#!/usr/bin/env python3 # -*- coding: utf-8 -*- r"""v802 -- GNET.MARTINGALE.LIMIT.01: the noncommutative-martingale route to the G_net GNS limit -- its finite hypotheses MEASURED, all met (15/15 checks, controls 4/4, ~7 s; discovery probe gnet_martingale_probe.py, 2026-08-06, verdict MARTINGALE-HYPOTHESES-MET). THE PROGRAMME: replace continuum heuristics by martingale convergence -- if the local-state defects along the certified Haar inductive system are SUMMABLE in trace norm, the limit state exists uniquely by elementary Banach telescoping; the operator-algebra content becomes a typed list of analytic steps with finite evidence in hand. S1 FILTRATION EXACT: Haar isometry V*V = 1 with support coherence iota(win(N,0,l)) inside win(2N,0,2l) EXACT (leak 0); the Watatani quasi-basis explicit at the predeclared depths (185 elements at l = 2, 3041 at l = 3, sum v v* = 4*1); martingale compatibility omega_k o E = omega_k at every level; window data literally level-independent (H_W identical at N = 48 and N = 3072 -- one index-4 expectation for the whole tower). HONEST TYPING: the omega_k are an ALMOST-martingale (approximate, with measured defects); the convergence mechanism is norm summability + dual-space completeness (Umegaki expectations; Cuculescu cited as context, not used). S2 SUMMABLE DEFECTS (load-bearing): delta_k = ||rho_{k+1} - rho_k||_1 geometric with fitted rates 0.250 / 0.263 / 0.265 per doubling for l = 1/2/3 -- ONE POWER BETTER than the 2^{-l} dim-law envelope (reported against 0.5, not forced; bar 0.55); envelope delta_k <= 8 delta_1 2^{-k} quantifies summability (l = 2 tail sum 2.9e-2). S3 TELESCOPING / UNIQUE LIMIT: the Cauchy tail bound holds on every rung against the Richardson extrapolant; the frozen 6-observable battery gaps fall at <= 0.6; extrapolation uniqueness within 2.25 delta_5. S4 SPLIT/NORMALITY PRECURSORS uniform along the tower: sector-resolved cross-window profiles anchored with shrinking tails; determinant witness D_k = det(1 - X X*) >= 0.985 and stabilizing at 0.992756 -- the uniform estimate a normality/split argument consumes. STEPS 5-7 TYPED, NOT EXECUTED (the analytic remainder): GNS reconstruction of omega_inf, local normality (S4 witnesses as the finite input), split property + the mu4 simple-current extension (GATE.METRIC.11) on the limit net. THE QGEO ARCHITECTURAL NOTE (typed, no gate): the open majorant residues (b) 'L2 uniform FH/RH' and (c) 'L3 uniformity' of QGEO.GRIDSUP.01/v745 are NOT needed for limit EXISTENCE (summable state-norm defects suffice) -- the route relocates them into the typed normality step: an architectural reallocation, not a proof of (b)/(c). CONTROLS: the shifted-bond isometry leaks out of the doubled window (0.707 > 0.3 -- the filtration breaks); the Z2-only average has lambda = 1/2 != 1/4; the scrambled sea gives O(1) defects (summability breaks); DECIMATION CONVERGES TO THE WRONG DEGENERATE LIMIT (||rho_dec - 1/16||_1 < 1e-3 vs Haar 0.1+) -- a summable filtration with the wrong bond data forgets the sea: the coherent Haar bond is load-bearing. Every measurement is finite-level evidence FOR the cited continuum steps, never the theorems. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. PROVENANCE: discovery probe gnet_martingale_probe.py (2026-08-06, 15/15, ~6 s, MARTINGALE-HYPOTHESES-MET; construction source + constants SHA-frozen in-run; no spec corrections disclosed); re-run identically at promotion. Promoted verbatim (the probe is function-structured with a guarded main); a module-level _LAST verdict capture inserted at the single verdict print (v791 precedent); numbers unchanged; run() encodes the pattern (v757 precedent). Original gnet_martingale_probe.py docstring (verbatim): G_net continuum strand -- GNET.MARTINGALE.LIMIT.01 (finite part): the Haar ladder as a filtration with index-4 expectations, and the noncommutative-martingale route to the GNS limit net, its finite hypotheses MEASURED. PROGRAMME. Replace continuum heuristics by martingale convergence: if the local-state defect sequence along the certified inductive system is SUMMABLE in the state norm, the limit state exists uniquely by elementary Banach telescoping; the operator-algebra content (GNS, local normality, split, simple-current extension) is then a typed list of analytic steps with finite-level evidence in hand. INPUTS (certified upstream, read-only): * gnet_gns_limit_probe.py (GNS-PRECURSORS-COHERENT): exact Haar isometry tower, exact commuting index-4 expectation squares, geometric state convergence, duality census dim B'/dim A' = 4. * gnet_arf_qsystem_probe.py: the exact dim law d_c(l) = 4^{l-1} + 2^{l-1} cos(pi c/2), obstruction gap 2^{-l}; uniform sector-resolved split bounds; 185-element quasi-basis at l = 2 with sum v v* = 4*1 exact. * GATE.QGEO majorant-lemma state (contracts 2026-08-04/05 notes): residue (a) grid-sup closed (v778); residues (b) "L2 uniform FH/RH [hard-technical]" and (c) "L3 uniformity [medium]" open. THE FILTRATION (step 1, formalized). A_k := iota^k(A(W)) for the fixed base window W = win(48, 0, l) (doubled), iota = the Haar CAR embedding (V_N: e_j -> (e_2j + e_2j+1)/sqrt2, an exact isometry whose one-particle image of win(N,0,l) is EXACTLY win(2N,0,2l): support coherence gate). On the common 2^{2l}-dim representation the tower states are the pullbacks omega_k = rho_k of the level-k chiral sea; the index-4 Watatani expectations E (mu4 average, quasi-basis explicit) are LITERALLY level-independent (window data identical at every N: slim recheck). DEPTH PREDECLARED: quasi-basis explicit at l = 2 (185 elements) and l = 3 (3041 elements); tower depth k <= 6 (N = 48*2^k up to 3072); defect windows l = 1, 2, 3. HONEST TYPING OF "MARTINGALE": the omega_k are an ALMOST-martingale (approximate martingale with measured defects), not an exact one; the exact E-compatibility at every level is the mu4 invariance omega_k o E = omega_k (gated at 1e-8). The convergence mechanism used is norm convergence under summable defects -- elementary completeness of the dual Banach space (telescoping); "martingale" names the filtration + expectation structure (Umegaki conditional expectations; cf. Cuculescu 1971 for the genuinely noncommutative a.e. statements, cited as context, NOT used as a lemma here). PREDECLARED NORM: the state norm on the finite-dimensional algebra = trace norm of the density-matrix difference, ||omega - omega'|| := ||rho - rho'||_1 (dual of the operator norm; = 2 x total variation). All defect gates use this norm. PREREGISTERED GATES (frozen before the first run): S1 FILTRATION EXACT: S1.1 V*V = 1 exact and support coherence ||(1-P_{W'}) V P_W|| = 0 (W' = the doubled window) at N = 48 and 96. S1.2 quasi-basis: 185 elements at l = 2 and 3041 at l = 3, each with sum v v* = 4*1 within 1e-6 (predeclared depth). S1.3 martingale compatibility: max_k ||[C_k(W), H_W]|| < 1e-10 and max_{k, battery} |tr(rho_k (E(a) - a))| < 1e-8 (omega_k o E = omega_k at every level). S1.4 window data (H_W) identical at N = 48 and N = 3072 (exact): one E for the whole tower (the commuting-square system). S2 SUMMABLE DEFECT (load-bearing): delta_k := ||rho_{k+1} - rho_k||_1, k = 0..5, for l = 1, 2, 3. S2.1 geometric tail: r_k = delta_k/delta_{k-1} <= 0.55 for every k >= 1 with delta_{k-1} > 1e-12 (rate <= 1/2 + 10% slack -- the dim-law rate 2^{-l} is the conjectured structural source; the fitted rate is REPORTED against 0.5, not forced). S2.2 envelope: delta_k <= C * 2^{-k} for all k >= 1 with the frozen anchor C := 4 * 2 * delta_1 (i.e. 2 x the k = 1 envelope value; no tail fitting) -- summability quantified. S3 TELESCOPING / UNIQUE LIMIT (l = 2 window): S3.1 tail bound on every rung: ||rho_proxy - rho_k||_1 <= min(2.25 * delta_k, 2 * C * 2^{-k}) + 1e-12 for k = 1..5, rho_proxy = Richardson extrapolant from the last two states (the exact comparison C*2^{-l}/(1 - 1/2) demanded by the programme, with the measured C). S3.2 frozen battery of 6 local observables: the gaps |tr((rho_k - rho_proxy) a)| fall with tail ratio <= 0.6 (floor 1e-12). S3.3 uniqueness of the extrapolation: plain-last vs Richardson proxies differ by <= 2.25 * delta_5. S4 SPLIT/NORMALITY PRECURSORS, uniform in the tower (k = 0..6): S4.1 sector-resolved cross-window profiles nu_cc'(k) (frozen battery of mu4-charge monomials, two disjoint l = 2 windows, sep = 6 base sites) stay <= max(2 max(nu(0), nu(1)), 1e-3) (the corrected two-level anchor of the arf-qsystem run) with shrinking tails. S4.2 determinant witness: D_k = det(1 - X_k X_k*) for the 4x4 cross-covariance block X_k: D_k >= 0.5 for all k and |D_6 - D_5| <= |D_5 - D_4| + 1e-12 (the uniform determinant estimate a normality/split argument consumes). STEPS 5-7 (typed, NOT executed; printed as the contract's analytic remainder): GNS reconstruction of omega_inf; local normality (nuclearity-type input = S4 witnesses); split property (standard inclusion); mu4 simple-current extension (GATE.METRIC.11) on the limit net. Finite evidence for each is named; nothing is claimed. CONTROLS (all must fire): C1 BOND SCRAMBLE: shifted-bond isometry e_j -> (e_{2j+1} + e_{2j+2})/sqrt2 (an isometry, but scale-incoherent): support leakage ||(1-P_{W'}) V_s P_W|| > 0.3 -- the filtration property A_k c A_{k+1} breaks. C2 Z2-ONLY average: lambda = 1/2 != 1/4 -- the expectations are not the index-4 system (martingale structure gone). C3 SCRAMBLED SEA (alternating 0.3/0.7 filling): defects O(1): min(delta_4, delta_5) > 0.05 -- summability breaks. C4 DECIMATION embedding: converges to the DEGENERATE limit: ||rho_dec(6) - 1/16||_1 < 1e-3 while ||rho_haar(6) - 1/16||_1 > 0.1 -- a summable-defect filtration with the WRONG bond data forgets the sea (the coherent Haar bond is load-bearing). RELATION TO GATE.QGEO (typed architectural note, no gate): the majorant-lemma residues (b) L2 uniform FH/RH and (c) L3 uniformity demand UNIFORM POINTWISE (sup-norm, uniform-in-refinement) control of kernel majorants. The martingale route needs NONE of that for the existence/uniqueness of the limit state: summable STATE-NORM defects (this probe's S2) close the limit by completeness of the dual; uniformity enters only later, in the typed normality/split step, where the S4 determinant witnesses are the finite input. The route thus bypasses (b)/(c) for the LIMIT-EXISTENCE step and localizes them in the normality step -- an architectural reallocation, not a proof of (b)/(c). VERDICT ENUM (frozen): MARTINGALE-HYPOTHESES-MET : S1-S4 all pass and all controls fire (the continuum paper's section 4 upgrades from "tightness cited" to "unique limit via summable defects, hypotheses measured"). MARTINGALE-DEAD : the defects are not summable (S2 tail ratios >= 0.9 or defects non-decreasing on the tail) -- the route dies, typed plainly. MARTINGALE-PARTIAL : anything else (steps named); any control failure flagged CONTROL-VOID. HONESTY: every measurement is finite-level evidence FOR the cited continuum steps, never the continuum theorem. No marker moves. Exploration only (tfpt-experiment firewall): NOT wired into run_all.py, no ledger row, no paper edit, no file writes. Run: experiments/tfpt-discovery/.venv/bin/python experiments/tfpt-discovery/gnet_martingale_probe.py """ import hashlib import inspect import sys import time import numpy as np _LAST = {} TOL = 1e-10 SEED = 20260806 N0 = 48 KMAX = 6 RATE_BAR = 0.55 TAIL_FACTOR = 2.25 # >= 1/(1 - RATE_BAR) DET_BAR = 0.5 CTRL_DEFECT_BAR = 0.05 LEAK_BAR = 0.3 CHECKS = [] def check(name, ok, detail=""): CHECKS.append((name, bool(ok))) print(("PASS " if ok else "FAIL ") + name + (" -- " + detail if detail else "")) return ok # ---------------- constructions (identical to the sibling probes) ---- def spower(N, k): P = np.zeros((N, N)) for j in range(N): P[(j + k) % N, j] = (-1.0) ** ((j + k) // N) return P def covariance_occ(N, occ): th = 2 * np.pi * (np.arange(N) + 0.5) / N th = np.mod(th + np.pi, 2 * np.pi) - np.pi th[np.isclose(th, -np.pi)] = np.pi F = np.exp(1j * np.outer(np.arange(N), th)) / np.sqrt(N) return (F * occ) @ F.conj().T def covariance(N): th = 2 * np.pi * (np.arange(N) + 0.5) / N th = np.mod(th + np.pi, 2 * np.pi) - np.pi th[np.isclose(th, -np.pi)] = np.pi return covariance_occ(N, (th < 0).astype(float)) def window(N, p, l): return [(p + i) % N for i in range(l)] + \ [(p + N // 2 + i) % N for i in range(l)] def haar_iso(N): V = np.zeros((2 * N, N)) for j in range(N): V[2 * j, j] = V[2 * j + 1, j] = 1.0 / np.sqrt(2.0) return V def shift_iso(N): """C1 control: shifted-bond isometry (scale-incoherent).""" V = np.zeros((2 * N, N)) for j in range(N): V[(2 * j + 1) % (2 * N), j] = 1.0 / np.sqrt(2.0) V[(2 * j + 2) % (2 * N), j] = 1.0 / np.sqrt(2.0) return V def decim_iso(N): V = np.zeros((2 * N, N)) for j in range(N): V[2 * j, j] = 1.0 return V def jw_ops(n): sm = np.array([[0, 1], [0, 0]], dtype=complex) sz = np.array([[1, 0], [0, -1]], dtype=complex) I2 = np.eye(2, dtype=complex) ops = [] for j in range(n): m = np.array([[1.0]], dtype=complex) for l in range(n): m = np.kron(m, sz if l < j else (sm if l == j else I2)) ops.append(m) return ops def gamma_partial(Hsub, idx, cops): mu, V = np.linalg.eigh(1j * Hsub) dim = cops[0].shape[0] U = np.eye(dim, dtype=complex) for i in range(len(idx)): d = sum(np.conj(V[j, i]) * cops[idx[j]] for j in range(len(idx))) n_i = d.conj().T @ d ev = -1j * mu[i] U = U @ (np.eye(dim) + (ev - 1) * n_i) return U def gaussian_rho(CW, cops): lam, V = np.linalg.eigh(CW) lam = np.clip(lam.real, 0.0, 1.0) dim = cops[0].shape[0] rho = np.eye(dim, dtype=complex) for i in range(len(cops)): d = sum(np.conj(V[j, i]) * cops[j] for j in range(len(cops))) rho = rho @ ((1 - lam[i]) * np.eye(dim) + (2 * lam[i] - 1) * (d.conj().T @ d)) return rho def sector_projs(U): return [sum((1j ** (-q * j)) * np.linalg.matrix_power(U, j) for j in range(4)) / 4 for q in range(4)] def lam_of(B, Efun): A = Efun(B) a, W = np.linalg.eigh((A + A.conj().T) / 2) keep = a > 1e-11 * max(a.max(), 1.0) Ws = W[:, keep] / np.sqrt(a[keep]) M = Ws.conj().T @ B @ Ws return 1.0 / float(np.linalg.eigvalsh((M + M.conj().T) / 2).max().real) def onb_of(P): w, W = np.linalg.eigh((P + P.conj().T) / 2) return W[:, w > 0.5] def quasi_basis(Ps): onbs = [onb_of(P) for P in Ps] dims = [o.shape[1] for o in onbs] vs = [np.eye(Ps[0].shape[0], dtype=complex)] for q in range(4): for qp in range(4): if q == qp or dims[q] == 0 or dims[qp] == 0: continue for a in range(dims[q]): for b in range(dims[qp]): vs.append(np.outer(onbs[q][:, a], onbs[qp][:, b].conj()) / np.sqrt(dims[qp])) return vs def tracenorm(A): return float(np.abs(np.linalg.eigvalsh((A + A.conj().T) / 2)).sum()) def sha_freeze(): fns = [spower, covariance_occ, covariance, window, haar_iso, shift_iso, decim_iso, jw_ops, gamma_partial, gaussian_rho, sector_projs, lam_of, onb_of, quasi_basis, tracenorm] blob = "".join(inspect.getsource(f) for f in fns) blob += repr((TOL, SEED, N0, KMAX, RATE_BAR, TAIL_FACTOR, DET_BAR, CTRL_DEFECT_BAR, LEAK_BAR)) return hashlib.sha256(blob.encode()).hexdigest() # ===================================================================== def main(): t0 = time.time() print("gnet_martingale_probe -- GNET.MARTINGALE.LIMIT.01 finite " "hypotheses\n") print(f"SHA-freeze (construction source + constants): " f"{sha_freeze()[:32]}\n") covs = {} def cov_of(N): if N not in covs: covs[N] = covariance(N) return covs[N] def pullback(iso_fun, cov_fun, kmax): out = [] chain = np.eye(N0) Nk = N0 for k in range(kmax + 1): out.append(chain.T.conj() @ cov_fun(Nk, k) @ chain) chain = iso_fun(Nk) @ chain Nk *= 2 return out Cts = pullback(haar_iso, lambda Nk, k: cov_of(Nk), KMAX) wins = {l: window(N0, 0, l) for l in (1, 2, 3)} cops = {l: jw_ops(2 * l) for l in (1, 2, 3)} rhos = {l: [gaussian_rho(Ct[np.ix_(wins[l], wins[l])], cops[l]) for Ct in Cts] for l in (1, 2, 3)} # ============ S1: the filtration, exact =========================== print("-- S1: the filtration (Haar embeddings + index-4 E) --") ok_iso, ok_supp = True, True for N in (48, 96): V = haar_iso(N) ok_iso &= np.abs(V.T @ V - np.eye(N)).max() < 1e-14 w_small = window(N, 0, 2) w_big = window(2 * N, 0, 4) P = np.zeros(2 * N) P[w_big] = 1.0 leak = np.abs(V[:, w_small] * (1 - P)[:, None]).max() ok_supp &= leak < 1e-14 check("S1.1 Haar isometry exact (V*V = 1) and support coherence " "iota(win(N,0,2)) inside win(2N,0,4) EXACT (leak 0)", ok_iso and ok_supp) HN = spower(N0, N0 // 2) HW = {l: HN[np.ix_(wins[l], wins[l])] for l in (1, 2, 3)} U4 = {l: gamma_partial(HW[l], list(range(2 * l)), cops[l]) for l in (1, 2, 3)} Ps = {l: sector_projs(U4[l]) for l in (1, 2, 3)} ok_qb = True for l, n_exp in ((2, 185), (3, 3041)): vs = quasi_basis(Ps[l]) ind = sum(v @ v.conj().T for v in vs) ok_qb &= len(vs) == n_exp ok_qb &= float(np.abs(ind - 4 * np.eye(4 ** l)).max()) < 1e-6 check("S1.2 Watatani quasi-basis explicit at predeclared depth: " "185 elements (l = 2) and 3041 (l = 3), sum v v* = 4*1", ok_qb) rng = np.random.default_rng(SEED) A_r = rng.normal(size=(16, 16)) + 1j * rng.normal(size=(16, 16)) A_r = (A_r + A_r.conj().T) / 2 c2 = cops[2] batt = [c2[0].conj().T @ c2[0], c2[0].conj().T @ c2[1] + c2[1].conj().T @ c2[0], c2[0].conj().T @ c2[2].conj().T + c2[2] @ c2[0], Ps[2][0], Ps[2][2], A_r] def E2(x): return sum(P @ x @ P for P in Ps[2]) dev_c = max(float(np.abs(Ct[np.ix_(wins[2], wins[2])] @ HW[2] - HW[2] @ Ct[np.ix_(wins[2], wins[2])]).max()) for Ct in Cts) dev_e = max(abs(np.trace(r @ (E2(a) - a))) for r in rhos[2] for a in batt) check("S1.3 martingale compatibility at every level: " "[C_k(W), H_W] = 0 and omega_k(E(a)) = omega_k(a)", dev_c < 1e-10 and dev_e < 1e-8, f"devs = {dev_c:.1e}, {dev_e:.1e}") H_big = spower(N0 * 2 ** KMAX, N0 * 2 ** KMAX // 2) w_big = window(N0 * 2 ** KMAX, 0, 2) check("S1.4 window data literally level-independent: H_W identical " f"at N = 48 and N = {N0 * 2 ** KMAX} (one E for the tower)", np.abs(H_big[np.ix_(w_big, w_big)] - HW[2]).max() < 1e-14) # ============ S2: the summable defect ============================= print("\n-- S2: state defects delta_k = ||rho_(k+1) - rho_k||_1 --") deltas = {l: [tracenorm(rhos[l][k + 1] - rhos[l][k]) for k in range(KMAX)] for l in (1, 2, 3)} print(f"{'k':>3} {'N->N*2':>12} {'l=1':>12} {'l=2':>12} {'l=3':>12}" f" {'r(l=2)':>8}") for k in range(KMAX): r = (deltas[2][k] / deltas[2][k - 1]) if k >= 1 else float("nan") print(f"{k:>3} {N0 * 2 ** k:>5}->{N0 * 2 ** (k + 1):<6}" f" {deltas[1][k]:>12.3e} {deltas[2][k]:>12.3e}" f" {deltas[3][k]:>12.3e} {r:>8.3f}") ok_rate, ok_env = True, True for l in (1, 2, 3): d = deltas[l] for k in range(1, KMAX): if d[k - 1] > 1e-12: ok_rate &= d[k] <= RATE_BAR * d[k - 1] C_l = 8 * d[1] # frozen anchor 4*2*delta_1 ok_env &= all(d[k] <= C_l * 2.0 ** (-k) + 1e-15 for k in range(1, KMAX)) rate_fit = {l: (deltas[l][KMAX - 1] / deltas[l][1]) ** (1 / (KMAX - 2)) for l in (1, 2, 3)} check(f"S2.1 geometric tail: every ratio <= {RATE_BAR} (fitted " "rates per doubling: " + ", ".join(f"l={l}: {rate_fit[l]:.3f}" for l in (1, 2, 3)) + " -- vs the dim-law 0.5; reported, not forced)", ok_rate) check("S2.2 envelope delta_k <= C 2^-k with frozen anchor " "C = 8 delta_1: summability quantified " f"(sum of tail defects l=2: {sum(deltas[2][1:]):.3e})", ok_env) s2_ok = ok_rate and ok_env # ============ S3: telescoping / unique limit ====================== print("\n-- S3: telescoping to the unique limit (l = 2) --") r_last = deltas[2][KMAX - 1] / deltas[2][KMAX - 2] rho_last = rhos[2][KMAX] rho_prev = rhos[2][KMAX - 1] rho_proxy = rho_last + (rho_last - rho_prev) * r_last / (1 - r_last) C2m = 8 * deltas[2][1] ok_tail = True for k in range(1, KMAX): lhs = tracenorm(rho_proxy - rhos[2][k]) bar = min(TAIL_FACTOR * deltas[2][k], 2 * C2m * 2.0 ** (-k)) ok_tail &= lhs <= bar + 1e-12 print(f" k={k}: ||rho_proxy - rho_k||_1 = {lhs:.3e} <= " f"min(2.25 delta_k, 2C 2^-k) = {bar:.3e}") check("S3.1 Cauchy tail bound holds on every rung (exact " "comparison with the measured C)", ok_tail) ok_batt = True for a in batt: g = [abs(np.trace((rhos[2][k] - rho_proxy) @ a)) for k in range(KMAX + 1)] for k in range(2, KMAX): if g[k - 1] > 1e-12: ok_batt &= g[k] <= 0.6 * g[k - 1] + 1e-12 check("S3.2 frozen 6-observable battery: gaps to the limit fall " "with tail ratio <= 0.6 (floor 1e-12)", ok_batt) dev_prox = tracenorm(rho_proxy - rho_last) check("S3.3 extrapolation uniqueness: Richardson vs plain-last " f"proxies differ by {dev_prox:.2e} <= " f"2.25 delta_5 = {TAIL_FACTOR * deltas[2][KMAX - 1]:.2e}", dev_prox <= TAIL_FACTOR * deltas[2][KMAX - 1] + 1e-12) print(" TYPED: summable defects => norm-Cauchy => unique limit " "state is ELEMENTARY (dual-space completeness, telescoping);" "\n the operator-algebra content (Umegaki expectations; " "Cuculescu-type a.e. results cited as context) sits in the" "\n GNS/normality steps 5-7 below, typed not executed.") s3_ok = ok_tail and ok_batt \ and dev_prox <= TAIL_FACTOR * deltas[2][KMAX - 1] + 1e-12 # ============ S4: uniform split/normality precursors ============== print("\n-- S4: uniform split bounds + determinant witness --") winB = window(N0, 6, 2) joint = wins[2] + winB c8 = jw_ops(8) ww, WW = np.linalg.eig(HW[2]) order = np.argsort(-np.imag(ww)) ww, WW = ww[order], WW[:, order] i = 0 while i < 4: j = i while j < 4 and abs(ww[j] - ww[i]) < 1e-9: j += 1 Q, _ = np.linalg.qr(WW[:, i:j]) WW[:, i:j] = Q i = j d1 = [sum(np.conj(WW[j, i]) * c8[j] for j in range(4)) for i in range(4)] d2 = [sum(np.conj(WW[j, i]) * c8[4 + j] for j in range(4)) for i in range(4)] def battc(d): return {0: [d[0].conj().T @ d[0], d[0].conj().T @ d[1]], 1: [d[0].conj().T, d[2]], 2: [d[0].conj().T @ d[1].conj().T, d[0].conj().T @ d[2]], 3: [d[0], d[2].conj().T]} b1, b2 = battc(d1), battc(d2) prof = {(c, cp): [] for c in range(4) for cp in range(4)} dets = [] for k in range(KMAX + 1): Cj = Cts[k][np.ix_(joint, joint)] X = Cts[k][np.ix_(wins[2], winB)] dets.append(float(np.linalg.det( np.eye(4) - X @ X.conj().T).real)) rho8 = gaussian_rho(Cj, c8) for c in range(4): for cp in range(4): m = 0.0 for a in b1[c]: for b in b2[cp]: val = abs(np.trace(rho8 @ (a @ b)) - np.trace(rho8 @ a) * np.trace(rho8 @ b)) m = max(m, float(val) / (np.linalg.norm(a, 2) * np.linalg.norm(b, 2))) prof[(c, cp)].append(m) ok_bar, ok_tl, n_zero = True, True, 0 for (c, cp), p in sorted(prof.items()): if max(p) < 1e-12: n_zero += 1 continue bar = max(2 * max(p[0], p[1]), 1e-3) ok_bar &= all(v <= bar for v in p) ok_tl &= abs(p[KMAX] - p[KMAX - 1]) \ <= abs(p[KMAX - 1] - p[KMAX - 2]) + 1e-12 print(f" nu({c},{cp}): " + " ".join(f"{v:.5f}" for v in p)) check(f"S4.1 sector-resolved profiles uniform along the tower " f"(two-level anchor; {16 - n_zero} nonzero, {n_zero} exact " "conservation zeros) with shrinking tails", ok_bar and ok_tl) print(" D_k = det(1 - X X*): " + " ".join(f"{v:.6f}" for v in dets)) ok_det = all(v >= DET_BAR for v in dets) and \ abs(dets[KMAX] - dets[KMAX - 1]) \ <= abs(dets[KMAX - 1] - dets[KMAX - 2]) + 1e-12 check(f"S4.2 determinant witness D_k >= {DET_BAR} uniformly and " "stabilizing (the estimate a normality/split step consumes)", ok_det) s4_ok = ok_bar and ok_tl and ok_det # ============ steps 5-7 typed ==================================== print(""" -- Steps 5-7 (typed, NOT executed): the analytic remainder -- 5. GNS RECONSTRUCTION of omega_inf on the quasi-local algebra (inductive-limit C*-algebra exists by S1 exactness; omega_inf exists uniquely by S2+S3). Finite evidence: this probe. 6. LOCAL NORMALITY of the GNS representation on each A(W) (nuclearity-type estimate; the uniform S4 witnesses -- sector profiles + determinant bound -- are the finite input). 7. SPLIT PROPERTY (standard split inclusion) and the mu4 SIMPLE-CURRENT EXTENSION (GATE.METRIC.11, [B:A] = 4 = |mu4|) on the limit net; finite evidence: S4 + the exact dim law d_c = 4^(l-1) + 2^(l-1) cos(pi c/2) with obstruction gap 2^-l (arf-qsystem run) as the asymptotic-crossed-product precursor. """) # ============ Controls ============================================ print("-- Controls (must fire) --") Vs = shift_iso(48) w_small = window(48, 0, 2) w_big4 = window(96, 0, 4) Pmask = np.zeros(96) Pmask[w_big4] = 1.0 leak = float(np.linalg.norm(Vs[:, w_small] * (1 - Pmask)[:, None], 2)) check(f"C1 fires: shifted-bond isometry leaks out of the doubled " f"window (leak {leak:.3f} > {LEAK_BAR}): the filtration " "A_k c A_(k+1) breaks", leak > LEAK_BAR) v2m = np.zeros(16, dtype=complex) for P in Ps[2]: o = onb_of(P) if o.shape[1]: v2m += o[:, 0] v2m /= np.linalg.norm(v2m) def EZ2(x): UU = U4[2] @ U4[2] return (x + UU @ x @ UU.conj().T) / 2 lam2 = lam_of(np.outer(v2m, v2m.conj()), EZ2) check("C2 fires: Z2-only average has lambda = 1/2 != 1/4 (not the " "index-4 expectation system)", abs(lam2 - 0.5) < 1e-8, f"lambda = {lam2:.10f}") rng3 = np.random.default_rng(SEED + 1) def scr_cov(Nk, k): frac = 0.3 if k % 2 == 0 else 0.7 occ = np.zeros(Nk) occ[rng3.permutation(Nk)[:int(frac * Nk)]] = 1.0 return covariance_occ(Nk, occ) Cts_scr = pullback(haar_iso, scr_cov, KMAX) rho_scr = [gaussian_rho(Ct[np.ix_(wins[2], wins[2])], cops[2]) for Ct in Cts_scr] d_scr = [tracenorm(rho_scr[k + 1] - rho_scr[k]) for k in range(KMAX)] check("C3 fires: scrambled sea gives O(1) defects " f"(delta_4 = {d_scr[4]:.3f}, delta_5 = {d_scr[5]:.3f} > " f"{CTRL_DEFECT_BAR}): summability breaks", min(d_scr[4], d_scr[5]) > CTRL_DEFECT_BAR) Cts_dec = pullback(decim_iso, lambda Nk, k: cov_of(Nk), KMAX) rho_dec = gaussian_rho(Cts_dec[KMAX][np.ix_(wins[2], wins[2])], cops[2]) dev_dec = tracenorm(rho_dec - np.eye(16) / 16) dev_haar = tracenorm(rhos[2][KMAX] - np.eye(16) / 16) check("C4 fires: decimation converges to the DEGENERATE limit " f"(||rho_dec - 1/16||_1 = {dev_dec:.1e} < 1e-3, Haar limit " f"{dev_haar:.3f} > 0.1): the coherent bond is load-bearing", dev_dec < 1e-3 and dev_haar > 0.1) # ============ QGEO note + verdict ================================= print(""" -- GATE.QGEO relation (typed architectural note, no gate) -- The open majorant residues (b) 'L2 uniform FH/RH [hard-technical]' and (c) 'L3 uniformity [medium]' demand uniform POINTWISE majorant control (sup norms, uniform in grid refinement). The martingale route needs only SUMMABLE STATE-NORM defects (S2) for the existence and uniqueness of the limit state -- dual-space completeness, no uniform-in-refinement constant. Uniformity re-enters only in the typed normality/split step (6-7), where the S4 determinant witnesses are the finite input. The route bypasses (b)/(c) for limit existence and localizes them in normality: an architectural reallocation, not a proof of (b)/(c).""") names_ok = dict(CHECKS) ctrl = all(v for n, v in CHECKS if n.startswith("C")) s1_ok = all(v for n, v in CHECKS if n.startswith("S1")) dead = any(deltas[l][k] >= 0.9 * deltas[l][k - 1] - 1e-15 and deltas[l][k - 1] > 1e-12 for l in (1, 2, 3) for k in range(2, KMAX)) if s1_ok and s2_ok and s3_ok and s4_ok and ctrl: verdict = "MARTINGALE-HYPOTHESES-MET" elif dead and not s2_ok: verdict = "MARTINGALE-DEAD" else: verdict = "MARTINGALE-PARTIAL" + ("" if ctrl else " (CONTROL-VOID)") n_pass = sum(1 for _, ok in CHECKS if ok) _LAST["verdict"] = verdict print(f"\n{n_pass}/{len(CHECKS)} checks pass -- VERDICT: {verdict}" f" ({time.time() - t0:.1f} s)") print(f""" FLAGS: S1 = {s1_ok}, S2 = {s2_ok} (fitted rates {rate_fit[1]:.3f}/""" f"""{rate_fit[2]:.3f}/{rate_fit[3]:.3f} per doubling), S3 = {s3_ok}, S4 = {s4_ok}, controls = {ctrl}. Typed: all finite-level evidence FOR the cited continuum steps; the limit-net theorems (GNS, normality, split, extension) remain typed steps 5-7. No marker moves; exploration only.""") return 0 if n_pass == len(CHECKS) else 1 def run(): """run_all entry point (v757 precedent): expected pattern 15/15 with verdict MARTINGALE-HYPOTHESES-MET.""" rc = main() n_pass = sum(1 for _n, ok in CHECKS if ok) fails = [n for n, ok in CHECKS if not ok] v = _LAST.get("verdict", "") ok = (rc == 0 and n_pass == len(CHECKS) == 15 and not fails and v == "MARTINGALE-HYPOTHESES-MET") print("\n[%s] PATTERN GATE: expected 15/15 with verdict " "MARTINGALE-HYPOTHESES-MET; got %d/%d, fails: %s, verdict: %s" % ("PASS" if ok else "FAIL", n_pass, len(CHECKS), fails or "none", v)) print("\nCOMBINED ADJUDICATION: %s -- MARTINGALE-HYPOTHESES-MET: the " "Haar ladder is an exact filtration with one index-4 " "expectation for the whole tower; the state defects are " "summable at measured rate ~0.25 per doubling (one power " "better than the 2^-l envelope), the unique limit follows by " "telescoping and is verified on every rung, and the " "split/normality precursors are uniform (determinant witness " ">= 0.99); decimation converges to the WRONG degenerate limit " "-- the coherent bond is load-bearing; the open QGEO majorant " "residues are relocated into the typed normality step, not " "proven. Steps 5-7 stay typed analytic remainder. " "NO RH claim." % ("PASS" if ok else "FAIL")) return 0 if ok else 1 if __name__ == "__main__": raise SystemExit(run())