A registered result in its source context

SEAM.CONTACT.UNIT.02

Original ledger classification: Completion of the computable side [E]/[C]/[O] (diagonal zero at the KMS scale; closed-form resummation; multiplicity dichotomy; residual = the SEAM.EQUIV.01 face alone)

Original claim text

THE DIAGONAL CHANNEL SETTLED -- THE MERGED TARGET COLLAPSES ONTO THE KEYSTONE (2026-07-14, deep-dive round III; completes the computable side of SEAM.CONTACT.UNIT.01/v484): [E] DIAGONAL ZERO AT THE KMS SCALE: the renormalised coincident-point Green function of |D| on a circle of circumference l (zero mode removed, flat short-distance subtraction) is G_reg(0; l) = (1/pi) ln(l/2pi) EXACTLY (symbolic limit + 30-digit numeric checks at l = pi, 2pi, 4pi) -- it vanishes iff l = 2 pi = 1/(4 c3), the v239 KMS seam circumference: the seam scale is the unique zero-self-energy scheme point. [E] THE LOG IS THE SCALE CHANNEL: G_reg(0; 2 pi kappa) = (1/pi) ln kappa -- matching the functional structure: F_U(1) carries exactly ONE log term (the transport 8 b1 c3^6 ln(1/phi_seam), v3) while the phi0 split (4/3)c3 + 48 c3^4 is log-free; [C] reading: the log term is the diagonal/scale channel, the log-free terms are contact data at the KMS point. [E] CLOSED-FORM RESUMMATION (the BFK/gluing route already in the suite, v151): for a finite-rank mark potential the zeta-det ratio reduces to the 4x4 mark determinant, and with the diagonal zero the v484 expansion RESUMS exactly: det(I - u C) = (1 - 4u)(1 + 2u)^2, u = eps c3 ln2, C = circ(0,1,2,1) -- closed form to ALL orders, the linear term absent precisely because the diagonal vanishes (Tr C = 0), series = v484's Tr(C^k) = 4^k + 2(-2)^k term by term. [E] WEIGHT DICHOTOMY (the multiplicity matching): the SAME 48 = 3 x 16 admissible states (v245) carry both multiplicities -- counted FLAT they give the phi0 puncture count 48 = Omega_adm = N_fam dim S+; counted with the U(1) response weights (Y^2 with the standard loop/GUT factors) they give the alpha budget 41 = 10 b1 = 40 fermionic (Tr_16 Y^2 = 10/3 exact, nu^c = 0) + 1 Higgs (Ginsparg (3/5)(41/6) = 41/10, v470) -- one state set, two response weights, no new number. [C]/[O] CONSEQUENCE: with v484 (unit + cycle sector) EVERY finite/computable piece of the merged ALPHA.QUILLEN.EXACT.01 + HYP.PHI0.PUNCTURE.01 target is proven; the single remaining [O] is the identification of the abstract seam zeta-det with this glued mark determinant -- a face of SEAM.EQUIV.01, exactly the v382 typing, now substantiated computationally. The merged residual collapses onto the keystone; no separate analytic unknown remains. No gate closes, no marker moves; no coefficient fitted (all integers forced by the square configuration and the SM charge table). UPDATED 2026-08-27 (v974): the BFK/gluing resummation named here is now EXECUTED (probe seam_bfk_conical_det_probe 21/21, promoted as v974): C_glue = 2^-N exactly (N = 1..4 symbolic, general interval lengths, Laurent-polynomial identity), massless reassembly det'_zeta(Delta_S1) = l^2 = 2^-N prod(2 a_j) (l/N) det'(R_0) with det'(R_0) = N l / prod a_j, jump matrix = inverse mark Green matrix R(m) = (G_m|_marks)^-1 exactly, and the diagonal/scale channel reproduced exactly through log det'Delta(l) - log det'Delta(1/(4 c3)) = (1/(4 c3)) G_reg(0; l). The keystone typing is unchanged: pillowcase moduli, the 8 b1 index budget and a |D|-level BFK as an operator identity stay open with the keystone.

Original classification detail

Computable side complete: diagonal = 0 at KMS scale, resummation closed-form, 48/41 = one state set under two weights; remaining [O] = the SEAM.EQUIV.01 face (v382 typing substantiated)

Dependencies as recorded

SEAM.CONTACT.UNIT.01; ALPHA.QUILLEN.EXACT.01; HYP.PHI0.PUNCTURE.01; HYP.PHI0.GEOM.01; EM.WARD.02; DYN.KMS.01; v3; v151; v239; v245; v434; v470; v483; v484