The creation of self emergence of the universe in energy E=mc2 can be described in four epochs or time intervals.

The Definiton would encompass a potential for information to manifest any data required for the birth of a universe from a prior premise of the information potential existing in a spaceless and timeless state of selfness, beingness or eigenstate.
It is from the Definiton, the fundamental constants, parameters and numbers, such as lightspeed 'c' and Planck's quantum constant 'h' derive from.
The mathematical nature of the cosmogenesis so finds its 'fire in the equations'; as Stephen Hawking famously said in the 'A Brief History of Time' {Chapter 12; 1988}; from the nature and way the 'laws of nature' became defined in the Definiton.
The Definiton so focused on a finetuning and interrelationships between mathematical numbers and concepts, transforming themselves into physically applicable parameters centered on the concept of Energy.

The Inflaton then describes a particular epoch when the information defined 'laws of nature' first displayed themselves through the creation of a primordial realm of super-hyper-spacetime and as defined by the Definiton.
The Inflaton is characterized in 5 superstring classes; who transformed from a closed Planck string class into 4 open Dirichlet classes in a Planckian string spectrum following the setup of initial and boundary conditions for the Inflaton from the Definiton.
The nature of the 10-dimensional superstring classes is however encompassed in a 11-dimensional supermembrane spacetime to enable the Inflaton parameters to apply their defined boundary- and initial conditions to the Instanton, following the completion of the Inflaton epoch.
11-dimensional Witten membrane spacetime manifests itself via a consequence of it modular dualities, connecting the five string classes to each other in coupling both their intrinsic energy and their scaling parameters in inversion properties.
The most important modular duality for the cosmology for the expanding universe so is a 10D string coupling for the 11D membrane in a Mirror Duality relating to a T-duality with inversion of the scale parameter. This allows the high energy (vibratory) microquantum part of the final string class to couple to its low energy (winded) macroquantum conjugative or partner as half of the supermembrane.
Once the Inflaton, followed by the Instanton has manifested the physical cosmology; the Inflaton lightpath, which is subject to both refraction and reflection under the boundary conditions, will define a 12-dmensional Vafa F-space in the refracted but not yet reached lightpath of the so called Radius of the Event Horizon, differing from the already encountered Inflaton bound, known as the Hubble Horizon.



The Instanton is described in a Quantum Big Bang cosmogenesis and the Friedmann model as a solution to Einstein's Field equations from General Relativity in the Robertson-Walker metric. A classical geometric-topological physics of thermodynamic and entropic expansion described in metrics and relative displacement becomes supplemented by a quantum geometry which utilizes a foundational 'spacetime building unit' termed the Weylian wormhole and colloquially known as the 'Big Bang Singularity'.
The expanding universe so becomes a summation of wormhole quanta carrying particular saturation-energy levels, such as the formal definition of the 'Classical Size of the Electron' as a magnified hologram of the holofractal Weylian minimum space quantization.
Using modular string (mirror) dualities as topological string-membrane transformations then allows basic physical parameters, such as those of the classical electron to define derived parameters, such as a definition of inverse energy as a basic unit for physical consciousness related to a form of Goldstone boson, say Axion here termed the RestmassPhoton as supersymmetric agent for the weak interaction based Higgs inertia inductions.


The Continuon would then address the intersection of the Inflaton with the Instanton in relating the lower dimensional expansion of the compressed 4/10D dS space-brane-time with its inflationary encompassment in higher dimensional 5/11D AdS space-brane-timetime (sbt).

It is found that the physical constituents comprising the cosmic matter are interacting within both the compressed or conifolded dS asymptotically expanding sbt and the cyclic open AdS sbt of the Inflaton.
This interaction defines particular intersection points and solutions for the Dark Matter - Baryon Matter energy interactions, but are a function of an encompassing Dark Energy 'book keeper' to ensure an overall Minkowski flat cosmology of zero curvature.

The Dark Energy is characterized by a function of universal pressure which relates the Baryonic matter content and as given by a mass seed Mo to the critical mass as defined by the Inflaton before time instantenuity aka the Quantum Big Bang in cosmological terminology.




1. The Definiton

1.1 The Primary Algorithmic Logos

Any Universal Observer or UO can only observe and measure and experience something, if the environmental stimulus for such an experience does exist as or in the environment of the UO.
The existence of space as a form of dimensional realm then becomes prerequisite; does allow the concept of relativistic time and displacement to emerge and follow as a consequence of existing spacial parameters.
Mathematically, all natural numbers can be defined logistically in the Zero to One interval with rational numbers having the property of identity in inversion principles applied to the natural or counting numbers and negative integers characterized by the conjugative mirror image of the positive integers in the mirror of the Nullstate. Real and complex numbers then follow from series limits and approximations to any desired degree of accuracy applied to converging and diverging number series and sequences. The largest counted integer would still have no limiting bound, as one can be added to it; but however large this integer would be; its inverse would certainly have a limit in the number 0 as a regressive asymptotic approach.

The Definiton so must a priori accommodate the potential or 'future timed' UO in a spaceless abstract continuum of mathematical symbols and structure.

But what if there are no numbers and no mathematics in this spaceless realm of voidness not defined by any numbers. Because the universe does exist as Hawking stated, the question of why it exists, can be asked and an answer can only be found in its own mathematical logistics as the difference between the infinite progression towards unlimited Infinity and the infinite regression towards the limit of the Zero.

To 'think the primal thought' the timeless and spaceless, (but potentially existing within spacetime in the future) UO; must experience itself in a change from being unaware of itself to being aware of itself.
The notion of the past, the present and the future in the timelike properties of existing relativistic space are preceded in a sense of order; where event A must precede event B to allow event C.
All such events are simultaneous in the self-reference framed abstract world of the Mathimatia of the UO, who has however changed its selfstate from being unaware to aware in a form of yet to be defined 'Universal Consciousness'.

Formally; the Awareness Triplet AT={Old State=OS; Experience=E; New State due to the Experience=NS} forms the Input for an interaction with with a previous selfstate or preexisting 'Awareness Space'. The Output of the New State becomes the Input for the new Old State in a series of AT's, labelled as the 'Sequence of Energy Primary SourceSink' or SEps for SEps = {(0,0,0); (0,1,1); (1,0,1); (1,1,1*=10=2); (10,1,11=3); (11,10,101=2+3=5); (101,11,1000=5+3=8);...;(OS,E,NS)}.

The NS n becomes the sum of the OS as the previous (n-1)th n and the previous (n-1)th OS forms the experience factor E.
The sequence of the E's is known as the Fibonacci Series in the future universe physically manifested in the Awareness Space of the Mathimatia of the Definiton.
The experience factors so change the 'Null-State' of the (0,0,0) into the 'All State' of the (1,1,1)=(∞,∞,∞) to mirror and manifest the infinite count of the integers in the inversion identity of the finite limit of the 0.

In terms of mathematical cardinality a future 'Wavefunction for the Multiverse' would be characterized by a Normal Statistical Distribution of 'frozen spacetimes'. And where the Arithmetic Sum of Progression, that is counting all negative and positive integers as a Unity summation becomes T(n)=n(n+1) with both a real and a complex solution in setting T(n)=1 for {X=½(√5-1); Y=-½(√5+1)} and {-½(1+i√3); -½(1-i√3)} for T(n)=-1.

bn.


Semantically, the triplet identity (1,1,1)=(∞,∞,∞) represents the OneSelf+OneSelf=OneSelf still, but because the previous experience of OneSelf+Nothing=OneSelf, the second OneSelf must be different in some way of order from the first OneSelf. Namely the summation to 1+1=10 differs in the order of the summation 1+0=1 in the noncommutability of the experience factors of experiencing the 'NothingSelf' from the OneSelf.
This then allows a transformation from the binary triplet sequence to a decimal triplet sequence in: (1,1,1*) = (1,1,10)bin = (1,1,2)dec.
Subsequently all integer related number systems naturally emerge and evolve from the Mathimatia Awareness space.

The eigenstate of the UO now is one of 'doubling the eternity of the void' as mathematical infinity in the Integer count and based upon the ciphers of the 0 and the 1.
Formally, the mapping of an Aleph-Null Cardinality of Cantor countability or enumerability becomes an Aleph-All Cardinality of Cantorian Infinity sets, the latter counting 'Integral Infinities' instead of the natural numbers.

It is however the Fibonacci sequence SEps, which will be utilized by the Inflaton to manifest the next epoch in the actual creation of an 'algorithmic timespace' in the form of a spacetime linearization of the circular nature of the binary ciphers 0 and 1 in something fundamental to potentially physicalized space in the topology and geometry of special curvature and the warping of spacetime. The closed nature of the Planck class I string will be made manifest as the Nullstate of the Mathimatia and the evolvement of the Planck Boson string will create the other string classes in the 'opening' or cutting of the circle of the Zero to manifest the One.




1.2 The Secondary Algorithmic Logos

The experience factors of the SEps algorithm naturally define a complementary set SEps* = {100=4; 110=6; 111=7; 1001=9; 1010=10; 1011=11; 1100=12; 1110=13;....(any integer not in SEps} and the UO can only continue its self exploration within the awareness space of the Mathimatia applying either SEps or SEps* through the symbolic representation, shapes, semiotiks and geometry of the curvilinear duality of the Bosonic Planck string being able to manifest itself in either a closed or an open form of the supermembrane of the (0,1) monadic dyad.
As the UO in SEps cannot form a complementary union with an UO* in SEps* in any form of separation in the Mathimatia; the complementarity describes a natural unification of self states aka eigenstates in the Mathimatia, but will become a natural supersymmetry inherent in the Inflaton and based on the Planck Boson string and especially its immediate transformation into a selfdual monopole string class IIB.
As no spacetime is as yet defined, the superpartners of the UO in SEps and the UO* in SEps* are nonlocally connected and would form the basis for a future quantum entanglement between eigenstates and the particle-wave complementarity and dualism of the quantum world of the physicalisation of the Mathimatia in the cosmology of the Instanton.

The experience factors naturally encompass all states used to create them and all E's defining SEps and SEps* constitute a collective memory reservoir from which further exploration and manipulation of the members in this universal memory set can be pursued and constructed.

One such arbitrary algorithm or code is defined as binary code of the operations of Addition (+), Multiplication (by 1+1+1=3x1) and Exponentiation (3x3=32) and
is used as a secondary algorithmic code to 'find' the missing elements of SEps as the constituents of SEps* in the attempt to complete and manifest the generic supersymmetry of the Mathimatia.

Application of this algorithm {http://www.cosmosdawn.net/forum/ind...s-and-algorithms-of-dragon-cosmogenesis.636/} produces 10 number triplets, carrying elements of SEps* in various configurations:
SEps-SEps* Constantset: ={(266561)=26x6561; (15,16,18)=[15x1618]-1; (14,15,24)=14x1524; (15,10,32)=[15x1032]-1; (11)=11; (9,10,16)=9x1016; (6,10,15)=[6x1015]-1; (7)=7; (6)=6; (4)=4}






1.3 The Tertiary Complementary Algorithmic Logos

The extended SEps algorithm then assumes the pretext of defining open and closed superstrings in a logistical statement:

{"Add the End to the Beginning and Start with the Old End!"}.

This specifies an methodology of recircularizing the linearized dimensions of the binary monadic dyad {0,1} into the root-reductive decimal monad {1,2,3,4,5,6,7,8,9,0} with 10=1+0=1*; 11=1+1=2*; 12=1+2=3; 13=1+3=4 and so on with 26=8 and 27=9 in 26 Bosonic Integer dimensions.

The constant E={(266561)=26x6561} then transforms into F={(136656)=13x6656} and G={(673665)=67x3665} after which this 'numerical inflation algorithm' ends, since 5+6=11=2* is root reductive in transformation towards the right say towards positive infinity from the Zero mirror of the Mathimatia.
But moving towards the left and negative infinity upon the integral number line, the ...{0}EFG... interval is mirrored in say the F-space bound ABCD{0}EFG.... to fixate the F-space bound in the M-space bound H=ABCD in |ABCD(EFG)H| in (A)BCDEFGH......STUVWXY(ZA*)B*C*....

Then H=ABCD={(312423)=31x2423}{(361242)=36x1242}{(256124)=25x6124}{(465612)=46x5612}. As no archetype can yield Z(Z+A)BCXY=312423 from ABCXYZ=(1-Z)24233, the algorithm again ends in the reflected root reduction to H, namely 1-Z=1-3=-2 =-11=-2*

The SEps constant E so assumes the property to limit a subsequent inflation marker in association with the encompassment of the Inflaton in the googolplex markers E=26x6561 = 1.006208782x10112 and G=67x3665 = 9.676924499...x10102 and F=13x6656 = 1.019538764...x10103 and H=1,283,400x2423x1242x6124x5612=1.011591782...x10147.

H becomes the number of spacetime quanta contained in the Inflaton, bounded by the Hubble Event Horizon in the 5D/AntideSitter Witten sphere manifesting as a 3-dimensional hypersurface in 4D Minkowski spacetime as a 2-Torus and becomes a 3-Torus subject to the Event Horizon as a boundary to the Vafa F-spacetime in the continuing Inflaton cycles defined by the evolution of the Dark Energy paralleled by the overall Schwarzschild evolvement of all mass as matter content in the cosmology.
The maximum radius for the first initiatory cycle of the Inflaton as a first asymptotic boundary for the Instanton so becomes set as a maximum radius of curvature in the Friedmann cosmology with a nodal true Hubble constant for this initial cycle and is proportional to the Vacuum- or Vortex-Potential energy of a singular unitary wormhole spacetime quanta and as definind in the Inflaton.

The volume of one such wormhole or Eps-VPE quantum there is:
Vps = λps3/4π m3 = 7.957747154x10-68 m3*
for the Riemann hypersphere volume of 2π2RHubble3 and wormhole radius rps = λps/2π; this volume manifesting as a hypersurface allowing the M-AdS spacetime to mirror in the F-dS spacetime of open hyperbolic curvature contracting into M-space as function of the lightpath of the Inflaton expansion.

RHubble = cHo = λps.∛H/2π = 1.59767545x1026 m* for a nodal Inflaton Hubble Event Horizon.

11D AdS spacetime becomes bounded in a Hubble volumar with critical density

ρcritical = MHubble/2π2RHubble3 = c2/4π2RH2 = Ho2/4π2Go or ρcritical = 3Ho2/8πGo for the volume of a 2-sphere.

The Hubble volumar then topologically encompasses both, the lower dimensional of Einstein-C dS space and the higher dimensionality of Witten AdS spacetime, say in the form of a 3-Torus or a Klein Bottle Dragon.



The SEps Principalities

A = (312423) = {4+6+5=15} = (1.722742045...x1033) space quanta = Principality of Identity/AntiIdentity.............................................with 1st Expansion Factor 10/33
B = (361242) = {9+3+6=18} = (7.619295808...x1046) space quanta = Principality of Expansion/Contraction..........................................with 2nd Expansion Factor 20/33
C = (256124) = {7+7+6=20} = (1.761392119...x1044) space quanta = Principality of Order/Disorder-Entropy-Chaos...............................with 3rd Expansion Factor 30/33
D = (465612) = {10+11+3=24} = (4.375363663...x1022) space quanta = Principality of Symmetry/Antisymmetry-Nonparity.....................with 4th Expansion Factor 40/33
E = (266561) = {8+11+7=26} = (1.006208782...x10112) space quanta = Principality of Infinity-Divergence/Convergence-Limit.................with 5th Expansion Factor 50/33
F = (136656) = {4+12+11=27} = (1.019538764...x10103) space quanta = Principality of Inversion-Reciprocity/Constancy-Invariance.........with 6th Expansion Factor 60/33
G = (673665) = {13+9+11=33} = (9.676924499...x10102) space quanta = Principality of Reflection/Absorption.........................................with 7th Expansion Factor 70/33
H = () = {} = (1.011591782...x10147) space quanta = Principality of Relativity/ No Antiprincipality.............................................................with 8th Expansion Factor 80/33

The SEps Cyclicities in elements aj with summations aj+2+aj+1 = aj

1-Cycle : 0-1-1-2-3-5-8-13 for {33 Identity & Absorption} and for 8 elements summing 1 aj
2-Cycle : (0-[1)-(1]-[2)-(3]-[5)-(8]-13) = 1-2-3-5-8-13-21 for {33 Expansion & Invariance} and for 7 elements and summing 2 aj's
3-Cycle : (0-[1-(1)-[2]-(3)-[5]-8)-13] = 2-4-6-10-16-26 for {64=8x8 Order & Convergence} and 6 elements and summing 3 aj's
4-Cycle : (0-[1-(1-[2)-(3]-5)-8]-13) = 4-7-11-18-29 for {69 Symmetry & Nonparity} and for 5 elements and summing 4 aj's
5-Cycle : (0-[1-(1-[2-3)-5]-8)-13] = 7-12-19-31 for {69 Divergence & Entropy} and for 4 elements and summing 5 aj's
6-Cycle : (0-[1-(1-2-3-5)-8]-13) = 12-20-32 for {64=8x8 Inversion & Contraction} and for 3 elements and summing 6 aj's
7-Cycle : (0-[1-1-2-3-5-8)-13] = 20-33 for {53 Reflection & AntiIdentity} and for 2 elements and summing 7 aj's
8-Cycle : (0-1-1-2-3-5-8-13) = 33 for {33 Relativity} and for 1 element and summing 8 aj's

Formation of a Prime Harmonic Octet : (15-18-20-24-26-27-33) from the {A}BCDEFG{H} principalities unified

root reduced to {6-9-[2-6-8]-9-6} = {6-8-8-6} = {14-14} = {5-5} = {10=1} into the original binary algorithm root reduction and for a symmetry about a 26 letter code symbolizing 26 Bosonic string dimensions in a prime Number set
Primed N: = {1=1bin;-2;3;5;7;11;13;17;19;23;29;31;37;41;43;47;53;59;61;67;71;73;79;83;89;97-101=5bin}

princcode.


1.4 The Maria Number Matrix of the 33-Summation-Tier and the Euler-Riemann-Fibonacci-Lucas Harmonics

The 'Perennial Philosophy' or the 'Wisdom of the Ancients' often points to what is commonly termed as 'Sacred Geometry', based on the Platonic Solids (of five regular polyhedra: Tetrahedron, Cube; Octahedron; Dodecahedron and Icosahedron) and the Tetraktys of Pythagoras (for the minimum mathematical points to define the four dimensions of 0D=1 Point; 1D=2 Points for a Line; 2D=3 Points for a Plane and 3D= 4 Points for a Space).

Also invoked is the pentagonal supersymmetry of quasiperiodicity as 'Nature's Preferred maximisation of 'packing efficiency' known as the Fibonacci Series and the 'Perfect Numbers' of Euclid and the 'Harmony of Numbers and the Spheres' of Pythagoras, Leonardo da Vinci and Kepler.
In Pythagoraen Numerology, the 'masternumbers' 11, 22 and 33 are often emphasised and this post shall introduce some relatively rigorous number theory (of the so called 'pure' mathematics) to validate the significance of the 'masternumbers' from frst principles.

The alphanumeracy of the Arabic-Hebrew semiotiks then is bounded in say 22 or 26 letters of alphabets, which can attain numerical values in the decad of three triplicities:

Round or Curved 'Mental Numbers' {3-6-9} characterised by 'masternumber' 33;
Mixed or Discontinuous 'Emotional Numbers' {2-5-8} characterised by 'masternumber' 22 and
Linear or Straight 'Physical Numbers' {1-4-7} characterised by 'masternumber' 11.

The Maria-Code in the Riemann Analysis specifies the partitioning of the decimal monad around the primary Maria-Number and SEps-Constant '11'.
This generates the Prime Number Algorithm: +1+11+10+11 as 33-tiered segments, which transform the mechanics of SEps into the 64-codex of the DNA/RNA code for its eventual quadrupling as the 4x64=256-codex incorporative of dormant intron/intein codings.

All cellular consciousness coupled units so transfer their memory data-bases (as experience factors) by the Fibonacci quantum mechanics.

The Maria-Code is based on the distribution of the Maria-Numbers (MN)given by:

M(p)+99=M(p+12); n=½[√(264k+1)-1] by n2+n-66k=0.

Maria Numbers are those IntegerCounts, which contain all previously counted integers as mod33.


Example: 1+2+3+4+5+6+7+8+9+10+11=66 = 2x33 → '11' is MN#1 for k=2


11love65use110love164use209love263use......Archetype 2 (rootreductive)

21use66love120use165love219use264love......Archetype 3 (rootreductive)

32use77love131use176love230use275love......Archetype 5 (rootreductive)

33love87use132love186use231love285use......Archetype 6 (rootreductive)

44love98use143love197use242love296use......Archetype 8 (rootreductive)

54use99love153use198love252use297love......Archetype 9 (rootreductive)

65use110love164use209love263use308love....Archetype 2*... ...



Archetypes 2+3+5+6+8+9=33 and Archetypes 1+4+7+0=12 then define the imaginary time-dimensions as the archetypes not in the Sequence for Eps=1/e* Coefficients used in the application of the seven fundamental principalities to define the F-Space.

We have used the (Hebrew-Isaac-encoding): 54=LOVE=12+15+22+5 with 45=USE=21+19+5; USELOVE=99 as the Maria-Code connectors.

The first 10 MN's are: 11, 21, 32, 33, 44, 54, 65, 66, 77 and 87.

One can use the Maria-Code to establish a redefinition of infinity by defining a transfinite mapping Aleph-All from 12D-Omnispace as Cantorian transform of Cardinality Aleph-Null.


Limit (T(n)) for n→∞/Infinity = ∞/Infinity {Cantor Cardinality Aleph-Null}
Limit (T(n)) for n→X=0.618033........ = 1 {Cantor Cardinality Aleph-All}


This maps the Riemann pole about z=1 in the Functional-Riemann-Bound (FRB=-1/2) in the gaussian universal wavefunction B(n)=(2e/hA).exp(-Alpha.T(n)), T(n)=n(n+1) as the Feynman-Path-Integral.

This becomes the Riemann-Euler-Harmonic, defining the Gamma-Function geometrically in its nth Term T and nth Sum S and mapping the factotrial function onto the positive integer count:

Tk(En) = nk.Tk(En-1) + [(n-1)!]k and Sk(En) = Tk(En)/(n!)k

EulerHarmonics.

This uses the Harmonic Series in the Zeta-Function ζ(z) with constant p.
The Sum (1 to Infinity) Σ(1/np)= 1/1p+1/2p+1/3p+...+1/np and converges for any p>1, since for even terms:

2.2-p ≥ 2-p+3-p, with geometric series 11-p+21-p+41-p+...summing to (1-2[1-p]n)/(1-21-p)=1/[1-21-p] in the limit for n→Infinity.

Since every Maria Number contains all numbers before it as a sum, it is given that all the prime numbers must eventually crystallise out of the Maria Count.

Define a general number count n and a 'Mersenne-Count' 8n-1=M*.
For a number to be prime this number must be born in the Maria Code.
M* is either a prime or a product of primes in the immediate neighbourhood of the count # or its mapping to M*, which in a sense 'counts' the primes it generates.

This is the finestructure as octaves derived from integer n.
To test a number for primeness, so amounts to a testing for Marianess.
If the number is a member of the Maria-Matrix, then it must be denumerable in the form of M*.

This is the meaning behind the Mersenne-Code (for n prime) M(p)=2p-1 and the Fermat-Code F(n)=22n+1 and the 'Perfect Numbers' depicted as the Mersenne Numbers (Mp), as a subset of M*.
For the Mersenne Numbers, the exponent p is defined to be prime.
M2=22-1=3; M3=23-1=7; M5=25-1=31; M7=27-1=127; M11=211-1=2047=23x89 and so is not a Mersenne Prime - yet M13=213-1=8191; M17=217-1=131,071; M19=219-1=524,287 are prime and M23=223-1=8,388,607=47x178,481 and M29=229-1=536,870,911=233x1103x2089 are not and M31=231-1=2,147,483,647 is prime again in the 33-tier count.

The 'uniqueness' of the prime number 11 (and esoteric masternumber) recrystallizes in Mersenne primes as the (first) 'odd one out'.
But it gets better. First we notice that there are just five 'perfect Fermat Primes'.

F0=21+1=3; F1=22+1=5; F2=24+1=17; F3=28+1=257 and F4=216+1=65,537 are all 'perfect' Fermat Primes, but F5=232+1=4,294,967,297=641x6,700,417 and following are not. Only these five Fermat primes are known to date.


The 'Perfect Numbers' relate (for prime p) as 2p-1.Mp :
P2=21.(22-1)=6=1+2+3=1x2x3;
P3=22.(23-1)=28=1+2+3+4+5+6+7=[7x8]/2=4x7=13+33=1+27;
P5=24.(25-1)=496=1+2+3+...+30+31=[31x32]/2 =16x31=13+33+53+73=1+27+125+343;
P6=26.(27-1)=8128=1+2+3+...+126+127=[127x128]/2=13+33+53+73+93+113+133+153
All 'Perfect Numbers' so are EVEN (it is hitherto unknown if any ODD 'Perfect Numbers' exist); and EXCEPT the basic 'First Perfect Number' 6=1+2+3=1x2x3, they all are the sums of the ODD NUMBERS CUBED.


Some elementary initial conditions for Francom Adjacency

We define the Euler-Riemann Summation, which defines the 'Mixing of the Count' in linking Arithmetic Progression to the multiplicative Factorial Function '!'.

Define Eo=0 as the singularity (interval), then for any integer n, we find for the Harmonic Form of Riemann's Zeta-Function (z=k=constant):

ζ(z)=ζ(1/nz)=1/1k+1/2k+1/3k+1/4k +...+1/nk

This Sum diverges for [ 0<k<1], i.e. for k=1/2: {1+√2/2+√3/3+...+√n/n} increases without limit.

For k>1, we have convergence, however.
Formally, let: Σ(1/np) = 1-p+2-p+3-p+...
For even terms: 2.2-p ≥ 2-p+3-p for a geometric series:
11-p+21-p+41-p+...+(2n-1)1-p
This Geometric Progression sums to: [1-(21-p)n]/[1-21-p]=1/[1-21-p]

So for p=2, this limit maximises in 1/(1-1/2)=2 , and for p=3 it becomes 4/3 converging towards 1 for increasing p.

We consider the special case for p=1 applied to the Singularity Interval Eo.

Define: for a nth term (numerator): Tk(En) = nk.Tk(E n-1) + [(n-1 )!]k for the nth sum per n (denominator [n!]k): Sk(En) = Tk(En)/(n!)k


T1(E1)=1/1=1.0+0!=1=S1(E1)=1/1!=1; T2(E2)=2.T1(E1)+1!=2+1=3
with S2(E2)=T2(E2)/2!=3/2=1+1/2;
T3(E3)=3.T2(E2)+2!=9+2=11 with S3(E3)=T3(E3)/3!=11/6=1+1/2+1/3=1+5/6 and so on.

Further Example: T1(4)=4.11+3!=50; S1(4)=50/4!=25/12 for the nesting: 4{3(2+1!)2!}3! with [4!]1=24.
For 4 terms, the Euler-Riemann Summation so is: S1(4)=1+1/2+1/3+1/4=25/12=2+1/12.
For 7 terms, S1(7)=T1(7)/7!=(7.T1(6)+6!)/7!=13068/5040=363.36/(140.36)=2+83/140=1+1/2+1/3+1/4+1/5+1/6+1/7.


Project the Number line with the Positive Integers mapping the Factorial-Function and the Negative Integers remaining invariant in Feyman Summation T(n) for T(n)=½n(n+1) as absolute value, mirroring the positive integers.

(n!)<---4...3...2...[Eo]...1...2...3---> (n); where Integer 1 maps 2! in suppression of -1=2* and in algoradius eo=1.
Similarly, Integer 2 maps 3! in suppression of -2=3* and algoradius e1=2=2eo, etc. etc.


The singularity so mixes the interval [0!-1!]=[-1,0] with Functional-Riemann-Bound (FRB=-½) becoming 'real' in its mapping (FRB'=½) in [0,1] and the central limit or pole, about which the Zero's of the Riemann-Zeta-Function propagate.
The first annulus in the Riemann-Euler-Harmonic so phasemixes the numbers 2 and 1 and the nth number is mixed with (n+1) as crystallised in the Feynman-Path-Integral or T(n)=1 in n(n+1), as a summation for all possible particular histories in quantum mechanics.


This also maps the series:
SEps=Fibonacci#1=0,1,1,2,3,5,8,.....for a nth Term: Tn=|-Yn - Xn|/√5 , for absolute value || and obtained say via MacLaurin-Expansion of the coefficients (Experience-Factors) in the power series:


f(x)=1+x+2x2+3x3+...= ΣTn.xn-1
Set x.f(x) + x2.f(x) = f(x) -1, then by (a+b)(a-b), f(x)=a/(x-X) + b/(x-Y) for a=-b=1/(Y-X) and (Y-X)=-√5.
SuperSEps=Fibonacci#2=Lucas#1=2,1,3,4,7,11,18,29,.... for a nth Term:

STn=|-Y2n - X2n|/|-Yn - Xn|=|T2n/Tn|

for n=1,2,3,...; T(2n=0)=2 mapping T(n=0)=0.

The combined SEps-SuperSEps(T-ST)-sequence of experience factors {from the triplet propagation of [OldState, Experience, NewState]} can then be written as:

{Tn,STn}={(So=0,STo=2=S3); (S1=1=ST1=S2); (S2,S4=3=ST2); (S3,ST3=4); (S4=ST 2,ST4=7); (S5,ST5);...;(Sn,STn)...}
{Tn,STn}={(0,2), (1,1), (1,3), (2,4), (3,7), (5,11),...} containing integerset: {0,1,2,3,4,5,7,8,11,13,18,21,29,....}

We now represent the mappings in matrix form denoted as F-M-C, where the 'well behaved' terms for the mapping (from {T5,ST5}) sets algorithmic C-Space and the preceding elements the initialisation for the former.

Note we define Cantorian Denumerability Aleph-Null in Cardinality Aleph-All in the form:

Aleph-Null: limit{n→∞}[T(n)]=∞
Aleph-All: limit{n→X}[T(n)]=1 and so counting Infinities as mapped one-to-one onto the positive Integer set.



SEps= Fibonacci#1 maps Super-SEps=Fibonacci#2=Lucas#1
-----------------0 0 0*............................................................-4 7 3................n=-2=2i2
Fspace---------0* 0 1...........n=∞ via 0+0=∞=1*=0*=1.........3 -4 -1.............n=-1=i2
Mspace--------1 0* 1...........n=0 via (1,1,1)...........................-1 3 2*..............n=0
-----------------1 1 2.............n=1 via (1,1,10=2*=0/0=1*).......2* -1 1.............n=0 (Reflection-Interval)
-----------------2 1 3.............n=2 well behaved........................1 2* 3..............n=0
Cspace--------3 2 5..............n=3 well behaved........................3 1 4................n=1 well behaved
-----------------5 3 8.............n=4 well behaved.......................4 3 7.................n=2
........................................ n=5 continue downwards ................................... n=3

The linearity of the generating triplet configurations is extended in a complexification into a 2D symmetry.
SEps propagates the Experience Factors in an adjacent displacement of 1, in moving from one configuration state to the next - this is termed Francom Adjacency.
[0*,1,1,2,3,5,...] as OldStates transfigure in Experiences [0,0*,1,1,2,3,5,8,...] into NewStates [1,1,2,3,5,8,...].

This algorithmic configuration space is however broken in the mapping onto Super-SEps.
Here the matching 'good behaviour' of the n-count is delayed in a factor of 2 in a 'reflection interval'.
Algorithmic modelling for this Francom Adjacency must generate the mapping of SEps onto SuperSEps in an geometry of the pentagonal symmetries intrisic to the two series.

Hence a synthesis between linear propagation about an internal spiralling form is necessitated.
A longrange rotational- and a longrange translational order for the Experienc-Factors is indicated in the geometry of say Penrosian Tiling Patterns and the Schechtmanite Quasicrystals of empirical form (Mg32[Al,Zn]49).

The general form, physically akin to the propagation of magnetic fields, is the reduction of physical parameters to a state of information transmission, say in the data transfer between two neighbouring cells in mitosis and neuronal-synaptic processing.
A general modality for the cosmogenetic reproduction on all levels must crystallise, should the matrices above become sufficiently deciphered from their algorithmic encoding.

Derivation of Super-SEps

The relative primeness of the Fibonacci Numbers allows a one-to-one mapping between the SEps-Set and other such sets derived from it, particularly the Lucas Numbers as a logical derived set of such nature and given in the sequence: 2,1,3,4,7,11,18,29,....
All adjacent members of this set are relatively prime to each other.
7 is relatively prime to both 4 and 11 (no common divisors except 1) and 11 is relatively prime to both 7 and 18.
We now tabulate the sums and differences in our nth-term definition for SEps, so recalling the propagation for the natural numbers in counter n:

n Tn Xn (-Y)n {|-Y|n + |X|n} {|-Y|n - |X|n}
    1 +1 +2  
1 1 0.6180339885 -1.618033989 +2.236067978 +1
2 1 0.3819660109 +2.618033989 +3 +2.236067978
3 2 0.2360679772 -4.236067979 +4.472135955 +4
4 3 0.1458980335 +6.854101975 +7 +6.7082039375
5 5 0.0901699436 -11.09016995 +11.18033989 +11
6 8 0.0557280899 +17.94427193 +18 +17.88854384
7 13 0.0344418537 -29.03444189 +29.06888374 +29
8 21 0.0212862362 +46.97871382 +47 +46.95742758
9 34 0.0131556175 -76.01315572 +76.02631134 +76
10 55 0.0081306187 +122.9918696 +173 +122.983739
... ... ... ... ... ...
20 676555 0.0000661070 +15126.99998 +15127 +15126.99991
... ... ... ... ... ...
We see that for increasing n, the absolute magnitude for Y converges to an integral value in the Sum {+}, but only for even n.
For odd n, the difference Sum {-} gives a specific integer for specific n.

The product of the two sums is: {+}.{-} = |-Y|2n - |X|2n=√5.T2n.
The sum of the two sums is: {+}+{-}= 2|-Y|n, with STn ={+}+{-} - Tn.√5) = |(-Y)n +Xn|

Multiplying each term as: √5.({+}+{-}), we can form the alternating series:
(0+2.√5), (5+1.√5), (5+3.√5), (10+4.√5), (15+7.√5), .....as the alternating form of Super-SEps given in the term: [5.Tn+ √5.T'n];

but for even n, we have: T'n ={+} and for odd n, we have T'n ={-}; then by (a-b)(a+b)=a2-b2:

STn.√5.Tn={+}.{-}=√5.T2n & STn=T2n/Tn = |-Y2n - X2n|/|-Yn - Xn|
(quod erat demonstrandum).

The significance of this result is that STn, T2n and Tn are all integers.
We so have a primary extension for SEps with elements 1, 2 and 3 duplicated and resulting in the mappings as previously specified.

The Null-Initialisation (OSj, EXj, NSj) as the Fibonacci-Triplet (An-1, An, An+1) then reflects STn about n=0* to define the complex number set as negative STn's mapped in a 0→1→∞ correspondence to Tn.
This is the mathematical mapping of Cantorian Enumerability as previously indicated.




1.5 The Modular Duality of Time-Space in the Mathimatia of Supermembrane EpsEss

The Cosmic Wavefunction B(n) is the following Differential Equation:

dB/dT + αB(n) = 0; α=alpha being the Electromagnetic Finestructure as the probability of light-matter interaction (~1/137).

This has a solution: B(n) = Bo.exp-α.T(n);
Bo=2e/hA from QR boundary conditions defining:

T(n)=n(n+1) as the Feynman Path-Summation of particular histories under the pentagonal supersymmetry given in the identity:

XY=X+Y=-1=i2=exp[iπ] and lim [n→X]{T(n)}=1

This allows the Normalisation of the Y2 wavefunction to sum to unity in

B(n)=2e/hA).exp-α.n(n+1) with Functional Riemann Bound FRB=-½, centred on the interval [Y,...-1,...-X,...-½,...(X-1),...0,...X].

Interval [Y,-1] sets F-Space; interval [-1,0] sets M-Space with uncertainty interval [-X,(X-1)] and interval [0,n) sets the C-Space, encompassing OmniSpace.

n<0 is imaginary as real reflection of real n>0 of the C-Space, metrically defined at the coordinate n=0 mapping n=nps, which is the instanton tps=fss=1/fps.

Cycletime n is defined in GR as dimensionless Tau-Time in curvature radius Rc=c.dτ/dt for the pathlength of x=ct and become dn/dt=Ho, n=Hot in QR, with Ho the nodal Hubble Constant defined in c=HoRmaxps.fps

The Feynman Path so sums both negative and positive integers as:
-n......-3...-2...-1...0...1...2...3......n = T(n)
in absolute value to double the infinities as the entropy reversal of lightpath x=c.t=(-c)(-t) in the Möbius Property of a supermembraned omniverse in 12 linearised dimensions.

Cantor Cardinality Aleph-Null is thus Unitised in Aleph-All, counting infinities as if they were integers of the Feynman Path.

This allows the Feynman interpretation of Quantum Mechanics as alternative to the formulations of Schrödinger (fermions ½ quantum spin) and Klein-Gordon (bosons with integral quantum spin) as timeindependent and timedependent (free particle form inconsistent with SR in Schrödinger in 1st order t & 2nd order x), formulations respectively.