TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/PROOF.txt
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000519 | 11.3 The new joint identity is not a fixed-field reconstruction theorem
000520 | 
000521 | This is an inverse-data test in the actual first TWO particle/hole excitation
000522 | pairs of the already supplied single chiral source. It is not a new source
000523 | ansatz, a locality proof or the P2 five-slot internal carrier. Keep
000524 |   epsilon=(alpha,1+alpha,1-alpha,2-alpha),
000525 |   Q=N_p0+N_p1-N_h0-N_h1,
000526 |   H_alpha=E_sea(alpha)I+sum_j epsilon_j N_j,
000527 | with the very SAME full prescribed Hurwitz sea term as in section10.
000528 | In this four-mode Fock space rotate ONLY the neutral two-particle states
000529 | |p0 h0> (excitation energy1) and |p1 h1> (energy3) by the real unitary block
000530 |   U=[[3/5,-4/5],[4/5,3/5]],
000531 | and take U=I on the other states. Define H'_alpha=U H_alpha U*.
000532 | 
000533 | This gives exact identities for ALL alpha in the chart and beta>0:
000534 |   UQ=QU,  U W_quarter=W_quarter U,
000535 |   Tr exp(-beta H'_alpha)=Tr exp(-beta H_alpha),
000536 |   partial_alpha H'_alpha=Q+partial_alpha E_sea(alpha).
000537 | Thus the full thermal partition and new eta/sea/thermal Ward identity agree,
000538 | including the absolute central phase; none has been renormalized away.
000539 | U fixes the vacuum AND the entire one-particle sector, so all vacuum TWO-field
000540 | time functions of the original fixed creators/annihilators also agree.
000541 | 
000542 | The four-field response distinguishes the fixed field dictionary. At t=log2,
000543 | remove only the COMMON factor exp(-t E_sea), and let A*=c_p0* c_h0*.
000544 |   <0|A exp(-t H_rel) A*|0> = 1/2,
000545 |   <0|A exp(-t H'_rel) A*|0> = (9/25)(1/2)+(16/25)(1/8)=13/50.
000546 | The difference is6/25. The original linear creator equation [H,c_p0*]=alpha c_p0*
000547 | fails for H' with the same creator. This is not a failure of positivity:
000548 | both positive-time evolution operators are positive and have identical spectra.
000549 | 
000550 | The assertion is about reconstruction with FIXED field/source labels. Under a
000551 | simultaneous conjugation of every field these two executions are equivalent.
000552 | Nor is this mode-space U proved local in the seam coordinate. Consequently this
000553 | is NOT a countermodel to the full geometrically local P1/P2 theory and does not
000554 | show the original source can contain arbitrary such deformations. It DOES show
000555 | that even the strengthened traced/one-particle comparison cannot select the
000556 | original fixed-field amplitude without its field/source identification.
000557 | 
000558 | 11.4 Consequence for the actual constructive target
000559 | 
000560 | Preserve the coupled eta/sea/thermal identity as a consistency check in its given
000561 | source class. Do not use it to infer an entire physical measure or field map.
000562 | The next source equation must retain actual insertions, for example the same
000563 | neutral four-field response, or the appropriate full Schwinger-Dyson identities
000564 | of the ORIGINAL interacting S_adm. Do not simply impose a free creator equation
000565 | on that interacting action: its higher vertices are explicitly retained there.
000566 | The search must also use an actual selfadjoint total operator/domain and the
000567 | Green-compatible geometric source pairing, not the false perturbative shortcut.
000568 | These corrections make a proposed derivation falsifiable at a concrete missing
000569 | response. They do not establish the requested complete physical P1/P2 solution.