Postulate 1

Introduction   Definition: Cantor-Cauchy sequence zn of rational numbers defines the real numbers, eg.,1. Fundamental insight: sequences zn Cantor-Cauchy = iteration sequences zn Mandelbrot set   

Problem we solve in this paper: We want to understand but Many Assumptions   means a lack of fundamental understanding. (eg., Standard Model and its hundreds of assumptions).  Solution One simple Assumption   Postulate 1 ->Mandelbrot set -> eigenfunctions z (efz) means ultimate understanding.                                                       Problem solved                                        

 See Appendix C Cauchy Sequence and Mandelbrot set definitions. Algebraic definition of 1:  z=zz+C (1) since 1=1X1 with z=1+δz, C=mδz defining real# (noise) uncertainty C defining the above Cauchy sequence number choices as real# (noise) uncertainty C since numbers here are also observables (eq.1.16a). Eq.1 Real 0=0X0 (0=z=1+δz so δz=-1) implies eq.1.5 (MS extremum) CM is the eq.1 big noise 2ndsolution. Define fractal Mandelbrot Set(MS): zN+1=zNzN+(1.1),  δC=0 (1.2) subset eq.1 (for |CM-1|<|C|<<|CM| in eq.1.1 then z1=z≡z with the only noniterative result) z=zz+C. Application Solve Realequation 1: z=1->lepton efz and noisier z=0->Boson efz and a fractal cosmology

Introduce Noise C To Equation 1 In Steps From 0->CM resulting in 3 Invariants ds We show that our single real set z=1,0 implies two islands of eigenfunctions z, one for the extremum of the set (z=0, so large noise δz+δzc=CM) and the other for small noise z=1. Define charge from the Fiegenbaum pt.=1.40115..=CM= ke2. Also using z=1+δz define mass m (=mL) from largest noise=CM=mdz. Substitute into eq. 1.2        . δC=δ(mδz)=δmδz+mδδz.        (1.3) Using the distributive law:  1+δz=1+δz+δzδz+C  and canceling the 1 and rearranging                                δzδz+δz+C=0                     (1.5) Solving eq.1.5: δz=[-1±sqrt(1-4C)]/2. If C>¼ allows for z=1:  . δz=dr+idt  (1.5a)  For C=¼, δz=-½. So for z=0,   δzt=-1-½=-1.5<CM=-1.4. So the eq.1.2 MS extremum Csolves eq.1 for the z=0 case on the dr axis. Put equation 1.5, 1.4 into 1.2 get   0=δC=δ(-dz-dzdz)=δ(δzδz) (1.6)             which with 1.5a is  then      . . δ[(dr+idt)(dr+idt)]=δ(dr2+i(drdt+dtdr)-dt2)=0     (1.7)                                                         The Imaginary part of eq.1.7 is from (eq.1.10)   δ(drdt+dtdr)=0  (1.8)         dr,dt positive then drdt+dtdr=ds3=0 is a minimum. Alternatively if dr,dt is negative then drdt+dtdr=0 is maximum instead for dr-dt solutions. In fact all dr,dt sign cases imply a single invariant extremum:                                       drdt+dtdr=0                          (1.9) Next factor the real part of eq.1.7 to get:                                                                                   δ(dr2-dt2)=δ[(dr+dt)(dr-dt)]=δ(ds2)= [[δ(dr+dt)](dr – dt))] + [(dr +dt)[d(dr – dt)]] =0. (1.10)   Solve eq. 1.10  and get   (->degenerate ±e)  dr+dt=sqrt2ds, dr-dt=sqrt2ds,                             (1.11) (->light cone v)    dr+dt=sqrt2ds, dr=-dt,                                      (1.12)             “        “              dr-dt=sqrt2ds,   dr=dt,                            (1.13)  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Appendix z=0 details Rotate z by 45+45 Longer summary of section 2 Thus we rotate this (small C) δz through 2 quadrants: (so S=½±½=integer) from the 4 different axis’ max extreme (of 1.15 and 1.11) branch cuts and so get the 4 results: the Z, +-W, photon Bosons of the Standard Model (We have derived it!) with correct field equations (see A6). This is the mother of all reality checks! Also this rotates SR into GR (So we have derived GR too!!) and 1.11 into eq.2. This same rotation also breaks those two 2D degeneracies of eq.1.11 creating our 4D. Three eq.9 components in 2P3/2 at r=rH give the baryons (PartII) and the tauon and muon are S½ excited states of eq.2. That 1040X fractal electron scale difference of the rH (so next smaller rH gravity from our SR rotation) in the (eq.1.11) eq.2 is at the Fiegenbaum point where δC=0 (i.e., ground state) even for 4D Large fractal baseline (mixed and pure lepton) metric quantization states are in partIII. Mond is a special case.

So we have merely solved for z=1 and z=0 eigenfunctions.

Why Is This Important? So we derived both physics and math, the most rational thing we could be doing. That 4D implies we got not more and not less than the physical universe.  Also given the fractalness, astronomers are observing from the inside of what particle physicists are studying from the outside, that ONE thing (eq.2) we postulated. Try looking up at a starry night sky and contemplating that some time. This illustrates the infinite simplicity of this idea. So by knowing essentially nothing (i.e., ONE) you know everything!  We finally do understand (just postulate 1).

References (1)Cantor: Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen,  “Ueber eine elementare Frage der Mannigfaltigkeitslehre” Jahresbericht der Deutschen Mathematiker-Vereinigung (2) Penrose recently implied in a Utube video that all of physics might be extracted from the Mandelbrot set.  Here we have merely shown how to do that and why it is important.

Table of Contents

      Postulate 1 


 Part I     1U1 states

Part II    1U1U1 states 

Part III   Mixed State

 Rest       Miscellaneous


Cantor’s Cauchy sequence via iteration: Note for example that Cantor’s Cauchy sequence is generated by iteration 1a: So for any initial rational finite z1 between 1 and -1 in iteration formula 1a gives rational finite odd Z2N+1=1-z2N+1,  even Z2n≡1-z2N both of which together make our Cauchy sequence ZN with limit 1 that we required. Note also that the Mandelbrot Set is derived uniquely from 1 {real set}. In that regard eq.1a is iteration zN+1=zNzN+C and eq.1beq.1b says that for some C that δC=0 allowing eigenvalues (giving eq.2AIA). Note if we solve eq.1a for noise C in δC=0 (1b) we get δ(C)=δ(zN+1-zNzN)=0 implying zN+1 is finite since ∞-∞ cannot equal 0 so that eq.1a,1b defines the Mandelbrot set {CM} since then z∞ cannot be infinity. Note also for limit 1,0 finite zN central limit →0(i.e.,small C) in eq.1a then z1=z=z=zz+C (eq.1). Eq.1 implies(1,0) corresponding to the dichotomy ‘set1 always with subset O’: so{1set,O}→(1,0) so OO=O→0∪0=0+0 in our whole number algebra. That makes this choice of 1a,1b the only one possible since it implies both the Cauchy seq, and the set 1,0.Also we define C from noise C=kδz if k in general arbitrary real. So in eq.1.1 if k is exactly a constant and/or k is small (i.e., k<<δC) then δr and δt in δz are trivially constant. So (in  δC=δ(kδz)=δkδz+kδδz so k=k1δz+k2) δk must be small but not zero here. Indeed δz is order 1 deep inside the Mandelbrot set limacon lobe so δz=0 inside.  But δz is constrained to be small (D) on the edge therefore allowing for a small varying k on the edge of any particular Mandelbrot set lobe (in δkδz>0, eg.,fig.1 eq.2AIA diagonal eq.2AI at the Fiegenbaum point. Also large k gives nontrivial results as well and so then δδz is very small. The Fiegenbaum point is defined to be the limit (given it’s the rH horizon, sect.4) of C on the [large (cosmological) and] small baseline fractal scale in our definition of eq.1. So the above subatomic limit is really near 0 if noise C small=D.