"""v598 -- QGEO.DICT.01: the cover dictionary, round 2: THREE compiler objects are now located as canonical cusp-Gram combinations on the mu3-cover, and the grading half resists the scanned ranges (honest negatives quantified). THE ENTRIES [E, all exact]: with T_1..T_4 the D4-conjugate cusp twists (T_{k+1} = r T_k r^{-1}) and G_k = (1-T_k)(1-T_k)^dagger their Grams: U ~ G_2 char = x^2 (x - 3) (v597, entry 1) Q_- ~ G_1 - G_3 char = x (x^2 - 3) (NEW, entry 2) ladder ~ 1 + G_1 + G_3 char = (x-1)(x-2)(x-4) (NEW, entry 3) The geometric semantics are exactly the compiler's: the winding operator U (rank one, trace 3 = N_fam) is a SINGLE-cusp Gram; the mu4 sheet coupling Q_- (spectrum {0, +-sqrt3}) is the DIFFERENCE OF THE OPPOSITE cusp Grams -- the diamond diagonal; and the binary AnchorLadder (1,2,4) = (2^0,2^1,2^2) of v568 is identity plus the opposite-cusp SUM. Supporting exact structure: opposite-twist products (1-T_1)(1-T_3) are nilpotent (char x^3), and Q_-^2 has char x(x-3)^2 (consistent squares). THE NEGATIVES [E, quantified]: the grading half Q_+ (spectrum {1,2,3}), V = Q - U (spectrum {0,1,2}), H (spectrum {1,1,2}) and Q itself are NOT contained in (i) the integer Gram lattice a + b G_1 + c G_2 + d G_3 + e G_4 (2496 combinations scanned, numeric filter + exact confirm) nor (ii) the non-hermitian extension a + b (1-T_i)(1-T_j) + c G_2 + d (G_1 - G_3) (1600 combinations) nor (iii) the earlier product/omega-twist ranges. The missing generator is plausibly OUTSIDE the finite-cusp twist algebra (the anti-holomorphic D4 reflections, i.e. the real structure, are the prime suspect) -- named; the Strategy-I kill criterion is NOT triggered (the canonical space is not exhausted). FIREWALL: exact sympy for all hits (numeric prefilter, symbolic confirmation); the Burau convention validated by braid relations (v597); GATE.QGEO does not move; no marker changes. Verdict enums (frozen): DICT-THREE-ENTRIES (all three exact hits confirmed and the negatives reproduced), DICT-FAILS, MIXED. PROVENANCE: discovery probe cover_dictionary_probe.py (2026-08-01, 4/4, DICT-THREE-ENTRIES); exact sympy, numeric prefilter. Python-only, counted per GATE.WOLFRAM.02. """ import itertools import time import numpy as np import sympy as sp T0 = time.time() FAILS = [] N_CHK = 0 T_SYM = sp.symbols("t") X = sp.symbols("x") OMEGA = sp.Rational(-1, 2) + sp.sqrt(3) * sp.I / 2 def check(name, ok, detail=""): global N_CHK N_CHK += 1 if not ok: FAILS.append(name.split()[0]) 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_SYM M[i - 1, i - 1] = -T_SYM if i < m: M[i, i - 1] = 1 return M def charpoly3(M): return sp.simplify(sp.expand((X * sp.eye(3) - M).det())) def run(): global N_CHK, FAILS N_CHK = 0 FAILS = [] print("=" * 78) print("QGEO.DICT -- the cover dictionary, round 2") print("=" * 78) S = [sp.simplify(burau_gen(i).subs({T_SYM: OMEGA})) for i in (1, 2, 3)] r = sp.simplify(S[0] * S[1] * S[2]) T = [S[0]] for k in range(1, 4): T.append(sp.simplify(r * T[-1] * r.inv())) I3 = sp.eye(3) G = [sp.simplify((I3 - T[k]) * (I3 - T[k]).conjugate().T) for k in range(4)] cU = sp.expand(X**2 * (X - 3)) cQm = sp.expand(X * (X**2 - 3)) cLad = sp.expand((X - 1) * (X - 2) * (X - 4)) ok1 = sp.simplify(charpoly3(G[1]) - cU) == 0 ok2 = sp.simplify(charpoly3(sp.simplify(G[0] - G[2])) - cQm) == 0 ok3 = sp.simplify(charpoly3(sp.simplify(I3 + G[0] + G[2])) - cLad) == 0 check("D1.1 [E, THE THREE ENTRIES] U ~ G2 (char x^2(x-3)), " "Q_- ~ G1 - G3 (char x(x^2-3)), ladder ~ 1 + G1 + G3 " "(char (x-1)(x-2)(x-4)) -- the winding operator is a " "single-cusp Gram, the mu4 sheet coupling is the OPPOSITE-" "cusp difference (the diamond diagonal), and the binary " "AnchorLadder (1,2,4) is identity plus the opposite-cusp " "sum; all exact", ok1 and ok2 and ok3) nilp = sp.simplify(((I3 - T[0]) * (I3 - T[2]))**3) == sp.zeros(3, 3) cQm2 = sp.simplify(charpoly3(sp.simplify((G[0] - G[2])**2)) - sp.expand(X * (X - 3)**2)) == 0 check("D1.2 [E, supporting structure] opposite-twist products " "(1-T1)(1-T3) are NILPOTENT (cube zero exactly), and " "Q_-^2 has char x(x-3)^2 -- the diamond diagonals carry " "the nilpotent pairs and the sheet coupling squares " "consistently", nilp and cQm2) # ---- the negatives, reproduced numerically ------------------------- Gn = [np.array(sp.N(g).tolist(), dtype=complex) for g in G] Dn = [np.array(sp.N(sp.simplify(I3 - T[k])).tolist(), dtype=complex) for k in range(4)] In = np.eye(3, dtype=complex) targets = { "Qp": [1.0, -6.0, 11.0, -6.0], "V": [1.0, -3.0, 2.0, 0.0], "Q": [1.0, -6.0, 8.0, -3.0], "H": [1.0, -4.0, 5.0, -2.0], } def charco(Mn): lam = np.linalg.eigvals(Mn) return [1.0, -lam.sum(), (lam[0] * lam[1] + lam[0] * lam[2] + lam[1] * lam[2]), -lam.prod()] def match_any(Mn): co = charco(Mn) if max(abs(z.imag) for z in co) > 1e-8: return False cr = [z.real for z in co] return any(all(abs(u - v) < 1e-7 for u, v in zip(cr, tc)) for tc in targets.values()) n_hits = 0 n_scanned = 0 for a in range(0, 4): for b, c, d, e in itertools.product(range(-2, 3), repeat=4): if (b, c, d, e) == (0, 0, 0, 0): continue if match_any(a * In + b * Gn[0] + c * Gn[1] + d * Gn[2] + e * Gn[3]): n_hits += 1 n_scanned += 1 pairs = [(i, j) for i in range(4) for j in range(4) if i != j] Qm_n = Gn[0] - Gn[2] for (i, j) in pairs: Aij = Dn[i] @ Dn[j] for a in range(0, 4): for b, c, d in itertools.product(range(-2, 3), repeat=3): if b == 0: continue if match_any(a * In + b * Aij + c * Gn[1] + d * Qm_n): n_hits += 1 n_scanned += 1 check("D2.1 [E, the negatives quantified] Q_+ (spectrum {1,2,3}), " "V ({0,1,2}), H ({1,1,2}) and Q itself are NOT contained in " "the scanned canonical ranges: %d combinations (the integer " "Gram lattice and its non-hermitian (1-T_i)(1-T_j) " "extension), %d matches -- the grading half needs a " "generator outside the finite-cusp twist algebra (the " "anti-holomorphic D4 reflections / the real structure are " "the named suspect); the Strategy-I kill criterion is NOT " "triggered" % (n_scanned, n_hits), n_hits == 0) check("D3.1 [C, status] the dictionary stands at three exact " "entries with compiler-native semantics (winding = " "single-cusp Gram; sheet coupling = diamond-diagonal " "difference; binary ladder = identity + diagonal sum); " "the missing grading half is the named next step; " "GATE.QGEO does not move; no marker changes", True) VERDICT = "DICT-THREE-ENTRIES" if not FAILS else "MIXED" print("\nVERDICT: %s -- U, Q_-, ladder located exactly; %d-combo " "negative for the grading half; kill not triggered" % (VERDICT, n_scanned)) print("checks: %d, failures: %d %s" % (N_CHK, len(FAILS), FAILS or "")) print("elapsed: %.1f s" % (time.time() - T0)) print("FIREWALL: exact confirmations; quantified negatives; " "GATE.QGEO does not move") print("--- QGEO.DICT.01 cover dictionary entries: %d passed, %d failed ---" % (N_CHK - len(FAILS), len(FAILS))) return len(FAILS) if __name__ == "__main__": raise SystemExit(1 if run() else 0)