"""v222 -- CM-norm duality: the two exceptional moduli give 41 (square) and 7 (hex). The seam carries the SQUARE modulus (mu4 = {1,i,-1,-i}, cross-ratio 2 => j=1728, CM by the Gaussian integers Z[i]; v214/v216). Its hexagonal partner is the modulus j=0 (tau=rho=e^{i pi/3}, CM by the Eisenstein integers Z[omega]). This script shows that the two CM rings are the two metric readings of the SAME carrier split (3,2)=(colour,weak) and the anchor (e1,e2)=(|mu4|,g_car)=(4,5), and that they deliver two central TFPT integers as ideal NORMS, not as loose sums: [E] SQUARE (Z[i], j=1728): N(g_car + i|mu4|) = 5^2 + 4^2 = 41 = 10 b1 (EM index) N(3 + 2i) = 3^2 + 2^2 = 13 = Delta_Q (carrier-split norm) [E] HEX (Z[omega], j=0): N(3 + 2 omega) = 3^2-3*2+2^2 = 7 = scalaron exponent So 10 b1 = 41 stops being "e1^2+e2^2" and becomes the Gaussian norm of the carrier-glue vector g_car + i|mu4| on the square seam; and the scalaron 7 is the Eisenstein norm of the same (3,2) split on the hexagonal partner. The single split (3,2) generates (5,6,7,13) by elementary ring operations: 3 + 2 = 5 = g_car, 3 * 2 = 6 = |R^+(A3)|, 3 - 2 = 1 = N_Phi, N_omega(3,2) = 7 = scalaron, N_i(3,2) = 13 = Delta_Q. Negative controls (the assignment is rigid, not a coincidence): the Eisenstein norm of (5,4) is 21 (not 41); the Gaussian norm of (3,2) is 13 (not 7); generic splits b+s=5 do not reproduce the whole (5,6,7,13) table. Status: [E] as ring arithmetic; [C] as the architecture bridge "square<->EM/mu4-clock, hex<->scalaron/CP-phase". Mirrored in wolfram/tfpt_readouts_extension.wl. """ from tfpt_constants import (check, summary, reset, g_car, N_fam, b1) MU4 = 4 # |mu4| = e1(a) N_PHI = 1 SCALARON = 7 DELTA_Q = 13 RP_A3 = 6 # |R^+(A3)| def N_gauss(a, b): """Norm in the Gaussian integers Z[i]: |a + b i|^2 = a^2 + b^2.""" return a * a + b * b def N_eisen(a, b): """Norm in the Eisenstein integers Z[omega], omega = e^{2 pi i/3}: |a + b omega|^2 = a^2 - a b + b^2.""" return a * a - a * b + b * b def run(): reset() print("v222 CM-norm duality: square gives 41 & 13, hexagon gives 7") # ---- the square seam (Gaussian Z[i], j = 1728) ---- check("SQUARE [E]: N_Z[i](g_car + i|mu4|) = 5^2 + 4^2 = 41 = 10 b1 (EM index); " "the EM index 41 IS the Gaussian norm of the carrier-glue vector on the " "square seam, not merely e1(a)^2 + e2(a)^2", N_gauss(g_car, MU4) == 41 == int(10 * b1) == MU4**2 + g_car**2) check("SQUARE [E]: N_Z[i](3 + 2 i) = 3^2 + 2^2 = 13 = Delta_Q (carrier-split norm)", N_gauss(N_fam, 2) == DELTA_Q == 13) # ---- the hexagonal partner (Eisenstein Z[omega], j = 0) ---- check("HEX [E]: N_Z[omega](3 + 2 omega) = 3^2 - 3*2 + 2^2 = 7 = scalaron " "exponent; the same (3,2) split read on the hexagonal CM ring", N_eisen(N_fam, 2) == SCALARON == 7) # ---- the (3,2) split is a small generator ---- check("(3,2) split generator: 3+2 = 5 = g_car", N_fam + 2 == g_car) check("(3,2) split generator: 3*2 = 6 = |R^+(A3)|", N_fam * 2 == RP_A3) check("(3,2) split generator: 3-2 = 1 = N_Phi", N_fam - 2 == N_PHI) check("(3,2) -> (5, 6, 7, 13) = (g_car, |R^+(A3)|, scalaron, Delta_Q) by " "sum / product / Eisenstein-norm / Gaussian-norm", [N_fam + 2, N_fam * 2, N_eisen(N_fam, 2), N_gauss(N_fam, 2)] == [g_car, RP_A3, SCALARON, DELTA_Q]) # ---- negative controls: the ring assignment is RIGID ---- check("NEG: Eisenstein norm of (5,4) = 21, NOT 41 (the square needs Z[i])", N_eisen(g_car, MU4) == 21 and N_eisen(g_car, MU4) != 41) check("NEG: Gaussian norm of (3,2) = 13, NOT 7 (the scalaron needs Z[omega])", N_gauss(N_fam, 2) != SCALARON) check("NEG: the two norms of (3,2) differ -- 7 (hex) vs 13 (square)", N_eisen(N_fam, 2) != N_gauss(N_fam, 2)) # generic splits {b,s} (b<=s) with b+s=5 do not give the whole table; # the norms are symmetric, so {1,4} and {2,3} are the only two multisets. splits = {(b, 5 - b): (N_eisen(b, 5 - b), N_gauss(b, 5 - b)) for b in range(1, 3)} # {1,4} and {2,3} hits = [bs for bs, pair in splits.items() if pair == (SCALARON, DELTA_Q)] check("NEG: among the splits {1,4},{2,3} only {2,3} gives (N_omega,N_i)=(7,13) " "-- {1,4} gives (13,17); the carrier split (colour 3, weak 2) is rigid", hits == [(2, 3)] and splits[(1, 4)] == (13, 17)) return summary("v222 CM-norm duality (square 41/13, hexagon 7)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)