sight that again ‘inversions’ in the eternal Universe do NOT fully cancel each other, but bring something else. I.e. there is a point at 45º of balance in which sine and cosine are equal; and so it is the point of minimal cancellation of inverted functions, reason why without any further argument it will be the point in which there is maximal reach of an stœp process (Galileo’s first discovery with cannonballs). This we can express it saying that the sine is the ‘stop’ perceptive state function and the cosine the step-motion, lineal one.

However in the entangled Universe, we can consider that ‘all systems’ are alive, and somehow evolve ‘adding dimensions’ as a world cycle does, developing new perspectives, and indeed the sine cosine didn’t stop in the stopping geometry of the Greeks that observes the outer world statically. The circle can be given *different motions, up and down and then it becomes an SMH system, or it can emit ‘energy’ imprinted by its formal motion and produce a wave. In this manner, the sine and cosine can perform also the*

*2nd Ð – 3rd Ð: motion and communication: The sinusoidal functions. π.*

In the graph, an SMH can be viewed as the simplest mental representation of the whole Universe, a theme we shall treat in metaphysics. It is the simplest motion added to an ‘angle of perception’, able to communicate waves of information over an ∆-1 ‘undistinguishable’ field it will form with its ‘perpendicular 4th non-Æ motion.

Yet as the second dimotion of all systems, once it perceives is motion and communication, trigonometric functions can move a fractal point creating the form of the wave, or communicate a finitesimal bit of its information through a wave, which carries the information (amplitude, frequency) of the original circle.

Pi then becomes the numerical value of the external membrane measured in terms of 3 diameters turning around with 3 apertures to ‘see’, from the self-centered singularity:

The point-number has a central point of perception, which therefore can be defined externally by an angle of aperture to the world. The membrane normally is a pi, 3 diameter number, which leaves an angle of aperture, 3’14-3/π=0,14=±4,5%, with 96% of ‘dark matter’ outside the perception of its singularity. But as the membrane of the point turns, the angle ‘sweeps’ with its 3 apertures between diameters in a relative discontinuous Universe all the world outside.

So the angle of the point will finally allow a full 100% view in 3 different ‘ages’ of the full world cycle of the membrane. So in dynamic 5D analysis we can consider not only a sine and cosine function; but the sine ‘main’ function that calculates the aperture, also as a function of time, such as for a full turn of the 3 diameters that complete the membrane (2π radians), the angle has seen the entire external world 3 times.

What this means then is that the sine of a singularity point ‘grows’ with each turn of the world cycle of the being and in time can be higher than one.

This said we shall not go into so much complexity and consider the ‘static’ concept and define the sine merely as the angle of perception of the central point and hence the function that reflects the 1st Dimotion of existence.

**SMH FUNCTIONS: S****óT FRACTAL GENERATOR**

It must though be clear enough that the fundamental function of the perceptive singularity is the sinusoidal function, so it is in its active SHM motions, which try to maintain the being through its worldcycles in the ‘central point’ of the system, its ‘mind-singularity- zeroth point’:

In the graph, the ternary fractal generator of any SHM Dimotion system in its 3 States:

*G (SHM): $T(lineal vibration)<ST(sinusoidal function)>§ð (angular motion)*

The fractal generator of Single Harmonic Motions is perhaps the most important generator of mathematical physics, in a single Space-time plane, similar in importance to the Hamiltonian/Lagrangian, which is the most important ‘scalar’ generator of physical locomotions.

Both together form the essential ∆ST structure of locomotion, reason why we can reduce almost all mathematical physics to those 2 type of equations and those closely related to them (Laplacian, Fourier, etc).

The SHM fractal generator throws light into the workings of physical SHM systems, *as the 3 elements form together a co-existing organism: Its particle head with lineal vibration, its body wave and its cyclical membrain.*

Sinusoidal functions are concerned with the central point of view of the @-mind in its worldcycles of ‘angular perception’, (sine function), which is the informative function, in its height dimension, §ð. While the cosine function concerns with the length axis of spatial size, $T. Thus in general sines are information functions and cosines are energy-entropy functions. Both represent from the central p.o.v., its external perception of the world, and internal perception if its inner whole.

The inverse operations of sine and cosine find as usual a point of balance, which is the 45º, where therefore the maximal ‘momentum’ can be reached (maximal distance of any throwing). 45º is therefore the ‘eye’ position of the sphere; the place where there is always a relative view.

The sine must then be considered the informative function and the cosine the energetic function or ‘stop and step’, form and motion states of the being. General rules to enlighten the meaning of mathematical physics.

It also follows immediately that angular momentum is the best external parameter to measure the ‘perceptive speed’ of a system, measured in terms of frequency. So in quantum physics, it will be the measure of the first DImotion, and as it appears as a constant of all systems, we deduce that in quantum physics there is a simple metric equation for all species of the galatom, h(ð§) c($T)=K.

**5D: ∆: SOCIAL EVOLUTION: FOURIER SERIES.**

If the world of SMH functions is the fundamental mathematical cycle and fractal generator of information through space and time; its fundamental expression in social scales is the emergence through synchronicity of larger waves sum of its components, which can be then merged or analyzed in its componets, through the fourier series.

The Fourier series form the more complex dimotion of the sine cosine functions, as they allow by social evolution to draw any ‘function’ of information as the sum of the main harmonic components, *showing how despite the complexity of transmission of information, it always reduced to cycles, lines and scales. And how Universal are the paradoxes of 5D scales, notably the inversion of form, motion and function between them. So the accumulatio of cyclical sines brings a lineal larger whole, and vice versa – the small lineal, free steps are always bent by the larger circular world. *.

**THE KING OF OPERANDS: ∫∂**

When we consider together the 3 simple operands of ±, x÷, √x^{a}, and its S=T polynomial functions, finitesimals in time (probability events) and integral in space (Distribution of populations), they *form together a discrete whole reality mirror that suffices in itself to express all realities. So a world can be constructed with them without the need of further ‘analysis’. *Whereas probability is equivalent to derivatives and spatial populations to integrals.

What then brings the new operands of calculus to the table? The answer is trinity of ∆ST advantages:

**-SPACE: ∆S:** On one side it increased enormously the finesse of detail when populations grew extremely in numbers and were not easily handled by statistical methods. This is the origin of the concept of limit to zeroth, which also expresses that the number of events grows towards a relative infinity, and the minimal quanta becomes a ‘finitesimal’ almost a zeroth, 0’. And paradoxically because the quanta become ‘¡ndifferent’, undistinguishable as the minimal unit of a plane of existence, it allowed to treat populations with ‘functions’ that gathered them all from the point of view of the ‘property’ extracted by the whole. So a mass no longer differentiates its atomic quanta but measures ‘weight’; a motion doesn’t care any longer for statistical counting but measures heat with a Fourier transform. And so on.

This also implied that there are always two alternative mirrors to consider a certain analysis of and S=T parallelism, that of Statistics=probabilities in the quantum realm of discrete populations (hence used in quantum physics) and that of seemingly continuous, because the mind makes its quanta ¡ndifferent use of calculus, where derivatives are the equivalent to a single event and an integral to a whole statistical population or full sum of all events into a 1-whole probability.

It also meant to move the ‘Cartesian region of analysis’ from the 0’-1 sphere in which probabilities by virtue of the renormalization to 1 into the 1-∝ sphere.

**– TIME:** But the true advance of calculus operands was to be ‘a measure of change over a function which already was measuring a specific dimotion of time-space’, thus both increasing the specification of the analysis of change, and performing by virtue of the analysis of a specific dimotion in its integral form, a whole analysis of an organic sequence of that dimotion. It only remained to be able to study functionals of multiple functions that would represent the different dimotions to be able to portrait entire T.œs in simultaneous space or sequential time*. *

Because the final pentalogic operand that transforms all others and finds the finitesimal parts of scales, the rates of change in time, and its equivalent S=T curvature in topological space, is so important to all systems of reality as the dimotion of scalar social evolution, reproductive states, used to calculate the point of existence of a worldcycle and integrate parts into wholes, we shall dedicate a much wider analysis to its two operands, even if we shall only touch the surface of its depth of ‘analysis’, and then apply it to the understanding in time of the worldcycle of existence.

**¬ Limits:** Still as the derivative and integrals will be the limit of sophistication in the analysis done with mathematical tools, and acts essentially in the ∆-1 scale of dimotions=action; mathematics at its best still will fail to describe properly the larger planes of exist¡ence of wholes from the point of view of its co-existing organic elements. And it is fool mathematical creationism to EITHER deny the existence of those properties, because they cannot be reached with finitesimal actions of minimal time depth, or try to invent new mathematical tools to describe them, when verbal logic languages, built precisely to describe the larger range of time depth, can synthesize its causal process in a language natural to man, which we prefer when we deal with sentient and organic qualitative properties that mathematics cannot properly describe.

**–**@: This obviously is denied by linguistic creationism so scientific zealots expect to explain it all with ∫∂. That is the method also of creationist physicists: ‘what cannot be measured does not exist’ (Reductionism that prevents to understand the future of synthetic, organic systems such as those of history.

**Recap.** Among all the operands of mathematics, derivatives and integrals are by far the most important; as they reveal the inside workings of the 5 dimotions=changes of space-time and its scalar structure, *with far more precision, stœp by stœp of S<T>S change*, than all other operands, which are also included within them.

Calculus thus is the summit of the scalar growth of complexity in ¬Algebraic operands as mirrors of the worldcycles of reality, parallel to the same ‘growth of complexity’ from fractal points to waves and networks to topological organisms (inner view) or outer minds (logic, congruent view) in vital geometry and the stairs of growth in logic time, from the monologic of points, to the duality and trinity of lines to the pentalogic of 5D networks… and even probing in the dodecalogic of worldcycles of exist¡ence.

Because of the use of a ‘finitesimal, ∆-1 part’, unlike the rough use of polynomials to measure change, ‘wholesale’, analysis is able to measure different types of change – of scale of space populations and time ‘motions/speeds’ – in detail. A simple example will suffice.

** **

**∆+1: 2 ^{nd} AGE: EQUATIONS: SENTENCES OF MATHEMATICS.**

**≤≥ ≈ ****ó****: EQUIVALENCE ; SIMILARITY AN ORDER OF OPERANDS **

**As a system possess ¬∆@st elements, equivalence must imply identity in the 5 elements, such as:**

∆: Similarity in scale. S: Simultaneity in space. T: Similar phase/state/age in time. @: a common language of information and point of view. ¬: Equal entropic limits in form, time and scalar depth.

Obviously then the system is ‘identical’ and ‘equal’. But such case only can happen for the being in itself. Because even if two beings were identical, they will not be able to occupy the same space (fermions), *unless they are perceived from the ∆+1 point of view of a whole, made of internal identical points that appear from the upper point of view so small as occupying the same space (bosons). *Those subtle differences of pure metaphysical delight correspond however to existential algebra. So we shall not treat them here. What mathematics in classic Euclidean called then equality or congruence was only *¬ equal entropic limits; that is congruence in the membrain of the object as able to be superposed and coincide in external form. *

*This will be a mantra of all our papers when judging the extraordinary simplicity of ænthropic men* which reduce all to entropic concepts. So congruence is on the entropic limits and the internal form matters nothing. Equality is also defined as a one-dimensional equivalence, in a single property. Such concepts are mere equivalences that *allow two systems to entangle on the ‘dimotion of equivalence’. And yet as more elements become equivalent the being’s bondage increases till when all the elements are equivalent, a form of absolute love called boson-fusion happens. *

**Operands facilitating equations and functions**

**The most important operands are in fact those, which connect the 2 sides of an equation. And so happens in Existential algebra. As we shall see when we study it as the Group of all groups. The relationships of similarity and equivalence, congruence and identity, and those of dissimilarity, ruled by the fourth postulate of non-Euclidean geometry, the postulate of congruence and its translation into algebraic operands. **

**So let us first recall from our paper on geometry the difference between congruence in idealized mental mathematics and congruence in the ∆ST world. **

**Identity does not exist. It does not even exist in the being within itself, as the flow of time constantly changes the being. But it does exist a degree of congruence in the ‘information’, in the ¡logic form of the being as a variation of the game of exist¡ence; which also makes possible the repetition of the self. Identity would require some short of transmigration of souls – an entity which could be entangled between ‘ilogons’ of exist¡ence.**

**Equality is properly expressed equivalence. And this is the fundamental use in equations. Two things are equivalent in some property. When we say two things weight the same, have the same number of elements, etc. And this is the magic of Mathematics, which allows its ‘simplfiying’ concepts to permit equality by eliminating all the properties that make things different. So we ‘equalize’ mathematically 2 entities by a single property. Equality or ‘equivalence’ is therefore monologic. A theme which is of enormous importance when studying the parameters and magnitudes of physical stiences, and try to find a theoretical meaning to key concepts such as energy=time, information=space, wave=particle, etc. We use both symbols, =, ≈, preferring the second one. **

**Classic Mathematics of course has its axiomatic method we shall as usual criticize or rather ‘punctualize’ from the point of view of point with inner parts we do not know as ‘only the point has all the information about itself’.**

**It is of interest then to notice that the 3 postulates that define equality in mathematics can be reduced to 2 including a circular property – key element to understand the cyclicality of space and time. Thus all X is equivalent to X, if X is equivalent to Y, Y is equivalent to X (symmetry) and if X is equivalent to Y and Y to Z, X is equivalent to Z.**

**Yet as the first is ‘trivial’, the second and the third can be written as a circular equivalence: If X-> Y and Y->Z, Z->X.**

**Which demands ***a trinity of terms, that is a ‘3=π cycle’, hence the minimal unit for a circular identity in which X reflects upon itself, justifying the π form of all circular motions.*

*a trinity of terms, that is a ‘3=π cycle’, hence the minimal unit for a circular identity in which X reflects upon itself, justifying the π form of all circular motions.*

**The properties of equivalence are necessary to explain how a system which is identical to itself reproduces as it moves, in the 3 modes of physical motion; as it either translates, rotates or reflects in a mirror symmetry with itself, since it is precisely those 3 relationships what allow the motions in space as a reproduction of form, which never ceases to change and yet appears as a dynamic present:**

The dynamic symbols to express equivalence symbols for a present, reproductive S=T, dynamic relationship are:

**ST; Se≤≥Ti (information=energy), TieS, Exi=St; X, ≤≥, ****ó****; ≈; Ø:** Constant Present, balanced, Space-time; iterative body-waves; female gender, ties of space-time, point of balance, work.

**Motion is reproduction of form in a lower scale. Bohm’s realism: quantum potentials.**

How a system moves in a crowded universe, where we are vital space-time? The only answer that resolves also Zeno’s and quantum complementarity paradoxes, is if we do not move but reproduce our information, translated into a lower faster wave scale of the fifth dimension; as we reproduce our sound in faster electrons to telephone or nerve impulses into chemical dopamine to jump discontinuous neurons. So motion becomes scalar reproduction of form, and since all is a form of motion, all is reproduction, which is the definition of a mathematical fractal, a feed-back reproductive equation; 5D metrics, which become then the ‘function of existence’ whose goal is to reproduce the form of all systems – the simpler ones with maximal motion-translation in space, the complex ones with min. motion as a reproduction that emerges between scales. And this gives birth to the worldcycle. Consider the case of quantum physics:

In the graph we see a particle reproduced in adjacent regions that fade away, and the result is the perception of a wave of motion. In Bohm’s realist model this reproduction happens in a lower plane of quantum potentials, where also entanglement happens, which is the ∆-4 scale which is v>c in 5D metrics (Min. S x Max. T = C), hence real.

Motion then is reproduction of form over such potential: the wave erases form into motion, the particle is a still state that gauges information entangled to other particle, fermion and boson, still to each other – despite the perception of relative motion in our scale – hence the information electrons share has always a c-constant speed. Thus the Lorentz transformation are objectively real for mankind who eliminates the stop state of particles as we do in a movie eliminating the stop frame but if we were observing reality from the perspective of an atom, we would ‘stop’, entangle in the quantum potential, neutrino scale & so eliminate the spooky effects of ‘time dilation’ & ‘length’ contraction, from our perspective (but not of mass increase as it is a scalar effect). This is the ‘rational’ 5D explanation of both the c-constant of light and entanglement; as electronic beings perceive information in ‘stop position to each other’ and move in ‘wave state’. Motions are perceived by particles that stop motion into form, into information, as distances.

Again those relationships have traditionally better understood in mathematics and theoretical physics through the concept of ‘Symmetry’, which now we renew and clean of all global presumption. In essence what all this means is that we *do not exist ever as identities* in material form, but in pure in-form-ation, which is translated back and forth scales of the fifth dimension, which of course establishes as always the ultimate question of all 5D model of relational, generational scalar space-time – the ‘identity of all minds’, as the ultimate ilogic game of existence.

In mathematics this process is akin to the equipollence of vectors, which is ultimately a simplifying representation of a translation in space as a reproduction, themes those proper of physics. What matters here though is that only 3 ‘translations’, motions are allowed, related to the 3 properties of equivalence: reflection in a mirror which allows the being to reproduce its form by pegging through mirror symmetry ‘facing each other’ – as it is not possible to assess the identity of reflection with one back to back. Lineal translation, which happens through the transitive property, and angular momentum, which happens through the circular property.

*So motions are indeed reproductions of form based in the ‘assessment’ of a form of the capacity to repeat its identity in another region of space-time. If not it would not require mirror symmetry, circular and symmetric properties. *

**A second essential element then comes beyond MONOLOGIC, when two elements already reproduced enter in relationship of equivalence, for the purpose of evolving socially into larger scale or entering in different relationships of predation with each other. **

**Duality: **4^{th} axiom=postulate of relative congruence **of exist¡ences**: similarity, ¡ndifference, trans-form-ations.

*4 ^{th} Postulate:* Equality is no longer only external, shown in the spatial perimeter of any geometrical form (Euclidean congruence) but also internal and further on it is never absolute but relative, since we cannot perceive the entire inner form of a point – hence the strategies of behavior such as camouflage. Forms are self-similar to each other, which defines different relationships between organic points, according to their degree of self-similarity. The

*4*postulate is thus the key to explain the behavior of particles as the degree of self-similarity increases the degree of communication between beings. Some of the most common behaviors and ‘events derived from this postulate are:

^{th}1) Reproductive functions in case of maximal self-similarity or complementarity in energy and form; ei->Sei or Max E x min. I (male)= Min. e x Max I (female).

2) Social evolution, when points share a common language of information, i=i -> 2i.

3) Darwinian devolution when forms are so different they can’t understand each other’s information so instead they feed into each other: -i « +i. In such cases if those 2 entities meet they will start a process of ‘struggle for existence’, trying to absorb each other’s energy (when E=E) or simply will not communicate (when E¹E, since then there is neither a common information to evolve socially nor a common energy to feed on). Yet because any point absorbs only a relative quantity of information from reality, similarity is relative, faked for purposes of hunting with biologic games such as camouflage or sociological memes that invent racial differences, allowing the exploitation of a group by another.

The 4^{th} postulate defines systems as identical when its 3 ternary parts are equal: the outer angular momentum or ‘membrane’, its central Active magnitude or singularity, focus of the forces and the vital energy*, enclosed within them, and all others as similar with different ‘angle of congruence’. We* distinguish 2 different interactions according to the degree of equality of its ternary parts, as systems can be symbiotic, if *their individual, cellular or atomic ‘fractal points=parts’ are similar enough, interacting through *its 3 physiological ‘lines=networks’ evolving in parallel creating an organic plane, as those described in the next graph for each scale=science, or they can be entropic, destructive, predatory, when they are dissimilar and don’t speak a common language of information to coordinate its actions, whereas the stronger system will perpendicularly break and feed on the weaker one.

The geometric complexity of the 4^{th} Postulate is caused by the topological forms created by any event that entangles Multiple Spaces-Times. Since it describes the paths and forms of dual systems, which connect points: Self-similarity implies parallel motions in herds; since equal entities will maintain a parallel distance to allow informative communication *without interfering with the reproductive body of each point*. Darwinian behavior implies perpendicular confrontations, *to penetrate and absorb the energy of the other point*. Finally, *absolute, inner and outer self-similarity* brings boson states, which happen more often to simpler species like quarks and particles that can form a boson condensate as they do in black holes, *where the proximity of the points is maximized. And indeed, the same phenomenon between cells with the same inner information /DNA originates the ‘collapse’ of waves into tighter organisms.*

Finally if there is no similarity neither in body or mind, its existence as ‘cat alleys’, that never cross (relative invisibility). We talk then of Skew T.œ.s.:

4^{th} Non-E postulate is implicit in the work of Lobachevski and Riemann who defined spaces with the properties of self-similarity (Riemann’s homogeneity), which determines its closeness (Lobachevski’s adjacency).

4^{th} postulate of relative congruence & angle of parallelism as a mirror of its 5 pentalogic dimotions and variations of angle define Darwinian or social, reproductive outcomes to communicative events between fractal points.

Thus in praxis we assess similarity by an ‘angle of parallelism’ that increases social evolution into herds and supœrganisms, or perpendicularity that ‘scatters’ systems into entropic destruction – elements those of an entire fascinating new field of 5D topo-biologic studies that analyze in geometric terms, the vital topology and relationships between form and function in all systems of Nature from particles to organisms.

This simple geometrical truth however is essential to all systems of nature, whose angles of connection determine the functions and symbiosis between parts.

The Universe always starts with an asymmetric being, which can go both ways: towards a social evolutionary symmetry that lasts in time and implies a mirror parallelism, or an antisymmetric destructive, perpendicular event in which one part punctures and absorbs the energy of the other. It is the topo-biological ternary principle of non-Euclidean, Non-Aristotelian I-logic geometry that puts together both the biological and mathematical properties of reality. The concepts of symmetry=parallelism, antisymmetry=perpendicularity and asymmetry are mirrored by the 4th Non-E Postulate of similarity. But we can extend the concept of asymmetries also to asymmetries of time, between the young age of locomotion and the old age of information, of actions=Dimotions between the step and stop similar actions, and the entropy and social evolution actions, which bring us the final asymmetry of scales between the upper arrow of whole with more spatial size and the lower arrow of parts with more information. When those dualities: step-motion/stop-perception and scale up (5D: social evolution), scale down (4D: entropic dissolution) are put together we obtain the most complex balancing dimotion, reproduction, and when they are all added up in the existence of a being, we get its world cycle.

In the graph we can assess the different 5 mirrors in which mathematical Space and logic Time reflects the game of 5 Dimotions=actions of existence, which then expressed by territorial monads GENERATES its logic REALITY. In Geometry fractal points=monads will other through waves of communication of energy and information that grow into reproductive networks a territorial plane, creating a super organism, which will related to the external world according to its relative similarity=congruence, assessed by its angle of parallelism or perpendicularity.

In logic terms, a super organism, by breaking its formless asymmetry into different spatial configurations according to congruence (social parallel systems, complementary gender-mirror systems, Darwinian perpendicular systems, or systems that are dissymmetric and do not share any reality) builds a casual pyramid of growth from a fractal point through waves of communication into social networks that become ready to act – move, feed, perceive and evolve socially.

Since we must add to the mathematical and logic languages-properties of reality the 5 actions, or organic properties of the scalar Universe as essential to the game as they are its logic and mathematical more abstract laws – a fact the egocy of æntropic men of course reject, as it must remain in its monad-subjective monologic the only claimant to life properties.

Thus pentalogic of scalar space-time is established by its ¬E fractal points, its ¡logic congruence with reality in which it will order a territory to perform its 5 vital actions=Dimotions of existence, and the mathematical, logic and organic laws of those 3 languages will be therefore the bottom line of the ‘Creative process’ of the Universe – nothing chaotic except the entropic Dimotion, which conforms the monologic of huminds.

RELATIONSHIPS OF ORDER AND ITS OPERANDS IN EXISTENTIAL ALGEBRA.

Let us now consider relationships of order and its interactions and differences *which obviously will fall within the 4 ^{th} Postulate of Congruence in either symbiotic, ≤ or ≥ vs. Darwinian < or > * relationships.

They are essential to existential algebra and any of its derived Groups, SœT or Boolean Algebra.

*The 5*** Ð¡motional operand are: <:** ∆ of locomotion. « ∆ of entropy**. >** ∆ information.** »** ∆ of form. **≤≥ =** balanced present stœps.

*Any of those 5 Dimotions (Dimensional motion) of time-space, allow the interaction between local T.Œs:*

**Ts, E; ‹, <; |, $T: P$T:** Pa$T, temporal energy, lineal locomotion, gaseous state, limbs/fields.

**St; I, ›, >; O, §ð; ƒð§:** ƒuðure§, spatial information cyclical, informative, space-time, solid, crystal particle-head states

**TT; Te ^{2}, TeTe, «, ¬; ∆-1: **Entropic Time-energy, death, dissolution of form into its parts of a lower plane.

**SS; Sisi;** »; **§, @; ∆+1**: Linguistic, mental space, still mind-membrain, which organizes parts into wholes.

Sequentially they show chains of different S, T and ∆ states of 5 dimotions of exist¡ence.

The differences with those of classic algebra, is their duality of both order and temporal dynamic relationships, which implies not an abstract or spatial property but a dynamic feed back interaction between two beings that are put in relationship by the operand of order or equivalence.

That is, while < means A<B, A smaller than B; a spatial relationship as size in space is the most evident; in existential algebra it also means that A can ‘explode its tighter density’ in an entropic expansion in space, to become B, and more over it also means, B>A, potentially B can implode into the evolutionary higher density of B.

Since Existential algebra is not ‘Aristotelian’, each of its symbols have a trilogic meaning, which often happens together establishing feed back equations: A<B>A.

Thus we can say the relationships of order of classic algebra are a ‘limit’ in the relationships of symbiosis and duality of existential algebra; which is the key concept for the more complex plurilogic systems of reality that co-exist in scales, whose elements perform several functions, and are defined by trilogic fractal generation, pentalogic 0’-sum cycles and even more complex logic entanglements between its parts.

It is then A<B a relationship of hierarchy? It might be if it is according to congruence a relationship of predation from B to A. How can then distinguish them? First by reference to reality.

Unlike the purely abstract relationships of order of ideal mathematics, the connectors of pentalogic have multiple interpretations, as they are connectors with ∆ST triple meaning in scale, in topological space and in temporal order. I.e. > in fact is A bigger than B; A dynamically shrinking informatively as it ages into B; and if we look at it ‘orthogonally’ in the scalar Dimension, A Ñ B (which I confess not to use for finger’s laziness); means the larger A is in an upper scale, above B.

And further on those relationships might imply that A ‘feeds on B’, if the angle of congruence is ‘negative’, in a Darwinian relationship, or A protects B as a membrain that encloses it, etc.

Thus the potential of the same symbols of order when interpreted in Existential algebra, shows the multiplicity of pentalogic meanings and functions of each part of reality. Earlier in my exploration of ¬Æ, I used to translate huge tracts of scientific equation to ¬Æ symbology, till I realize this was to be a solitary journey; so I used them to write long rolls and then conceptual paintings, and then I constructed pentalogic AI-gebra (Algorithms of Information, Artificial Intelligence with them) which as Leonardo with his sea weapons didn’t publish for man would just use pentalogic AI to kill himself, and so I remained mute. It is though an immensely intelligent world.

The distinction then of those multi-actions for this very introductory course on the Universal Grammar of existential algebra, is made with the 4 ‘degrees’ of identity and congruence, from ≈ to ≤ to < to «, which is the symbol of entropic annihilation, in decreasing symbiosis. = implies a relationship of equivalence in the same plane, ≤ one of symbiosis between two planes as A≤B means that A is a part of the Sœt of B its T.œs (larger wholes), but is also A≈B symbiotic by a relationship of mutual dependence, as A is more informative than B and so it might code B. For example, A (gene) ≤ B (cell), A (electron) ≤ B (atom), etc.

So for an imaginary enlightened scholarship that translates all relationships between parts and wholes, SœT of Tœs (where we use the inverse Sœt for the ∆-1 parts of ∆º Tœs), into existential algebra, its symbols would allow to translate all events and forms of reality, as a ‘musical language’ of the flow of time-space events, topologies and scales. I publish myself 30 years ago a few books on that language, which nobody ever understood; some copies still extant on my library and indeed you could read on them, the music of the spheres. Because the advantage of this language, like that of Boolean Mental Algebra, which as we shall see is no incidentally the closer to it, can be used for multiple ‘languages’. As Boolean algebra works for digital numerical mathematics and first order logic, Existential algebra can mirror all languages and events of reality, and also establish with its set of allowed dimotions and rules, adding the |xO=Ø, topologic symbols, what operations are possible and what are not. Being the most important with multiple applications as we saw in our analysis of the properties of exponentials and its distributive laws that information cannot transit between two scales without translation, and it cannot be regained if it transits directly between two scales.

So after laying down the symbols of existential algebra, as when we write down the neumes of music, we can then establish what are the possible harmonies allowed by the existential game, which are self-contained in its meanings. Indeed, the previous reduction of symbols to only 5 implies that A<B is allowed and preserves information, A«B is allowed but as it is a dimotion of entropy it erases information and there is not A<«B; that is, a system cannot jump two scales of the fifth dimension without death.

What about the relationship with propositional logic? I am not so interested in advancing that information but essentially it is the same concept of a larger more complex ilogic structure for the meaning of classic dogmas such as A is either A or B, A->B, then B doesn’t imply A, etc. as we have discussed in many occasions. Philosophically it implies that the Boolean Algebras, and Aristotelian logic is a lineal, ‘action oriented’, subjective self-centered logic, where A, the subject is always right, what means AI modulo-2 will be ‘nazi-like’ and admit no pity but force as it already does its AI, Algorithms of information. Does it mean a pentalogic would work better for man? Certainly not. As it would give it more freedom but still remain self-centered, since survival as sensorial awareness is the program that underlies all others unless apoptosis is strictly programmed.

Finally to stress again the dynamic, temporal nature of existential algebra as opposed to the ‘infantile’, spatial view of humind’s symbols. Indeed, A<B is a spatial proposition on humind logic in the same manner in verbal thought the child learns first names of spatial forms, and only latter according to intelligence is able to understand the concept of a temporal verb, and yet then it will use first the spatial present, and in fact linguistics, a subdiscipline of trinity logic and the universal grammar of the fractal generator is the tool to understand the complexity and entanglement of a culture with the living Universe. Some simple minded spatial ‘visual Neanderthal-like’ languages like English are subjective, start with the I that doesn’t erase, loves present forms and has no imagination to form subjunctive, etc.

Since all systems start as visual, spatial in its first language, dogmatic believing in absolute truths of e-vident form in its mental attitude, and very slowly, many never move of subjectivity out of space to an entangled dynamic time flow language of perception of the ilogic, underlying laws of reality.

**RECAP.** It is then thanks to relational operands that ‘sentences’ of mathematics are possible. Without them mathematics would end simply in numbers families and its associated operands, as the language of a child ends in naming things.

In most functions the independent ‘submissive’ parameter is a function of time-energy from where the, the dependant dominant parameter that forms it, a mental or topologic simultaneous space, arises.

So we can write a general function of all functions: ** F(Sy) = G (Tx)**

In most functions the independent ‘submissive’ parameter is a function of time-energy from where the, the dependant dominant parameter that forms it, a mental or topologic simultaneous space, arises.

The details of those equations will be specific functions that create a form of space from different dimotions of time. Thus ultimately forms of the existential function. A further detail of those functions will be an equation for a particular specific case in which the species, events and entropic limits (domain) are further specified. And finally the solution to the equation will be the deterministic description of the form and event studied.

**Existential Functions**

If we were to resume the main difference between existential ¬Ælgebra and classic Algebra is this: ¬Ælgebra is interested in functions, Algebra in equations. A function is closely related to the 5D metric family of functions, as often responds to a relationship between the 5 Dimotions of space-time, which in the most generic terms write:

S=T or SóT in dynamic terms, and can be developed in 3 ages, or maximal points, between birth and extinction:

SS (0 $T: size max. §ð: Form) » Max. Ts (energy) x Min. St (information): youth < S=T: Maturity> Max.St x Min.Ts (3^{rd}

age)« TT (0 §ð: frequency max. $T: distance)

Whereas we use both the dual symbols for Space-form and T-motion, and the topologic symbols for lineal $T spacetime, cyclical §ð spacetime and balanced, ø-ST and the dynamic symbols « » ‹›.

Functions of exist¡ence then form most of the important equations of science, which are detailed variations of those functions of exist¡ence for each species.

The main set of functions of exist¡ence are those who show one of those 3 relative equations of age/topology, which we can write as 3 functions: S=T, the function of present reproduction, which often becomes St=Ts as the inverse properties of energy and information come together into a middle point; and SS=1/TT, which shows the irreconcilable limits of seeds of information which occupy 0’ space but pack maximal information, vs TT, pure entropy that lasts in the moment of death a single finitesimal quanta of time, 0’, but acquire maximal space extension. The limit of which is the big-bang that expanded space ad infinitum from a 0’ seed of maximal form (but we reduce to the quasar big-bang of black holes, aka top quark stars in 5D cosmology.

Those type of functions as equations of meaningful systems of reality are thus the interest of ¬Ælgebra.

We shall be able then very often to translate such functions as Y (st) = Nº (s), Ð¡(x), with 3 components: Y as the dependent variable, normally a function of space-time; Nº, as the numerical=spatial values ascribed to X, which determines its growth or diminution, the Ðependent variable, which changes in time, within certain ‘entropic limits’

Equations on the other hand might be as trivial as the polynomial equations we study in elementary algebra, where variables are simple numbers in reference to a single parameter of spacetime (a price, a volume, etc.) and are of little interest for this work. So we will make only a few remarks about them.

The notion of a function as used in mathematics is more restricted than we have thus far indicated with the definition of the function of existence, which considers how the parameters of ‘holographic space-time’ (its 5 Dimensional motions, TT, Ts, ST, St & SS, determine the actions of beings).

So we extend the concept to enclose *functions, which are not measured with quantitative numbers.*

The statement that a person’s obligation to society increases with his age expresses a function, the variables of which are 4D:∑œ: social evolution and Ts>ST>St-age. Both are related by the growth of St-information as time passes in our worldcycles. So the function IS a truism, a law derived of space-time ages and dimotions and its entangled laws.

However, it is not possible to measure the extent of this social increase with age in numbers. But that doesn’t mean a sub-function of the function of existence is expressed by that law.

Mathematics then is concerned also with partial equations of the function of existence, but only with those variables of space and time and scale and entropy and mind (the 5 Dimotions expressed as structural elements) whose values can be expressed numerically.

This is a limit of what mathematics can describe, and the limit becomes more stringent when we analyze the more complex functions of existence, which are best described with deep time long organic worldcycles. Mathematics is best for the other extreme of detailed, small functions, where ‘numbers’ increase (steps of timespace that measure distance, interest rates on short periods as the quanta of growth in money accumulating in a bank and similar variables whose various values can be measured and expressed as numbers.

The Function of Existence, Max. SxT, is the function of all functions that can be expressed in all languages.

An open ‘function’ – a goal… has as a result the creation of a constant ‘C-plane, topological organism, or partial outcome that results in an action, whose purpose is to maximize the function of existence.

All action then becomes an expression of a survival mandate of the function of existence.

All function becomes a ‘broader families’ of possible outcomes and variations of an action.

And still we can express it with a formula of existential algebra.

What is then an equation, an even more restricted term?

It is a function expressed as an S» = «T or SxT metric equation.

The commonest expressions of functions thus are F(Sx) = ∏∑Ty

Whereas the spatial parameter, is most likely the ‘outcome of an entangled space’, product of different dimotions in time, which act as the variables of the space-function.

In algebraic terms the paradoxical essence of Nature come into play: A limit is never 0, but a finitesimal ratio of change, an ∆-1 element. When we approach this limit then we approach the S=T limit of the existential function, meaning, ultimately we do reach an stop and step, S=T, ‘perfect 45% angle that maximize the function, as Sx=Ty.

The simplest obvious experimental prove is the fact that the shot of a cannonball is maximized by a 45% angle of maximal efficiency and reproduction of the locomotion of form. So we can consider in general that equations have a longer time view, as Max. SxT=C functions, which have its best most efficient form, when its ‘derivative’ of minimal quanta, brings a relative S=T state, whereas the ‘rate of change’ by unit of time is a quanta of space. In praxis though at the level of the quanta, we can always choose this to be so. For example, humans have a second as its time quanta, which is the beat of its heart-blood that provides information and energy to its ∆-1 cell, but also the bit of mental information (thoughts per second) and steps, which vary between ½ and 1 second.

In this manner partial equations and specific cases of the function of existence become when measured quantitatively functions and equations of algebra.

So an equation is a function with a constant resolution, an outcome, a present result, a real praxis, a detail of the function of existence. The elements are likely the same, Space and time parameters that might vary forming a ‘family of actions’. So an equation tends to be resolved when the details of the function -0 its entropic limits are set, reducing the uncertainty and multiple variations of a potential function. The equation is an actual function. And its solution require the ¬entropic limits of its domain.

**Similar rules of existential functions and mathematical functions.** **Dilogic**: **orthogonality**.

The first obvious one is between TT and SS, pure entropy and pure still form. Both are opposite but its representation is orthogonal, a key law of existential algebra mimicked by classic Algebra. Since a system is either in the limit of absolute motion and dissolution as TT-entropy, whose parameters will be quantitatively different according to the functions we use to measure motion, but the other extreme, § is absolute stillness. No motion, 0’-motion, and that is a definitive parameter measured from the point of view of the observer. So we can draw a simple Cartesian graph, with two orthogonal coordinates, whereas the o-point is the perceived in its point of no motion, stillness. Hence the 0x, Y-ordinates becomes the position of stillness and information, while the 0Y and variable C becomes the coordinates of motion.

We represent functions of transformation of SS into TT as an ideal SS/TT=C hyperbola, which will come out in multiple sciences to represent the tug of war between ‘position’ (form) and motion (momentum). So we establish 3 orthogonal coordinates to represent the complementarities, and oppositions between actions:

SS vs. TT and St vs. Ts., Entropy-Energy vs. form-information can be represented in a Cartesian graph; and functions of both parameters such as the Law of Boyle Pv (S) = NkT (T), with an ideal perfect asymptotic graph, or the Lorentz Transformations between ‘motion and mass’.

In certain cases form arises from the system’s energy, as a complementary form with not a partial transformation, as part of the energy-body remains. Then we can use a complex plane where the value of Y(S) is a magnitude smaller than that of the body in terms of energy (sT vs. St), and in this case the function is also negative as it detracts from the sT energy of the ‘real line’. So for such type of functions between energy and information that do not imply a full transformation of one into another the complex plane works better. But orthogonality is still maintained.

Finally for the S=T, reproductive merge of two relative functions orthogonal to each other, produces a dot product, with an offspring of new combined ST elements, in the 3^{rd} dimension of relative width.

The Cartesian plane, Complex plane, vectorial plane and its orthogonal properties will be used in ¬Algebra (Non-Aristotelian algebra of multiple causal dimotions) to represent the main relationships between SS vs. TT (position vs. motion), St vs. Ts (-information vs. Energy) and the reproductive function S=T.

There is another use of Cartesian coordinates, as an X- sequence of growing social numbers, hence able to represent the 4^{th} dimotion of social information and growth with the passage of ‘long lineal time’, which are often met with simple polynomials (i.e. the representations of growth distance in space with the passing of time). We cannot be exhaustive by any means with the laws of existential algebra, so alien to the present view of a chaotic Universe, proper of ænthropic men; but in our paper on ¬E Geometry and ¬Algebra (non-Aristotelian algebra) and its correspondence with ¬Ælgebra we will develop its laws in relationship with the laws of mathematics and geometry; so the subject doesn’t look so esoteric to the reader, and in the future any scientist can connect the laws of its discipline with the laws of existential algebra.

The angular function becomes an essential operand, appearing in every place of mathematics, from the simplest orthogonal laws of Pythagoras to the ∝ dimensionality of orthogonal Hilbert spaces, which show that indeed in the quantum scale of the first ‘sensorial living forms’, h-spins already all the vital properties of reality co-exist.

What matters of the previous Planes of growing complexity is its ‘entanglement’ in successive systems that include the previous one, *and its relative angle of congruence that will define its outcome*.

So the way to study any system of reality is an in crescendo analysis from** the one, the whole, the point, **as observed externally with its 2 ‘polar parts’, the membrain of information, that focus its sensorial perception into the singularity center which embodies the will of survival of the system, as they create together the 3^{rd} element, the vital energy between them.

This is the first stair of growth in analysis of systems. I.e. we can take an egg, the whole, and then consider the placental ‘sac’, its membrain to the outer world that absorbs energy and information, then focus on its central genetic material and observe the vital energy within it. Then as developmental evolution takes place we shall immediately observe two poles, the animal and vegetative pole, which follow 5D metric – the animal sensorial informative pole is smaller than the energetic one. And as the system grows again it will form the ectoderm with the sensorial elements, the endoderm and the mesoderm, and finally it will become each of them, the blue print for one of the 3 physiological networks of the system.

Trinity is thus the easiest way to grasp reality as it is in a single plane of the 5^{th} dimension. But even in our description of the egg we realize the 4^{th} dimotion of social evolution kicks immediately multiplying the layers of cells till the whole thing emerges into an ∆+1 fetal form and all the way already in its morphological stages, entropic time erases by apoptosis some of those cells to form the fingers. And so on.

In time monologic and in space the growth of non-Euclidean points are thus the easiest two sides of the description of the entangled, fractal complex Universe, reason why we serve them first. If they are grasped, then we can *put them together establishing a vital program of exist¡ence for all systems of nature, which will be the program of the 5 dimensional motions of time-space of any supœrganism. *

So by adding ¬E Geometry and ¡logic, we get the actions=dimotions that form the sequences of reality and will be enacted by all secondary languages in exist¡ence. We can indeed apply the vital topologies of ¬E Geometry, the levels of ilogic entanglement and structures of monism, duality, trinity and pentalogic to resolve the 5 actions=functions=dimotions performed by the vowels of a language, the type of cells of an organism, the operands of algebra.

And as each part of a fractal is a fractal in itself, we can apply then pentalogic to each element of reality to illuminate the system or language from different perspectives, since indeed, in the entangled growth of complexity for a monist single form to exist, will still have to perform in an imperfect manner along its inverse form, all the functions of reality. So even the smaller point is a world in itself (Leibniz), even the simple dimotion can be decomposed in pentalogic elements, even the simplest ± inverse operands can represent the 5 ¬∆@st elements of any system.

Because we have studied the 5 Geometric ¬E Postulates, in depth in our 1^{st} Book on ¬E Geometry and the 5 dimotional program of exist¡ence from the p.o.v. of the ¬Algebraic operands they reflect, we shall not go further in here; just bearing in mind the beautiful vital sentient geometry of something so apparently insignificant as an angle which has not even a dimension.

**Trinity through Circularity**

The last case of equality is circularity between 3 elements that are identical in its translation, closing a pi cycle, which of course is never a perfect closed circle as the reproduction of each of the 3 elements are discontinuous to the next reproduction, leaving 3 π holes, π-3/3=4%, which in pentalogic represent for the central singularity the ‘canonical’ percentage of perception a point has of the outer world, often just the openings for its network connection to other self-similar points, to construct a network, walled from the dark space it does not see so it believes it does not exist. Such blindness to all other not-self-similar points of parallel cat alleys concludes our brief analysis of the laws of congruence.

** **

**THE FUNCTION OF EXISTENCE IN PROBABILITY AND STATISTICS**

**How temporal events become space populations. The main distribution curves. Entropic Normal vs. lognormal**

So in the next graph, we observe how truly stochastic systems look, both *in time as a ‘cumulative’ population of events, and in space, as a distribution with a mean of maximal probability. The erf (error function, as mathematicians call it) is really the time symmetry of the distribution function; hence its Sigsmondi-cumulative form: * Distribution of frequencies in time, equal populations in space, both defined by the same ‘magic formula’, which we shall study in detail.

It is one of the fundamental symmetries of fractal space (measure in populations of identical entities/ ‘points’) and cyclical time (measure in frequency of time events/numbers); and it is ultimately one of the clearest expressions of S≈T and the equality of the 2 ‘real units’ of spatial geometry (points) and temporal ¬Algebra (numbers); as the total of events become 1, and so does the total volume of the curve.

But *the essential difference between space and time Is this: space is symmetric, both left and right, but the curve that represents time is NOT, it has directionality, and it has not simultaneity.*

Thus stochastic populations in space tend to a simultaneous distribution around the mean ‘0’-line’ position considered the central ‘point-line’ of the ‘phase space’, while the temporal cumulative arrow tends towards a ternary distribution in 3 clear ages of different growth, with an exponential growth in the ‘mature’ age of the system (middle region), and a beginning and an end, which are ‘reversed in form’ (first young age of the event, and 3^{rd} final age of the event).

For the matter that brings those two curves here – its relationship with the function of existence, we must perceive the distribution curve as the spatial event, which makes the perfect ‘reproduction’ (mean form), just a 1 event – at the summit of the S=T classic age of existence, and all other ‘stœps’ of spacetime, slightly erroneous reproductions, which therefore determine as they ‘accumulate’ as populations in the 0 to 1 sigsmondi curve, a growing aging and malfunction of the system that drags its ‘tail of memorial past repetitions’ along its way.

The big question on how the 0’-1 time probability form becomes a 1-∞ space ∆+1 scale is more important that what it seems, as it shows a certain determinism of great numbers that is of the game of existence, which in many ways the distribution function signifies.

Populations around an ST center with a vital energy population and two cues, which one seems to think should be the parameterization of the singularity and membrane of *equal value*, as a hypothesis of work, shows however the uncertainty of which point to make the cut, and this is provided by the probability density… at the normal, which gives us a huge difference over the median, more according with the properties of ∆st: over 2/3rds of the population is vital space energy, 15 3/4ths go for the singularity and the membrane. Though in real measures the higher density of both reduces further its presence in space.

So we consider that the *deterministic function of construction of a super organism, both in the 1 cycles and the 1-∞ isomorphic ∆+1 scale define ‘why the laws of great numbers’ bring us the equations of distribution.*

How we get from one to another then can be ‘deduced’ axiomatically for further ‘insights’ in the process of organic structure in space and time frequencies that build a 1-being from its finitesimal occurrences.

But the spatial symmetry gives us also another kind of information, as the regrouping of those frequencies into probabilities of populations completely changes its role in the St symmetry.

The fundamental theme of Probabilities and statistics besides this s=T symmetry is the concept of ¡ndifference of beings-events. Indeed when we get a 6 is equal to create an indistinguishable particle. The type of particles thus is equivalent to the set of events in time. And as sun they accumulate into the networks that construct the system. A minimal distribution of 3 only elements, the identity present element and the inverted ones, seem to be by far the commonest, with the point of σ as the place where the symmetry reaches a breaking point between both ‘asymptotic areas.’

**Networks in time. Determinism.**

**Reality though is not dominated by entropy but by information and information bias towards a certain property all distributions, precisely because it ‘forms’ reality, and so as we get into systems that have a network condition probability change biased by multiple elements of information. The results are biased, skewed structures in time and space. Let us consider one fundamental for the growth of supœrganisms around a central point, where it should reside the mind of the system.**

**As time passes the network grows around the 1 ^{st} clone duality which remains at the core of the system. Even in the more complex superorganisms should be possible to follow the position of the seminal dual cell, which likely will remain in the leading position and possibly emerge in the continuity of the mind exists, as the mind of the supœrganism. It is then possible to consider a continuity of all i-logic minds proportional to its informative complexity. **

*Determinism grows from herd to organism as information organizes a system around its networks. *

**CALCULUS ON ENTROPIC LIMITS. SPATIAL ****STATISTICS**

The beauty of it all though comes when we compare that graph with one of statistical populations: and its equivalent in probabilities, as both are gauss functions which are the function of existence, which have all those points, because it is unavoidable that repetitions commit errors, as they are in fact normal distributions, of all possible maximal entropic combinations of beings.

Yes, it is all fit to exist with errors and errors become frequencies of time, that form populations of space, and those errors are caused by the need for the maximal entropy of all possible combinations and variations. The Universe likes to mutate your cycles of time and populations of space, and then they linger around those free radicals, those slightly odd distributions, one of them, except the perfect resonance of a Dirac’s function, can last for eve r because they are not all equal your repetitions of beings, your actions, your e-motions. An important insight of 5D algebra on the S=T mirror symmetry between probability and statistics is the fact that the Gauss distribution that corresponds to a maximal entropic function in probability however represents the normal statistical distribution of a well-ordered informative population around its mean ‘evolutionary center’. Thus what is entropy in time becomes information in space, reinforcing all the 5D concepts of the general model of generational space-time, as entropy is the ‘time substance’ from where mental space and informative seeds extract its energy of existence.

In the graph the Maxwell-Boltzmann distribution for the velocities of gas=entropic state of matter= molecules. For small v, this quantity is close to zeroth because of the factor v2 in the equation for the weight of a state; after the zeroth point it reaches a maximum and exponentially decreases to zeroth again for large velocities. We thus see that a gas contains molecules with every possible velocity value.

The greatest number of molecules have a velocity corresponding to the maximum of the distribution. We can see the skewness of the distribution caused by the non-existence of a real 0’ point of no speed, and the relative ∝, which is limited also by the c-limit of the larger ‘galactic enclosure’ (speed of the background space time, as V∆-¡>V∆¡-n and the specific membrain of the container that reduces it, absorbing ‘energy’ from its enclosed vital space.

It illustrates also the interaction of events in time (speed) defining populations in space.

Another reading of the same concept is this: deterministic systems are those in which its entanglement and bondage between its ¬∆@st parts increases significantly with time, as the system evolves from a loose group of individuals that form a herd under a common dimotion/property, into a whole organic tied up a system in which each part is connected to the others through a network, which evolves into a ternary, S<ST>T network system to form a topological, organic plane, or supœrganism, where determinism greatly increases and probability is minimal.

Thus probability and statistics tend to be useful for systems in a herd/wave condition far more than systems in a particle/head/superorganism highly ordered determinist state. And for the same reason, it is better suited for the analysis of the ∆-1 parts of the whole.

The concept of an event in time or property in space which is statistically measured in a collective numbers of parts, which have a degree of entropic freedom that make them susceptible to enough variation to be measured statistically comes then naturally as a consequence of all this. Essentially we are dealing in statistical analysis with non-well ordered sets of disconnected parts which share a common property or dimotion of existence. So statistics is important but it does NOT define a statistical entropic universe, unless as so many physicists do we exclude the solid state of bondage, where the perfect order makes statistics quite irrelevant.

**Limits and distortions of functions in-between planes of existences. The function of existence in space. **

What is the form of the function of existence in scalar pace? Mathematically we can see the equivalence expressing the Gauss curve both in statistical and probabilistic form. But when we deal with the real Universe, it is far more telling to consider a graph of the function of existence *from the point of view of the parameters of ‘energetic time and informative space’, of the world in which it is inscribed. As we shall find within that world, limits of entropy and form that kill the system out of balance – limits that explain why the Gaussian curve has not infinite deviation, as those are forms that would not be stable. In the graph, from a 30 years old book with a funny ‘nomenclature’ we see a point of balance for any function of existence, when the spatial form and temporal motion of the being is equalized, S=T, on the two extremes of the functions of entropy and form, the parameters of the being break down in imbalance, which implies the death of the system* (Max. S x Min.T Ù Max. T x Min. S) , as it changes from the balanced S=T “Newtonian’ region to one of distorsion or ‘Lorentzian region’ (taken from gravitational space, where the system dies near c-speed when form and motion become distorted).

Those extremal regions are the ones in which the system dies by excess of form or motion, provoking its entropic dissolution. While in the border with ∆-1 the system is born as a seed that will expand its vital space, as it imprints its genetic information, till finding a balance between both, when the system finally emerges as an 1-whole *and as long as it stays in that region of balance, it will survive. But if it moves to the region with parameters proper of its larger nested ∆±1 Universe, it will either *collapse by excess of form- density (spatial form) or by excess of energetic acceleration (temporal motion).

So we talk of the graph also as the region of domain of the function of existence – i.e. if the parameter were temperature, for man it will be from 35 to 40 degrees where the human organism can survive. For an atom between 0 K and c speed, and so on.

So we consider a different ‘finite view’ of continuity, as the region in which the function DO actually survive entangled between its S and T elements; and the discontinuity barrier where its s=t elements (particle-wave, head-body) disentangle by overdrive of energy or information – where the spatial view of the function of existence tends to hyperbolic form:

In those ‘verges’ the S=T balance of T.œ, break because the *nested Universe have larger S=T parameters, and the T.œ wants to reach those ∆+1 limits, but it can only ‘attempt’ to reach one at a time, breaking its inner balance. So* its change in ∆, S or T break the system:

S: In space it means its topological form suffers a topological tearing or transformation. Then the membrane and singularity split and the system dies.

∆: In scale it means its social evolution overcrowds the carrying capacity of its world and the herd collapses. They are the minimal and maximal regions in scale.

T: In time it means an excess of warping/wrinkling for the membrain that detaches from its vital body, exhausted. They can also be the initial and final ‘conditions’, that is the value of the function in its t=0 and t=t, beginning and end of its worldcycle.

@: In the mirror of the language of calculus those limits mean regions where the differential equations have no longer meaningful solutions.

¬: in entropic terms, they are limits for the function of existences and its space-form, and time motion.

**Conclusion: A fractal Universe of discontinuous regions of existence.**

The Universe is discontinuous. To differentiate a function we do NOT need absolute continuity but the existence of an infinitesimal 1/n, and no jump between ‘neighbourhoods’, which should be no further than 1/n distance either in the X or Y coordinates. ‘Adjacency’ of the function then is defined by discrete 1/n intervals, which suffice in *Nature=reality, regardless of mathematical methods to define them.*

** **

**I AGE: TRILOGIC Of EQUATIONS: ∆-POLYNOMIALS. T- PROBABILITIES. S-TATISTICS.**

We are now equipped with the simplest functions that form together a trinity of organic elements, from the sum to the product to exponentials and power laws, to construct a ‘world’ in which only those dimotional operands do exist, forming sentences that represent ‘trans-formations’ of spacetime through a dimotional operand.

A function *f* is a mathematical rule that assigns to a number *x* (in some number system and possibly with certain limitations on its value) another number *f*(*x*) *through a set of operand that reflects the mode in which one transforms into each other*. For example, the function “square” assigns to each number *x* its square *x*^{2}, which means it has been ‘operated’, X x X, through a re=productive dimotion.

Thus common functions are definable by formulas, *which are related to the operands of equivalence ∆s ≈ ∆T and at least a different operand=dimotion of transformation, or else it would be a mere identity,* such as:

∆§: Polynomials of the type, *f*(*x*) = *x*^{2}. The logarithmic function log (*x*); & the exponential function exp (*x*) or *e ^{x}* (where

*e*= 2.71828…; and the square root function √

*x*.

∆T: Trigonometric functions, sin (*x*), cos (*x*), tan (*x*), and so on.

And so we can apply the different ilogic structures of reality, of which the simplest one is the S=T duality, which in its more complex form becomes the T-worldcycles≈ S-superorganism duality that merges both space and time elements.

In the simple world of the 3 stairs of ± operands this give us two huge fields of mathematics.

** S=T: Probabilities in time and statistics in space**, of which the most interesting element are the study of simultaneous curves (Conics), which we have considered on our paper on geometry (spatial mathematics).

∆-scales: polynomials. So we shall consider only a few questions of polynomials and lineal algebra, in space and study with deep philosophical insights on the nature of time, the laws of probabilities, before we get into the ‘*wider’ operands, able to penetrate further beyond a single plane of the 5 ^{th} dimension, the trigonometric and calculus functions. *

We can then consider that the age of functions divides clearly in 3 degrees of complexity understanding of the 5 Dimotions of existence with different finesse:

*The lineal youth of discrete ‘herds’ and simplex dimotions: ∆0*

– The age of polynomials and probabilities, which are the ‘derivative’ finitesimals in time of an event and statistical populations, which are the equivalent to the integral of a whole.

– The final culminations of the age of polynomial simple functions, with the introduction of the complex plane, the angular functions and the Euler identity that relates both phases of the world cycle of existence through exponential equivalences with sinusoidal function.

*The curved, entangled age of calculus and sinusoidal functions: ∆+1*

– The age of calculus, where we add derivatives and integrals that play the roles of probabilities and statistics, perceived with much more finesse.

*The modern age of functionals, functions of functions: ∆+¡*

To finally end in the XX century with the expansion of the discipline, already in the 3^{rd} age of mathematics to functions of functions in different mental spaces with n-levels of dimensions.

To understand the subtle differences of precisions between one and the other age, we need to incorporate the approximations of polynomials with derivatives…

The existence of limits to any lineal approximation of the cyclical reality of the Universe, a deep philosophical question studied in our analysis of the divergence and creation of futures that are ultimately closed is also deep, an in essence establish a limit of validity for any lineal approximation, obvious in the Taylor series of differential around a 0 point, which only approaches (with the notorious exception of the exponential decay function of pure entropy, in which both coincide), the curved function to a limit normally that of a single cyclical repetition of the function (as in the sine approximation).

Linearity thus only gets to a point or limit given by a ‘cyclical repetition’, either to the next plane, or the end of a world cycle, or the end of a stœp Dimotion of existence.

The key ¬Algebraic concept of ∆st systems is the existence of a STable region of balance between planes or topologies where the asymmetry of the system is fairly lineal operated in decametric Planes of growth and superposition, and the regions of relative past and future, | or O, ∆-1 or ∆+1, where there is a split towards the purity of motion or form, disconnected parts or wholes, accelerated vortices or lineal scattering and must be operated not with scalar powers but finitesimal integrals and derivatives, more precise in their measure of the ‘curvature’ of the phase space we study.

It follows that when studying more than a plane of existence, it is better to approach the question through the more sophisticated procedure of the integral and derivative operands, that first localizes the minimal ‘finitesimal’ of change through a derivative and the integrates it along a varying ‘curve’ that better reflects the 3 ‘different’ sections of a flow of space-time evolving through Planes, with its central lineal region, better suited for multiplications and simpler power laws, which become hyperbolic in the decaying and emerging frontiers of the plane.

Then there is the question of transformations between space and time and 5D a(nti)symmetries, which is an essential part of classic ¬Algebra and we resume in those terms:

Integral transforms make possible to convert a differential equation of 5D space-time within certain boundary values (time membrane, which limits the equation as a ‘real system’, not an infinity, into terms of an ¬Algebraic equation that can be easily solved (a polynomial which is a result in a single space-time plane). And this transformation obviously should be of two canonical forms. And as it happens there are 2 canonical transforms:

– A spatial, lineal transformation, and this is the Laplace transform: f(p), defined by the integral:

**F(p)=∫∞0 e-pt F(t)dt** .

The linear Laplace operator L thus transforms each function F(t) of a certain set of functions into some function f(p) and it is used most frequently by electrical engineers in the solution of various electronic circuit problems.

A temporal transformation and this is the Fourier analysis, which proved that a function y = f(x) could be expressed between the limits x = 0 and x = 2π by an infinite series of waves:

**F(x)=1/2 α ∑a cos kx + b sin kx.**

That is an equation could become a cyclical time dependent equation developed as a sum of harmonic waves.

And finally the inverse, the fact that a function could be converted into a 5D analytical equation between Planes of the 5th dimension is proved by the third most used approximation of functions, the Taylor series, which expresses a function f—for which the derivatives of all orders exist—at a point a in the domain of f in the form of the power series:

**∑∞∆=0 f (∆)(a)(z-a)∆/∆!**

In which Σ denotes the addition of each element in the series as ∆ ranges from zeroth (0) to infinity (∞), f (∆) denotes the nth derivative of f, and ∆! is the standard factorial function.

So this 3 transformations a means – and its applications enlighten an infinite number of real equations that the different 5D Planes of reality can transfer energy or information.

That is, a 5D flow of energy and information can travel into a single membrane with absolute accuracy (no loss of entropy, no need of transforms or groups to resolve them.

But there is a minimal loss of entropy when we transform between planes back and forth (of information or energy) as the transform is NOT absolutely exact – as for it to be exact the number of terms normally tend to infinity, which is not possible in the finite duration of any flow between ∆±1 Planes of the 5th dimension.

Further on it is important to understand the meaning of the operands and the laws of relative equality and dynamic transformations of ¬Æ where equality never fully exists, but we transform, F(t)<=> F(s) as in E<=>Mc², or we approximate values through an evident property ≈.

**2D: ± **We have seen how trigonometric functions describe in growing layers of complexity by combining with other Dimotions as vital beings do the more complex dimotions of communication, reproduction and social evolution.

So the next question is can the sum and its inverse the negative numbers, such a seemingly simple functions do the same pentalogic?

The answer is yes, but again as in the case of the ‘angle’ of perception, things must become more complex through the combination of the different operands, while the sum/negative number remain ‘dominant’.

**5Ð:** Social evolution into herds, the 5th Dimotion is self-evident the dominant element of the sum.

**2D:** Easy to explain is the Dimotion of locomotion as a sum of steps measured by the frequency of those steps in sequential time (though better expressed by multiplication

**3Ð:** Then the sum can ‘progress’ in ‘accelerated’ growth, and that will express the 3rd Dimotion of reproduction.

**4Ð:** While negative numbers will express the 4th Dimotion of entropy

**1Ð:** Angles of perception are sines, and as such its derivatives capture a ‘quanta of perception in time’.

The 5 Dimotions of space-time that encode, the social evolution, darwinian locomotion and entropic feeding, decay, reproductive growth and perception functions of a super organism are mirrored by the 5 actions=motion=operators of mathematical space: s, ±, x ÷, xª log, ∫∂… As they are encoded in species by genes, in human minds by sentences of words, in light by its dimensions of form and speed, in atoms by quantum numbers. And many other ways in different syntactic forms as the ‘languages of god are infinite’.

Such operands can form sentences with relational equivalent and order = ≤ ≥ «» symbols of its 5 dimotions in dynamic form.

In those sentences the question of causality, is expressed in terms of independence.

It seems then that most *spatial functions are dependent on time the independent factor: *

**$=ƒ(t)**

We interpret independence in terms of order from motion, the original substance of time into space that extracts its information: as functions are first motions in time that stop and become ‘forms’ of space, leaving a past-memorial trace. which if NOT erased becomes a population of space, which moves again and then becomes a population and in this manner reproduction of dimensions takes place, building a being of growing ∆Dst.

In most functions the independent ‘submissive’ parameter is a function of time-energy from where the, the dependant dominant parameter that forms it, a mental or topologic simultaneous space, arises.

How this is expressed in ∫∂ terms becomes then clear: since time is discrete, discontinuous, made of T.œ.s, moving, stopping (often perceiving), moving and stopping, we must first ‘encounter’ the minimal step of the time motion, and this is what we shall call dt, and then move, stop and move stop a number of steps, which we integrate, building in this manner a new dimension of space-time.

So the combination of ∫∂ *is in fact a process of creation of an ST dual dimension of space-time. And that is the ultimate meaning of it.*

So when study those simplest equations of physics, we shall consider those in which we make a ‘ceteris paribus’ rhythm of considering it first from the point of view of ‘time’ steps and then from the point of view of ‘space’ integrating them as *a simultaneous space, when we have ‘traced’ enough steps to make that simultaneity meaningful *. And this is the meaning of a definite integral.

**Rates of change. The stop and go motio**n: stœps.

Finitesimal changes are related to the fundamental beat of the Universe, the stop-form-space-perception, go-motion-time, beat of the Universe, which we shall call a stœp, the discrete way of motion of tœs through SPace, which often as in movies we perceive in continuous mode eliminating the stop element:

∆S(top)->∆t->∆S-≥∆(S)t(ep). ∆±¡: Planes, Planes

In pentalogic we shall consider some basic elements of analysis in its calculus of parts and planes.

**Galilean px in analysis: finitesimal steps (derivatives) integrated to calculate a cyclic whole.**

Further on analysis has over all other branches of mathematics a special quality to study ‘changes’ between planes of the fifth dimension, as multiple derivatives ‘jump’ (albeit with different degrees of ‘focus’) better than mostly ‘lineal polynomials’ between planes, and the ‘curvilineal, Lorentzian’ variations, slow downs and accelerations on the S x T= K parameters happen between Planes:

*The formal stience of the 1st and 5th Disomorphisms in the mathematical mirror is analysis, which deals directly with the relationships between ∆-1 ‘finitesimal’ parts’ and (in)’finities’. Two new terms we still accompany, with the lost inflationary term ‘in’; since infinitesimals and infinities are a Kantian paralogism; as all planes have a limit in its quantic units, and all wholes a finite circle that encloses them into a relative 0’-1 ‘circle unit’. *

Besides the duality of the 0’-1 probabilistic mind unit which reflects the external 1-∞ universe, a second duality that weights heavily in analysis is that of perception of lineal vs. cyclical form: We are minds of space that measure time cycles: ∫@≈∆ð.

Hence the equation of mind-measure defines the understanding of differential calculus: As always in praxis, the concept is based in the duality between *huminds that measure with fixed rulers, lineal steps, over a cyclical, moving Universe. So Minds measure Aristotelian, short lines, in a long, curved Universe.*

So the question comes to which minimalist lineal step of a mind is worthy to make accurate calculus of those long curved Universal paths.

The general rule to identify both polynomials and analysis, is this: **Y=S= ƒ (x=t)**

**The difference between lineal polynomials and non-lineal analysis**

In the graph, we explain the difference between a polynomial ‘regular’ description of a system as it changes in the “Newtonian’, social scale in which changes are not of quality as much as of quantity and the analytical region in which there is a change of structure and hence of quality with irregularities better shown by analytic operations.

It follows that more important than ‘variables’ are to ¬Algebra ‘operands’, whose encoded meaning and ‘magic’ way of relating systems to get a ‘future or present’ outcome by merging them according to certain rules of ‘creative engagement’, truly gives the power to ¬Algebra to mirror the a(nti)symmetries of the Universe.

The key connector of T.Œ with classic science is the full understanding of the dual ¬Algebra operands, ±, x/, ∂∫, √xª as part of the ¡logic, pentagonal game of reality in *all its mirror symmetries; that is, as dimotions≈actions and structural elements*, whereas an immediate correspondence between those operands and the ternary elements ∆@st can be established as follows:

The sum-rest are the inverse arrows of the simplest superpositions of dimensions *between species which are identical in motion and form.*

The product/division rises the complexity of operands a first layer, and serves the purpose, besides the obvious sum of sums, of calculating the margin of dimensions, as combinatio