TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/historical-dynamics-source-recovery-20261103/HISTORICAL-THREE-HYPOTHESES-AND-WHOLE-SOURCE.txt
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000048 | 3. H2: KEEP THE COMPLETE SOURCE RESPONSE, INCLUDING NONLOCAL CONTACTS
000049 | Use the already derived actual regulated historical source
000050 |  L_E=A S²-S.Q+Lrest, Q=iB da+(3/4)J5E, A=3/(4kappa_phys²).
000051 | The real paired current is the one in TORSION20261104E, all Y/MR retained.
000052 | At fixed remaining fields an additional ORIGINAL S-dependent factor F stays
000053 | inside this same integral. No new F or physical measure is chosen here.
000054 | 
000055 | With an external insertion ell.S, W=log Z,
000056 |  d_ell W=<S>, d_ell² W=G_SS,c, d_ell²(-log Z)=-G_SS,c.
000057 | The external source-independent Zref[0] has no ell derivatives. Varying the
000058 | INTEGRATED field S is a different operation and inserts Q and F derivatives.
000059 | For all external fermion coefficients the unnormalized integration-by-parts
000060 | identities are
000061 |  2A Z_x=I[Q_x F+F_x],
000062 |  4A² Z_xy=2A delta_xy Z+I[Q_x Q_y F+Q_x F_y+F_x Q_y+F_xy].
000063 | Boundary/domain/selector contributions must be added when present. No
000064 | division by F is needed, including at Pfaffian zeros. With F nonzero locally,
000065 |  Y_x=delta logF/dS_x,
000066 |  G_SS,c=delta/(2A)+[Cov(Q+Y,Q+Y)+<deltaY/dS>]/(4A²).
000067 | This equality is for the WHOLE joint state, not a frozen fermion background.
000068 | If fermions have been integrated into F, remove their explicit J5 from Q.
000069 | 
000070 | For F=1, constant couplings and co-closed interior TEST sources the original
000071 | transverse response is
000072 |  G_STST,c=(2kappa_phys²/3)P_T+(kappa_phys^4/4)G_J5T J5T,c.
000073 | Subtract only the genuinely local term; separate kinematic projectors.
000074 | Where the actual original time/reflection map identifies the Hermitian
000075 | fields and sources, this implies the corresponding separated-time spectral
000076 | relation rho_S=(kappa_phys^4/4)rho_J5T in this minimal parent. It gives no
000077 | computed mass, pole or continuum support because the whole G_JJ is unevaluated.
000078 | It does not assert div J5=0 and does not select a new physical OS parity.
000079 | 
000080 | For general actual F, F_xy is a functional second variation, not necessarily
000081 | a distribution supported at x=y. It can carry separated-time response.
000082 | This distinction is relevant to a full spectral profile: the existing full
000083 | ordinary heat jet in HISTORICAL-REAL-TORSION-OPERATOR.txt:65-116 contains
000084 |  Fheat(t p²)=int_0^1 exp[-t p² a(1-a)]da.
000085 | Its nth Taylor coefficient is (-t)^n n!/(2n+1)!, nonzero for all n.
000086 | It is not a finite polynomial in momentum and therefore is not a finite
000087 | local contact distribution. This diagnoses the already calculated profile
000088 | response; no single heat atom is selected as physical input. A local a4
000089 | truncation does not retain the full nonlocal profile. Integrate only against
000090 | the actual original profile when that representation applies. This is not
000091 | a computation of arbitrary primitive Brel or moving-domain contributions.
000092 | 
000093 | Thus H2 is established in the declared minimal-parent response sense. Its
000094 | extension to the entire primitive source must include the actual reference
000095 | and other field forces, their mixed correlations and nonlocal second
000096 | variations. The stronger claim that all full-source torsion spectral weight
000097 | comes from J5 alone is not established. This narrows the hypothesis without
000098 | discarding the proven shared-current contact or any earlier result.
000099 |