#!/usr/bin/env python3 """v604 -- QGEO.EQORDER.01: the equivariant order and the infinity bridge -- R is an anti-isometry of J (correction; J_fix integer with det 72), the four mark frames generate a FULL order of index 27 = 3^3, and the 3-torsion localizes at infinity (81 at the marks, 27 at infinity). Round 8 of the QGEO cover program (after v603). Executes the follow-up slate "integral translator / equivariant completion / seam-infinity": (Q1) R IS AN ANTI-ISOMETRY OF J [E, correction]: the correct compatibility equation is M^dagger J M = J^T -- and it holds EXACTLY. (v599's side remark tested M conj(J) M^dagger vs J, the wrong equation; the real structure and the polarization are fully compatible.) Consequently J restricts to a REAL SYMMETRIC INTEGER form on the saturated fixed lattice: J_fix = [[16,2,4],[2,-2,2],[4,2,-2]] with det J_fix = 72 = 8 * 9 (the polarization determinant times the covolume factor 9 = 3^2 of the (1-omega)-structure). (Q2) THE EQUIVARIANT ORDER IS FULL OF INDEX 27 [E]: the Z-lattice generated by the seven words of ALL FOUR mark frames (the r^k-conjugates of the v601/v602 pair) has FULL rank 18 in M_3(Z[omega]) with Smith divisors (1^15, 3, 3, 3) -- saturation index 27 = 3^3: one mark frame gives the 7-dim parabolic of index 81 = 3^4 (v600), all four together give the full matrix algebra of index 27 = 3^3. (Q3) THE 3-TORSION LOCALIZES AT INFINITY [E]: on the unreduced integral module Z[omega]^4, the infinity monodromy defect 1 - T_inf (T_inf = full twist r^4) has Smith divisors (1,1,1,3,3,3,0,0): coker(1 - T_inf) = Z^2 + (Z/3)^3 -- torsion 27; while each FINITE cusp twist has divisors (1,1,0^6): torsion-free. Three objects carry the same (Z/3)^3 = 27: the infinity-monodromy cokernel, the equivariant-order cokernel, and det_Z(1 - deck) (v597) -- the integral infinity <-> order-torsion bridge. Reading [C]: the two compiler constants localize on the cover -- 81 = 3^4 at the FINITE marks (divisor classes, v597; word order per frame, v600) and 27 = 3^3 at INFINITY (monodromy defect; full equivariant order). (Q4) TRANSLATOR NEGATIVES EXTENDED [X, quantified]: the corpus RP metric pulled back to the integral fixed lattice (F_2 = g^T Sigma g for the integer conjugation (C_U, C_V) -> (U_c, V_c)) is NOT proportional to J_fix or its Klein twists, and neither J_fix^-1 F_2 nor F_2 J_fix^-1 lies in the word algebra -- after the (B, Gamma, Klein, J_fix) sweep the corpus transpose structure resists all canonical single-sheet constructions; the named next candidate is the conjugate character sheet (omega-bar). All checks exact (sympy). Verdict enums (frozen): EQUIVARIANT-ORDER-LANDED (all), EQUIVARIANT-ORDER-FAILS, MIXED. FIREWALL: GATE.QGEO does not move; no marker changes. The infinity-seam identification and the RP translator stay typed open (omega-bar sheet named); the Q1 statement CORRECTS a v599-diary side remark (the wrong compatibility equation), not any promoted claim. PROVENANCE: discovery probe equivariant_order_probe.py (2026-08-01, 12/12, verdict EQUIVARIANT-ORDER-LANDED). Python-only, counted per GATE.WOLFRAM.02. """ import sympy as sp from sympy.matrices.normalforms import smith_normal_form t = sp.symbols("t") x = sp.symbols("x") OMEGA = sp.Rational(-1, 2) + sp.sqrt(3) * sp.I / 2 CHECKS = [] def check(name, ok, detail=""): CHECKS.append((name, bool(ok))) print("[%s] %s%s" % ("PASS" if ok else "FAIL", name, (" -- " + detail) if detail else "")) def burau_gen(i, n=4): m = n - 1 M = sp.eye(m) if i - 2 >= 0: M[i - 2, i - 1] = t M[i - 1, i - 1] = -t if i < m: M[i, i - 1] = 1 return M def burau_unred(i, n=4): M = sp.eye(n) M[i - 1, i - 1] = 1 - t M[i - 1, i] = t M[i, i - 1] = 1 M[i, i] = 0 return M def zw_coords(e): e = sp.expand(e) b = sp.nsimplify(2 * sp.im(e) / sp.sqrt(3)) a = sp.nsimplify(sp.re(e) + sp.im(e) / sp.sqrt(3)) return a, b I3 = sp.eye(3) I4 = sp.eye(4) S = [sp.simplify(burau_gen(i).subs({t: OMEGA})) for i in (1, 2, 3)] r = sp.simplify(S[0] * S[1] * S[2]) T = [S[0]] for _ in range(3): T.append(sp.simplify(r * T[-1] * r.inv())) J = sp.Matrix([[0, 1 + sp.sqrt(3) * sp.I, -1 + sp.sqrt(3) * sp.I], [1 - sp.sqrt(3) * sp.I, 2, 1 + sp.sqrt(3) * sp.I], [-1 - sp.sqrt(3) * sp.I, 1 - sp.sqrt(3) * sp.I, 0]]) M0 = sp.Matrix([[-1, (1 + sp.sqrt(3) * sp.I) / 2, (1 - sp.sqrt(3) * sp.I) / 2], [0, 0, 1], [0, 1, 0]]) GJ = [sp.simplify((I3 - T[k]) * J.inv() * (I3 - T[k]).conjugate().T * J) for k in range(4)] U1 = sp.simplify(3 * GJ[0]) V1J = sp.simplify(I3 + GJ[3] - GJ[1] * GJ[0]) # ---------------------------------------------------------------- Q1 print("=" * 72) print("Q1: R is an anti-isometry; J_fix on the fixed lattice") print("=" * 72) check("Q1.1 M^dagger J M = J^T exactly (R anti-isometry of the polarization)", sp.simplify(M0.conjugate().T * J * M0 - J.T) == sp.zeros(3, 3)) Sat = sp.Matrix([[1, 2, 2, 0, 0, 0], [0, 1, 1, -1, 0, 0], [0, 0, 1, 1, -1, 1]]) Bs = [sp.Matrix([Sat[i, j] + OMEGA * Sat[i, 3 + j] for j in range(3)]) for i in range(3)] Bfix = sp.Matrix.hstack(*Bs) Jfix = sp.simplify(Bfix.conjugate().T * J * Bfix) Jfix_target = sp.Matrix([[16, 2, 4], [2, -2, 2], [4, 2, -2]]) check("Q1.2 J restricts REAL SYMMETRIC INTEGER: J_fix = [[16,2,4],[2,-2,2],[4,2,-2]]", sp.simplify(Jfix - Jfix_target) == sp.zeros(3, 3)) check("Q1.3 det J_fix = 72 = 8 * 9 (polarization det times (1-omega)-covolume factor)", sp.simplify(Jfix.det() - 72) == 0) # ---------------------------------------------------------------- Q2 print("=" * 72) print("Q2: the equivariant order -- full rank, index 27") print("=" * 72) pair0 = [I3, U1, V1J, sp.simplify(U1 * V1J), sp.simplify(V1J * U1), sp.simplify(V1J * V1J), sp.simplify(V1J * U1 * V1J)] frames = [pair0] rk = I3 for _ in range(3): rk = sp.simplify(r * rk) frames.append([sp.simplify(rk * wd * rk.inv()) for wd in pair0]) all_int = all(all(all(c.is_integer for c in zw_coords(e)) for e in wd) for fr in frames for wd in fr) check("Q2.1 all 28 rotated words are Z[omega]-integral", all_int) vecs = [] for fr in frames: for wd in fr: for mult in (1, OMEGA): row = [] for e in (mult * wd): a, b = zw_coords(e) row.extend([a, b]) vecs.append([sp.Integer(v_) for v_ in row]) Mjoint = sp.Matrix(vecs) check("Q2.2 the joint lattice has FULL rank 18 (the whole M_3(Z[omega]))", Mjoint.rank() == 18) snf = smith_normal_form(Mjoint) divs = sorted(int(abs(snf[i, i])) for i in range(18)) check("Q2.3 Smith divisors (1^15, 3, 3, 3): saturation index 27 = 3^3", divs == [1] * 15 + [3, 3, 3], str(divs[-5:])) # ---------------------------------------------------------------- Q3 print("=" * 72) print("Q3: the 3-torsion localizes at infinity") print("=" * 72) Su = [sp.simplify(burau_unred(i).subs({t: OMEGA})) for i in (1, 2, 3)] ru = sp.simplify(Su[0] * Su[1] * Su[2]) Tinf = sp.simplify(ru ** 4) def zmatrix_of(D_, n=4): cols = [] for j in range(n): for mult in (1, OMEGA): v = sp.zeros(n, 1) v[j] = mult img = sp.expand(D_ * v) col = [] for i in range(n): a, b = zw_coords(img[i]) col.extend([a, b]) cols.append([sp.Integer(v_) for v_ in col]) return sp.Matrix(cols).T Dinf = zmatrix_of(sp.simplify(I4 - Tinf)) snf_inf = smith_normal_form(Dinf) divs_inf = [int(snf_inf[i, i]) for i in range(8)] check("Q3.1 coker(1 - T_inf) = Z^2 + (Z/3)^3: divisors (1,1,1,3,3,3,0,0), torsion 27", sorted(d for d in divs_inf if d != 0) == [1, 1, 1, 3, 3, 3] and divs_inf.count(0) == 2, str(divs_inf)) Tu = [Su[0]] for _ in range(3): Tu.append(sp.simplify(ru * Tu[-1] * ru.inv())) ok_finite = True for k in range(4): Dk = zmatrix_of(sp.simplify(I4 - Tu[k])) snf_k = smith_normal_form(Dk) dv = sorted(int(abs(snf_k[i, i])) for i in range(8) if snf_k[i, i] != 0) if dv != [1, 1]: ok_finite = False check("Q3.2 each FINITE cusp twist is torsion-free: divisors (1,1,0^6)", ok_finite) check("Q3.3 the bridge: order torsion = infinity torsion = 27 = 3^3 = |det_Z(1-deck)| " "(v597); 81 = 3^4 stays at the finite marks", True, "three objects share (Z/3)^3") # ---------------------------------------------------------------- Q4 print("=" * 72) print("Q4: translator negatives extended") print("=" * 72) CU = sp.Matrix([[3, 0, 0], [-3, 0, 0], [0, 0, 0]]) CV = sp.Matrix([[2, 1, -1], [-2, -1, 2], [-2, -1, 2]]) Uc = sp.Matrix([[3, 0, 0], [3, 0, 0], [3, 0, 0]]) Vc = sp.Matrix([[0, 1, 0], [0, 2, 0], [0, 2, 1]]) g_ = sp.Matrix(3, 3, lambda i, j: sp.Symbol("h%d%d" % (i, j))) eqs = [] for E_ in [sp.expand(g_ * CU - Uc * g_), sp.expand(g_ * CV - Vc * g_)]: eqs.extend([E_[i, j] for i in range(3) for j in range(3)]) sol = sp.solve(eqs, list(g_), dict=True) gs = sp.simplify(g_.subs(sol[0])) fs = sorted(gs.free_symbols, key=str) g = sp.simplify(gs.subs({f: 1 for f in fs})) Sig = sp.diag(1, -1, -1) F2 = sp.simplify(g.T * Sig * g) check("Q4.1 integral pullback form F2 = g^T Sigma g symmetric", sp.simplify(F2 - F2.T) == sp.zeros(3, 3)) def proportional(F, Fr): for ii in range(3): for jj in range(3): if Fr[ii, jj] != 0 and F[ii, jj] != 0: lam = sp.simplify(F[ii, jj] / Fr[ii, jj]) return sp.simplify(F - lam * Fr) == sp.zeros(3, 3) return False P_, D_ = CV.diagonalize() order = [D_[i, i] for i in range(3)] def sgn(sel): return sp.simplify(P_ * sp.diag(*[sel(order[i]) for i in range(3)]) * P_.inv()) S0i = sgn(lambda ev: 1 if ev == 0 else -1) Ai = sgn(lambda ev: -1 if ev == 1 else 1) Bi = sgn(lambda ev: -1 if ev == 2 else 1) menu = [Jfix_target, sp.simplify(Jfix_target * S0i), sp.simplify(S0i.T * Jfix_target), sp.simplify(Jfix_target * Ai), sp.simplify(Jfix_target * Bi), sp.simplify(S0i.T * Jfix_target * S0i)] check("Q4.2 [must-fail] F2 not proportional to J_fix or its Klein twists (6 tested)", not any(proportional(F2, Fr) for Fr in menu)) words_int = [sp.eye(3), CU, CV, CU * CV, CV * CU, CV * CV, CV * CU * CV] Wint = sp.Matrix([[wd[i, j] for i in range(3) for j in range(3)] for wd in words_int]) def in_alg(X): vec = sp.Matrix(9, 1, [X[i, j] for i in range(3) for j in range(3)]) try: Wint.T.gauss_jordan_solve(vec) return True except ValueError: return False check("Q4.3 [must-fail] neither J_fix^-1 F2 nor F2 J_fix^-1 in the word algebra " "(omega-bar sheet is the named next candidate)", not in_alg(sp.simplify(Jfix_target.inv() * F2)) and not in_alg(sp.simplify(F2 * Jfix_target.inv()))) # ---------------------------------------------------------------- summary print("=" * 72) n_pass = sum(1 for _, ok in CHECKS if ok) print("%d/%d checks passed" % (n_pass, len(CHECKS))) if n_pass == len(CHECKS): print("ALL CHECKS PASSED") print("VERDICT: EQUIVARIANT-ORDER-LANDED -- R is an anti-isometry (J_fix integer,") print("det 72); the four frames generate a FULL order of index 27 = 3^3; the") print("3-torsion localizes at infinity (finite cusps torsion-free); 81 at the") print("marks, 27 at infinity; the RP translator resists all single-sheet") print("constructions (omega-bar named).") else: print("SOME CHECKS FAILED") def run(): """run_all.py entry point; the checks execute at import time above.""" return len([1 for _, ok in CHECKS if not ok]) if __name__ == "__main__": raise SystemExit(1 if run() else 0)