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The displayed arguments below are the proof-bearing source. The current repository regression is v1035_ matter_ coupling.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_matter_coupling.tex · lines 2 to 37
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We state the spatial regulator of the preceding classical and quantum artifacts as the intended real-space interpretation. The checker below instantiates these declared masks algebraically, but does not import a repository implementation. On a periodic cubic lattice of spacing [formula in source] , a component with mask [formula in source] lives at [formula in source] , and [formula in source] Thus [formula in source
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tex-artefacts/toe_round7_matter_coupling.tex · lines 38 to 100
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Let [formula in source] be a symmetric c-number source on the scalar, vector, and tensor staggered cells. Its two Ward equations are [formula in source] At [formula in source] , add the spatial metric coupling and deform both constraints: [formula in source] The lapse and shift multipliers multiply [formula in source] and [formula in source] in the total Hamiltonian. Thus (8) is not merely the isolated [formula in so
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tex-artefacts/toe_round7_matter_coupling.tex · lines 101 to 211
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The local curvature variables remain [formula in source] They commute strongly with the gravitational parts of all four constraints: [formula in source] and [formula in source] . The c-number shifts in (8) have zero bracket with them, so [formula in source] remain gauge invariant. They obey [formula in source] and, on [formula in source] , [formula in source] The retarded solution with vanishing past data is therefor
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tex-artefacts/toe_round7_matter_coupling.tex · lines 212 to 272
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Let matter have canonical variables [formula in source] and an uncoupled Hamiltonian [formula in source] . If its zeroth-order densities obey (7), then [formula in source] obeys (9) modulo [formula in source] . This is the precise first-order feedback result. It is not exact nonlinear closure. Indeed, even before computing the [formula in source] -correction to the matter stress, [formula in source] plus analogous cu
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tex-artefacts/toe_round7_matter_coupling.tex · lines 273 to 299
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It suffices to examine a collinear sector; failure there is failure of a purported universal cubic-lattice identity. Set [formula in source] on a periodic one-dimensional row and write [formula in source] The site energy and link flux [formula in source] obey the exact energy equation [formula in source] The continuum-looking averaged-gradient stress [formula in source] does not obey momentum continuity. An exact sam
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tex-artefacts/toe_round7_matter_coupling.tex · lines 300 to 382
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A second collinear test uses a free chiral fermion whose one-particle Hamiltonian is nearest-neighbour, with dispersion [formula in source] . This is a finite-spacing Ward test, not a claim of a doubler-free [formula in source] dimensional chiral theory. For a bilinear transferring momentum [formula in source] , set [formula in source] The exact divided-difference identity is [formula in source] Consequently the ener
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tex-artefacts/toe_round7_matter_coupling.tex · lines 383 to 420
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The preceding free tensor construction removed the complete [formula in source] canonical block to eliminate its negative trace direction. At [formula in source] , [formula in source] , so its vacuum local constraints vanish identically. Equation (8) would reduce to [formula in source] A positive-energy matter state normally has [formula in source] , and a state may also have [formula in source] . For [formula in sou
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tex-artefacts/toe_round7_matter_coupling.tex · lines 421 to 442
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Equation (13) is an exact classical Green-function solution of the conditional staggered Fourier equations and is gauge invariant and Ward compatible there. A smooth compact-time c-number source is expected to define linear forcing of the already-constructed oscillator Fock space, but the CCR import and Kubo commutator needed to prove that quantum statement are not executed here. Thus the generic prescribed-source fi
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tex-artefacts/toe_round7_matter_coupling.tex · lines 443 to 468
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The repository certificate v1035_ matter_ coupling.py imports no proof text. It checks the generic prescribed-source polynomial identities with their full complex Fourier signs, a conserved plane-wave witness, the mandatory failure of the old restored- [formula in source] sign, paired Fourier reality, all three declared Bloch transitions, the classical retarded TT step solution, the two collinear Ward complexes and m
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tex-artefacts/toe_round7_matter_coupling.tex · lines 469 to 505
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tex-artefacts/toe_round7_parent_mirror.tex · lines 1 to 1
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The displayed arguments below are the proof-bearing source. The current repository regression is v1034_ parent_ mirror.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_parent_mirror.tex · lines 2 to 46
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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 script identifier v1008_master_assembly_scaffold.py
literal short identifier uniquely resolves in local script registry
- explicit script identifier v1002_spin10_mirror_projector.py
literal short identifier uniquely resolves in local script registry
- explicit script identifier v1027_signed_det_car_wall.py
literal short identifier uniquely resolves in local script registry
- explicit file citation v1034_parent_mirror.py
literal source filename/path in TeX
- cross reference join tfpt_round7.wl
TeX and evidence source explicitly name the same Python identifier; statement equivalence not inferred
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The certificate reads, but does not import, the following landed files at the audited repository tip: lll source & exact object used here & status relevant to this note\\ v1002 & [formula in source] on 16 complex modes & onsite projector only\\ v1008 & [formula in source] , [formula in source] & momentum-cell scaffold\\ v1024 & positive seam/number-square parent & separate construction\\ v1027 & [formula in source] ,
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tex-artefacts/toe_round7_parent_mirror.tex · lines 47 to 89
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The v1008 routine edge_cluster_hamiltonian is The scan uses [formula in source] . This is a separate Fock cell at every sampled momentum, not a real-space hopping Hamiltonian. In particular, a standard symmetric nearest-neighbour term [formula in source] has dispersion magnitude [formula in source] (up to a momentum/phase convention). Matching the v1008 dispersion amplitude would therefore give [formula in source] ,
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tex-artefacts/toe_round7_parent_mirror.tex · lines 90 to 104
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For [formula in source] , every occupation- [formula in source] state has energy On [formula in source] , after removing [formula in source] , the matrix is with eigenvalues
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tex-artefacts/toe_round7_parent_mirror.tex · lines 105 to 124
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[formula in source] vanishes on occupation sectors [formula in source] , which gives rank-one matrix specified by polynomial gives
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tex-artefacts/toe_round7_parent_mirror.tex · lines 125 to 141
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Let [formula in source] be any finite set of momentum cells, with arbitrary dispersions [formula in source] , and define the graded tensor-product Hamiltonian [formula in source] . Then [formula in source] has a unique product ground state and The same number is the uniform lower edge of canonical one-fermion spectral support.
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tex-artefacts/toe_round7_parent_mirror.tex · lines 142 to 154
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For [formula in source] , differentiation of minimum and substitution gives [formula in source] . The other empty/full eigenstate costs at least [formula in source] , and the [formula in source] sector also costs at least [formula in source] , so neither undercuts tensor product and energies add. Any state orthogonal to the product ground excites at least one cell, proving the same bound independently of [formula in
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tex-artefacts/toe_round7_parent_mirror.tex · lines 155 to 175
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Put two DET cells on a bond and let This is a bare-bond theorem. The numerical choice [formula in source] below is a newly chosen stress analogue, not a real-space coefficient derived from v1008. For even [formula in source] , the full-Fock CAR calculation contains only [formula in source] nonzero output monomials and does not require construction of the [formula in source] matrix.
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tex-artefacts/toe_round7_parent_mirror.tex · lines 176 to 189
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For every even [formula in source] and [formula in source] ,
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tex-artefacts/toe_round7_parent_mirror.tex · lines 190 to 200
These are recorded citation relations. A source citation or topic match does not establish a mathematical or physical result.
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