TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/RELATIVE-PAIRING-REVIEW.txt
Original file SHA-256: ef280950f70a108ecff5dde9f57c67d097494e3824fdd65aea2ca176b5b83a34
Inclusive original lines: 314-409
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.

000314 | 6. The original electromagnetic readout is one-sided
000315 | -----------------------------------------------------
000316 | 
000317 | The current source does select one oriented collar side before doubling. Its
000318 | starting datum is
000319 | 
000320 |   T_boundary=(A_+,H_+,D_+,J,Gamma,B_Sigma),
000321 |   Spec Z(A_+)=M_+,
000322 |   D_+=gamma_n(partial_n+B_Sigma).
000323 | 
000324 | This is stated in
000325 | 
000326 |   documents/archive--paper-latex--tfpt-42/sections/
000327 |   003-subsection-primitive-relative-spectral-dynamics.tex:3-24.
000328 | 
000329 | Only afterwards does the source form
000330 | 
000331 |   M_tilde=M_+ union_Sigma M_+^op
000332 | 
000333 | to obtain the deck involution, the doubled operator, and the Calderon projector
000334 | of the APS problem for D_+; see the same file at lines 37-88. Thus the double is
000335 | derived comparison and reconstruction data. It is not the definition of the
000336 | domain on which every later observable must be traced.
000337 | 
000338 | The relevant plus signs have distinct types. M_+, H_+, and D_+ name the chosen
000339 | oriented collar side. The boundary projector P_C selects Cauchy data of interior
000340 | D_+ solutions. In the primitive complex,
000341 | 
000342 |   P_Seam,+=(I+iota_C)/2=P_C,
000343 | 
000344 | selects the seam-even Calderon polarization. Gamma is the real-even spectral
000345 | grading compatible with the boundary restriction. None of these statements
000346 | identifies Gamma with a position-face projector. The source itself warns that
000347 | deck reflection, Calderon polarization, and the finite carrier involution may
000348 | only be related by the displayed restrictions and transports.
000349 | 
000350 | The electromagnetic formula is also singly directed. It defines
000351 | 
000352 |   U_Sigma^min=|psi_Sigma><psi_Sigma|
000353 | 
000354 | from the normalized primitive seam mode and evaluates
000355 | 
000356 |   -B gamma tr_adm log(1-q(alpha) U_Sigma^min).            (10)
000357 | 
000358 | Source:
000359 | 
000360 |   documents/archive--paper-latex--tfpt-42/sections/
000361 |   007-subsection-carrier-form-of-the-electromagnetic-closure.tex:20-48.
000362 | 
000363 | The formula in (10) is therefore singly directed: it contains one retained
000364 | rank-one seam transfer and no reflected inverse partner. This is compatible
000365 | with the oriented one-sided starting datum, and the source does not instruct
000366 | one to adjoin the reflected inverse determinant before taking the trace. But
000367 | the notation tr_adm is not defined as a trace on H_+ itself. The same source
000368 | defines
000369 | 
000370 |   H_adm=Ran(P_adm) subset H_OS
000371 | 
000372 | and defines the physical net by restriction to this invariant admissible
000373 | subspace; see 003-subsection-primitive-relative-spectral-dynamics.tex:248-285.
000374 | Thus P_adm is an admissibility selector after OS reconstruction, not a
000375 | position-face projector. The full-pair calculation in Sections 1 and 4 is a
000376 | type check on a possible doubled reinterpretation, not an obstruction to the
000377 | written singly directed EM formula; conversely, one must not identify
000378 | tr_adm with a chiral or face trace without another map.
000379 | 
000380 | Historical A3 uses two boundary cycles plus the eta gluing term to normalize
000381 | the cover curvature, and A4 adds the discrete topological correction. Those
000382 | statements fix phi_base and delta_top; they do not define the operator domain of
000383 | tr_adm or turn (10) into a full doubled vector determinant:
000384 | 
000385 |   documents/archive--paper-latex--tfpt-theory-fullv21/sections/
000386 |   001-section-introduction-the-problem-of-parameters.tex:165-221, 258-276.
000387 | 
000388 | The one-sided selection does not yet supply the KMS marginal used below. The
000389 | weakened ambient object
000390 | 
000391 |   (A_+,H_+,D_+,J,Gamma,B_Sigma,tau_t,omega)
000392 | 
000393 | assumes quasilocal positive dynamics, reflection positivity, and an admissible
000394 | vacuum sector, but it neither states that omega is a quasi-free beta-KMS state
000395 | nor gives
000396 | 
000397 |   C_adm,Sigma=(1+exp(beta B_Sigma))^(-1).                (11)
000398 | 
000399 | Source:
000400 | 
000401 |   documents/archive--paper-latex--tfpt-42/sections/
000402 |   034-subsection-weakened-ambient-class.tex:3-18, 107-120.
000403 | 
000404 | Nor is there a displayed isometry from the recovered old circle to the
000405 | admissible boundary marginal that intertwines B_a, u, tau_t, and the state.
000406 | Equation (10) instead defines the EM response directly from the already
000407 | selected rank-one transfer U_Sigma^min. Therefore the side selection is
000408 | sourced, while (11) is the exact remaining state/operator identification. No
000409 | additional chiral projection and no equality Gamma=P_C is needed.