#!/usr/bin/env python3 """v635 -- PRIME.PCANON.01: P-canonicity + finer Hodge-chamber invariants. Part A (canonicity of the v624 congruence matrix P): A0 sanity guards: P^T J_det P = J_fix exactly, det P = -6, det J_det = 2, det J_fix = 72; the v600/v604 integer conjugation pair (C_U, C_V) on the R-fixed lattice has the compiler characteristic polynomials (copied-matrix guard). A1 Smith normal forms of J_det, J_fix, P. A2 word-algebra role: is P in the Q-span (or Z-span) of the 7-word algebra {1, C_U, C_V, C_U C_V, C_V C_U, C_V^2, C_V C_U C_V}? Are P's columns (+-) word-images of the fixed-lattice basis vectors? (Deck acts omega-multiplicatively and R acts as identity on the fixed lattice, so the words in (C_U, C_V) are the available integral operator combinations.) A3 does P conjugate the word generators into integer matrices (P C_U P^{-1}, P C_V P^{-1} integral)? A4 the census: ALL integer 3x3 Q with entries in [-4,4] and Q^T J_det Q = J_fix (columns satisfy q^T J_det q = 16/-2/-2 and the three cross products 2/4/2); count, det distribution, and the distinguisher scan: minimal Frobenius norm, operator compatibility (both conjugations integral), column sign patterns. Part B (finer chamber invariants; the coarse chamber does not separate, v627 H4): B1 for all complete windows: theta = projective angle of z = P^{-1} y to the positive eigendirection of J_fix; q(z)/||z||^2; the spacelike component ratio c1/c2 and spacelike plane angle phi. NOTE: J_fix has Lorentz signature (1,2) -- ONE positive eigendirection; the task's 'ratio of the two J_fix-positive components' is read as the ratio of the two SPACELIKE (negative-eigenvalue) components. B2 scramble controls: seeds (1,2,3) at the reference window h = 540 and at 5 spread windows; per-invariant separation statistics (global range test + per-window shift). Verdict enums (frozen): P-SELECTED-MOD-SYMMETRY + FINE-INVARIANTS-SEPARATE (as measured), FINE-INVARIANTS-PARTIAL, FINE-INVARIANTS-NO-SEPARATION, MIXED. FIREWALL: no marker changes; no positivity claim beyond the measured surface, no RH statement; the [-4,4] census is a box census (the full O(J_det,Z) torsor is infinite, typed). PROVENANCE: discovery probe p_canonicity_hodge_fine_probe.py (2026-08-02, 18/18, verdicts P-SELECTED-MOD-SYMMETRY + FINE-INVARIANTS-SEPARATE). Machinery: v563 read-only. Python-only, counted per GATE.WOLFRAM.02. """ import itertools import math import os import sys import time from collections import Counter import numpy as np import sympy as sp from sympy.matrices.normalforms import smith_normal_form _here = os.path.dirname(os.path.abspath(__file__)) for _cand in (_here, os.path.join(_here, "..", "..", "verification")): if os.path.exists(os.path.join(_cand, "v563_paper2_readouts.py")): sys.path.insert(0, os.path.abspath(_cand)) break T0 = time.time() 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 "")) # ================================================================== A print("=" * 78) print("PART A: canonicity of P") print("=" * 78) x = sp.symbols("x") Jdet = sp.Matrix([[0, 1, 0], [1, 0, 0], [0, 0, -2]]) Jfix = sp.Matrix([[16, 2, 4], [2, -2, 2], [4, 2, -2]]) P = sp.Matrix([[3, 0, 0], [3, 0, 2], [-1, 1, -1]]) CU = sp.Matrix([[3, 0, 0], [-3, 0, 0], [0, 0, 0]]) CV = sp.Matrix([[2, 1, -1], [-2, -1, 2], [-2, -1, 2]]) check("A0.1 congruence guard: P^T J_det P = J_fix, det P = -6, " "det J_det = 2, det J_fix = 72", sp.simplify(P.T * Jdet * P - Jfix) == sp.zeros(3, 3) and P.det() == -6 and Jdet.det() == 2 and Jfix.det() == 72) char_ok = (CU.charpoly(x).as_expr() == sp.expand(x ** 2 * (x - 3)) and CV.charpoly(x).as_expr() == sp.expand(x * (x - 1) * (x - 2)) and (CU + CV).charpoly(x).as_expr() == sp.expand((x - 1) * (x ** 2 - 5 * x + 3)) and sp.trace(CU * CV) == 3) check("A0.2 (C_U, C_V) copied-matrix guard: char C_U = x^2(x-3), " "char C_V = x(x-1)(x-2), char(C_U+C_V) = (x-1)(x^2-5x+3), " "tr(C_U C_V) = 3 (the v600 J1 identities)", char_ok) snf_det = smith_normal_form(Jdet) snf_fix = smith_normal_form(Jfix) snf_P = smith_normal_form(P) d_det = [int(abs(snf_det[i, i])) for i in range(3)] d_fix = [int(abs(snf_fix[i, i])) for i in range(3)] d_P = [int(abs(snf_P[i, i])) for i in range(3)] check("A1.1 Smith normal forms computed", True, "SNF(J_det) = %s, SNF(J_fix) = %s, SNF(P) = %s" % (d_det, d_fix, d_P)) # ---- A2: word algebra role words = [sp.eye(3), CU, CV, CU * CV, CV * CU, CV * CV, CV * CU * CV] Wmat = sp.Matrix([[w[i, j] for i in range(3) for j in range(3)] for w in words]) vecP = sp.Matrix(1, 9, [P[i, j] for i in range(3) for j in range(3)]) rank_W = Wmat.rank() rank_aug = sp.Matrix.vstack(Wmat, vecP).rank() in_span = (rank_aug == rank_W) coeffs_str = "not in Q-span" if in_span: a = sp.symbols("a0:7") sol = sp.solve(sp.Matrix(sum((a[i] * words[i] for i in range(1, 7)), a[0] * words[0]) - P), list(a), dict=True) coeffs_str = str(sol) check("A2.1 [measured] P in the Q-span of the 7-word algebra of " "(C_U, C_V)? answer: %s" % in_span, True, "rank(words) = %d, rank(words+P) = %d; %s" % (rank_W, rank_aug, coeffs_str)) word_cols = set() for w in words: for j in range(3): col = tuple(int(w[i, j]) for i in range(3)) word_cols.add(col) word_cols.add(tuple(-c for c in col)) p_cols = [tuple(int(P[i, j]) for i in range(3)) for j in range(3)] hits = [c for c in p_cols if c in word_cols] check("A2.2 [measured] P columns as (+-) word-images of basis vectors: " "%d of 3 columns are word columns" % len(hits), True, "P columns %s; word-column hits: %s" % (p_cols, hits)) # ---- A3: conjugation integrality PCU = P * CU * P.inv() PCV = P * CV * P.inv() int_CU = all(e.is_integer for e in PCU) int_CV = all(e.is_integer for e in PCV) check("A3.1 [measured] P C_U P^-1 integral? %s; P C_V P^-1 integral? %s" % (int_CU, int_CV), True, "P C_U P^-1 = %s, P C_V P^-1 = %s" % (sp.nsimplify(PCU).tolist(), sp.nsimplify(PCV).tolist())) # ---- A4: the census rng4 = range(-4, 5) cols16 = [c for c in itertools.product(rng4, repeat=3) if c[0] * c[1] - c[2] * c[2] == 8] # q^T J_det q = 16 colsm2 = [c for c in itertools.product(rng4, repeat=3) if c[0] * c[1] - c[2] * c[2] == -1] # q^T J_det q = -2 def bil(u, v): return u[0] * v[1] + u[1] * v[0] - 2 * u[2] * v[2] sols = [] for q1 in cols16: c2s = [q2 for q2 in colsm2 if bil(q1, q2) == 2] c3s = [q3 for q3 in colsm2 if bil(q1, q3) == 4] for q2 in c2s: for q3 in c3s: if bil(q2, q3) == 2: Q = tuple(tuple(int(v) for v in (q1[i], q2[i], q3[i])) for i in range(3)) sols.append(Q) P_rows = tuple(tuple(int(P[i, j]) for j in range(3)) for i in range(3)) check("A4.1 census: %d integer solutions Q with entries in [-4,4] and " "Q^T J_det Q = J_fix; P is among them: %s" % (len(sols), P_rows in sols), P_rows in sols, "(column candidates: %d with norm 16, %d with norm -2)" % (len(cols16), len(colsm2))) def det3(A): return (A[0][0] * (A[1][1] * A[2][2] - A[1][2] * A[2][1]) - A[0][1] * (A[1][0] * A[2][2] - A[1][2] * A[2][0]) + A[0][2] * (A[1][0] * A[2][1] - A[1][1] * A[2][0])) def adj3(A): return [[A[1][1] * A[2][2] - A[1][2] * A[2][1], A[0][2] * A[2][1] - A[0][1] * A[2][2], A[0][1] * A[1][2] - A[0][2] * A[1][1]], [A[1][2] * A[2][0] - A[1][0] * A[2][2], A[0][0] * A[2][2] - A[0][2] * A[2][0], A[0][2] * A[1][0] - A[0][0] * A[1][2]], [A[1][0] * A[2][1] - A[1][1] * A[2][0], A[0][1] * A[2][0] - A[0][0] * A[2][1], A[0][0] * A[1][1] - A[0][1] * A[1][0]]] def matmul3(A, B): return [[sum(A[i][k] * B[k][j] for k in range(3)) for j in range(3)] for i in range(3)] CU_l = [[3, 0, 0], [-3, 0, 0], [0, 0, 0]] CV_l = [[2, 1, -1], [-2, -1, 2], [-2, -1, 2]] def conj_integral(Q, C): d = det3(Q) A = adj3(Q) M = matmul3(matmul3([list(r) for r in Q], C), A) return all(m % d == 0 for row in M for m in row) dets = Counter(det3(Q) for Q in sols) frob = {Q: sum(v * v for r in Q for v in r) for Q in sols} fmin = min(frob.values()) at_min = [Q for Q in sols if frob[Q] == fmin] op_ok = [Q for Q in sols if conj_integral(Q, CU_l) and conj_integral(Q, CV_l)] col1pos = [Q for Q in sols if all(Q[i][0] > 0 for i in range(3))] col1nn = [Q for Q in sols if all(Q[i][0] >= 0 for i in range(3))] check("A4.2 census det distribution: %s (torsor under the infinite " "O(J_det,Z); box census only)" % dict(dets), True) check("A4.3 distinguisher: minimal Frobenius norm^2 = %d, attained by %d " "solutions; P (norm^2 = %d) among them: %s" % (fmin, len(at_min), frob[P_rows], P_rows in at_min), True, "min-norm solutions: %s" % (at_min if len(at_min) <= 8 else str(len(at_min)) + " sols")) check("A4.4 distinguisher: operator compatibility (BOTH P C P^-1 " "integral for C_U and C_V): %d of %d solutions; P among them: %s" % (len(op_ok), len(sols), P_rows in op_ok), True, "op-compatible: %s" % (op_ok if len(op_ok) <= 8 else str(len(op_ok)) + " sols")) check("A4.5 distinguisher: strictly positive first column: %d sols; " "nonnegative first column: %d sols; P has col1 = (3,3,-1) " "(mixed sign)" % (len(col1pos), len(col1nn)), True) frob_at_op = sorted(frob[Q] for Q in op_ok) if op_ok else [] check("A4.6 combined distinguisher scan executed", True, "Frobenius norms of op-compatible sols: %s" % (frob_at_op[:20] if frob_at_op else "none")) # A4.7: classes modulo the NATURAL prime-side symmetries (all in # O(J_det,Z)): global sign, the X11 <-> X22 swap, the X12 sign flip T_swap = [[0, 1, 0], [1, 0, 0], [0, 0, 1]] T_sgn = [[1, 0, 0], [0, 1, 0], [0, 0, -1]] T_min = [[-1, 0, 0], [0, -1, 0], [0, 0, -1]] G8 = [[[1, 0, 0], [0, 1, 0], [0, 0, 1]]] frontier = [G8[0]] while frontier: g = frontier.pop() for t in (T_swap, T_sgn, T_min): h = matmul3(t, g) if h not in G8: G8.append(h) frontier.append(h) iso_ok = all(matmul3(matmul3([list(r) for r in zip(*g)], [[0, 1, 0], [1, 0, 0], [0, 0, -2]]), g) == [[0, 1, 0], [1, 0, 0], [0, 0, -2]] for g in G8) def orbit_class(Q): return frozenset(tuple(tuple(r) for r in matmul3(g, [list(r_) for r_ in Q])) for g in G8) classes = {} for Q in sols: classes.setdefault(orbit_class(Q), []).append(Q) P_class = orbit_class(P_rows) min_is_P_class = set(at_min) == set(P_class) & set(sols) op_classes = {orbit_class(Q) for Q in op_ok} check("A4.7 modulo the order-%d natural prime-side symmetry group " "(global sign, X11<->X22, X12-sign; all verified in O(J_det,Z)): " "%d classes; the minimal-Frobenius set IS exactly P's class: %s; " "op-compatible classes: %d of %d" % (len(G8), len(classes), min_is_P_class, len(op_classes), len(classes)), len(G8) == 8 and iso_ok, "class sizes %s" % sorted(len(v) for v in classes.values())) verdict_A = "P-CANONICITY-OPEN" if op_ok == [P_rows] or at_min == [P_rows]: verdict_A = "P-SELECTED" elif min_is_P_class and P_rows in op_ok: verdict_A = ("P-SELECTED-MOD-SYMMETRY (unique minimal-Frobenius " "class, op-compatible)") elif P_rows in op_ok and len(op_ok) <= 8: verdict_A = "P-NARROWED" # ================================================================== B print("=" * 78) print("PART B: finer chamber invariants vs scramble") print("=" * 78) import v563_paper2_readouts as core # noqa: E402 (READ-ONLY import) PinvN = np.array(sp.Matrix(P).inv().evalf(20).tolist(), dtype=float) JfixN = np.array(Jfix.tolist(), dtype=float) w_, V_ = np.linalg.eigh(JfixN) for j in range(3): k = int(np.argmax(np.abs(V_[:, j]))) if V_[k, j] < 0: V_[:, j] *= -1.0 check("B0.1 J_fix eigenvalues (ascending): %s -- signature (1,2), ONE " "positive direction (the task's 'two positive components' is read " "as the two SPACELIKE components)" % np.array2string(w_, precision=6), w_[0] < 0 and w_[1] < 0 and w_[2] > 0) def invariants(S): y = np.array([S[0, 0], S[1, 1], S[0, 1]], dtype=float) z = PinvN @ y nz = float(np.linalg.norm(z)) c = V_.T @ z theta = math.degrees(math.acos(min(1.0, abs(float(c[2])) / nz))) qn = float(z @ (JfixN @ z)) / nz ** 2 ratio = float(c[0] / c[1]) if c[1] != 0 else float("inf") phi = math.degrees(math.atan2(float(c[1]), float(c[0]))) % 180.0 return theta, qn, ratio, phi, float(np.linalg.det(S)) u_max = float(core.U_ALL[-1]) rows = [] for kz in core.frame_a_zones(): r = core.build_window(kz) if r["h"] == 1292 or 2.0 * r["alpha"] > u_max: continue rows.append((kz, r["h"], r["S"])) true_inv = {kz: invariants(S) for kz, h, S in rows} qz_resid = 0.0 for kz, h, S in rows: y = np.array([S[0, 0], S[1, 1], S[0, 1]], dtype=float) z = PinvN @ y qz_resid = max(qz_resid, abs(float(z @ (JfixN @ z)) - 2.0 * float(np.linalg.det(S)))) check("B1.1 transport bookkeeping: %d complete windows, max " "|q(z) - 2 det S| = %.2e" % (len(rows), qz_resid), len(rows) == 67 and qz_resid < 1e-9) names = ["theta_deg", "q/|z|^2", "c1/c2", "phi_deg"] tvals = {n: [true_inv[kz][i] for kz, h, S in rows] for i, n in enumerate(names)} for n in names: v = tvals[n] print(" true %-9s: min %.6g max %.6g mean %.6g std %.6g" % (n, min(v), max(v), float(np.mean(v)), float(np.std(v)))) check("B1.2 true-family invariant ranges recorded (67 windows)", True) # scramble windows: reference h=540 + 5 spread picks kz_ref = [kz for kz, h, S in rows if h == 540][0] order = sorted(range(len(rows)), key=lambda i: rows[i][1]) pick_idx = [order[0], order[len(rows) // 4], order[len(rows) // 2], order[3 * len(rows) // 4], order[-1]] extra = [] for i in pick_idx: if rows[i][0] != kz_ref and rows[i][0] not in extra: extra.append(rows[i][0]) scr_kz = [kz_ref] + extra SEEDS = (1, 2, 3) scr_inv = {} for kz in scr_kz: for seed in SEEDS: rs = core.build_window(kz, scramble_seed=seed) scr_inv[(kz, seed)] = invariants(rs["S"]) h_of = {kz: h for kz, h, S in rows} print(" scramble table (window h | seed | theta, q/|z|^2, c1/c2, phi " "| true values):") for kz in scr_kz: ti = true_inv[kz] print(" h=%-5d true : theta %.6g qn %.6g ratio %.6g phi %.6g" % (h_of[kz], ti[0], ti[1], ti[2], ti[3])) for seed in SEEDS: si = scr_inv[(kz, seed)] print(" h=%-5d s=%d : theta %.6g qn %.6g ratio %.6g " "phi %.6g" % (h_of[kz], seed, si[0], si[1], si[2], si[3])) n_scr = len(scr_inv) sep_stats = {} for i, n in enumerate(names): lo, hi = min(tvals[n]), max(tvals[n]) outside = sum(1 for v in scr_inv.values() if not (lo <= v[i] <= hi)) sep_stats[n] = outside check("B2.1 separation census: scrambles outside the true 67-window " "range, per invariant: %s (of %d scrambles each)" % (sep_stats, n_scr), True) shift_stats = {} for i, n in enumerate(names): tstd = float(np.std(tvals[n])) shifts = [abs(scr_inv[(kz, s)][i] - true_inv[kz][i]) / (tstd if tstd > 0 else 1.0) for kz in scr_kz for s in SEEDS] shift_stats[n] = (min(shifts), float(np.median(shifts)), max(shifts)) check("B2.2 per-window shift in units of the true-family std " "(min/median/max): %s" % {k: tuple(round(v, 3) for v in vv) for k, vv in shift_stats.items()}, True) sep_verdicts = {} for n in names: if sep_stats[n] == n_scr: sep_verdicts[n] = "SEPARATES" elif sep_stats[n] == 0: sep_verdicts[n] = "NO-SEPARATION" else: sep_verdicts[n] = "PARTIAL(%d/%d)" % (sep_stats[n], n_scr) verdict_B = ("FINE-INVARIANTS-SEPARATE" if any(v == "SEPARATES" for v in sep_verdicts.values()) else ("FINE-INVARIANTS-PARTIAL" if any(v.startswith("PARTIAL") for v in sep_verdicts.values()) else "FINE-INVARIANTS-NO-SEPARATION")) # ================================================================== summary print("=" * 78) n_pass = sum(1 for _, ok in CHECKS if ok) print("%d/%d checks passed" % (n_pass, len(CHECKS))) print("VERDICT A: %s" % verdict_A) print("VERDICT B: %s -- per invariant: %s" % (verdict_B, sep_verdicts)) print("elapsed: %.1f s" % (time.time() - T0)) if n_pass == len(CHECKS): print("ALL CHECKS PASSED") 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)