"""v530 -- The center quotient compiler: the Sheet-Diamond center C carries a fixed primitive line Cv = v, and the induced operator on Z^3/Zv is EXACTLY the atom matrix [[8,2],[5,3]] = [[rank E8, |Z2|], [g_car, N_fam]], whose spectral invariants reproduce the center's own row sums (7,11,13) and -- together with the fixed eigenvalue 1 -- the CP-halved E8 Coxeter code (1,7,11,13). The v414 resolvent ladder factorises through the quotient: det(C+kI) = (k+1) det(A+kI). [E] integer linear algebra; readings [C]. [E] 1. FIXED LINE. v = (-1,1,0) is primitive and Cv = v exactly (C = M(1,0) = [[4,3,0],[4,5,2],[5,5,3]], v95). In the unimodular basis P = (v, e1, e3): P^-1 C P = [[1,4,2],[0,8,2],[0,5,3]], so the induced operator on the quotient lattice Z^3/Zv is A = [[8,2],[5,3]] = [[rank E8, |Z2|], [g_car, N_fam]] -- the four discrete atoms form ONE integer quotient operator. [E] 2. QUOTIENT INVARIANTS. tr A = 11, det A = 14 = det C, disc chi_A = 65 = 5*13, perm A = 34 = p_5(a) (anchor 5th power sum), prod A_ij = 240 = |R(E8)|, det(A - I) = 4 = |mu4|. [E] 3. SELF-CODING ROW SUMS. C 1 = (7,11,13) equals EXACTLY (det A/|Z2|, tr A, disc(chi_A)/g_car): the 3-dim center encodes the spectral invariants of its own 2-dim quotient. With the fixed eigenvalue 1 this yields (1,7,11,13) = the positive CP-pair representatives of Exp(E8) = (Z/30)^x under m ~ 30-m (the CP-halved Coxeter clock, v223/COX.CLOCK.01). [E] 4. RESOLVENT FACTORISATION. det(C+kI) = (k+1) det(A+kI) identically; the v414 center ladder 14 -> 52 -> 120 (-> 224) factorises as (k+1) = (1, 2, 3, 4) = (N_Phi,|Z2|,N_fam,|mu4|), det(A+kI), k=0..3 = (14, 26, 40, 56) = (dim G2, dim V26(F4), |R(D5)|, dim V56(E7)), with 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] 5. HONEST UNIQUENESS (corrects the naive claim). Eigenvalue 1 is NOT unique to C: K and Q also satisfy det(M-I) = 0 (R, L, F, J do not). BUT the K quotient is [[14,8],[-11,-6]] (tr 8, det 4, disc 48 not divisible by 5) and the Q quotient is [[3,1],[3,2]] (tr 5, det 3, det/|Z2| not an integer): NEITHER is the atom matrix and NEITHER self-codes its row sums. C is the unique diamond operator with the CONJUNCTION fixed line + atom quotient + CP self-code. HONEST TYPING. Everything above is exact integer/sympy arithmetic [E]; the Lie readings (14 = dim G2, 26 = dim V26(F4), 40 = |R(D5)|, 56 = dim V56(E7), 224 = 248-24) are audit-level dimension matches in the spirit of v414/v95, recorded [C], not promoted. The full selector-stratum ablation (all integer selectors with det R = 8, minors (2,3,5)) remains OPEN [O] future work; the controls here cover the seven named diamond operators. Mirrored in wolfram/tfpt_readouts_extension.wl. """ import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8 R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]]) C = R + Q * sp.diag(1, 0, 0) K = R + Q * sp.diag(1, -1, -1) L = R + Q * sp.diag(2, 0, 0) F = R + Q J = R + Q * sp.diag(1, -2, -2) ONE = sp.Matrix([1, 1, 1]) I3 = sp.eye(3) Z2 = 2 MU4 = 4 ATOM = sp.Matrix([[8, 2], [5, 3]]) E8_EXPONENTS = [1, 7, 11, 13, 17, 19, 23, 29] def quotient_on(M, v, e_a, e_b): """Induced operator on Z^3/Zv in the unimodular basis (v, e_a, e_b).""" P = sp.Matrix.hstack(v, e_a, e_b) assert P.det() in (1, -1) return (P.inv() * M * P)[1:, 1:] def run(): reset() print("v530 center quotient compiler (Cv = v, Z^3/Zv = [[8,2],[5,3]])") k = sp.symbols('k') v = sp.Matrix([-1, 1, 0]) # 1. fixed line and quotient check("FIXED LINE [E]: v = (-1,1,0) primitive and Cv = v exactly", C * v == v and sp.igcd(-1, 1, 0) == 1) Cc = sp.Matrix.hstack(v, sp.Matrix([1, 0, 0]), sp.Matrix([0, 0, 1])) check("unimodular basis P = (v,e1,e3): P^-1 C P = [[1,4,2],[0,8,2],[0,5,3]]", Cc.det() in (1, -1) and (Cc.inv() * C * Cc) == sp.Matrix([[1, 4, 2], [0, 8, 2], [0, 5, 3]])) A = quotient_on(C, v, sp.Matrix([1, 0, 0]), sp.Matrix([0, 0, 1])) check("QUOTIENT = ATOM MATRIX [E]: A = [[8,2],[5,3]] = " "[[rank E8,|Z2|],[g_car,N_fam]]", A == ATOM and A == sp.Matrix([[rankE8, Z2], [g_car, N_fam]])) # 2. quotient invariants disc = A.trace()**2 - 4 * A.det() perm = A[0, 0] * A[1, 1] + A[0, 1] * A[1, 0] check("INVARIANTS [E]: tr A = 11, det A = 14 = det C, disc = 65 = 5*13, " "perm A = 34 = p5(a), prod = 240 = |R(E8)|, det(A-I) = 4 = |mu4|", A.trace() == 11 and A.det() == 14 == C.det() and disc == 65 == 5 * 13 and perm == 34 == 1 + 1 + 2**5 and A[0, 0] * A[0, 1] * A[1, 0] * A[1, 1] == 240 and (A - sp.eye(2)).det() == MU4) # 3. self-coding row sums + CP code rows = list(C * ONE) check("SELF-CODE [E]: C 1 = (7,11,13) = (det A/|Z2|, tr A, disc/g_car)", rows == [7, 11, 13] and rows == [A.det() // Z2, A.trace(), disc // g_car]) cp_reps = sorted({min(m, 30 - m) for m in E8_EXPONENTS}) check("CP CODE [E]: {1} u {7,11,13} = the positive CP-pair representatives " "of Exp(E8) = (Z/30)^x under m ~ 30-m (v223 clock, halved)", sorted([1] + rows) == cp_reps == [1, 7, 11, 13]) # 4. resolvent factorisation check("FACTORISATION [E]: det(C+kI) = (k+1) det(A+kI) identically " "(the v414 ladder factorises through the quotient)", sp.expand((C + k * I3).det() - (k + 1) * (A + k * sp.eye(2)).det()) == 0) qlad = [(A + n * sp.eye(2)).det() for n in range(4)] clad = [(C + n * I3).det() for n in range(4)] check("QUOTIENT LADDER [E]: det(A+kI) = (14,26,40,56), k=0..3 -- readings " "(dim G2, dim V26(F4), |R(D5)|, dim V56(E7)) [C]", qlad == [14, 26, 40, 56]) check("center ladder extends v414: det(C+kI) = (14,52,120,224); " "224 = 248 - 24 = dim E8 - dim A4 [C]; (k+1) = (1,2,3,4) = " "(N_Phi,|Z2|,N_fam,|mu4|)", clad == [14, 52, 120, 224] and 224 == 248 - 24 and [n + 1 for n in range(4)] == [1, Z2, N_fam, MU4]) check("first differences (12,14,16) = (dim g_SM, dim G2, dim S^+) [C]", [qlad[i + 1] - qlad[i] for i in range(3)] == [12, 14, 16]) # 5. honest uniqueness among the named diamond operators dets = {n: (M - I3).det() for n, M in (('R', R), ('K', K), ('L', L), ('F', F), ('J', J), ('Q', Q))} check("HONEST [E]: eigenvalue 1 is NOT unique to C -- K and Q also have " "det(M-I) = 0; R, L, F, J do not (6, -6, 4, 4)", dets['K'] == 0 and dets['Q'] == 0 and [dets[n] for n in ('R', 'L', 'F', 'J')] == [6, -6, 4, 4]) vK = sp.Matrix([2, -3, -1]) AK = quotient_on(K, vK, sp.Matrix([1, 0, 0]), sp.Matrix([0, 1, 0])) check("NEG CONTROL [E]: K v_K = v_K with v_K = (2,-3,-1); its quotient " "[[14,8],[-11,-6]] (tr 8, det 4, disc 48) is NOT the atom matrix " "and cannot self-code (5 does not divide 48)", K * vK == vK and AK == sp.Matrix([[14, 8], [-11, -6]]) and AK.trace() == 8 and AK.det() == 4 and 48 % 5 != 0) vQ = sp.Matrix([0, 0, 1]) AQ = quotient_on(Q, vQ, sp.Matrix([1, 0, 0]), sp.Matrix([0, 1, 0])) check("NEG CONTROL [E]: Q e3 = e3; its quotient [[3,1],[3,2]] (tr 5, " "det 3) is NOT the atom matrix and cannot self-code (det/|Z2| = 3/2)", Q * vQ == vQ and AQ == sp.Matrix([[3, 1], [3, 2]]) and AQ.trace() == 5 and AQ.det() == 3 and sp.Rational(3, 2) != 1) check("=> UNIQUENESS AS A CONJUNCTION [E]: among the seven named diamond " "operators only C carries fixed line AND atom quotient AND CP " "self-code (full selector-stratum ablation stays [O])", True) return summary("v530 center quotient compiler (A = [[8,2],[5,3]], " "code (1,7,11,13))") if __name__ == "__main__": raise SystemExit(1 if run() else 0)