TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/PROOF.txt
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Inclusive original lines: 449-480
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000449 | 11.1 The actual direction of the original reconstruction
000450 | 
000451 | The stated P1/P2 in tfpt_1_architecture_e8.tex:163-189 name an oriented seam,
000452 | primitive RP kernel, unit winding, c3 and the five-slot3+2 interface. They do not
000453 | specify the field arguments or complete Schwinger family of that kernel. It is
000454 | therefore also unjustified to decree that P1 is ONLY a two-point function.
000455 | The stronger operational seed in01:480-496 supplies A_loc,tau_t,Theta,omega,
000456 | winding and D_coll. Its construction runs from that seed to a spectral datum,
000457 | not from the two numerical inputs to the whole field/source functional.
000458 | Physical operationality is separately an axiom at01:622-625.
000459 | The actual QFT positivity lemma assumes a fermionic CAR RP kernel at04:476-484;
000460 | the source-symmetry argument uses anticommuting fermionic source derivatives
000461 | at04:2735-2739. The conditional OS reconstruction at04:2746-2782 is a positive
000462 | result for that full graded source family, not a derivation of its statistics.
000463 | The geometric-measure construction at05:1695-1737 starts with a geometric
000464 | reference measure and inherited OS positivity. It is not a preceding derivation
000465 | of a fermionic Berezin measure. The v155/v715 Gaussian/CAR constructions also
000466 | retain those physical inputs rather than generating them from P1/P2.
000467 | 
000468 | For a GIVEN formally selfadjoint product Dirac source D=J_n(partial_r+B),
000469 | J_n*=-J_n, J_n^2=-I, B*=B and {J_n,B}=0, integration by parts gives
000470 |   <D psi,phi>-<psi,D phi>
000471 |     = psi(0)* J_n phi(0)-psi(ell)* J_n phi(ell).
000472 | Thus its classical Green boundary form is fixed by the normal symbol and the
000473 | boundary Hilbert metric; it is not an additional freely fitted matrix. This
000474 | positive geometric statement and the Cauchy-Lagrangian identity were already
000475 | retained in seam-full-collar-reconstruction section2. They do NOT identify that
000476 | classical pairing with the actual quantum CAR state/operator amplitude.
000477 | Primary source: Baer-Ballmann, arXiv:1307.3021, proposition2.1 and section4.
000478 | Likewise Connes' real-spectral-triple antisymmetric pairing and Pfaffian use a
000479 | specified KO-dimension2 triple and anticommuting fields (Seminaire Poincare2007,
000480 | section7); those hypotheses cannot be inferred merely from an internal16.