Postulate 1

It has been suggested(2) that all of physics might be found in the Mandelbrot set.  Here we  merely show how to do that and why it is important.   

Many assumptions means a lack Of Fundamental Understanding. In that regard the real numbers (eg.,1) are defined from Cantor’s Cauchy sequence zN of rational numbers. My fundamental insight is that these sequences also define the Mandelbrot set iteration formula sequences zN. So our One simple assumption is (postulate)1 real set” getting us the Mandelbrot set (zN+1=zNzN+C (1a) and δC=0 (1b)) and thereby the physics and so fundamental understanding, a worthy goal indeed thereby showing why it is important. 

 Summary of Physics from the Mandelbrot set C Small  C:  (i.e.,C→0) So z1=z≡z in eq.1a  has as the only noniterative result: z=zz+C  (eq.1).              (we use z≡1+δz, C≡kδz and note limit 1,0 =z binary Real#math).  We note that equation 1 (given eq.1b) gives nontrivial results (and also δδz=0) only at the small baseline (10-40X smaller) Mandelbulb boundaries. So plugging eq.1 into 1b at this boundary, we have the resulting quadratic equation (-δz =δzδz+C), which in general has complex solution dz=dr+idt. Taking the variation using eq.1b thereby gets δ(δzδz)=0 (eq.2). Factoring this variation gives eq.2A: ds2=dr2-dt2 (SpecialRelativity SR) and Clifford algebra drdt+dtdr=0 (as eq.2B extreme). After also factoring eq.2A we get degenerate eq.2AI, light cone 2AII (S=½ electron and neutrino leptons in each quadrant.) and associated 2AIA (Hermitian) observables. So we have merely solved for z.

Large C: Implies two successive 80° rotations (eg.,d(dr), eq.4.2b)) of this (small C) dz through 2 quadrants: (so S=½±½=integer) from the 4 different axis’ max extreme (of 2AIA and 2AI) and so gives the 4 results:  the  Z, +-W, photon Bosons of the Standard Model (We have derived it!!). Also this rotates SR into GR (So we have derived GR too!!) and 2AI into eq.9. This same rotation also breaks those two 2D degeneracies of eq.2AI 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.9. That 1040X fractal electron scale difference of the rH (so next smaller rH gravity from our SR rotation) in the (eq.2AI) eq.9 is at the Fiegenbaum point where δC=0 (i.e., ground state) even for 4DLarge fractal baseline (mixed and pure lepton) metric quantization states in partIII which also explains the large galaxy halo velocities without dark matter. So we have merely rotated z. Summary: So postulate 1 (real set) set (zN+1=zNzN+C) with the small C limit getting lepton observables and big C rotation the Bosons.

Notes 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.9) we postulated. Try looking up at a starry night sky and contemplating that some time.  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 

Introduction

 Part I     1U1 states

Part II    1U1U1 states 

Part III   Mixed State

 Rest       Miscellaneous

Appendix:

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 requiredNote 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=O00=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 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 edge contributions, called Mandelbulbs here). Also large k gives nontrivial results as well and so then δδz is very small. 

Cantor’s Cauchy Sequence via Iteration . 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}. But 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 which both of which together make the Cauchy sequence ZN with limit 1 that we required.

So in that case the 1set in the postulate of 1(1∪1≡1+1)whole numbers rational numbersCauchy sequence ZN of rational numbers(same as eq.1a,1b)real number 1). Eq.1a,1b imply for z1-zfor C0 that z=zz eq.1 so eq.1 implies (1,0) corresponding to the dichotomy ‘1set always contains nullset O‘ so {1set,O}(1,0) so OO=O00=0+0 in our whole number algebra. That makes this choice of 1a,1b the only one possible since it implies both the Cauchy sequence and 1,0. Eq. 1a,1b also implies the math of observables (eq.2AIA). Also this ‘self contained’ circular derivation guarantees we don’t pull in postulates other than