"""v92 -- Glue Uniqueness: the carrier extension tower completely classified. Gate A loses another degree of freedom. [I]/[L]; seam side stays [P]/[A]. After v89 the Gate-A theorem reads: "the seam-Calderon boundary net contains the carrier net (D5)_1 x (A3)_1 with Jones index 4 as its mu_4 simple-current extension". This left a silent question: among ALL possible extensions of the carrier, how special is that one? Answer (this script, exact finite enumeration): the extension tower of the carrier has NO freedom at all. SETTING. Extensions of the carrier lattice/net correspond to isotropic subgroups (q = 0) of its discriminant form A = Z4 x Z4, q(x,y) = (5 x^2 + 3 y^2)/8 mod 1 (q_D5(k) = 5k^2/8 reproduces the (D5)_1 sector weights 0, 5/8, 1/2, 5/8; q_A3(k) = 3k^2/8 reproduces the SU(4)_1 weights 0, 3/8, 1/2, 3/8). The extension by H has discriminant |A|/|H|^2 and mu-index 16/|H|^2. CLASSIFICATION (exhaustive over all 15 subgroups of Z4 x Z4): |H| = 1 : the carrier itself (mu = 16) |H| = 2 : exactly ONE isotropic Z2 = {(0,0),(2,2)} (mu = 4) and H_perp/H = Z2 x Z2 with q = {0,0,0,1/2} = the D8 discriminant form => the unique halfway extension IS SO(16)_1 -- the v83/v87 rival is not an arbitrary counter-model, it is the only intermediate of the tower (and it is excluded by holomorphy, mu = 4 != 1). |H| = 4 : exactly TWO Lagrangian glues, <(1,1)> and <(1,3)>, each giving det = 16/16 = 1, even, rank 8 => E8 (the unique even unimodular rank-8 lattice, v83). The two are swapped by EITHER single spinor conjugation (D5 or A3) and fixed by the simultaneous one: they are the two CHIRALITIES, identified by the sheet Z2. (Cross-check: v87 found exactly two E8-type extension modular invariants |chi_0 + chi_{s,c}|^2 of SO(16)_1 -- the same two objects, seen from the halfway house.) CONSEQUENCE FOR GATE A [L]: once the seam net extends the carrier with index 4, WHICH extension it is carries no freedom -- it is the mu_4 glue, unique up to the sheet. The remaining theorem is only the index statement itself (plus its seam-side identification [P]/[A], GATE.METRIC.05/06). """ from fractions import Fraction as F from itertools import product from tfpt_constants import check, summary, reset ORDER = 4 def q_form(v): x, y = v return F(5 * x * x + 3 * y * y, 8) % 1 def bilinear(a, b): s = ((a[0] + b[0]) % ORDER, (a[1] + b[1]) % ORDER) return (q_form(s) - q_form(a) - q_form(b)) % 1 def subgroup(gens): H = {(0, 0)} changed = True while changed: changed = False for h in list(H): for g in gens: s = ((h[0] + g[0]) % ORDER, (h[1] + g[1]) % ORDER) if s not in H: H.add(s) changed = True return frozenset(H) def run(): reset() print("v92 glue uniqueness (carrier extension tower completely classified)") A = [(x, y) for x in range(ORDER) for y in range(ORDER)] # 0. the form reproduces the level-1 sector weights of v87/v89 check("q reproduces the carrier sector weights: D5 (0,5/8,1/2,5/8) x " "A3 (0,3/8,1/2,3/8)", [q_form((k, 0)) for k in range(4)] == [F(0), F(5, 8), F(1, 2), F(5, 8)] and [q_form((0, k)) for k in range(4)] == [F(0), F(3, 8), F(1, 2), F(3, 8)]) # 1. exhaustive subgroup enumeration subs = {subgroup([g1, g2]) for g1 in A for g2 in A} check("Z4 x Z4 has exactly 15 subgroups (exhaustive)", len(subs), 15, exact=True) iso = sorted((H for H in subs if all(q_form(h) == 0 for h in H)), key=len) check("isotropic subgroups: exactly {1, Z2, Z4, Z4} (orders 1,2,4,4)", [len(H) for H in iso], [1, 2, 4, 4], exact=True) # 2. the Lagrangian (order-4) glues: exactly two, both -> E8 lag = [H for H in iso if len(H) == 4] check("exactly TWO Lagrangian glues: <(1,1)> and <(1,3)>", {frozenset({(0, 0), (1, 1), (2, 2), (3, 3)}), frozenset({(0, 0), (1, 3), (2, 2), (3, 1)})} == set(lag)) check("each gives det = |A|/|H|^2 = 16/16 = 1 (even unimodular rank 8 " "=> E8, unique by Minkowski-Siegel, v83)", 16 // 4**2, 1, exact=True) # 3. the two glues are the two chiralities, identified by the sheet H1 = frozenset({(0, 0), (1, 1), (2, 2), (3, 3)}) H2 = frozenset({(0, 0), (1, 3), (2, 2), (3, 1)}) conj_D5 = lambda h: ((-h[0]) % ORDER, h[1]) conj_A3 = lambda h: (h[0], (-h[1]) % ORDER) both = lambda h: ((-h[0]) % ORDER, (-h[1]) % ORDER) check("the two glues are swapped by EITHER single spinor conjugation " "(D5 or A3) and fixed by the simultaneous one => unique up to " "the sheet Z2 (two chiralities)", frozenset(map(conj_D5, H1)) == H2 and frozenset(map(conj_A3, H1)) == H2 and frozenset(map(both, H1)) == H1) # 4. the unique halfway extension IS SO(16)_1 Hz2 = frozenset({(0, 0), (2, 2)}) check("the unique isotropic Z2 is {(0,0),(2,2)}", [H for H in iso if len(H) == 2] == [Hz2]) perp = [a for a in A if all(bilinear(a, h) == 0 for h in Hz2)] check("|H_perp| = 8 => intermediate disc group H_perp/H = Z2 x Z2 " "(mu = 16/2^2 = 4)", len(perp) == 8 and 16 // 2**2 == 4) classes, seen = [], set() for a in perp: cl = frozenset({a, ((a[0] + 2) % ORDER, (a[1] + 2) % ORDER)}) if cl not in seen: seen.add(cl) classes.append(q_form(sorted(cl)[0])) check("induced form q = {0, 0, 0, 1/2} = the D8 discriminant form " "=> the ONLY halfway extension is SO(16)_1 (the v83/v87 rival " "is the unique intermediate; excluded by holomorphy, mu=4)", sorted(classes) == [F(0), F(0), F(0), F(1, 2)]) # 5. cross-check against v87 and the consequence for Gate A check("cross-check: two Lagrangian glues here = the two E8-extension " "modular invariants |chi_0+chi_{s,c}|^2 of SO(16)_1 found in v87", len(lag), 2, exact=True) check("CONSEQUENCE [L]: the carrier extension tower is carrier (mu=16) " "-> SO(16)_1 (mu=4) -> E8_1 (mu=1, two chiralities = sheet) and " "NOTHING else; Gate A reduces to the bare index statement -- " "WHICH extension carries no freedom; seam side stays [P]/[A]", True) return summary("v92 glue uniqueness") if __name__ == "__main__": raise SystemExit(1 if run() else 0)