TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/PROOF.txt
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000368 | 10.4 The thermal absolute factor and its central phase are separately typed
000369 | 
000370 | For the given sea-relative chiral source, an exponential Abel prescription
000371 | has E_epsilon=d/depsilon[exp(-(1-alpha)epsilon)/(1-exp(-epsilon))].
000372 | Its expansion is -1/epsilon^2+B2(1-alpha)/2+O(epsilon). Removing that specified
000373 | divergence gives E_sea^zeta=B2(1-alpha)/2=-1/96 at alpha1/4.
000374 | This is a REGULARIZATION CHOICE, not recovered from normalized G or P1/P2.
000375 | For beta=2ell=8log4 put q_beta=exp(-beta)=1/65536 (not q_ell=1/256).
000376 |   Z_beta^zeta[0]=exp(-beta E_sea^zeta)
000377 |      product_(n>=0)(1+q_beta^(n+alpha))(1+q_beta^(n+1-alpha)).
000378 | For finite mode sources S the full functional is this scalar times
000379 | exp(bareta_S (I+R_beta)^(-1)_SS eta_S). Existing section8 trace-tail bounds
000380 | control the omitted product at this SAME beta. Keeping only n=0 gives
000381 |   exp(-beta E_sea^zeta)=2^(1/6),
000382 |   Z_beta,0^zeta[0]=2^(1/6) 69649/65536.
000383 | The source-field conjugation still has phases i/-i. But exp(2pi i H_absolute)
000384 | additionally has central phase exp(2pi i E_sea^zeta)=exp(-i pi/48).
000385 | It must not be silently equated to the old relative quarter implementer.
000386 | Cancellation by an original eta/bulk determinant line is NOT proved here.
000387 | This is why the search must retain reference eigenvalue and central data rather
000388 | than fixing absolute vacuum amplitude to1 and then claiming a recovered source.
000389 | 
000390 | Within this SAME stipulated Hurwitz scheme, the angular and thermal quantities
000391 | are not independent fit parameters. On the zero-free chart0<alpha<1,
000392 |   E_sea=-zeta_H(-1,1-alpha)=eta_D(0)^2/8-1/24,
000393 |   partial_alpha E_sea=-eta_D(0)/2.
000394 | For the finite N-pair product with the full prescribed sea counterterm,
000395 |   -(1/beta) partial_alpha log Z_(beta,N)
000396 |     =-eta_D(0)/2 + sum_(n=0)^(N-1)
000397 |         [f_beta(n+alpha)-f_beta(n+1-alpha)],
000398 | where f_beta(e)=1/(1+exp(beta e)). Thus the derivative of the SAME unnormalized
000399 | functional equals its state/occupation response including the sea term.
000400 | For alpha1/4,beta8log4,N1 it is exactly
000401 |   -1/4+1/17-1/4097=-53329/278596=-3137/16388.
000402 | An independent four-state Fock trace checks this identity and its conjugate
000403 | and alpha1/2 controls. The full-mode limit uses the same positive-beta trace
000404 | tails: the omitted charge response is bounded in absolute value by
000405 |   exp(-beta N) [exp(-beta alpha)+exp(-beta(1-alpha))]/[1-exp(-beta)].
000406 | This also gives locally uniform convergence of the derivative in this chart.
000407 | These are source-coordinate spectral/thermal Ward identities, not the
000408 | angular i pi contact response itself or a P1/P2-derived physical event law.
000409 | Other counterterms can change them; matching the prescribed spectral scheme
000410 | is a premise. Across integer alpha the zero-mode chart and spectral flow must
000411 | be treated anew. The quarter branch is retained throughout this calculation.