A registered result in its source context

SEAM.CONTACT.UNIT.01

Original ledger classification: Unification/reduction [E]/[C]/[O] (the two analytic targets merge; contact unit = KMS seam unit, derived for the finite cycle sector; diagonal renormalisation + multiplicity matching stay open)

Original claim text

THE SEAM CONTACT UNIT -- THE MERGED RESIDUAL OF THE TWO ANALYTIC TARGETS (2026-07-14, deep-dive round II): the shared 'c3 per boundary insertion' rule of ALPHA.QUILLEN.EXACT.01 (c3-ladder {0,3,6}, v3/v341/EM.WARD.02) and HYP.PHI0.PUNCTURE.01 (per-mark weight c3, v483) is ONE identity and NOT a new unknown: it is the KMS seam unit 2 pi = 1/(4 c3) with 1/4 = 1/|mu4| (v239) -- one bare boundary propagator ORBIT-AVERAGED over the four marks. [E] UNIT CHAIN: 1/(2 pi) = 4 c3 = |mu4| c3 = |Z2| chi(S^2) c3. [E] MECHANISM DERIVED ON THE SEAM CIRCLE: G(d) = -(1/pi) ln|2 sin(d/2)| (polylog identity, 40 digits) takes INTEGER multiples of c3 ln2 at the mu4 separations (G(pi/2) = -4 c3 ln2, G(pi) = -8 c3 ln2), so G_marks = -4 c3 ln2 x circ(0,1,2,1) with spec {4,-2,0,-2}; the mu4-invariant (orbit-averaged) insertion carries (GV)|_marks = -(c3 ln2) C exactly -- the log-det contact expansion is EXACTLY graded in c3 per insertion, cycle combinatorics Tr(C^k) = 4^k + 2(-2)^k integer. [E] SUITE-WIDE GRAMMAR: F_U(1) ladder {0,3,6} (v3); phi0 ladder {1,4} (v37); Lambda prefactor 3/(4 pi^2) = (8 pi)^2 delta_top = 48 c3^2 (v60) -- the SAME multiplicity Omega_adm = 48 as the puncture at TWO insertions; the g-2 seam loop divides by the same 2 pi = 1/(4 c3). [C]/[O] THE MERGE: the two tracked [O] targets collapse into ONE named analytic step -- the diagonal (self-energy) zeta-renormalisation at the marks + the state-multiplicity matching (48 = Omega_adm = N_fam dim S+, 41 = 10 b1); the finite cycle sector is PROVEN. Anti-numerology: the check integers (circulant, spectrum, traces) are forced by the square mark configuration; no coefficient of either target is claimed. No gate closes, no marker moves. UPDATE 2026-07-14 (v485/SEAM.CONTACT.UNIT.02): the diagonal channel is settled -- G_reg(0) = 0 exactly at the KMS scale, the log is the pure scale channel, the mark determinant resums closed-form, and the 48/41 multiplicities are one state set under two response weights; the merged residual collapses onto the SEAM.EQUIV.01 face. UPDATED 2026-08-27 (v974): the finite cycle sector now has its computed Delta-side partner -- the BFK jump circulant B = circ(2,-1,0,-1) shares the mu4 Fourier eigenbasis with the mark circulant C = circ(0,1,2,1), the |spec| multisets are equal {0,2,2,4}, Tr B^k = 4^k + 2(+2)^k mirrors Tr C^k = 4^k + 2(-2)^k with the sign flip = the kernel swap (B kills the mu4-trivial gauge/zero-mode channel, C the mu4-sign channel), and B + C = 2(I + S). On the doublet channel R_0|_d = (4/ln2) G_marks|_d exactly, with the c3 grading CANCELLING in the ratio -- both matrices carry the same 'c3 per insertion' unit. No global proportionality: the merged residual is sharpened to the channel-swap question, not closed.

Original classification detail

Unification: 'c3 per insertion' = KMS seam unit (bare 1/(2pi) x orbit average 1/|mu4|), cycle sector derived; the two [O] targets are ONE remaining analytic step

Dependencies as recorded

ALPHA.QUILLEN.EXACT.01; HYP.PHI0.PUNCTURE.01; HYP.PHI0.GEOM.01; EM.WARD.02; DYN.KMS.01; LAMBDA.BRANCH.01; v3; v37; v58; v60; v239; v341; v408; v483