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Use the sigma- [formula in source] basis [formula in source] , [formula in source] , [formula in source] . The jump is [formula in source] (zero-based indices), and [formula in source] for [formula in source] . At the 512 midpoint nodes, 256-bit Acb validated eigensystems strictly separate eight occupied and eight unoccupied eigenvalues. The outward matrix DFT [formula in source] obeys the exact midpoint alias formul
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tex-artefacts/toe_round4_proofs.tex · lines 175 to 242
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CHIRAL4D.MIRROR.SIGNED.CAR.01. v1027_ signed_ det_ car_ wall.py checks the following algebra and its finite countermodels. The old Hubbard operators [formula in source] satisfy [formula in source] , not CAR, and [formula in source] . For even [formula in source] (in particular [formula in source] ), write [formula in source] and define [formula in source] This is a two-dimensional rotation and identity elsewhere. It
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tex-artefacts/toe_round4_proofs.tex · lines 243 to 303
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- explicit file citation v1027_signed_det_car_wall.py
literal source filename/path in TeX
- explicit claim identifier status_ledger.csv
exact local ledger claim identifier
- cross reference join tfpt_round4.wl
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PARENT.GAUSS.D4.ROUND4.01. v1028_ gauss_ d4_ cap.py uses the full exterior character, not a low-occupation truncation. Let [formula in source] consist of [formula in source] , [formula in source] , [formula in source] , [formula in source] , [formula in source] , [formula in source] , with charges [formula in source] . For color weights [formula in source] and weak weights [formula in source] , set [formula in source
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tex-artefacts/toe_round4_proofs.tex · lines 304 to 374
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- explicit file citation v1028_gauss_d4_cap.py
literal source filename/path in TeX
- cross reference join tfpt_round4.wl
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- cross reference join tfpt_round4.wl
TeX and evidence source explicitly name the same Python identifier; statement equivalence not inferred
- explicit claim identifier status_ledger.csv
exact local ledger claim identifier
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GRAV.SPIN2.LOCAL.CONSTRAINT.01. v1029_ local_ tensor_ constraints.py implements a quadratic target, not a TFPT embedding. For six orthonormal symmetric tensor coordinates, let [formula in source] , [formula in source] , [formula in source] , [formula in source] . Set [formula in source] with constraints [formula in source] , [formula in source] . Direct polynomial algebra gives [formula in source] The four constraint
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tex-artefacts/toe_round4_proofs.tex · lines 375 to 422
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SEAM.MMST.ALG2.FRAME.01. v1030_ alg2_ sector_ frame.py does not restore the refuted Vandermonde implication. Suppose, explicitly, [formula in source] and the actual neutral algebra is [formula in source] . Join [formula in source] if [formula in source] . Then [formula in source] precisely when this undirected sector graph is connected: the neutral commutant is [formula in source] , and each nonzero corner forces [fo
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tex-artefacts/toe_round4_proofs.tex · lines 423 to 457
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- explicit claim identifier status_ledger.csv
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- explicit file citation v1030_alg2_sector_frame.py
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- cross reference join tfpt_round4.wl
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tex-artefacts/toe_round7_charged_disorder.tex · lines 1 to 1
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The displayed arguments below are the proof-bearing source. The current repository regression is v1033_ charged_ disorder.py; its finite checks audit the implementation and do not replace the universal steps in the proof.
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tex-artefacts/toe_round7_charged_disorder.tex · lines 2 to 41
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The construction below uses the hopping matrix literally implemented in verification/v988_psi_lambda_reduction.py. With [formula in source] the one-particle QWZ Hamiltonian [formula in source] has mass [formula in source] , open [formula in source] boundary, and on the bond [formula in source] the forward hopping is [formula in source] . Every other longitudinal bond has [formula in source] . The executable v988 camp
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tex-artefacts/toe_round7_charged_disorder.tex · lines 42 to 98
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Let [formula in source] and let [formula in source] be the canonical number-conserving CAR Fock space. Adjoin one four-level link register [formula in source] at the chosen periodic seam. In the basis [formula in source] define [formula in source] If [formula in source] is the forward seam hopping and [formula in source] , set [formula in source] Thus the register is not a decorative sector label: its clock occurs in
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tex-artefacts/toe_round7_charged_disorder.tex · lines 99 to 128
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Choose an arc [formula in source] with [formula in source] . Let [formula in source] be the one-particle onsite gauge [formula in source] Define [formula in source]
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tex-artefacts/toe_round7_charged_disorder.tex · lines 129 to 207
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For a normalized complex chiral fermion, a twist by [formula in source] has minimal bosonized weight [formula in source] , with [formula in source] . The actual common phase [formula in source] therefore fixes, for the eight complex channels in the source, [formula in source] It has order four but nontrivial spin [formula in source] . The target (1.3) instead has integer spin and weight one. Norm is invariant under a
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tex-artefacts/toe_round7_charged_disorder.tex · lines 208 to 266
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There is a finite-lattice tradeoff that prevents another overclaim. Portability amendment, 2026-09-06. The [formula in source] one-particle operator has an exactly degenerate zero-energy space. Filling the first half of a numerical eigenbasis silently selects a platform-dependent pure vacuum. Let [formula in source] and [formula in source] denote its negative and zero spectral projectors, let [formula in source] , an
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tex-artefacts/toe_round7_charged_disorder.tex · lines 267 to 342
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The repository certificate v1033_ charged_ disorder.py audits the frozen sources, proves the finite Weyl/parity/charge identities with rational and symbolic arithmetic, imports the executable v988 QWZ builder and compares it entrywise, verifies the endpoint commutator on the actual [formula in source] hops, and reports finite energy diagnostics.
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tex-artefacts/toe_round7_charged_disorder.tex · lines 343 to 363
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- explicit file citation v1033_charged_disorder.py
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- explicit script identifier v988_psi_lambda_reduction.py
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tex-artefacts/toe_round7_covariant_curvature.tex · lines 1 to 1
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The displayed arguments below are the proof-bearing source. The current repository regression is v1032_ covariant_ curvature.py; its finite checks audit the implementation and do not replace the universal steps in the proof.
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tex-artefacts/toe_round7_covariant_curvature.tex · lines 2 to 26
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Use [formula in source] and a covariant null momentum [formula in source] All tensor components in this note carry lower indices. For a symmetric metric amplitude [formula in source] , define the linear curvature map It is the Fourier transform of [formula in source] No positive covariant quantum metric potential is assumed. Instead introduce the numerical tensor In a Euclidean orthonormal basis of symmetric [formula
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tex-artefacts/toe_round7_covariant_curvature.tex · lines 27 to 124
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Index the independent pair-symmetric bivector entries by [formula in source] . Define the two-point distribution The Bianchi identities make one of the 21 algebraic components redundant; retaining it with its null relation avoids an arbitrary choice of twenty coordinates.
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tex-artefacts/toe_round7_covariant_curvature.tex · lines 125 to 230
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The Round 6 continuum construction has, on the physical TT range, [formula in source] with [formula in source] . In radiation gauge Consequently the spectral electric-electric covariance is [formula in source] , electric-time-derivative covariance is [formula in source] at equal time in the order [formula in source] , and derivative-derivative covariance is [formula in source] . These are precisely one quarter of the
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tex-artefacts/toe_round7_covariant_curvature.tex · lines 231 to 273
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The repository certificate v1032_ covariant_ curvature.py is standalone SymPy code. Its 34 exact checks include the all-momentum polynomial identity, gauge and Bianchi identities, null Ricci contraction, positivity factorizations, a nontrivial rational Lorentz boost, the full null little group, and the exact electric and mixed time-derivative covariance normalization. Four rational fibres supplement, but do not repla
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tex-artefacts/toe_round7_covariant_curvature.tex · lines 274 to 304
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tex-artefacts/toe_round7_matter_coupling.tex · lines 1 to 1
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