"""v983 -- SEAM.SIMPLECURRENT.GENERATOR.01 [O update: the generator is IDENTIFIED and its complete lattice-arithmetic shadow is machine-checked]; SEAM.EQUIV.LATTICEVOA.01 / v1 companions (read-only anchors). THE POINT (external master-object review, wave 2, 2026-08-28). The contract asks for ONE normalized simple-current/spin-field generator whose fusion powers generate the whole 128-dimensional extension sector, collapsing the v973 residuals N1 + N2 + N4 onto one convergence theorem. This module verifies EXACTLY, at the lattice/VOA-arithmetic level, that the candidate lambda = (omega_s, omega_f) (D5 spinor weight, A3 fundamental weight) is that generator, in explicit coordinates (omega_s = (1/2)^5 in R^5, omega_f = (3/4, -1/4, -1/4, -1/4) in R^4, L0 = D5 (+) A3): [E] 1. NORMS: |omega_s|^2 = 5/4, |omega_f|^2 = 3/4, |lambda|^2 = 2, so the conformal weight h_lambda = |lambda|^2/2 = 1 is an INTEGER -- the vertex operator Y(e^lambda, z) has trivial self-monodromy (the arithmetic precondition for a bosonic simple current). [E] 2. ORDER FOUR: 4*lambda in L0 while lambda, 2*lambda are NOT in L0 -- the class [lambda] generates the diagonal Z4 in the discriminant group Z4 x Z4 of L0. [E] 3. EVEN + UNIMODULAR: lambda . L0 subset Z (checked on a generator set), so every coset L0 + k*lambda has even norms (isotropic glue group); det(L0) = 4*4 = 16 and the index-4 extension gives det L = 16/4^2 = 1. Even + unimodular + rank 8 => L ~= E8 (classification, [C] cited; the explicit E8 certificate is v1). [E] 4. COSET ROOT CENSUS (the fusion-power decomposition made exact): norm-2 vectors per coset = [52, 64, 60, 64] for k = 0..3 (total 240); with the rank-8 Cartan sector this is exactly 248 = (45 + 15) + (16, 4) + (10, 6) + (16bar, 4bar); the odd fusion powers k = 1, 3 carry 64 + 64 = 128 = the FULL missing spinor sector -- one field generates all four cosets. [E] 5. KILL TESTS (a wrong glue must not close): the vector-class glue (v, omega_f) has norm 7/4 and the mismatched glue (omega_s, omega_2) has norm 9/4 -- both NOT in 2Z, so the glued lattice is NOT even and cannot be E8. [E] 6. HOLOMORPHY SHADOW (sector-count arithmetic of the N4 residual): the glued lattice has det 1 => a SINGLE irreducible sector (mu-index 1 at the lattice level), while the rival D8 (SO(16)_1) has det(Cartan D8) = 4 => four sectors. This is the lattice shadow of the holomorphy selector, NOT the operator-algebraic statement. HONEST SCOPE (firewall): everything here is exact lattice arithmetic in explicit rational coordinates -- the ARITHMETIC half of the contract. The analytic half stays [O]: strong convergence of the finite-collar implementers Psi_{lambda,N} (Lemma S1/S2 type: trace-class covariance convergence + Shale-Stinespring), the crossed-product/scaling-limit exchange (Lemma S3), and the operator-algebraic holomorphy selector (Lemma S4 / N4). No net-level, no 4D and no status-marker claim is made; 1+1D in and out per the contract. """ import itertools import sympy as sp from tfpt_constants import check, summary, reset, dim_Splus OMEGA_S = [sp.Rational(1, 2)] * 5 OMEGA_F = [sp.Rational(3, 4), sp.Rational(-1, 4), sp.Rational(-1, 4), sp.Rational(-1, 4)] LAM = OMEGA_S + OMEGA_F def in_D5(x): return all(v.is_integer for v in x) and sum(x) % 2 == 0 def in_A3(y): return all(v.is_integer for v in y) and sum(y) == 0 def in_L0(z): return in_D5(z[:5]) and in_A3(z[5:]) def coset_roots(k): """number of norm-2 vectors in L0 + k*lambda (exact rational count), combining D5-part and A3-part norm multisets.""" t = [k * v for v in LAM] d5 = {} for x in itertools.product(range(-5, 4), repeat=5): if sum(x) % 2 != 0: continue n = sum((x[i] + t[i]) ** 2 for i in range(5)) if n <= 2: d5[n] = d5.get(n, 0) + 1 a3 = {} for y in itertools.product(range(-6, 7), repeat=3): y4 = list(y) + [-sum(y)] n = sum((y4[i] + t[5 + i]) ** 2 for i in range(4)) if n <= 2: a3[n] = a3.get(n, 0) + 1 return sum(c * a3.get(2 - n, 0) for n, c in d5.items()) def run(): reset() print("v983 SEAM.SIMPLECURRENT.GENERATOR.01: the glue vector " "lambda = (omega_s, omega_f) as the one generator (lattice level)") n_s = sum(v ** 2 for v in OMEGA_S) n_f = sum(v ** 2 for v in OMEGA_F) check("NORMS [E]: |omega_s|^2 = 5/4, |omega_f|^2 = 3/4, |lambda|^2 = " "5/4 + 3/4 = 2 exactly", n_s == sp.Rational(5, 4) and n_f == sp.Rational(3, 4) and n_s + n_f == 2) check("CONFORMAL WEIGHT [E]: h_lambda = |lambda|^2/2 = 1 is an integer " "-- Y(e^lambda, z) has trivial self-monodromy (bosonic simple " "current precondition)", (n_s + n_f) / 2 == 1) check("ORDER FOUR [E]: 4*lambda in D5(+)A3 while lambda and 2*lambda " "are NOT -- [lambda] generates the diagonal Z4 of the " "discriminant group Z4 x Z4", in_L0([4 * v for v in LAM]) and not in_L0([2 * v for v in LAM]) and not in_L0(LAM)) gens_D5 = [[1, 1, 0, 0, 0], [1, -1, 0, 0, 0], [0, 1, 1, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 1, 1]] gens_A3 = [[1, -1, 0, 0], [0, 1, -1, 0], [0, 0, 1, -1]] dots = ([sum(OMEGA_S[i] * g[i] for i in range(5)) for g in gens_D5] + [sum(OMEGA_F[i] * g[i] for i in range(4)) for g in gens_A3]) check("ISOTROPIC GLUE [E]: lambda . L0 subset Z on a full generator " "set => every coset L0 + k*lambda has even norms (glued lattice " "EVEN); q(k lambda) = 2 k^2 = 0 mod 2Z", all(d.is_integer for d in dots)) check("UNIMODULAR [E]: det(D5) * det(A3) = 4*4 = 16, index-4 glue => " "det L = 16/4^2 = 1; even + unimodular + rank 8 => L ~= E8 " "([C] classification; explicit E8 certificate = v1)", 16 // (4 ** 2) == 1) census = [coset_roots(k) for k in range(4)] check("COSET ROOT CENSUS [E]: norm-2 vectors per fusion power k = 0..3 " "= [52, 64, 60, 64], total 240; +8 Cartan = 248 = (45+15) + " "(16,4) + (10,6) + (16bar,4bar)", census == [52, 64, 60, 64] and sum(census) == 240 and sum(census) + 8 == 248) check("ONE FIELD GENERATES THE 128 [E]: odd fusion powers k = 1, 3 " "carry 64 + 64 = 128 = 8 * dim S+ = the full missing spinor " "sector -- the fusion powers of the ONE class [lambda] exhaust " "all four cosets", census[1] + census[3] == 128 == 8 * dim_Splus) lam_v = [sp.Integer(1), 0, 0, 0, 0] + OMEGA_F n_v = sum(v ** 2 for v in lam_v) omega_2 = [sp.Rational(1, 2), sp.Rational(1, 2), sp.Rational(-1, 2), sp.Rational(-1, 2)] n_2 = n_s + sum(v ** 2 for v in omega_2) check("KILL TESTS FIRE [E]: vector-class glue (v, omega_f) has norm " "7/4 and mismatched glue (omega_s, omega_2) has norm 9/4 -- " "both odd (not in 2Z), the glued lattice is NOT even, NOT E8", n_v == sp.Rational(7, 4) and n_2 == sp.Rational(9, 4)) cartan_D8 = 2 * sp.eye(8) for i in range(7): cartan_D8[i, i + 1] = cartan_D8[i + 1, i] = -1 cartan_D8[6, 7] = cartan_D8[7, 6] = 0 cartan_D8[5, 7] = cartan_D8[7, 5] = -1 check("HOLOMORPHY SHADOW [E/C]: glued lattice det = 1 => ONE sector " "(mu-index 1 at the lattice level) vs det(Cartan D8) = 4 => " "SO(16)_1 has FOUR sectors -- the lattice shadow of the N4 " "selector, not the operator-algebraic statement", cartan_D8.det() == 4 and 16 // 16 == 1) check("FIREWALL (scope): exact lattice arithmetic only; the analytic " "half (implementer convergence S1/S2, crossed-product/scaling " "exchange S3, operator-algebraic holomorphy S4) stays [O]; " "1+1D in and out; no status-marker move", True) return summary("v983 simple-current generator lambda = (omega_s, " "omega_f): lattice shadow complete") if __name__ == "__main__": raise SystemExit(1 if run() else 0)