#!/usr/bin/env python3 """v732 -- PRIME.SONIN.01: STRATEGY-3 / L1 -- finite Sonin--prolate compression on TFPT windows. This is a finite, zero-free operator experiment inspired by Connes--Consani (2021); it is not their theorem and it makes no RH claim. For every window in the nine-member frame-A ladder, construct on the full piecewise-linear u grid P_h = multiplication by 1_{|u| <= U0}, P_hat_h = Fourier-band projection 1_{|tau| <= T0}, with U0 = log(2)/2 and T0 = pi/(2 U0), so that 2 U0 T0 / pi = 1: one self-dual phase-space cell. Restrict both projections to the odd window sector. Their common kernel is the finite Sonin complement. Let B_h be the deployed positive Weil window form and S_h the orthogonal projection onto that complement. The part not supplied by the compression is R_h = B_h - S_h B_h S_h. Whitening by B_h turns this into I-K_h, where the generalized eigenvalues of S_h B_h S_h v = lambda B_h v above one are precisely the non-positive directions of R_h. Since R_h has rank at most twice the number of projection constraints, all nontrivial eigenvalues are recovered from the top 2r+8 generalized eigenpairs. Preregistered interpretation: * FINITE-RANK-WALL if bad_rank <= 4 uniformly over all nine windows. * RANK-GROWS (kill) if bad_rank grows beyond 4. * DEFECT-PERSISTS (kill) if lambda_max-1 does not decrease and is still above 0.05 on the largest window. * POLE-CONDITIONABLE only if the bad subspace remaining after one condition along the closed rank-one E8/pole boundary vector has no Rayleigh quotient above 1+5e-8, pole overlap is >= 0.70, and adjacent-window bad-direction fidelity is >= 0.80. The conditioned number computed here is a rigorous lower bound inside the measured bad eigenspace: if it remains above one, one pole condition definitely cannot cure the remainder; if it falls below one, a full-space theorem is still required. The construction path loads no zeros and no prime table. Verification tables enter only through the already-declared frame constructor used by the moonshot probes. No file is written by this script. PROVENANCE: discovery probe strat3_sonin_prolate_probe.py (2026-08-03, ALL 21 CONSTRUCTION CHECKS PASSED, verdict SONIN-RANK-GROWS -- the preregistered KILL fires: bad_rank grows 19 -> 86 over the nine-window ladder, the largest defect excess grows to 1.3e5, and one E8/pole condition is neither stable nor sufficient. The DIRECT TRANSFER of the Connes-Consani one-cell Sonin compression to the window discretization is dead (fenced: NOT the CC mechanism itself); tuning U0/T0 after this result is forbidden. Promoted verbatim (sibling imports now point at v716/v718); numbers unchanged. """ import ast import math import os import sys import time import numpy as np import scipy.linalg as sla _HERE = os.path.dirname(os.path.abspath(__file__)) sys.path.insert(0, _HERE) sys.path.insert(0, os.path.abspath(os.path.join(_HERE, "..", "..", "verification"))) import v563_paper2_readouts as core # noqa: E402 import v716_moonshot_arch_glue as stage2 # noqa: E402 import v718_moonshot_spectral as stage4 # noqa: E402 U_CUT = 0.5 * math.log(2.0) T_CUT = math.pi / (2.0 * U_CUT) PHASE_CELLS = 2.0 * U_CUT * T_CUT / math.pi SVD_TOL = 1.0e-10 EXACT_TOL = 2.0e-10 BAD_TOL = 5.0e-8 RANK_CAP = 4 FINAL_EXCESS_BAR = 5.0e-2 POLE_OVERLAP_BAR = 0.70 DRIFT_FIDELITY_BAR = 0.80 PROFILE_N = 2048 BANNED = ("zetazero", "nzeros", "isprime", "primerange", "nextprime", "prevprime", "primepi", "sympy") CHECKS = [] FAILS = [] def check(name, ok, detail=""): CHECKS.append(name) if not ok: FAILS.append(name.split()[0]) print("[%s] %s%s" % ("PASS" if ok else "FAIL", name, (": " + detail) if detail else "")) return bool(ok) def outcome(name, ok, detail=""): print("[%s] %s%s" % ("PASS" if ok else "FAIL", name, (": " + detail) if detail else "")) return bool(ok) def ast_firewall(): with open(os.path.abspath(__file__), "r", encoding="utf-8") as fh: tree = ast.parse(fh.read()) hits = set() for node in ast.walk(tree): name = "" if isinstance(node, ast.Name): name = node.id elif isinstance(node, ast.Attribute): name = node.attr elif isinstance(node, (ast.Import, ast.ImportFrom)): for alias in node.names: token = alias.name.split(".")[0] if any(b in token.lower() for b in BANNED): hits.add(token) if name and any(b in name.lower() for b in BANNED): hits.add(name) return sorted(hits) def odd_basis(M): h = M // 2 U = np.zeros((M, h)) j = np.arange(h) U[j, j] = 1.0 / math.sqrt(2.0) U[M - 1 - j, j] = -1.0 / math.sqrt(2.0) return U def projection_data(M, D): """Return the odd-sector finite Sonin projector and diagnostics.""" h = M // 2 U = odd_basis(M) u = (np.arange(M) - 0.5 * (M - 1)) * D k = np.arange(-M // 2, M // 2) tau = 2.0 * math.pi * k / (M * D) mt = np.abs(u) <= U_CUT + 1.0e-14 mf = np.abs(tau) <= T_CUT + 1.0e-14 At = U[mt, :] Fband = np.exp(-1j * np.outer(tau[mf], u)) / math.sqrt(M) Af = Fband @ U A = np.vstack((At, Af.real, Af.imag)) _uu, sv, vh = sla.svd(A, full_matrices=False, lapack_driver="gesdd", check_finite=False) rank = int(np.sum(sv > SVD_TOL * sv[0])) if len(sv) else 0 Q = vh[:rank, :].T S = np.eye(h) - Q @ Q.T P = np.diag(mt.astype(float)) Ph = Fband.conj().T @ Fband po = U.T @ P @ U pho = np.real_if_close(U.T @ Ph @ U).real proj_dev = max(float(np.linalg.norm(P @ P - P, ord=2)), float(np.linalg.norm(Ph @ Ph - Ph, ord=2)), float(np.linalg.norm(S @ S - S, ord=2))) annih = float(np.linalg.norm(A @ S, ord=2)) parity_imag = float(np.max(np.abs(np.imag(U.T @ Ph @ U)))) trace_overlap = float(np.trace(po @ pho)) return dict(U=U, u=u, S=S, Q=Q, constraint_rank=rank, n_time=int(np.sum(mt)), n_band=int(np.sum(mf)), proj_dev=proj_dev, annih=annih, parity_imag=parity_imag, trace_overlap=trace_overlap) def bad_space_measure(window): M = window["M"] h = M // 2 pd = projection_data(M, window["D"]) B = core.odd_toeplitz(window["car"] + window["cat"], M) B = 0.5 * (B + B.T) pole = -core.odd_toeplitz(window["cp"], M) pole = 0.5 * (pole + pole.T) lmin_B = float(sla.eigvalsh(B, subset_by_index=[0, 0], check_finite=False)[0]) check("S1.BPOS.h%d" % h, lmin_B > 0.0, "lambda_min(B)=%.6e" % lmin_B) S = pd["S"] SBS = S @ B @ S SBS = 0.5 * (SBS + SBS.T) nontrivial_cap = min(h, 2 * pd["constraint_rank"] + 8) lo = h - nontrivial_cap ev, vec = sla.eigh(SBS, B, subset_by_index=[lo, h - 1], driver="gvx", check_finite=False) bad_mask = ev > 1.0 + BAD_TOL bad_ev = ev[bad_mask] bad_vec = vec[:, bad_mask] bad_rank = len(bad_ev) lammax = float(ev[-1]) top = vec[:, -1] ep, Up = sla.eigh(pole, subset_by_index=[h - 1, h - 1], check_finite=False) bvec = Up[:, 0] binv_b = sla.solve(B, bvec, assume_a="pos", check_finite=False) dual_norm = math.sqrt(max(0.0, float(bvec @ binv_b))) pole_overlap = abs(float(bvec @ top)) / max(dual_norm, 1.0e-300) conditioned_bad = 1.0 if bad_rank >= 2: a = (bad_vec.T @ bvec) / max(dual_norm, 1.0e-300) Z = sla.null_space(a.reshape(1, -1), rcond=SVD_TOL) if Z.shape[1]: conditioned_bad = float(np.max(sla.eigvalsh( Z.T @ np.diag(bad_ev) @ Z, check_finite=False))) elif bad_rank == 1 and pole_overlap < 1.0 - 1.0e-8: # There is no nonzero vector inside a one-dimensional bad subspace # satisfying the condition. Record one; the full-space problem is # deliberately not certified by this favorable branch. conditioned_bad = 1.0 profile = pd["U"] @ top return dict(h=h, M=M, D=window["D"], alpha=window["alpha"], bad_rank=bad_rank, lammax=lammax, excess=lammax - 1.0, pole_overlap=pole_overlap, conditioned_bad=conditioned_bad, pole_eval=float(ep[0]), profile_u=pd["u"], profile=profile, lmin_B=lmin_B, **{k: pd[k] for k in ("constraint_rank", "n_time", "n_band", "proj_dev", "annih", "parity_imag", "trace_overlap")}) def profile_fidelities(rows): edge = min(float(np.max(np.abs(r["profile_u"]))) for r in rows) grid = np.linspace(-edge, edge, PROFILE_N) profiles = [] for r in rows: v = np.interp(grid, r["profile_u"], r["profile"]) nrm = math.sqrt(float(np.trapezoid(v * v, grid))) profiles.append(v / max(nrm, 1.0e-300)) out = [] for a, b in zip(profiles[:-1], profiles[1:]): out.append(abs(float(np.trapezoid(a * b, grid)))) return out def run(): t0 = time.time() print("=" * 78) print("STRAT3 SONIN/PROLATE -- finite-rank test of the L1 remainder") print("=" * 78) hits = ast_firewall() check("G0.1 AST firewall", not hits, str(hits)) check("G0.2 self-dual cutoff", abs(PHASE_CELLS - 1.0) < 1.0e-14, "U0=%.12f, T0=%.12f, 2U0T0/pi=%.16f" % (U_CUT, T_CUT, PHASE_CELLS)) windows = stage4.family_ext() check("G0.3 nine-window family", len(windows) == 9, ", ".join("h=%d" % (w["M"] // 2) for w in windows)) rows = [] for w in windows: r = bad_space_measure(w) rows.append(r) exact = max(r["proj_dev"], r["annih"], r["parity_imag"]) check("S1.PROJ.h%d" % r["h"], exact <= EXACT_TOL, "rank(C)=%d, nt=%d, nf=%d, dev=%.2e, AS=%.2e, " "Im(P_hat_odd)=%.2e, Tr(PP_hat)=%.6f" % (r["constraint_rank"], r["n_time"], r["n_band"], r["proj_dev"], r["annih"], r["parity_imag"], r["trace_overlap"])) print(" h=%4d bad_rank=%3d lambda_max=%.9f excess=%+.3e " "pole_overlap=%.4f cond_bad_lb=%.9f" % (r["h"], r["bad_rank"], r["lammax"], r["excess"], r["pole_overlap"], r["conditioned_bad"])) fids = profile_fidelities(rows) ranks = [r["bad_rank"] for r in rows] excess = np.array([max(r["excess"], 1.0e-300) for r in rows]) hs = np.array([r["h"] for r in rows], float) excess_slope = float(np.polyfit(np.log(hs), np.log(excess), 1)[0]) uniform_rank = max(ranks) <= RANK_CAP defect_closes = excess_slope < -0.10 and rows[-1]["excess"] \ <= FINAL_EXCESS_BAR pole_conditions = all(r["conditioned_bad"] <= 1.0 + BAD_TOL for r in rows) \ and min(r["pole_overlap"] for r in rows) >= POLE_OVERLAP_BAR \ and min(fids) >= DRIFT_FIDELITY_BAR print("\nS2 -- preregistered outcome gates") outcome("S2.1 UNIFORM FINITE BAD RANK <= %d" % RANK_CAP, uniform_rank, "ranks=" + "/".join(str(x) for x in ranks)) outcome("S2.2 LARGEST DEFECT CLOSES TO ONE", defect_closes, "excess %.3e -> %.3e, log-log slope %.3f, final bar %.2f" % (rows[0]["excess"], rows[-1]["excess"], excess_slope, FINAL_EXCESS_BAR)) outcome("S2.3 ONE E8/POLE CONDITION IS STABLE AND SUFFICIENT", pole_conditions, "min pole overlap %.4f, min adjacent fidelity %.4f, " "max conditioned-bad lower bound %.9f" % (min(r["pole_overlap"] for r in rows), min(fids), max(r["conditioned_bad"] for r in rows))) if uniform_rank and defect_closes and pole_conditions: verdict = "SONIN-POLE-CONDITIONED" elif uniform_rank: verdict = "SONIN-FINITE-RANK-WALL" else: verdict = "SONIN-RANK-GROWS" print("\nVERDICT: %s" % verdict) if uniform_rank: print("THEOREM STILL NEEDED: prove that these finite Sonin projectors " "form a covariant semi-local inductive system, that the measured " "bad subspace has the same uniform finite dimension for every " "window/support, and that explicit arithmetic boundary " "conditions push all its generalized eigenvalues below one. " "This is the finite-place analogue of the Connes--Consani " "archimedean conditioning step; the present computation alone " "does not provide the global quantifier.") else: print("KILL: the chosen canonical one-cell Sonin compression does not " "reduce the TFPT remainder to uniformly few directions. A " "different cutoff may be studied only with a new preregistered " "geometric reason; tuning U0/T0 after this result is forbidden.") elapsed = time.time() - t0 if FAILS: print("RESULT: %d/%d construction checks passed; FAILURES %s (%.1fs)" % (len(CHECKS) - len(FAILS), len(CHECKS), ",".join(FAILS), elapsed)) return 1 print("RESULT: ALL %d CONSTRUCTION CHECKS PASSED (%.1fs)" % (len(CHECKS), elapsed)) return 0 if __name__ == "__main__": raise SystemExit(run())