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tfpt research contracts · tfpt research contracts · Research contracts
Only the [formula in source] and [formula in source] components of the product DET vacuum can hop. Each has amplitude [formula in source] . The two hopping orientations and [formula in source] flavours produce [formula in source] mutually orthogonal states, each of modulus [formula in source] , which proves the norm in after the hop, hence both local projectors [formula in source] annihilate the image. [formula in so
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tex-artefacts/toe_round7_parent_mirror.tex · lines 201 to 215
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tex-artefacts/toe_round7_parent_mirror.tex · lines 216 to 222
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If [formula in source] would have zero expectation in [formula in source] by [formula in source] . Since [formula in source] , this would say [formula in source] , contradicting [formula in source]
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tex-artefacts/toe_round7_parent_mirror.tex · lines 223 to 252
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tfpt research contracts · tfpt research contracts · Research contracts
The bare theorem above makes no Gauss-sector claim. For the [formula in source] analogue and [formula in source] , now adjoin an oriented link basis [formula in source] and define This operator is implementable on matter Fock space tensored with the finite link space; it is fermion-even because each summand is a matter bilinear. Take the endpoint Gauss constraints to be [formula in source] and [formula in source] . T
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tex-artefacts/toe_round7_parent_mirror.tex · lines 253 to 297
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tfpt research contracts · tfpt research contracts · Research contracts
The following repair is included to make the result actionable, not to present a new source claim. On a bond let It obeys [formula in source] . For the flavour-diagonal hopping [formula in source] with unit coefficient, [formula in source] exactly. Therefore On a graph of maximum degree [formula in source] , replacing ordinary hopping by its projected version yields For the newly chosen periodic-1D bond analogue [for
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tex-artefacts/toe_round7_parent_mirror.tex · lines 298 to 345
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tfpt research contracts · tfpt research contracts · Research contracts
The repository certificate v1034_ parent_ mirror.py is read-only. Its JSON separates 29 exact algebra checks, six exact source-contract checks, and one floating-grid regression. The printed tag identifies each class. The checker performs an AST source-contract audit, exact [formula in source] onsite algebra, exact symbolic minimization, and a sparse CAR-monomial two-cell calculation. It does not build the [formula in
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tex-artefacts/toe_round7_parent_mirror.tex · lines 346 to 368
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tex-artefacts/toe_round7_proofs.tex · lines 1 to 4
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This supplement records five written proofs and their machine regressions: [formula in source] passing checks. The Round 6 prerequisite contributes [formula in source] checks ( [formula in source] exact and [formula in source] floating regressions) v1031_ quantum_ tensor_ curvature.py. Round 7 contributes [formula in source] checks: [formula in source] exact, [formula in source] source-contract, and [formula in sourc
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tex-artefacts/toe_round7_proofs.tex · lines 5 to 45
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tfpt research contracts · tfpt research contracts · Research contracts
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tex-artefacts/toe_round7_quantum_tensor_curvature.tex · lines 1 to 1
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The displayed arguments below are the proof-bearing source. The current repository regression is v1031_ quantum_ tensor_ 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_quantum_tensor_curvature.tex · lines 2 to 31
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Let a periodic cubic lattice have [formula in source] cells and spacing [formula in source] . The six orthonormal symmetric-tensor coordinates are ordered as [formula in source] , with the off-diagonal basis tensors normalized by [formula in source] . Component [formula in source] lives at [formula in source] , where [formula in source] We use the physical-position Fourier phase [formula in source] . The staggered de
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tex-artefacts/toe_round7_quantum_tensor_curvature.tex · lines 32 to 93
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For [formula in source] , let [formula in source] . The orthogonal spin-two projector on symmetric tensors is [formula in source] It has rank two, [formula in source] , and its range is [formula in source] .
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tex-artefacts/toe_round7_quantum_tensor_curvature.tex · lines 94 to 137
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Let [formula in source] be the real finite-volume smearing space of curvature pairs [formula in source] , with the homogeneous fibre absent and with the same reality/Bloch conditions as the fields. Define [formula in source] where [formula in source] and the brackets denote the real lattice inner product. Canonical quantization means [formula in source] The null space of (7) is [formula in source] . We therefore defi
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tex-artefacts/toe_round7_quantum_tensor_curvature.tex · lines 138 to 266
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Write [formula in source] . Physical-position Fourier coefficients obey [formula in source] and similarly for [formula in source] . Changing [formula in source] by [formula in source] reverses [formula in source] and gives [formula in source] Also [formula in source] . Therefore (7), (10), and (15) descend to the Bloch bundle and respect the real involution. No polarization vectors are chosen, so there is no hidden g
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tex-artefacts/toe_round7_quantum_tensor_curvature.tex · lines 267 to 296
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At fixed [formula in source] , extend the covariance symbols by zero at [formula in source] . The estimates [formula in source] show that all symbols are continuous on the compact Brillouin torus. A finite-support smearing has a trigonometric-polynomial Fourier transform. Consequently every entry of (7) and (10) is a Riemann sum of a continuous periodic function. As [formula in source] it converges to the correspondi
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tex-artefacts/toe_round7_quantum_tensor_curvature.tex · lines 297 to 333
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Take the thermodynamic limit first. For [formula in source] in [formula in source] , let [formula in source] The band-limited interpolation of a lattice kernel is the tempered distribution obtained by integrating its multiplier over [formula in source] . Equivalently, lattice fields with [formula in source] are scaled as [formula in source] distributions, so their smearings are [formula in source] . For Schwartz test
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tex-artefacts/toe_round7_quantum_tensor_curvature.tex · lines 334 to 433
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The repository certificate v1031_ quantum_ tensor_ curvature.py is standalone and imports no proof text. It is a separate implementation check, not an external independent review. Its roles were fixed before execution: exact SymPy polynomial algebra checks (3), (5), (6), and the separate index formula (4); exact rational-momentum fibres, characteristic polynomials, TT action, and the factorization (12); exact Bloch t
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tex-artefacts/toe_round7_quantum_tensor_curvature.tex · lines 434 to 482
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Research contracts · tfpt universalraum synthesis 2026-09-12
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tex-artefacts/toe_universalraum_synthesis_20260912.tex · lines 1 to 1
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Research contracts · tfpt universalraum synthesis 2026-09-12
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tex-artefacts/toe_universalraum_synthesis_20260912.tex · lines 2 to 4
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Research contracts · tfpt universalraum synthesis 2026-09-12
This section consolidates the local primitive-transfer investigation and the latest two available answers from each of four ChatGPT research tasks. It is not an independent rerun of all external certificates, an exhaustive conversation archive, or a derivation of a theory of everything. All T1–T8 and TFPT.TOE.COMPLETE.01 remain open. The local checkout base is 66b91e40e245569f06ab440ead80f446c9be0ee5; additional loca
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tex-artefacts/toe_universalraum_synthesis_20260912.tex · lines 5 to 21
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