TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/FULL-SOURCE-CORRECTION.txt
Original file SHA-256: 7d6f7e992b4e1f7290ce418b09e7d00dbd28c71fe8d2e16f8d6eeb7154f6f2e9
Inclusive original lines: 61-90
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.

000061 | Let I_N be the SAME original regulated integration functional, including
000062 | its actual field tuple, gauge/Berezin ordering, domains and Jacobians.
000063 | Let w_A,N and w_R,N be the actual admissible and reference weights,
000064 | Z_A=I_N(w_A), Z_R=I_N(w_R), both nonzero. Define
000065 | omega_R(F)=I_N(w_R F)/Z_R and omega_A(F)=I_N(w_A F)/Z_A.
000066 | In a fully abstract noncommuting I_N these are normalized linear
000067 | functionals; positivity needs the actual original trace/state contract.
000068 | The algebraic quotient identities do not assert it automatically.
000069 | With R_N=w_R^-1 w_A on a justified integrable branch, the exact identity is
000070 | 
000071 |                  Z_rel,N[F]=omega_R(R_N F).              (1)
000072 | 
000073 | Conversely, using the original ordered inverse ratio,
000074 | 
000075 |         Z_rel,N[F]=omega_A(F)/omega_A(w_A^-1 w_R).        (2)
000076 | 
000077 | In the classical/graded-even functional-integral class the even weights
000078 | commute and these are the usual density quotients R_N andR_N^-1.
000079 | At a noncommuting operator level retain the displayed order; a quantum
000080 | Gibbs perturbation requires the actual time-ordered expansional and is
000081 | not automatically exp(-(S_A-S_R)). Grassmann polynomials are not declared
000082 | positive probability densities. Integrability, invertibility/support
000083 | and the original phase branch are stated hypotheses of the given caller.
000084 | At zeros use the original integral rather than an invalid inverse ratio.
000085 | 
000086 | Proof of(1): I_N(w_R R_N F)=I_N(w_A F). Divide byZ_R.
000087 | Proof of(2): omega_A(w_A^-1 w_R)=Z_R/Z_A, so division reconstructsZ_A/Z_R.
000088 | These equations include every allowed mixed insertion and the null-source
000089 | factor. A second free measure or arbitrary normalization is unnecessary
000090 | once these ACTUAL original weights and common basis are given.