TFPT scientific source excerpt Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/NEUTRAL-DEFECT-PAIRS-AND-VECTOR-WARDS.txt Original file SHA-256: e798405b02c979e0f9e77e223a097ea9ae269c3bd677a501f2becc8f00c0dda5 Inclusive original lines: 159-205 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. 000159 | 6. A useful exact counting-statistics identity, with its regulator exposed 000160 | 000161 | Consider only the EXPLICIT FINITE FILLED-BAND COMPARISON 000162 | m=-8ell,...,-1 of the original single complex Fourier source. This is not 000163 | substituted for the full QWZ state or for an infinite chiral Dirac sea. 000164 | Because 1_E has Fourier support8Z, the occupied matrix for Gamma(sigma_E) 000165 | has eight identical residue blocks. If D_ell(I) is the one-block Toeplitz 000166 | determinant with symbol 1-2 1_I, then 000167 | _finite band = D_ell(I)^8. (10) 000168 | The common quarter phase cancels in these density matrix elements. 000169 | 000170 | For I=(0,pi), this symbol is antiperiodic under translation by pi, so its 000171 | Fourier coefficients vanish at even indices. The ell-by-ell matrix is 000172 | bipartite by index parity; odd ell has unequal bipartite block sizes and 000173 | therefore D_ell=0 EXACTLY. Examples: 000174 | D1=0, D2=-4/pi^2, D3=0, D4=256/(9pi^4). 000175 | 000176 | The finite occupied band has TWO Fermi edges: the original sea interface 000177 | and the artificial lower cutoff. It cannot establish a single-chiral h1 000178 | field limit. At parity counting angle the leading Fisher-Hartwig branches 000179 | are degenerate and must both be retained. Abanov-Ivanov-Qian, 000180 | arXiv:1108.1355, AppendixB.3, equation41, explicitly gives their cosine sum 000181 | for the stated nonchiral lattice Fermi gas. That exponent/branch formula 000182 | is NOT applied as a theorem to the QWZ matrix symbol or to a single chiral 000183 | source. Ivanov-Levkivskyi, arXiv:1507.07896, distinguishes sharp discrete 000184 | counting from smoothed density; changing the smoothing regime is a 000185 | changed limiting contract, not automatic equality of endpoint fields. 000186 | 000187 | Rehren-Tedesco, arXiv:1205.0324, constructs vacuum-preserving multilocal 000188 | fermionization with CHANGED localization and nonintertwined conformal 000189 | motions. Its existence is positive comparison evidence, not proof that 000190 | our fixed source/time/quarter dictionary already has a separated spinor. 000191 | 000192 | 7. Actual full QWZ control and what it decides 000193 | 000194 | The checker imports the ORIGINAL v1033 QWZ builder read-only, Nx32,Ny8, 000195 | mass1,register sectors1 and3. All256 negative-energy modes in each 000196 | 512-dimensional one-body operator are retained. It conjugates the full 000197 | covariance by the exact periodic mask, with no edge projection or Wick 000198 | replacement of a mixed Ramond zero-mode state. 000199 | 000200 | At both y-boundaries, the cross-endpoint two-point block is exactly sign- 000201 | reversed by the algebraic mask and is numerically nonzero in both actual 000202 | quarter states. Its change has Frobenius norm about0.644506838591787. 000203 | Both quarter seam phases remain unchanged. These are finite ORIGINAL 000204 | controls, not extrapolation of a continuum scaling dimension, a proof of 000205 | a monodromy OPE, or an independent right-edge absorption.