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Original repository path: experiments/theory-contracts/paper-source-reconstruction-20261025/MICROSCOPIC-TO-OCTIC-COMPOSITION.txt
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000001 | MICROSCOPIC QWZ TO NEUTRAL OCTIC COMPOSITION
000002 | =============================================
000003 | 
000004 | Purpose and verdict
000005 | -------------------
000006 | 
000007 | This note composes two existing proved maps:
000008 | 
000009 |   (i) the filled QWZ top edge to one complex chiral CAR channel, including
000010 |       fields, adjoints, vacuum, charge and time on a common finite-energy
000011 |       core; and
000012 | 
000013 |   (ii) the unitary eightfold reindexing of that same CAR channel, followed by
000014 |        the fixed current map A=U_+^T O^T M and the neutral octic I8.
000015 | 
000016 | Verdict: PROVED for the already fixed QWZ collar and the already fixed fold
000017 | dictionary.  The neutral F8 operator and its level-eight Ward amplitude are
000018 | microscopic scaling limits of operators in the original filled QWZ Fock
000019 | space.  The construction uses one CAR channel.  It adds neither eight
000020 | independent source copies nor the odd E8 charge sectors.  It therefore does
000021 | not require an abstract B_N, an E8 root-field exhaustion theorem or FE-GEN.
000022 | 
000023 | The remaining antecedent is the local operator and time identification of
000024 | the primitive seed with the boundary source represented by this QWZ collar.
000025 | The lattice parameters may be regulator coordinates rather than fundamental
000026 | P1 constants. Once that boundary-source identification is supplied, the
000027 | neutral field, state and time map is already present. For a full graded CAR
000028 | embedding in the same positive-energy vacuum representation, the state is
000029 | then forced by the time law, as GROUND-STATE-SOURCE-REDUCTION.txt proves.
000030 | This statement does not identify the whole 3+1 dimensional P1 bulk with a
000031 | free CAR theory.
000032 | 
000033 | 1. The microscopic theorem being composed
000034 | -------------------------------------------
000035 | 
000036 | For the width-eight, mass-one, holonomy-r=1 number-conserving QWZ strip, let
000037 | P_N be its rank-8N negative-energy projector and Omega_N its filled sea.  Put
000038 | 
000039 |   D_N=N h_N/(2 pi),
000040 |   H_N=dGamma(D_N)-Tr(P_N D_N),
000041 |   Q_N=Number-8N.
000042 | 
000043 | For a smooth periodic boundary test function g, the top-edge source field is
000044 | 
000045 |   Psi_N(g)=a_N(R_N g),
000046 | 
000047 |   (R_N g)(x,y)=N^(-1/2) exp(-i pi x/(2N)) g(x/N)
000048 |                delta_(y,7) chi_+.
000049 | 
000050 | These are operators on the full microscopic QWZ Fock space.  They obey the
000051 | exact CAR, have charges -1 and +1 for Psi_N and Psi_N^*, and keep the sampled
000052 | support of g.  The polarized one-particle isometry J_N satisfies
000053 | 
000054 |   P_N J_N=J_N P_cur.
000055 | 
000056 | Its filled-sea second quantization gives a comparison contraction V_N from
000057 | the continuum excitation Fock space into that same microscopic Fock space.
000058 | On every fixed finite-energy sector V_N is eventually an isometry.
000059 | For every finite-energy vector psi and every smooth g,
000060 | 
000061 |   ||Psi_N(g)V_N psi - V_N a(g)psi|| -> 0,
000062 |   ||Psi_N(g)^*V_N psi - V_N a(g)^*psi|| -> 0.              (1)
000063 | 
000064 | The same statement holds for every fixed finite word, including timed words,
000065 | and the generators satisfy on every fixed energy cutoff E
000066 | 
000067 |   ||(H_N V_N-V_N H) 1_(H<=E)|| -> 0.                       (2)
000068 | 
000069 | The transported continuum time law is
000070 | 
000071 |   g_t(x)=exp(i t/4) g(x-t/(2 pi)).                          (3)
000072 | 
000073 | Equations (1)-(3) are strong vector convergence on the common dense union of
000074 | finite-energy spaces.  They are stronger than convergence of compressed
000075 | matrix elements.  The construction writes V_N=I_N Gamma(Pi_M).  On any fixed
000076 | finite-energy space, including the energy-64 space used below, Pi_M is the
000077 | identity for all sufficiently large N.  I_N then intertwines every required
000078 | truncated particle and hole mode and its adjoint exactly.  The actual local
000079 | top-edge modes differ from those polarized modes by the proved vanishing
000080 | edge-mode residual.  Thus the limiting statement concerns actual QWZ local
000081 | operators.  It does not claim strong convergence uniformly on the entire
000082 | infinite-energy microscopic Fock space.
000083 | 
000084 | Primary registered theorem source:
000085 | 
000086 |   paper-source:7296b452273b7022082ac2e2
000087 |   tex-artefacts/toe_research_20260909.tex:16-96.
000088 | 
000089 | The full-core and adjoint estimates are expanded in
000090 | experiments/theory-contracts/microscopic-charged-car-limit/README.md,
000091 | sections 2, 3 and 5.
000092 | 
000093 | 2. One-channel eightfold fold and the quarter convention
000094 | ---------------------------------------------------------
000095 | 
000096 | On the continuum one-particle space define
000097 | 
000098 |   (F f)_r(x)=f((x+2 pi r)/8)/sqrt(8),  r=0,...,7.          (4)
000099 | 
000100 | This is a unitary reindexing of one field.  If
000101 | 
000102 |   f_(n,alpha)(theta)=exp(i(n+alpha)theta)/sqrt(2 pi),
000103 |   n=8l+k,  mu_k=(k+alpha)/8,
000104 | 
000105 | then exactly