Postulate1

INTRODUCTION

                                           Postulate I According to the mainstream the more we discover the more complicated the universe appears to be. I could site many well known examples. In contrast I am finding that the more we discover the simpler it is! I.e., 1. But we need mathematical rigor (2) so we:  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>   Postulate 1 algebraically as   min(z-zz)           >>>>>>>>>>>>>>>>>>>>>>(That is the whole shebang.) Notes:(1)min(z-zz) is rewritten as z-zz=C (1.1.1), δC=0 (1.1.2), (minonly(2), implies real#) which (given z=1+δz) is then δ(δz+δzδz)=0 (1.1.6). δz+δzδz=C=0 has a complex solution for noised C>¼ so in δ(δz+δzδz)=0 the real part is special relativity, the imaginary part is the Clifford algebra and combining them is the operator formalism, Dirac equations for electron e and neutrino v. Eq. 1.1.6 also gives 4D. The composite of e,v is the Standard electroweak Model (sect.2.7) and composite 3e solves particle physics (partII). It doesn’t get any better than that.

(2)Why min(z-zz)? Completeness and Choice  (since they imply z is a real number) Yes, ONE indeed is the simplest idea imaginable. But unfortunately we have to complicate matters by algebraically defining it as universal min(z-zz) and so as the two most profound axioms in real# mathematics: “completeness” (someminsup) and “choice” (choice function f(z)=z-zz). But here they are mere definitions (of “min” and “z-zz”) since z=zz, so no 1z=z field axiom for multiple z, implies our one z (See z»1 result below.). We did this also because that list-define math (appendix C PartI) replaces the rest (i.e., the order axioms, mathematical induction axiom (giving N) and the rest of the field axioms); Thus we have algebraically defined the real numbers thereby implying the usual Cauchy sequence of rational numbers(5) definition of the real#. The  Solution to z-zz=-δz-δzδz=C in general however is complex δz=dr+idt so we must provide this real Cauchy sequence CN as a BC=circle=C1 lemniscate sequence CN+1= CNCN+C1 (C1=dr2+dt2 circle) infinite iteration. This is because all these lemniscates share our one universal minC value of minCN+1 lemniscate=CM2 (Mandelbrot set(4) Fiegenbaum point CM, see figure 4.). Then divide both sides by B=mCM (CM=mδz) getting back z-zz=C=CM/m and so (after substituting z=1+δz) getting δz+δzδz=CM/m=C (1.1.6) and implying fractalness so cosmology, GR and gravity (sections 1.2, 7.4). Also δCM= δ(mδz)= δm(δz) +m(δδz) =0 so given δz=0 then m big so δδz1δ(iδt)»0 and δm=0 stability and z=1. Thus we have derived both physics and mathematics from the postulate of 1.  The universe is indeed infinitely simple. See summary below.

Summary: Postulate 1->min(z-zz)-> z-zz=C, δC=0 & real# -> δ(δz+δzδz)=0 & fractal physics So given the fractalness astronomers are observing from the inside of what particle physicists(1) are studying from the outside, that z»1 ONE object we postulated at the beginning, the new pde electron. Contemplate that as you look up at the starry night sky (See PartI, PartII and PartIII for backups.).

(4) Penrose in a utube video implied that the Mandelbrot set might contain physics. Here we merely showed how to find it. The fractal neighborhood of the Fiegenbaum point is a subset.

(5) Cantor: Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, “Ueber eine elementare Frage der Mannigfaltigkeitslehre” Jahresbericht der Deutschen Mathematiker-Vereinigung. 

Derivations of eq.1.9-1.16 from eq.1.1&1.2: Substitute z=1+δz into eq.1.1 and get 1+δz=(1+δz)(1+δz)+C (1.3). Thus    δzδz+δz+C=0   (1.4) : Solving eq.1.4:  δz=[-1±sqrt(1-4C)]/2. So for C>¼ in general  δz=dr+idt  (1.5)  (So we derived space-time.). eq.1.1,1.2 holds only at CM so δ(idt)=0 (stability). So plug eqs.1.4,1.5 into eq.1.2 to get the amazing equation:  δC =δ(dz)+δ(dzdz)=0 (1.6)                                                                                                                     For   δ(δzδz)=δ[(dr+idt)(dr+idt)]=δ(dr2+i(drdt+dtdr)-dt2)             (1.7)   If δ(idt)=0 (stability) in δ(dz) in1.6 then the Imaginary part of eq.1.7 is  δ(drdt+dtdr)=0 (1.8)         If 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   (our 1st invariant, sect.2.5)       (1.9)  Since we have two independent variables r,t in relδ(δzδz)=δ(dr2-dt2) =. δdr2-δdt2+δk2=δ(dr2-dt2+k2)=δ(dy2+dz2)=-Realδ(dz) still with two independent variables y,z because we can choose k2 randomly. Thus real eq.1.6 Real(δ(dzdz)+δ(dz)=0 becomes δ(dx2+dy2+dz2-dt2)=0 with 1.9 still holding(since δ(idt)=0). So we have 2+2=4 dimensions and dr2=dx2+dy2+dz2. Note in general dr,dt could be any two of these 4 independent variables implying eq.1.9 defines a Clifford algebra. Next factor the real part of eq.1.7 to get (our 4D universal 2nd invariant ds2=dr2-dt2): ds2=dr2-dt2=                                                                                                                                                                                                                     [[δ(dr+dt)](dr – dt))] + [(dr +dt)[d(dr – dt)]] =0. (1.10) Solve eq. 1.10 and get     (->±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)     (->vacuum)            dr=dt,           dr=-dt)          (1.14) Equation 1.10 gives Special Relativity ds2=dr2-dt2 (note natural unit constant 12 (=c2) in front of the dt2) and eq.1.9 gives the Clifford algebra (sect.2.5). A third invariant in section 1.2 changes these operators into derivatives thereby making these Dirac equations. So 1.11-1.13 are (neutrino v &stable electron e) leptons. All the rest are composites* of e,v. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 

*composites of e,v are Bosons (eg.,The SM: Z,W), and composite  3e baryons (eg.,proton), and Fiegenbaum Point fractal e cosmology (GR&gravity).  Excited states of e are u,T in appendix.   We finally do understand.      

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 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.

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) 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. The Fiegenbaum pt neighborhood is a subset.

Table of Contents

      Postulate 1 

Introduction

 Part I     1U1 states

Part II    1U1U1 states 

Part III   Mixed State

 Rest       Miscellaneous

Appendix:

Mandelbrot set derivation : Note 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.