TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/PAIR-TYPING.txt
Original file SHA-256: ff91e3075261d7381b594789ecc19de46770872d6530dff9714de40ba065b13e
Inclusive original lines: 201-303
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 | 
000202 | The old complex unitary U is also compatible with the existing charged
000203 | selfdual chart: C0=iN, so C0 E=iF. The charged map
000204 |  T(v,bar(w))=E v+C0 E w differs by that fixed i phase on its conjugate
000205 | second slot. Preserve the conjugate-space type; C0 is not relabeled J_F.
000206 | The actual original real structure is transported as U*J U and the full
000207 | adjoint/charge sectors remain present. These identities do not derive
000208 | an additional physical KO, Pfaffian, or OS time identification.
000209 | 
000210 | 5. EXACT MODULAR RETURN, INCLUDING THE REFERENCE
000211 | 
000212 | For the established normal-independent product collar, apply the same
000213 | graph construction to each member of the pair. Its actual second-face
000214 | occupation is
000215 |  C(h)=(I+exp(2ell h))^-1,
000216 |  C_ref=(I+exp(2ell h_ref))^-1.
000217 | Both are strict contractions with their genuine spectral logarithm
000218 | domains. Spectral calculus gives
000219 |  log((I-C)C^-1)=2ell h,
000220 |  log((I-C_ref)C_ref^-1)=2ell h_ref.                        (7)
000221 | On the circle Dom(h)=Dom(h_ref)=H1 for the smooth bounded coefficient
000222 | class. The logarithms are not asserted bounded or defined by numerical
000223 | eigenvalue clipping on all of L2.
000224 | 
000225 | The COMPLETE existing v258 reader includes its odd projection:
000226 |  D_F,ind=mu_geo Pi_odd^gamma log((I-C)C^-1).
000227 | Thus it and its actual transported reference give
000228 |  R_Phi=-2ell mu_geo i gamma Phi,
000229 |  R_Phiref=-2ell mu_geo i gamma Phi_ref,
000230 |  R_Phi-R_Phiref=-2ell mu_geo i gamma(Phi-Phi_ref).         (8)
000231 | With the already fixed U_gamma=exp(-i*pi*gamma/4),
000232 |  Phi=U_gamma* R_Phi U_gamma/(2ell mu_geo),
000233 |  Phi_ref=U_gamma* R_Phiref U_gamma/(2ell mu_geo).           (9)
000234 | Retaining A reconstructs h and h_ref from these finite coefficients.
000235 | No mu_geo=1/(2ell) normalization is chosen.
000236 | 
000237 | These are THREE distinct targets with explicit maps:
000238 |  the bulk pair (D,D_ref), the tangential pair(h,h_ref), and the bounded
000239 |  internal reader pair(R_Phi,R_Phiref). The old D_rel corresponds to
000240 | their DIFFERENCE in(3)/(8), not automatically to the whole h. The current
000241 | odd reader of the full face retains Phi, not the old difference alone.
000242 | This is the needed typed old-to-current comparison, not a notation fix.
000243 | 
000244 | Normal-dependent source profiles require the actual normal propagation,
000245 | not a new pointwise exponential. If K_ref is the actual closed injective
000246 | dense-range Cauchy transfer, its graph has second-face block
000247 |  C_ref,2=K_ref(I+K_ref* K_ref)^-1 K_ref*
000248 |         =K_ref K_ref*(I+K_ref K_ref*)^-1.
000249 | Consequently on its actual spectral domain,
000250 |  log((I-C_ref,2)C_ref,2^-1)=-log(K_ref K_ref*).             (10)
000251 | For a finite regulator the given normal equation determines K_ref by
000252 | partial_r K_ref(r,s)=-h_ref(r)K_ref(r,s), K_ref(s,s)=I.
000253 | The continuum statement uses the original well-posed Cauchy/graph
000254 | realization; existence of a global bounded normal semigroup on all L2
000255 | is not inferred from selfadjoint h with two-sided spectrum.
000256 | Only for the autonomous positive transfer K_ref=exp(-ell h_ref) does
000257 | (10) reduce to2ell h_ref. The full graph crossing retains the polar part
000258 | of K_ref as well; the diagonal face alone retains K_ref K_ref*.
000259 | Thus no normal profile, phase or polar transport is silently lost.
000260 | 
000261 | 6. SOURCES, REVIEW AND CONTROL
000262 | 
000263 | The E functor, its normal matrices and actual domains were read in full:
000264 |  research-source:251a551b4d69f1b19f606881:1-103,
000265 |  graded_boundary_functor/FINITE-COUPLED.txt,
000266 |  SHA256 61d3d524801aff0147d393c518ad554e3ab550271165e2000d9efbf613b90234;
000267 |  research-source:327a34bf933d5eceddd1a9b8:1-146,
000268 |  LOSSLESS-CARRIER-ODD-READER-DICTIONARY-REVIEW.txt,
000269 |  SHA256 b6f457488f84a06f5d97288c36210fc06ee2d94eb709ed90118e0fb27e40b767;
000270 |  research-source:59dc78be6ccd2e22cab708cc:1-105,
000271 |  CURRENT-CARRIER-ODD-READER-WHOLE-SOURCE.txt,
000272 |  SHA256 d45db71f3a493d6699473be195bea821a402523c4cf1c2c4cb102dafc39e0d05.
000273 | The complete v258 definition including Pi_odd was freshly read at
000274 |  source:3547205653ff5459c8c2:1-50,
000275 |  verification/v258_dirac_covariance_induction.py,
000276 |  SHA256 16c60e831dceba773f3629466895f1009654827286e5b760ae01bd755fd5c0b1.
000277 | 
000278 | Independent joint_origin review verified U, the full block operator(1),
000279 | and graph logodds(10). It explicitly required the shared normal-symbol
000280 | form before applying the theorem to D_ref; R032(4) supplies that premise.
000281 | The symbolic native control reuses the unchanged entire96 constructor
000282 | inputs of graded_boundary_functor/check_finite_typing.py, hash-read as
000283 | research-source:e5ac5dd24a510e809f86f91e:1-74. Only its existing constructor
000284 | prelude is executed; canonical outputs are not written. Independent
000285 | symbols encode the old normal derivative, tangential momentum, and the
000286 | even/odd radial Higgs directions of both R032 branches. The result is
000287 | recorded in pair-typing-check.json. This control checks signs and the
000288 | actual complete coefficient space. The analytic argument above proves
000289 | the general noncommuting operator/domain result, not a sample of Yukawas.
000290 | All19 exact controls pass, including the actual S/T fiber actions and
000291 | their direct reference half-sums. The first structural comparison of
000292 | one unexpanded radial polynomial returned false, although its expanded
000293 | whole-pair identity already passed. Expanding BE-Eh gives exactly zero;
000294 | no physical formula changed. The first raw diagnostic is retained in
000295 | pair-typing-check-first.json.
000296 | 
000297 | The new positive outcome is an explicit jointly transported bound pair,
000298 | its actual h_ref, and its full modular return. It does not identify a
000299 | bulk pair trace with the current S_rel[(D_rel+A_Sigma)/chi] call by name,
000300 | select the primitive lift, replace the physical P_F, or produce the full
000301 | unnormalized integration weight. Those conclusions require the original
000302 | corresponding caller; they do not follow from another representation
000303 | calculation. No further absence search or new model is performed.