TFPT scientific source excerpt Original repository path: experiments/theory-contracts/paper-source-reconstruction-20261025/MICROSCOPIC-TO-OCTIC-COMPOSITION.txt Original file SHA-256: eb0d6d436abeab7ace1ff094740c4b0d4748d3599331bfb3454dcb21e328ef5a Inclusive original lines: 1-200 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 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 000106 | 000107 | F f_(8l+k,alpha) 000108 | =U_alpha(:,k) exp(i(l+mu_k)x)/sqrt(2 pi), 000109 | 000110 | U_alpha(r,k)=exp(2 pi i(k+alpha)r/8)/sqrt(8). (5) 000111 | 000112 | Consequently the folded covariance is 000113 | 000114 | U_alpha^* F C_alpha F^* U_alpha 000115 | =diag_k C^-_(mu_k) (6) 000116 | 000117 | in the eigenchannel chart. In the eight-arc chart the off-diagonal 000118 | correlations U_alpha diag(C^-_(mu_k)) U_alpha^* remain. The diagonal 000119 | eigenchannel representation has the derived graded tensor-product 000120 | factorization of this fixed state. It adds no freely chosen source copies. 000121 | 000122 | The apparent alpha mismatch is only the stated occupation and orientation 000123 | change. The QWZ edge uses modes exp(i(j-1/4)theta) filled for j>=1. With 000124 | n=j-1 this is the positive-filled alpha=3/4 chart. Reversing the boundary 000125 | orientation and writing m=-n-1 gives 000126 | 000127 | exp(-i(n+3/4)theta)=exp(i(m+1/4)theta), m<=-1, (7) 000128 | 000129 | which is the negative-D alpha=1/4 sea used by the native Ward calculation. 000130 | Since m=-j, its one-particle generator is 000131 | 000132 | m+1/4=1/4-j=epsilon_j, 000133 | 000134 | exactly the QWZ limiting energy appearing in (3), including its sign. 000135 | The channel label reverses k -> 7-k, and the point-split counterterms obey 000136 | 000137 | -delta_k^(3/4)=delta_(7-k)^(1/4), 000138 | delta_k^(alpha)=(k+alpha)/8-1/2. (8) 000139 | 000140 | The minus sign in (8) is accompanied by the creator/annihilator order and 000141 | the current-charge convention. Flipping only the displayed c-number would 000142 | not define the same representation. With this complete transport, the QWZ 000143 | time law (3), the folded source time and the adjoint operation describe one 000144 | and the same filled-sea representation. 000145 | 000146 | Registered fold source: 000147 | 000148 | research-source:1a8c7c3f539e779301a7377d 000149 | FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt:53-152. 000150 | 000151 | 3. The actual neutral octic in the QWZ source 000152 | ---------------------------------------------- 000153 | 000154 | Let j_sea=(j_0,...,j_7) be the eight eigenchannel currents formed as 000155 | 000156 | j_k(z)=:chi_k^*(z) chi_k(z):_sea 000157 | 000158 | from the folded version of the one QWZ limit field. Retain the original 000159 | matrices M and O and the J-positive polarization U_+. Put 000160 | 000161 | A=U_+^T O^T M, 000162 | A A^*=I_4, 000163 | A A^T=0, 000164 | b_alpha=A delta_alpha. (9) 000165 | 000166 | With 000167 | 000168 | I8(x)=sum_a x_a^8 000169 | +14 sum_(a