TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ACTUAL-DECK-PULLBACK-INSERTION-REVIEW.txt Original file SHA-256: 348b7851bf816476609679058a5c922b6392c81b9b09b28f037ebe710b2efc1f Inclusive original lines: 1-87 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. 000001 | [OMITTED: non-scientific conversation or execution metadata] 000002 | [OMITTED: non-scientific conversation or execution metadata] 000003 | [OMITTED: non-scientific conversation or execution metadata] 000004 | [OMITTED: non-scientific conversation or execution metadata] 000005 | 000006 | 1. Original operation and reused results 000007 | The actual original is documents/archive--tfpt-45--source_extracts--01_boundary_ 000008 | kernel_source/sections/001-section-primitive-core-and-carrier-interface-handoff.tex: 000009 | 65-79,94-110,122-129,172-179,302-311. Its hash is 000010 | 7d6e673d99c29d950dc601283cd51c1cc0b113567eeb79c8396fec9c82a02c79. 000011 | Registered sources research-source:71d9b0e9792e4c99d88c2544 and 000012 | research-source:b54c34c09388e6044a7dd009 cover the split and action. 000013 | 000014 | The original tau is a map of the doubled MANIFOLD. Thus tau* includes the 000015 | base pullback and the compatible bundle/domain transport. It is used in 000016 | Dref=(D+tau*D tau*)/2 and Drel=(D-tau*D tau*)/2. The original explicitly 000017 | forbids literal identification of geometric reflection, Cauchy polarization 000018 | and the restricted finite-carrier involution at172-179. 000019 | 000020 | Reuse, not a new theorem: 000021 | finding:geometric-deck-not-calderon-polarization, and 000022 | finding:mass-only-deck-relative-heat-action-fails-free-infinite-trace. 000023 | The latter's source DECK-REFERENCE-AND-ACTION.txt:section3 already treats 000024 | the WHOLE product-reflection lift with the same finite grading. Its free 000025 | trace obstruction is not repeated here. 000026 | 000027 | 2. Smallest actual-source test; no global boundary choice needed 000028 | Use the existing signed collar coordinate r of the original double. For a 000029 | smooth central multiplier f supported inside that collar, a genuine bundle 000030 | lift U_tau over normal reversal must satisfy 000031 | U_tau M_f U_tau^-1=M_(f composed with tau). 000032 | For an odd nonzero f(r), the right side is -M_f. 000033 | But the proposed fiber F=S tensor gamma_F satisfies 000034 | F M_f F=M_f. 000035 | These differ on smooth compactly supported test sections in the doubled 000036 | collar. Such sections are a common interior core regardless of the later 000037 | closed boundary realization. This tests the actual coordinate action, not 000038 | an arbitrary replacement domain. F cannot itself be the original tau*. 000039 | 000040 | 3. Keep both the derivative symbol and the actual coupled coefficient 000041 | With the original matrices N,Z,S=NZ, the same existing expression is 000042 | D_s=N partial_r+S A+Z Phi_s, 000043 | Phi_s=(1+s)D_Y+D_MR, {gamma_F,Phi_s}=0. 000044 | Pure fiber conjugation gives 000045 | F D_s F=-N partial_r+S A+Z Phi_s. 000046 | It therefore gives the claimed Drel=N partial_r, Dref=S A+Z Phi_s, 000047 | but ONLY under that non-geometric fiber operation. 000048 | 000049 | Include the missing base pullback P_r:r->-r. On the same constant family, 000050 | tau_F=F P_r, tau_F D_s tau_F^-1=D_s, 000051 | Drel=0, Dref=D_s. (1) 000052 | Here the two sign reversals of the normal derivative cancel. Equation(1) 000053 | is the already established product-reflection covariance, not an assertion 000054 | that tau_F is the uniquely selected primitive double lift. For a transported 000055 | nonconstant background the actual Phi(-r,theta) must also be retained. 000056 | 000057 | The exact prior source is original_insertions/PROOF.txt:180-188: it says 000058 | R_prod preserves D_Phi TOGETHER WITH r reversal and explicitly does not 000059 | claim its domain, reference or admissibility identification. The later 000060 | EVEN-REFLECTION-POSITIVITY.txt:18-35 uses the two-face swap with this fiber. 000061 | Its whole Clifford/HS state transport remains a positive established result. 000062 | That two-face product reflection is not automatically the deck of source001. 000063 | 000064 | 4. Precise operator/domain status and the retained next source operation 000065 | The original source001 states reflection-regular doubling and D_+^op, but 000066 | the cited construction does not supply the spinor transition that identifies 000067 | its actual bulk tau* with F alone, or with the existing full tau_F. It cannot 000068 | do so for F alone because section2 contradicts its central-coordinate action. 000069 | No new choice of a closed APS extension repairs that local contradiction. 000070 | An operator equality for the full tau additionally carries the actual given 000071 | closed domain; the interior identities above do not independently choose it. 000072 | 000073 | The concrete retained task is therefore to use the actual original doubled 000074 | bundle transition together with its base pullback in the EXISTING bound 000075 | reference/relative split, along this same Phi_s. The known source graph and 000076 | whole reflection/state maps need no replacement. The earlier whole Loewner 000077 | response is a valid background-response arrow, not the completed Sprim call. 000078 | 000079 | Original chi_seed is lambda_1^+(|B_Sigma|)^-1 at source001:42-55. Even within 000080 | the failed fiber-only calculation, a Phi-independent unscaled N partial_r 000081 | does not imply a Phi-independent f(N partial_r/chi_s). If B_Sigma has actually 000082 | been identified with the present product B_s and g_s is its isolated positive 000083 | gap, chi_s=1/g_s and delta(Drel/chi)=dot(g) N partial_r in that calculation. 000084 | Without that B_Sigma identification, it is a conditional chart statement. 000085 | The original normalization, declared f, eta difference and domain remain 000086 | part of the Sprim comparison. No determinant/normalization cancellation or 000087 | bosonic background average follows from a fiber conjugation identity.