TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/old_source_embedding/REVIEW-CLAUDE-OLD-SOURCE.txt Original file SHA-256: df2871ace6df72300e56f7377368f21625f74441c8eab4f214ab0a3fea92f4dd Inclusive original lines: 201-271 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 | support I to -I. The actual old B_cov has instead the same orientation in 000202 | B_cov=2(A_1/4 direct_sum A_3/4) under W_0/W_1. 000203 | 000204 | More decisively, the original cover winding u_cov=exp(i theta) has the 000205 | W_0/W_1 matrix 000206 | u_cov=[[0,u_base],[I,0]], u_base=exp(i x). 000207 | It exchanges Quarter sectors and carries the extra winding in one off- 000208 | diagonal block. The newly given scalar multiplier u_base I_F preserves 000209 | F+ and F-. These are not the same supplied operation. Already 000210 | [B_cov,u_cov]=u_cov, 000211 | but 000212 | [2h_0,u_base I_F]=2u_base g 000213 | on the common smooth core. Thus matching the second spectral multiset is 000214 | not a complete original cover/mark/locality dictionary. 000215 | 000216 | One can explicitly pull u_cov back through R_sp and two chosen coefficient 000217 | lines, obtaining off-diagonal blocks u_base R_sp and R_sp. This constructs 000218 | an abstract global cover action, with reflected support on the minus line. 000219 | It does not identify that action with the actual supplied pointwise 000220 | multiplier, deck or marked original source operation. Such an identification 000221 | requires its real support and winding map, not another equality of spectra. 000222 | The single F+ Quarter embedding of section 1 is unaffected. 000223 | 000224 | For the free finite multiplicity, CAR of a one-particle direct sum gives a 000225 | graded tensor product of source algebras, not a direct sum of density states. 000226 | The positive-grade multiplicity alone is not an intertwiner of the actual 000227 | P2/D5/Spin(10) group action. The separate representation review handles that 000228 | claim; this operator/source proof does not use dimension as its substitute. 000229 | 000230 | 6. SOURCE-REVIEWED CONFIRMATION OF THERMAL-RIGIDITY.txt 000231 | 000232 | For constant Phi, a=alpha I and fixed unit e in F+, the reviewed formula is 000233 | 000234 | =1/2[1-p]. 000235 | It follows from h(p)^2=p^2+Phi^2 and the fact that the odd term maps F+ to 000236 | F-, so its expectation on e vanishes. Let f(x)=tanh(beta x/2)/x. 000237 | It is strictly decreasing on x>0, because tanh(y)-y sech^2(y)>0, with 000238 | derivative 2y sech^2(y)tanh(y)>0 and value0 at y=0. 000239 | Functional calculus of Phi^2>=0 therefore proves the stated one-sided 000240 | occupation inequality: >=the old free value for p>0, <= for p<0, with 000241 | equality precisely when Phi e=0. Quarter momenta have no zero value. 000242 | 000243 | Accordingly the three claims in that file are indeed equivalent: 000244 | one nonzero-momentum old free occupation is reproduced; 000245 | Phi e=0; 000246 | the FIXED local channel carries the entire old free graph/state/time map. 000247 | For an already given positive classical probability measure on constant 000248 | Phi with e,p,beta fixed, the occupation differences cannot cancel. Equality 000249 | of their average holds precisely for Phi e=0 almost surely. In that case 000250 | all conditional restricted scalar word functionals agree, hence so do 000251 | their averages. This proves the full stated scoped conclusion without a 000252 | Gaussian assumption on the averaged source. 000253 | 000254 | No corresponding assertion is proved for nonconstant backgrounds, 000255 | changing connections or channel maps, complex Pfaffian weights, interacting 000256 | quantum bosons, or an obligation that all physical TFPT remain free. The 000257 | classical constant-background proof in THERMAL-RIGIDITY.txt is accepted. 000258 | 000259 | FINAL SOURCE STATUS 000260 | 000261 | The free scalar Quarter source is explicitly and locally present in the 000262 | given positive-grade free Clifford source, with full two-face correlations, 000263 | reflection and SAME Quarter time. Its charges, eightfold source and retained 000264 | delta dictionary follow under the displayed actual reference/fold maps. 000265 | That positive result must not be left as an unknown charge law in THIS 000266 | stronger source class. The more general neutral-KMS caveat remains valid. 000267 | 000268 | The underlying physical P1 geometry/quantization and original marked 000269 | insertion map remain separate from this conditional source embedding. The 000270 | oppositely oriented finite grade does not by itself provide the old whole 000271 | NS winding tuple. No complete fundamental physical solution is asserted.