TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/REWEIGHTING-REVIEW.txt
Original file SHA-256: aa7479fc2d4bb9eb11fd071e81550906bd4105e43a19af759da7054efbc3d49a
Inclusive original lines: 201-291
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000201 | 
000202 | state
000203 | 
000204 |   S_def(alpha)=S_def(0)+alpha J,                        (20)
000205 | 
000206 | and infer
000207 | 
000208 |   delta_top(alpha)=delta_top exp(-alpha J).             (21)
000209 | 
000210 | The texts then impose J=d J_1, J_1=SF, d=2 and SF=1. The cover-degree
000211 | statement supports the coefficient d once J is an integrated deck-invariant
000212 | constant response. It does not by itself freeze Z_triv(alpha) or the residual
000213 | determinant and measure ratio.
000214 | 
000215 | There are therefore two source-faithful readings:
000216 | 
000217 |   1. If (20) is intended to mean that the ONLY relative alpha deformation is
000218 |      the constant topological term, then (21) is exact. This is the natural
000219 |      intended reading of the words "defect sector reweighting". Equation (7)
000220 |      should be part of the hypothesis or of the definition of the normalized
000221 |      defect weight.
000222 | 
000223 |   2. If (19) denotes complete sector partition functions with otherwise
000224 |      unrestricted alpha-dependent actions or measures, the correct theorem is
000225 |      (6). Equation (21) then additionally requires the Ward identity (8), in
000226 |      the concrete form (12) or (14).
000227 | 
000228 | This is a missing hypothesis in the written lemma, not evidence that its
000229 | chosen spectral-flow normalization is wrong.
000230 | 
000231 | 6. Relation to the recovered rank-one covariance determinant
000232 | -------------------------------------------------------------
000233 | 
000234 | FOCK-DEFECT-REVIEW.txt proves the directed algebraic identities
000235 | 
000236 |   det_F[(I-zC)(I-z(C+Pi))^(-1)]=(1-z)^(-1),
000237 |   det_F[(I-z(C+Pi))(I-zC)^(-1)]=1-z.                    (22)
000238 | 
000239 | Neither identity proves (8). For example, setting
000240 | 
000241 |   z=1-exp(-alpha d s)
000242 | 
000243 | makes the second factor in (22) equal exp(-alpha d s), but this is a parameter
000244 | substitution, not a derivation of the original action response. To use (22)
000245 | inside the old partition ratio, the original source must specify z as its
000246 | kernel or source deformation and show that every residual action, measure and
000247 | determinant contribution has the ratio R(alpha)=R(0). That last assertion is
000248 | exactly the Ward identity under review.
000249 | 
000250 | Conversely, the old reweighting argument does not need an identification of
000251 | alpha with heat time. If the same original source supplies the relative
000252 | action law (3), the constant d s and the cancellation (7), the exponential
000253 | follows immediately from its Euclidean sector weights. No equality with the
000254 | circle energy, charge, occupation number or positive heat semigroup is
000255 | required.
000256 | 
000257 | 7. Scope and repaired statement
000258 | --------------------------------
000259 | 
000260 | The strongest statement supported by the displayed general hypotheses is:
000261 | 
000262 |   For measure-compatible sectors with a sectorwise constant topological
000263 |   insertion d s,
000264 | 
000265 |     q(alpha)=exp(-alpha d s) R(alpha).
000266 | 
000267 |   The original formula q(alpha)=q(0)exp(-alpha d s) holds exactly on a given
000268 |   interval if and only if the normalized residual sector ratio R is constant
000269 |   on that interval.
000270 | 
000271 | For the existing original contract d=2 and s=SF=1, this becomes
000272 | 
000273 |   q(alpha)=exp(-2 alpha)R(alpha),                        (23)
000274 | 
000275 | and reduces to the archived exp(-2 alpha) law when R(alpha)=R(0).
000276 | 
000277 | This review removes an unnecessary proposed heat-time identification. It
000278 | does not complete the fundamental source claim. That claim still requires
000279 | showing that (3), the sectorwise constant spectral-flow insertion and (7)
000280 | hold for the two sectors and measures of the SAME original source whose
000281 | response enters the electromagnetic closure.
000282 | 
000283 | Source hashes at review time
000284 | ----------------------------
000285 | 
000286 |   fullv27 CFE section:
000287 |   8ae0867de9ddbd50e3684aa9b9b023d58791252c3fa146da7fc7b8ddf9cc1dd4
000288 | 
000289 |   fullv28 geometric self-consistency section:
000290 |   e2aa38f61b269e4c907559084ac6df3bf811961516c0cc72dd70d3285fd59081
000291 |