A registered result in its source context

DIAMOND.CENTER.QUOTIENT.01

Original ledger classification: Identity [E] (exact integer linear algebra: fixed line, unimodular quotient, self-coding row sums, resolvent factorisation, honest uniqueness controls -- sympy exact, Wolfram-mirrored) + [C] (the Lie-dimension readings of the quotient ladder, audit-typed like v414) + [O] (full selector-stratum ablation)

Original claim text

THE CENTER QUOTIENT COMPILER (2026-07-23; external review round, all identities machine-confirmed before promotion; 14 checks, <1 s, sympy exact): the Sheet-Diamond center C = M(1,0) (v95) carries the fixed primitive line Cv = v with v = (-1,1,0), and the induced operator on the quotient lattice Z^3/Zv (unimodular basis P = (v,e1,e3), P^-1 C P = [[1,4,2],[0,8,2],[0,5,3]]) is EXACTLY the atom matrix A = [[8,2],[5,3]] = [[rank E8,|Z2|],[g_car,N_fam]] -- the four discrete atoms form ONE integer quotient operator. [E] INVARIANTS: tr A = 11, det A = 14 = det C, disc chi_A = 65 = 5*13, perm A = 34 = p5(a) (anchor 5th power sum), prod A_ij = 240 = |R(E8)|, det(A-I) = 4 = |mu4|. [E] SELF-CODE: the row sums C.1 = (7,11,13) = (det A/|Z2|, tr A, disc(chi_A)/g_car) EXACTLY -- the 3-dim center encodes the spectral invariants of its own 2-dim quotient; together with the fixed eigenvalue 1 this gives (1,7,11,13) = the positive CP-pair representatives of Exp(E8) = (Z/30)^x under m ~ 30-m (the CP-halved Coxeter clock; complements COX.CLOCK.01/v223). [E] RESOLVENT FACTORISATION: det(C+kI) = (k+1) det(A+kI) identically -- the v414 ladder 14 -> 52 -> 120 acquires its inner structure: (k+1) = (1,2,3,4) = (N_Phi,|Z2|,N_fam,|mu4|) times the quotient ladder det(A+kI) = (14,26,40,56) with [C] readings (dim G2, dim V26(F4), |R(D5)|, dim V56(E7)), first differences (12,14,16) = (dim g_SM, dim G2, dim S^+), and the new endpoint det(C+3I) = 224 = 248-24 = dim E8 - dim A4. [E] HONEST UNIQUENESS (corrects the review's naive claim): eigenvalue 1 is NOT unique to C -- K and Q also satisfy det(M-I) = 0 (R,L,F,J give 6,-6,4,4); but the K quotient [[14,8],[-11,-6]] (tr 8, det 4, disc 48 not divisible by 5) and the Q quotient [[3,1],[3,2]] (tr 5, det 3, det/|Z2| not integer) are NOT the atom matrix and CANNOT self-code -- C is unique in the CONJUNCTION fixed line + atom quotient + CP self-code among the seven named diamond operators. [O] the full selector-stratum ablation (all integer selectors with det R = 8, minors (2,3,5)) stays open future work. Discovery provenance: center_quotient_coxeter_modular_disc7_probe.py (76/76) in experiments/tfpt-discovery/ (2026-07-23). NO marker moves.

Original classification detail

[E] the center quotient A = [[8,2],[5,3]] (atoms), self-code (7,11,13), CP code (1,7,11,13), factorisation det(C+kI) = (k+1)det(A+kI); [C] Lie readings; [O] stratum ablation; no marker moves

Dependencies as recorded

DIAMOND.CENTER.01; DIAMOND.CENTER.RESOLVENT.01; COX.CLOCK.01; v95; v223; v414