TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/VARIABLE-REVIEW.txt
Original file SHA-256: 6c8dd202a698c018ed3ecddfb38a0bdd1676101b9cf15b110547e08914095837
Inclusive original lines: 201-254
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.

000201 | If an original positive H measure is later used and is supported on this
000202 | bounded class, the uniform estimates pass through that existing integral.
000203 | No such support restriction or new measure is selected here. Unbounded
000204 | background ensembles need their own actual integrability control.
000205 | 
000206 | The same zero-order Hermitian V also preserves the original circle
000207 | finite-propagation argument: it cancels in the energy-current identity.
000208 | This retains the given cylinder locality where the R080 spacelike support
000209 | hypotheses hold; it derives no physical spatial dimension or Lorentz
000210 | invariance. The Hilbert-Schmidt step uses the given one-dimensional
000211 | compact Dirac resolvent and is not a3+1 source theorem.
000212 | 
000213 | R080 already supplies the S2 constant-background det2 source, centering,
000214 | trace correction, whole ordered words and supported circle net. The
000215 | present useful extension is the direct variable-background anticommutator
000216 | bound and its uniform exponential-moment control. It does not repair or
000217 | rederive the primitive H law, absolute Z_adm/Z_ref, Euclidean physical OS
000218 | reflection, Γ background normalization or full TFPT origin. Those claims
000219 | are outside the theorem, not negative conclusions from this review.
000220 | 
000221 | Independent review performed analytically. Root owns native controls;
000222 | none were duplicated here. No canonical or Atlas write.
000223 | 
000224 | 8. Review of the saved root proof
000225 | 
000226 | VARIABLE-BACKGROUND-PROOF.txt:1-188 was read in full after it was saved.
000227 | Its corrected htilde_N=gammaD0+P_N V P_N, explicit weak-form domain
000228 | argument, positive-matrix MGF proof, ordered complex-Cauchy step and
000229 | Galerkin boundary-defect treatment agree with this independent review.
000230 | No blocking mathematical error was found. The stated limited physical
000231 | scope and preservation of the original common S/Z route are appropriate.
000232 | 
000233 | One minor bookkeeping distinction is already implicit in its formulation:
000234 | the bound using ||A_F|| and ||A_F||_2 directly applies to A_N=P_N A_F P_N.
000235 | For a recomputed Galerkin A_N, use its own norms in(8), or the common
000236 | f0 and(6) bounds, rather than claiming its norms are bounded by the exact
000237 | untruncated A_F norms. Its limit and common uniform bound are proved by
000238 | the separate Galerkin argument. The root proof explicitly distinguishes
000239 | the two cutoffs; this is not a failure of the theorem.
000240 | 
000241 | [OMITTED: non-scientific conversation or execution metadata]
000242 | [OMITTED: non-scientific conversation or execution metadata]
000243 | [OMITTED: non-scientific conversation or execution metadata]
000244 | 
000245 | Primary formulas checked live
000246 | 
000247 | Klich, cond-mat/0209642v1, Section2 Eq8 and following multi-factor remark:
000248 | finite Fock traces become ordered one-particle determinants.
000249 | https://arxiv.org/html/cond-mat/0209642v1
000250 | Britz et al,2007.12834v3, Eqs1.3-1.5: det2 definition, continuity and
000251 | the trace correction in its product formula.
000252 | https://arxiv.org/html/2007.12834v3
000253 | The variable-background domain argument and inequality(8) are the explicit
000254 | calculations above; they are not attributed as TFPT results of these papers.