Fundamental Constants from First Principles and a Pre-Big Bang ...

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Tony Bermanseder

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Subject: Fundamental Constants from First Principles and a Pre-Big Bang Algorithm Date:

Fri, 2 Oct 2015 17:09:14 +1000

Fundamental Constants from First Principles The Mother of all Boundary Conditions must be the statement of: "No Boundary conditions apply!" So we consider a NullTime and a NullSpace in an undefined, say Platonic-Penrosian 'Space' of mathematical abstraction. This is like a 'thought' or an 'idea'; how can IT become real or even manifest as a physical and quantifiable universe? In hindsight one knows how to answer this. An idea is born from observations; mental energy is linked to passion and the emotions, breeding the inventiveness to engage in a course of experimentation and to bring the idea (say a lightbulb, following the observation of a lightning strike) into physical manifestation. But how can one apply foresight, when no observational reference frame exists in the NullState of the NAKED SINGULARITY or the GENESIS BOSON? This then is the 'Boundary Condition' for the NullState, nothing exists and everything requires to become invented. The Everything is the Nothing as the All and so one can model this state onthe 'idea' of an infinite selfrepeating computer loop, representing the 'Father of all Algorithms'. This can only be one thing, namely a mathematical representation of the Nothing, say in the numeral 0 representing circular closure, which linearises in the cipher 1. Angular multivalued Complex Infinity is linearised in negative and positive polarity of a Doubled-Infinity. The hindsight then defines the closed circular NullState continuumas a closed looped superstring class I of the PlanckScale, also containing the openended members of that same family. But that wasn't the beginning program of the primary algorithm, butthis is a secondary algorithm, which can be stated as: "ADD THE END TO THE BEGINNING AND START WITH THE OLD END"! This specifies the methodology of RECIRCULARISING the linearised dimensions of the binary dyad (0,1) into the rootreductive decimal monad (1,2,3,4,5,6,7,8,9,0) (or some other number system). Here then we define: 10=1+0=1*; 11=1+1=2*; 12=1+2=3* and so on until 26=2+6=8** and 27=2+7=9** give a kind of closure in the doubled or squared application of the algorithm, reflecting the 'Doubling of Infinity' by the identity x²=2x. The hindsight then one day discovers, that there exists a mathematical entity, called the 26-dimensional Bosonic Superstring, which folds up the Ricci-Metric in Calabi-Yau manifolds in 6- and in Joyce manifolds in 7 circularised dimensions. A An heterosis of 26-dimensional counterclockwise rotating vibration patterns becomes 'married' to such states of frequency, rotating in 10 dimensions in a clockwise manner. A DEFINITION of TIME then became effected, once the first algorithmic definitions had reset themselves in the pattern: CIRCULARISED - LINEARISED - RECIRCULARISED - ENERGY=FALSE VACUUM- QUANTUM TUNNELLING OF TEMPERATURE WELL = QUANTUM BIG BANG. Curling up the 16-dimensional anticlockwise patterns in special toroidal geometries, then would allow a definition of Heterotic classes of superbranes (HE(8x8) and HO(32)), one of whom would become the sought after MINIMUM condition for the Naked Singularity to manifest itself in its escape from the infinity-loop. The primary algorithm results from the application of 10 elementary principles/antiprinciples which defined the algorithm

as itself. And because this algorithm engages a fivefolded symmetry, there wouldbe five classes of 10-dimensional superstrings, upon whom the secondalgorithm could then act. The ten principalities are: 1. Identity/Antiidentity 2. Expansion/Contraction 3. Order/Disorder or Entropy or Chaos 4. Symmetry/Asymmetry or Distortion or Nonparity 5. Divergence or Infinity/Convergence or Limit 6. Inversion or Reciprocity/Constancy or Invariance 7. Reflection/Absorption 8. Relativity/No Antiprinciple 9. Quantization/No Antiprinciple 10. New Identity/No New AntiIdentity, because of the application of the other 9 principalities to effect the transformation of One Self into Two Selves, yet as the One Unity of One extended Self. The purpose for the establishment of the first algorithmic program is its cosmogenetic self-reproduction. Adding Self/Infinity to Self/Infinity still gives Self/Infinity,but somehow, the second Self is different (more), than the first Self. The first algorithm becomes the mapping of two integral numberseries onto each other, complementing each other in a filling inthe gaps, created by one of the sequences as a one-to-one correspondence (like even and odd numbers for example, together forming the set of integers). Now, this one-to-one correspondence resulted in the following data set: ={(266561); (151618); (141524); (151032);11; (091016); (061015); 7; 6; 4}={E°; 1/k; A²; 1/h; 11; c²; 1/Lo; 7; 6; 4} and with a natural limit of 12, set and generated by the principalities as boundary condition for the selfiterative programming. The secondary algorithm then utilised the initiatory number constant E=(266561) and generated the continuation, which would allow the Temperature Gradient or Energy-Wall to manifest in the linearisation of the physical parameters defined by the permutations of the code from the first algorithm. The secondary code became (with initialising counter E): BEGIN; ABCDEFG(H=ABCD) with E=(266561); F=(136656); G=(673665); STOP GENERATION because 5+6=11=2* is rootreductive; CONTINUE D=(465612);C=(256124); B=(361242); A=(312423); STOP GENERATION because no monadic archetype can yield Z(Z+A)BCXY=312423 from its precursor ABCXYZ=(1Z)24233, ending in root reduction (1-Z)=1-3=-2=-11=-2*; END of 2nd algorithm. The first algorithm applied the Ordering Principality to DEFINE TIMESPACE and the ordering of the constants as given. Constant#1=E became the Number of Spacequanta defining the compactified 16 dimensions of the superbrane scenario upon initialisationof the second algorithm. All principalities are engaged and the ORDER of the Constant application uniquely sets the physical parameters manifest in the orderspecified; (i.e. Temperature before Radiation before Restmass, etc.). Constant#2 defined ENERGY=(Constant#2).TEMPERATURE. Constant#3 defined ENERGY in terms of ELECTROMAGNETIC CHARGES and in a Universal wavefunction, normalised about the pentagonal symmetry of the first algorithm. Hindsight would discover that a local application of this wavefunction would result in summation of integer states or probabilities, which upon normalisation and the 'squaring' of the statewould reproduce the Unity of the original wavefunction in the pentagonal symmetry. These would be known as the wave equations of quantum mechanics.

In particular one summation: T(n)=n(n+1) would, upon unitising; define the universal wavefunction explicitely in a Feynman-Path-Summation and allow the following identity to engage the timespace evolution of certain parameters, including the linking of Constant#3 to Constant#6. The all encompassing Identity is: XY=X+Y=-1=i²=℮ip for the generating waveequation: B(n)=[2e/hA].℮(-Alpha.T(n)). Here A²=Constant#3 and the roots X and -Y form the Bounds for the NullTime definition in both real-positive and imaginary-negative linearised time parameters about the normalised Mean of B(n) for n=0 becoming reflective in n=-½. C-Space is n-real from n=0 and F-Space is n-imaginary from n=-1 towards the negative integral count. In between in the interval [-1,0], is the M-Space further symmetrised about the Functional-Riemann-Bound: FRB=-½ and as the Point of Origin. Three orthogonal axes so define 3D-Space, about which a 6D-Twistor-Space indicates two further degrees of freedom, each of the arbitrary up-down, left-right and in-out linear axes setting the NullTime Space as defined by B(n) allowing a clockwise or anticlockwise chirality in the compactification of the circularised Linespace. This extended definition for 3DLine-Space into 6D-Hyper-Space subsequently transforms into 9D-Quantum-Space in introducing the degree of stretching/contacting by the application of Principality #2, following the Identity/Antiidentity initiation. All principalities are engaged in their defined Order, indicating that 9D-Quantum-Space manifests the identity of an expanding and ordered and selfcontained continuum within the symmetry principality. 9D-Quantum-Space then diverges/converges in the setting of certain boundary conditions, which must incorporate the principles of inversion and of reflection with their antiprinciples. Henceforth, 9D-Quantum-Space sets a limiting condition, which hindsight would define in all manners of temporal parameters. The relativity principle then allows the perception of being located at either of two observation points, namely the principalled or antiprincipalled vantage point, given in the kaleidoscope of all the combined principalities. Thus, a 10DOmni-Space is created, which reflects/absorbs itself under engagement of inversion/invariance, becomes relative and subject to the 9th principality of the quantisation. This defines Omni-Space to quantize itself in 3D-Line-Space, following the definition of the root-reductive decimal monad, so birthing the New Identity as Antiidentity to itself in the completion of the selfiterative and selfrelative programming of the first algorithm. Omni-Space hence requires to be 12-dimensional in the construction of the mastercode, becoming physically implemented in the Universal Cosmic Wavefunction B(n). The interval [-X,X-1] specifies the maximum for B(n) as a probability interval with B(-½)=[2e/hA].℮[sup(¼Alpha)[/sup] and a standard deviation about the first moment by the multiplier √(Alpha/2p. Hindsight would discover this as the Schwinger numbers to extend the QED-Feynman diagrams to account for multiple particle interactions as permutational possibilities in perturbation dynamics. Constant#4 defines ENERGY=(Constant#4)FREQUENCY. Constant#5=11 defines the Identity-Relation between Constant#4 and Constant#6 in Equality by DimensionCounter 11. Constant#6 defines ENERGY=RESTMASS(Constant#6) as the Parity-Mirror. Hindsight would define: (1) hf = moc² (2) Energy=hf iff mo=0 (3) Energy=moc² iff fo=flimit=fmin=fss. Hence all restmass carrying entities must possess a minimum eigen-frequency to exist and this existence relies upon

the definition of TIME as the Inverse of some frequency: TIMEINSTANTENUITY=tps=1/fps; and where fps is the MAXIMUM FREQUENCY modular dual to the MINIMUM FREQUENCY defined in fss. One then applies the Schwarzschild-Solution to the HE(8x8) Supermembrane. Hindsight would discover a best fit for the observed and experimentally measured physical parameters, as given by that superstring. The Coupling Constants between the vibratory (MAXIMUM) part of that supermembrane (Eps for Energy of primary source) and the winded or (MINIMUM) part (E ss for Energy of secondary sink) are: EpsEss=h² with Eps/Ess=fps² and lps with lss=fps.fss=1. A direct consequence is the Hawking Black Hole for a minimum temperature condition from the inverse proportionality between Mass and Temperature for the Black Hole spectra. Wavelength lps=Wormhole-Perimeter=2prps and rps=1/rss in modular duality of 2plss=2GoMHawking/c² in the Schwarzschild metric solution. This is the manifestation of the mathematical modular duality for the B(n) definition in: X=1/-Y=1/(1+X). Constant#7 defines a transformation from the Radiative Energy Stateinto the Restmassive one given above and the integers 7, 6 and 4 are dimension counters (#'s 8, 9&10), limited by the counter 12 as the holographic image for the summed lower dimensionalities, i.e in 4+6=10 & 4+7=11. We have seen, that the concept of dimensional time was unnecessary to manifest the threefolded multispace of LineHyper-Quantum-Space in 9 dimensions. This is mirrored in the decimal monad, allowing redefinition of the first principality in the tenth as an Identity/Antiidentity transformation into a Double-Identity. The principalities became defined in the first subroutine of thegenerating master algorithm, which could be stated in generalisedform in the following manner to DEFINE THE (ABSTRACT) CONFIGURATIONSPACE in the form of AWARENESS-TRIPLETS represented by: {OLDSTATE, EXPERIENCE, NEWSTATE} and selfiterative in the following algorithmic statement: GENERATE NEWSTATE IN ADDING THE SECOND NEWEST OLDSTATE AS EXPERIENCETO THE NEWEST NEWSTATE AS THE NEWEST OLDSTATE. ...BEGIN (0,0,0)-DEFINE SELFSTATE=(1,0,1)-REDEFINE BINARY EIGENSTATE=(01,01,01)=(0+1,1,1+0)=(1,1,10)= (1,1,2)-(2,1,3)-(3,2,5)-...CONTINUE INITIALISE: N=0; LIMIT=[REDEFINITION OF EIGENSTATE]={GOOGOLPLEX E} -FOR (N=N+1 TO LIMIT) GOTO -SUBROUTINE (DEFINE PARAMETERS FOR EIGENSTATE FOR M=M+1) GOTO -SUBROUTINE (ALGO[M]) GOTO -SUBROUTINE (...) GOTO -SUBROUTINE (...) GOTO -SUBROUTINE (SEARCH FOR LIMIT=[!])-STOP GOTO -SUBROUTINE (!- GOTO BEGIN -REDEFINE SELFSTATE IN [!]) GOTO -CONTINUE FOR O=M FOR LIMIT=[EIGENCODE IN !]={GOOGOLPLEX E} GOTO -REPEAT FOR SUBROUTINE (O- [O]=[!], [!]=[!+1], [?]=[!]) GOTO -REPEAT FOR LIMIT GOTO BEGIN...

The Pre-Big Bang Setting of Creation We return to foresight and find ourselves equipped with a newpossible definition for TIME as: MINIMUM/MAXIMUM=CYCLETIME (n). Algorithm #2 has defined the limiting boundary for a potentialexpansion of 10D into 11D in spacequanta counter E=Constant#1. Quantum Geometry, applied to the concept of representing a 4D-Volume as a 3D-Surface defines the Riemannian Hypersphere with a toroidal volume 2p²R³, R taking the linearised form of the MINIMUM condition as ack]lps to define a spacequantum in the potential manifestation of monopolic space, hindsight would term 'The Vacuum'. This Vortex-PE or VPE is still contained in the NAKED SINGULARITY of the GENESIS BOSON, whose eigendefinition in terms of physical parameters can now proceed. Those quantities are necessarily BELOW the MINIMUM condition, manifesting as the last and most physically applicable form of the superbrane families in the EpsEss=HE(8x8) heterosis. And so a subcontinuum, a dimensional divide becomes established to allow the 10D-universe, manifesting as a 3DSurface, to expand into the 11D-universe as a 4D-Volume, the latter given as ½p²R4 in the 4-Sphere. The 10D-Universe must be hyperbolic, hindsight measurements would detect and hindsight would also conclude, that the overall universe would be Euclidean flat and of zero curvature. Eventually, hindsight would understand, that the closed geometry of 11D-M-space mirrors the open geometry of 10D-C-space onto itself as a mapping into the holographic image of the 12D-F-space. So just like one can observe both open and closed curvatures in the multiconnectedness of a 2D doughnut/torus; the inner hole curving hyperbolically encompassed in a spherical envelope, so is the universe multiple-connected within its own unity. But the superbrane scenario maps the MINIMUM onto itself and becomes the MAXIMUM, the dimensional expansion of C-Space into the M-space in the duality perception of being located at the Point of origin as the Big Bang Singularity or as the Event Horizon Singularity at the wavefront of the selfsame expansion. The vibrating Eps becomes the MINIMUM SCALE and the winding Ess the MAXIMUM SCALE in that subcontinuum. Defining the toroidal volume generator as a consequence of the integral googolplexed H-count: H={Rmax /rps}³ then define Rmax as a 10D/11D boundary. Hindsight would calculate: R max =1.59767545x1026 metre-units. Hindsight would also discover that a dimensional DIM-factor would assume the value 7.56 at the time of its discovery; implying that so 7.56 10D-universes would 'fit' into the 11D-universe at that time, defining a discrepancy of matter in the cosmological calculations.We shall find that the baryonic matter is just 2.803% of thecritical density at the Big Bang; an inertial mass seed, which would grow to 3.68% at the time of discovery, adding to a kind of 'shadow' or 'dark' matter amounting to 27.8% of the necessary total for its Euclidean closure. This is a function of the deceleration parameter, which as a scaleratio defines not only the architectural scale (Gravitationally bounding Sarkar Constant centred in the supercluster interval from236.1-472.2 Million lightyears), but also the Black Hole spectrumbounded by the critical density. So Foresight defines TIME=lps/Rmax =6.259093473x10-49 =nps, the cycletime at time-instantenuity, after the second algorithm had implemented the cosmic interaction of the dimensionalities, which hindsight would term de Broglien hyperinflation, proceeding with a phasespeed of Rmax .fps and a phase acceleration Rmax .fps². And here the 'False Vacuum' develops to energise the process.

TEMPERATURE=ENERGY/k=(h/k).FREQUENCY=h/(k.TIME)=hR max /klps and this calculates as 7.544809x1037 Temperature units at Genesis Boson Time, which is the superbrane epoch. RESTMASS becomes: Energy/c²=hRmax /c²lps=0.011834633 and leading to a definition of Avogadro's Constant upon application of Unified Field parameters, which span 8p radians of the 4-folded lps-wavelength of quadrupled 3DLineSpace as Line-Hyper-Quantum-OmniSpace and integrating as the toroidal spacequantum in its sinusoidal propagation in: ¼p∫{sin(3x/2)-cos(3x/4)}²dx for the interval [-600°,840°]and the VPE-sourcesink quantum in: VPE-Quantum=EVacuum-ZPE/V=4pEps/lps³=2.51327..x10 64 J/m³ as the energy desity in (Energy/Volume)-units. Since 12 magnetopolic current 'knots' specify this field, the vacuum is energised by the magnetopolic energy. This energy is given as [c³] unified F-space units and manifests in the lower dimensions as the Current/Linespace quantum [ec] or as precisely 27x1024 eV, which is a monopole mass of [ec=4.82x10-11] kg-units. The VPE of the vacuum so becomes defined in wormhole parameters for the Naked Singularity putting 'some clothes' on. In terms of TEMPERATURE however, the nps cycletime is lower than the Genesis Boson Time in 2.93516x10 36 Temperature units.

The 'False Vacuum' so induces the tunneling of the higher temperature to its lower value. This becomes the decisive parametrisation for the mechanism and dynamics of the tunneling matter wave inflation. The de Broglie phaseacceleration calculates as 1.437908x1087 (m/s²) and the Temperature-Field of the universe as a 'Black Body' gives the Quintessence or Einstein-Lambda as the Schwarzschild expression: GoMo/lps²=2.015x1085 (m/s²), with Mo a baryonic mass-seedling from the first algorithm defined as a function of E: Mass-Seedling: Mo=√E. (mc .m P/m e)=1.813713x1051 kg-units. mc is the nucleonic prototype; mP is the Planck-Mass and me is the effective electronmass. mc and me derive from the couplings between superbrane classes, one of which gives the scale relation between the electromagnetic and the gravitational interactions as: [AlphaGR]=[AlphaEMR]18. There are exactly 10 10 lps/360 wavelengths in one electronic radius Re=ke²/mec² as a consequence of linearising rps=lps/2p into lps over a 2p or 360 'degree' count. The 360-degree count is not arbitrary, because the dimensionless unification for the electric permittivity (e o=1/120cp) with the magnetic permeability (µo=120p/c) specifies a half rotation inthe geometric definition of the number p as a trigonometric expansion and sets 180 as that limit. Now the deceleration parameter, aptly so called in 'reducing' the de Broglie Phase acceleration to the instanton of the classical and metricated cosmic expansion of the quintessence calculates as: qo=GoMo/c²Rmax =0.01401506=Mo/2Mcritical=½Ωo. The critical Mass for closure for all definitions produced by the first algorithm is given in considering the universe as a 11D-Mother-Black Hole (F-M-C-Space=Father-Mother-Child-Space). Then, using the linearised time definition from cycletime n=H ot and the Nodal Hubble-Definition: Ho=c/Rmax , we obtain the critical volume as: 2p²Rmax ³ and the critical density as ρc =H o²/4p²Go. Using the spherical 3D-volume would give the critical density as ρc =3Ho² and which differs in the Feigenbaum-ChaosDelta δF=3p/2 as dimensional unifier for the curvatures. Mcritical, so becomes explicitely defined as the Mass-Total for the universe and as so 6.446x1052 kg-units. If we set the instanton time as tps=fss (modular), then this gives the MINIMUM SCALAR HIGGS FIELD POTENTIAL

from INFLATION MAXIMUM into the superbrane epoch as tinflatonlimit=3.946x10-32 seconds*. The instanton relates as fss=1/fps=3.333...x10-31 seconds*. TIME as linear expression for the multivalued circular singularity spectrum 'BEGINS" at TIMEINSTANTENUITY=tps=fss (modular). Now Newton's Gravitational constant 'G' was not defined in the first algorithm, so where did it derive from? We begin with the Gravitational Finestructure being a scaled version of the Electromagnetic Finestructure Alpha, with a generalised formulation: Dimensionless Constant=(Constant).(Charge)²/hc, a 'ForceCharge' taking either the form of Coulombic Charge 'e' or of Restmass Charge 'm', say. We then relate the Force- or Interaction fields in the identity [ke²]=[G om²], and where the Coulomb-Charge 'e' becomes defined (say in a Newton-Raphson Iteration) in the magnitude of the wavefunction B(n) for B(0)=Bo. The primordial algorithm specifies A=h/(2e) as the manifestation of the supermembrane class IIA and that in modular duality to the neighbouring EpsEss heterotic supermembrane class HE(8x8). This relates to superconductive states in the Josephson Constant, the Conductance Quantum (2e²/h) and the Quantum Hall Effect, all derivatives from the FSpace ACTION-LAW of ACTION=CHARGE². Generally, all mass definitions are defined in terms of the magnetic monopole, manifesting as class IIB. Hence all mass is rendered magneto-monopolic at this stage of the definition process. The following expressions for the magnetocharge e* (measured in the StarCoulomb C'*) are applied in the mapping of magnetopole charge 'e*' in F-space unto the electropole charge 'e' in C-space. This transition relates to the superbrane epoch before timeinstantenuity. (a) 2pGo.mc²/hc=Alpha18 and mc =mP defines mP for Alpha18=1. (b) Eps=hfps=Ess.fps²=√ol]pGo.m e²/4Alpha.hc.e²]=1/e*. (c) 2e.Re.=e*.l Planck√Alpha. (d) Eo=e√(E/Alpha)=1.886461127x1038 C* (Charge-Seed) (e) Mo/m c =1.813713x1051/9.92474572x10-28 .....Mo/m c =1.827469354x1078.......(No of Seed-Nucleons) .....Mcritical/m c =6.519663x1079.......(No of Nucleons) (f) OlPlanck=lPlanck.√Alpha=e/c² (Stoney Units). The last expression is the Planck-Length-Oscillation, unifying Gravitation with Electromagnetism. For unitary consistency in C-space the dimensionless nature of the Force-Charges must become linked to the monopolic spectrum and that is achieved in the mapping of e* onto e. The C-Space relation 2Re.c² maps the F-Space relation OlPlanck.c² and the unitary symmetry is the relation between the electric permittivity and gravitation in the identity: Go=4peo=1/30c (dimensionless) via unitary unity: [metre/Farad=Joule.metre/Coulomb²]=[kg²/Newton.metre²] and for [StarCoulomb*]=[metres³/seconds²]= [Volume.Angular Acceleration]. This defines 'The Equivalence Principle' to relate 'Gravitational Mass' to electric permittivity via the interaction charges for a prior existence of (classical) electromagnetic mass before the inertial eigenstate induced per acceleration as spacetime curvature in geometric General Relativity. The C-Space relation 2Re.c² so crystallises in the diameter of the electronic spacial extent in manifesting the energy of F-space in magnetopolic equivalence to the electropolic size of an atomic nucleus in 3 dimensions.

We note the independence of the primordial Energy from the restmass. Substituting k=1/Go=1/4peo=30c into the electric constant from the Maxwell Constant gives the 1/30c numeracy, independent from the [kg] unit. The generation of the masterconstants demands the establishment of a cosmic-standardised unitary system (*) and the only constant shared between both the (*) and the SI-system is the magnetic permeability constant µo=120p/c in the finestructured form. Amongst other things, this becomes decisive in determining the lower dimensional 'dip' in the Alpha-Constant, which remains invariant in the higher dimensional settings, where Alpha is independent of c and so e o

The Algorithm of the Cosmogenesis in Strings The Algorithm for MasterConstant Generation. The eigendefinition of Eps; {originally standing for Energy-primary-source but today identifiable as the sourcesink supermembrane EpsEss of heterotic class HE(8x8), based on the maximised octonion composition algebra of exceptional lie group E8; also gnostically or esoterically Eps=GoddoG and Ess=DoggoD as GOD and AntiGOD=DOG as the coupling of vibratory (GOD) and winded (DOG) superstring parts under modular symmetry, say mirror or Msymmetry} in the pre-O-time universe of omnispace ]{10D-13D=1D}, was found to be a consequence of the awareness triplets of (Old State,Experience,New State) and generally coded triplets of the form (a,b,c) became effective to quantumize integer definitions into the natural laws from quantum-space {7D-10D} into hyper-space {4D-7D} into spacetime {or linespace 1D-4D}. We recall the sequence of experience factors: SEps={0,1,1,2,3,5,8,13,21,34,55,89,...} and generalised via: SEps(n)=[Yn-Xn/√5; n=0,1,2,3,4...; X=½(1-√5); Y=½(1+√5). The aim is now to generate the complementary set to SEps, namely SEps* in the application of the primary seven fundamental principles. We so define the two main {binary} operations of ADDITION and MULTIPLICATION: (a,b)+=a+b and (a,b)x=axb, where both a and b are elements of SEps. If the value obtained by either operation is a member of SEps, then it is rejected, otherwise it is accepted and becomes an element of CONSTANTSET, a subset of SEps*. The principles of IDENTITY and EXPANSION are herewith fulfilled and the ORDER principle becomes the main mechanism for constant generation. We further define 'Pure Number Constants' or PNC's, which will be integers of SEps* and Proportionality Constants or P-Constants of the form (a,b,c)=(axbc )j and with j=+1 or j=-1. Starting with (a,b)+ an algorithm is applied until a generated constant is OOO or OUT OF ORDER. This OOO-Constant becomes then the first element a1 of the first P-Constant. Then the operation changes to (a,b)x until another OOO-Constant is found, with this second OOO-Integer becoming assigned to be b1 of the first P-Constant ONLY if it is greater than a1; if it is not, it becomes a2 for the second PConstant. If a b-element is defined, then an a-element must also be defined in the same triplet and the next element would become c and finalise the triplet and the P-Constant.

The principle of SYMMETRY is effected whenever a triplet element is generated OOO or when a prospective b or c must be reset as an a. All P-Constants must be symmetric, with a's followed by b's followed by c's. Whenever an a and a b are defined within any particular triplet, the operation of + or x is not altered and the next generated constant, being either pure or impure=OOO, becoming either the next PNC or finalising the next P-Constant. The overall ORDERING must however always prevail with PNC's generated in ORDER, as well as the elements of the P-Constant triplets. This is the general concept of the algorithm. We must however also account for the DUPLICATION of numbers, MULTIPLE IRREGULARITIES and INCOMPLETE TRIPLETS. To apply a FINITE algorithm to SEps, we must define the ORDERED SUBSET NSTEP. NSTEP-k=(a1,a2,a3,...,ak)=(0,1,1,2,3,...,ak) and the order is k, with ak the limiting element of SEps. The LIMIT for constant generation now becomes that application of NSTEP, which does not generate at least one constant in SEps* and not having been generated in an earlier NSTEP application. The complementary set to NSTEP is INSEQUENCE=(b1,b2,b3,...,bk')=(4,6,7,9,10,...,bk'). A necessary initialisation for NSTEP is so NSTEP-5=(0,1,1,2,3) with a5=3 and b1=4 with b1=4 the first element of INSEQUENCE to be generated by the algorithm, either as PNC or as a1 of a P-Constant#1. NSTEP-5 is so limited by b1=4 and the algorithm ends if this number 4 is not generated via NSTEP-5. Should however the first k' elements of INSEQUENCE be obtained, say 4,6,7,9,10 with k'=5, then NSTEP-6 is required to produce b6=11 for the continuation of the algorithm. This process does not exclude the generation of any other missing numbers however and we define the set TOTALCONSTANTSET of which CONSTANTSET-k is a subset. The algorithmic iterations must so be increased by a minimum of one for every application of NSTEP. Formally then: NSTEP-kÌSEps, such that (a1,a2,a3,...,ak)=(0,1,1,...,ak) with a(k+1)>ak " k]; INSEQUENCE-k'Ì SEps*, such that (b1,b2,b3,...,bk')=(4,6,7,...,bk') with b(k'+1)> bk' " k'. Let us now define DUPLICATIONSET=DS as the set of all duplications, to which all new duplications of a particular application of NSTEP are added. 1+5=2x3=6 and 2+8=2x5=10 and the resulting numbers 8 or 10 can be obtained in a number of ways. Say, we have 10 as a PNC and is duplicated in (7,10,12). This represents an 'IRREGULARITY' and rectifying steps are applied via DS. INCOMPLETE TRIPLETS at the end of an algorithm apply a subroutine in comparing the a-elements of all generated triplets and setting ascending ORDER for a's, b's and c's. This regulation ensures redefinition of the largest a-triplet. The principle of INFINITY {as the DIVERGENCE being antiprincipled in CONVERGENCE or LIMIT as primary principality #5} is utilized via the principle of INVERSION by RECIPROCATING the 'IRREGULARITIES' of DUPLICATION. Every application of NSTEP resets CONSTANTSET to EMPTY after including all previously generated constants in TOTALCONSTANTSET. All constants in DUPLICATIONSET are omitted from the next NSTEP.

There is only one exception to the strictly increasing nature of triplet elements a< b< c and this is the occurrence of MULTIPLE IRREGULARITIES and INCOMPLETE TRIPLETS. For a single duplication the reciprocal is taken with adherence to ORDER. The first OOO-Constant becomes a1 in the first P-Constant. If the next OOO< a, then that OOO becomes a2 in PConstant#2, irrespective of duplication. Yet if b>a and if b is duplicated as well as a, then this DOUBLE-DUPLICATION of a and b is ORDERED and a single inversion suffices. Two ordered irregularities form a regular irregularity and a single rectification - inversion, is satisfactory. Similarly, when a c is duplicated besides a and b, but a< b< c, then ordered duplication is a single irregularity. Thus ORDER rules supreme and we must consider the possibilities of one or more triplets being INCOMPLETE at the end of a particular NSTEP with MULTIPLE IRREGULARITIES of uncompleted triplets. In this situation, the triplets are formed not in terms of ORDER but in terms of their INITIALIZATION, which is another form of ORDER. All a's are utilised before b's before c's and the remaining numbers are totalled and are combined via modulus 3 or modulus 2. Example #1: (23,40, ) and (14,16, ) and (15, ) and (10,12, ) have members: a1=23, b1=14, c1=15, a2=10, b2=40 and the final exponent=(16+12)/2=14. Example #2: (23,40 ) and (14,16, ) and (15,16, ) has members: a1=23, b1=14, c1=15, a2=40, b2=16 and c2=16 with the last triplet becoming (40x1616)-1, because of the duplication of element 16. These Examples therefore yield P-Constants 23x1415 and 10x4014 and 23x1415 and 1/(40x1616). A multiple duplication not in order could occur in the generation of the final exponent and represents an irregular irregularity and so a regularity. The initialisation process of a's before b's before c's has made the irregularity regular however and a single inversion negates this disorder. The constants obtained in computing the final exponent are to be included in TOTALCONSTANTSET=TCS, as is the integer part of their result. A final c=13+2/3=(12+12+17)/3=41/3 includes so 12, 17, 41 and 13 in TCS via CS. It remains now to redefine all those duplicated P-Constants, which were not inverted, since in the process of their generation, their duplications were not members of CS, in which case they would have been inverted. These triplets are 'purer', with the only irregularity being the inclusion of a number, which would become duplicated LATER within the same application of NSTEP. So INVERSION is not justified, except in the case of incomplete triplets, with duplication becoming just one of the disorders. If (9,12,17) and (14, ) and (12, ), then a1=9 and b1=14 and c1=41/4=13+ 2/3 and the three irregularities are the duplicated 12 and the modulus 3 of 41/3 and the disorder of the resulting (9,14,13+2/3). We must apply a single inversion for the constant (9x14{13+2/3})-1. This would delete 9x1217 from CS and TCS, but adding the triplet and 9,12,13,14,17,41 to TCS. The {seventh and completing} principle of REFLECTION defines the triplets, which contain duplications, but are not inverted. Say the triplet (7,16,20) is generated, but later in the same NSTEP, the element 16 is duplicated. We set the Reflection-Function R(N)=(P-Constant)2, i.e. we reflect +1 in -1 and define Unity (-1)2=(+1)2 =1.

20.

The resulting constant then is 'squared': (P-Constant)2=7x16

General Structure of the Algorithm: INITIALIZATION MAIN LOOP (UNDEFINED=UD, INSEQUENCE=ISQ, (RML-CO=RC) SEps?=NO->XÎ ISQ?=NO->a UD?=NO->b UD?=NO->c>b?=NO->c>a?=NO=FIX a-RC |................................................................................................|YES=FIX b-RK |................................................................................|YES=FIX c&triplet-RK |.........................................................|YES=FIX b-(RML-KO)=RK IF b>a ELSE FIX a-RK |...........................................|YES=FIX a-RML-CHANGE OPERATION=CO |........................|YES=PNC-RML-KEEP OPERATION=KO |YES=RETURN TO MAIN LOOP=RML