Manuscript XVI-P@p23 · XVI-P.tex · section · 9b0037a7
Counterterm domain and solve
P.5 identity step (p19): the projected one-loop κ 2 ch-identity defect › Counterterm domain and solve · lines 509–520
Flag definitions. projected_defect_computed: Delta = P_ch[(S_Psi, Gamma_Psi)]_{hbar kappa\^{}2} assembled from every evaluated input of the three surviving rows (none Missing), exact in d, pole + nonlocal + MS-bar local parts, all seven external assignments. projected_identity_verified: Delta = 0 for the REGULATED one-loop functional Gamma1: pole, nonlocal and local parts vanish for all seven assignments, exactly in d, with all controls true and every mutation changing the outcome. projected_restoration_verified: p20 definition (replaces p19's, which follows from the identity + locality): FULL sigma_0 closure of the COMPLETE one-loop kappa\^{}2 MS-bar counterterm S_ct = -(pole) in the declared domain, h.h + c.K + c.cb (ghost self-energy Sigma held fixed at its P15 value, orientation converted, derived relative factor +1), in every ghost +1 sector (h.c, c.b, c.cbs), together with projected_identity_verified. Scope: the one-loop kappa\^{}2 counterterm only; NOT bulk symmetry, NOT O.1. bulk_symmetry_restored: hard stop: False regardless (not claimed; O.1 NOT_CLOSED).
K1 (corrected in p20; erratum to p19). derived relative factor 1; orientation ratio (route-G (c,cb) over P15 (cb,c)) ['-1', '-1']; Sigma pole in (c,cb) coordinates ['3/8', '-1/4']; full sigma_0 image of the complete MS-bar counterterm: {'closed': True, 'entries': 0, 'sectors': []}. ERRATUM to p19: the p19 statement 'closure fails with the derived factor, needs -1, unresolved' was a Grassmann-orientation bookkeeping error (P15 (cb,c) coordinates placed in route-G (c,cb) slots). Corrected: derived +1, full closure holds (finding K1 of the p19 independent rebuild).
J1 (finding about the p14r contract). FINDING ABOUT THE p14r CONTRACT (contract NOT modified): the p14r counterterm-domain counts (Im sigma_0 3, quotient 8, D cap ker 5, P_ch sigma_0 rank 4) are artefacts of a re/im split over Q (the densities carry explicit i for odd derivative counts, so i-multiples count as independent). The counts over Q(i) – sigma_0 rank 5, D cap ker 6, D cap Im 6, cohomology 0, quotient 5, P_ch sigma_0 rank 3 (= Route C) – equal the real dimensions under a compatible real structure. The p14r certificate stays as released; both are reported