Read the result in its research context.

Result

NativeRgeo preserveseachfixedsourcegraph. ComposingwithchiralE0givesNEWcomplexlinearBogoliubovmapstilltargetingPhiK,orbarPhi aftertoruscompensation. NativeTheta doesnotconjugateexternalcouplingsagain. Composition squaresminusoneoneparticle,fermionparityonCAR,identityonlyonevenobservables. Linearcircletransporthastargettime-t;nativeHSantiunitaryalonepreservessameparameter. Fullactualmatrixcontrolrejectsgeneric samePhi.

Required assumptions

• Alternative chiral comparison of q columns with actual v252 P2 carrier

  For this explicitly alternative comparison identify actual G columns(L0,L2,L1) with supplied generation slots through whole signed P2 dictionary. D_E acts on gamma_F positive,barD_E on negative. Retain unchanged full_D with Y and symmetricMR,F5 structure,faces,graph state and given time. Do not silently replace earlier particlewise hypothesis. No P1 selection,global trivialization,new state/time or physicalE8gauge action assumed.

• Geometric Clifford boundary with explicit changed complex structure

  Assume the existing free product class B=Gamma A or actual smooth tangential odd finite coefficient B_Phi=Z A-S Phi with total grading Z tensor gamma_F. Apply the standard Clifford boundary functor on the underlying real space with NEW complex structure Ical=i_old Gamma_tot and real structure C0=i_old gamma_n; the dual face has -C0. C0 is old-linear and new-antilinear,not identified with original charge J. This differs from full complex CAR quantization of both independent normal copies. State conclusions use the full graph as the actual quantum covariance in this functor. Its physical selection from P1/P2 is not presumed proved; Phi is a fixed classical background. Keep the fixed normal-independent product collar; a(theta) is smooth Hermitian and even,Phi(theta) smooth Hermitian and odd. No quantization of these bosonic backgrounds.

• Complete data of both product-collar faces

  A selfadjoint B occurs in an exact normal-independent product equation. The full orthogonal L2 Cauchy graph projection of both labeled faces, their actual fiber identification and independently known positive normal length ell are supplied. Selection of these data from P1/P2 is not presumed.

• Do not identify source types

  Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.

Original evidence

• CLAUDE29-NATIVE-THETA-COMPOSITION-REVIEW.txt lines 1–195

  Source checked at this snapshot

  Original location and fingerprint

  experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE29-NATIVE-THETA-COMPOSITION-REVIEW.txt

  da497fe88766278e77cebfbd4c344a9f91b8664f4a5811500450bdd449e10bc3

• CLAUDE29-CHIRAL-FINITE-REVIEW.txt lines 1–200

  Source checked at this snapshot

  Original location and fingerprint

  experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CLAUDE29-CHIRAL-FINITE-REVIEW.txt

  3446a9b90d1cbc7bd558c9b860bf10f8822e56cfce2169cab40423b29f095059

finding:native-theta-composition-retains-conjugate-source-and-parity-square