#!/usr/bin/env python3 """v717 -- PRIME.MOONSHOT.03: MOONSHOT stage 3 -- the positivity STRUCTURE of the glued object: is the Weil functional a STATE on the groupoid algebra? Question (Hilbert-Polya form of theorem (5), the RH level): a trace functional coming from a genuine unitary representation (vector/KMS state) is automatically positive on f * f~. This probe does NOT try to prove positivity (that would be RH); it TYPES the structure on truncations: which representation the state would come from, what is measurable, and where exactly the gap sits. Surfaces used (declared): * the 5-window family of L1/5b (v563 build == v643 W1 Weil-measure reference), the deployed odd window form B = odd_toeplitz(car+cat) with H^1_0 Galerkin metric Gm; * the v701 KEYSTONE decomposition B = odd_toeplitz(p) + P, p = c + pole_lags, P = -odd_toeplitz(pole_lags) rank-1 PSD; * the stage-1/2 groupoid comb (gaussian sieve + sigma-quotient + log generator; NO prime table in the construction path); * the v668 ground-truth labs verbatim: ihara_ground_truth_probe (Ramanujan graphs: T MUST be a state there) and epstein_firewall_probe (x^2+5y^2, h=2: T must NOT be a state). E3.1 [convolution positivity, exactly posed]: on the test algebra (odd piecewise-linear window functions Phi_V on the half-integer lattice, v643 P1(b); f~ = conj(f(-.))), the Weil functional is T(f * f~) = int int f(v) f(w) K(v - w), K = the unified Weil kernel of stage 2 (2cosh pole - rho arch - atom comb; tent reads == deployed lags at 2.2e-16). EXACT binding identity verified on a battery (window vectors + random f + the generalized minimizer): T(f*f~) = u^T Toeplitz(c) u = 2 V^T odd_toeplitz(c) V with u the odd extension of V -- the discrete convolution algebra evaluation EQUALS the deployed W1 window form (the keystone v701 consistency, bound here at the quadratic-form level). Plus the measured PD margins (the W3 numbers): lambda_min(B) plain and generalized (B, Gm) per window. E3.2a [the natural representation]: (i) CONSTRUCTIVE STATE: Levinson on the unified moment vector p (all reflection coefficients |k| < 1 == Toeplitz(p) PD) exhibits T as the vector state of the GNS/CMV shift representation of the truncated translation flow -- the SAME flow whose scaling limit is the L1 continuum operator (rates measured in L1/M2). Fejer symbol >= 0 (state side) vs RAW symbol sigma_p (may dip negative: the finite-truncation correction profile -- location + size reported). (ii) THE TOWER delta_e COMPARISON: the naive regular-representation vector state of the finite tower + trivial mode, phi_reg(V) = phi_pole(V) + phi_orb(V), phi_pole = 2 K_D e^{-(M-1)D/2} (a_+ . u)^2 (closed rank-1 pole square == the keystone P, the 'Trivialmoden' state), phi_orb = sum_k mu_k Phi_V(log k)^2 (orbit-diagonal, beta = 1/2 GNS amplitudes, geo comb), does NOT equal T: the difference Delta = T - phi_pole - phi_orb = (arch flow term, NSD on the odd sector) + (atom orbit interference minus diagonal) is measured: sign profile, size relative to the W3 margin, structure (the trace-formula boundary = the RH substance). E3.2b [KMS variant]: the degree grading as time evolution (Bost-Connes). Measured Gibbs structure of the unified kernel: ONE global half-density e^{-u/2} across all three sectors -- rho(u) e^{u/2} (1 - e^{-2u}) == 1 (flow partition), 2cosh(u/2) e^{u/2} == e^u + 1 (boundary characters), mu_n sqrt(n) == 2 Lambda_geo(n) (orbit counting), i.e. counting side M(u) = W(u) e^{u/2}; evenness of W (the functional equation) == detailed balance M(-u) = e^{-u} M(u): KMS-beta = 1, the BC critical temperature; the s = 1/2 axis is the symmetric GNS splitting. Sharpness: the half-density exponent is MEASURED (-1/2 exact; -1/4 and -1 misfit reported). The full KMS condition needs the noncommutative BC crossed product -- typed as a missing theorem, no claim. E3.3 [controls]: IHARA lab (Petersen, Ramanujan): trace formula EXACT (integer path counts == spectral side), Toeplitz forms PSD at every depth, the representation EXISTS (finite unitary) -- this is what outcome (i) looks like when the trace formula is a theorem; Levinson rank-saturates at the support size F (states with margins -> 0, the v668 recalibration). Violator prism C16 x K2 breaks at K* = 15. EPSTEIN side (x^2+5y^2, disc -20, h = 2): genus identity exact, Lambda_E has negative sites (first n = 36: the orbit measure itself is NOT positive -- no Gibbs state candidate), and the identical window machinery gives lambda_min << -floor and Levinson BREAKDOWN: T is NOT a state there. The construction measures something. E3.4 verdict enum: STATE-ON-TRUNCATIONS / STATE-PLUS-CORRECTION / NOT-A-STATE, + PRIME.Z1.MOONSHOT.03 contract note + the final ordered list of missing theorems. RADICAL HONESTY: no RH claim in any outcome; outcome (i) only moves the quantifier into the limit. RESULTS (run 2026-08-03, 21/21 PASS; family h = 184, 344, 606, 1027, 1433 with X = 256 .. 262144): * E3.1: binding identity max rel dev 1.0e-12 over 80 battery vectors x 5 windows (two float summation orders); keystone B = odd(p) + P at 2.8e-16, P rank 1 PSD (lam2/lam1 2.9e-11), closed pole square == P-form at 1.5e-10; W3 margins lambda_min(B, Gm) = 2.679e-2 .. 1.516e-3 > 0 on all 5 windows (plain 4.04e-4 .. 6.60e-6), Levinson PD full depth (max |k| = 0.2376, min headroom 1-|k| = 0.76). * E3.2a: Fejer symbol min +1.39e-2 .. >= 0 on all 5 windows (the state side); RAW sigma_p dips NEGATIVE on every window (mins -4.39 .. -11.57 at tau* = 48.8 .. 221.1, window-dependent, band-localized in the zero-ordinate band; PD holds anyway); tower delta_e comparison (windows h = 184/344/606, geo comb X = 29929): sector additivity exact (1.6e-11), phi_pole, phi_orb >= 0 on 48/48, arch origin cell car[0] = 2.7763 > 0 carries the positive Pf/delta_0 mass; Delta has MIXED sign on the battery (negative on 3/48 = exactly the three generalized minimizers) and at the minimizer the conspiracy carries: T = 5.36e-2/1.31e-2/6.81e-3 vs Delta = -2.79/-4.58/-5.32, |Delta|/T = 52/350/781, |Delta|/margin = 104/699/1563 -- the delta_e-regular reading does NOT give T; the GNS/shift state gives T exactly. * E3.2b: Gibbs identities at 2.2e-16 / 5.7e-14 / 1.8e-15; half-density exponent -1/2 measured (residual 8.9e-16; -1/4 misfit 2.577, -1 misfit 5.153); detailed balance M(-u) = e^{-u} M(u) at 1.1e-16 (KMS-beta = 1 = BC critical). * E3.3: Ihara/Petersen trace formula rel dev 2.7e-9 over m <= 39 (integer counts to ~7e6), kstar = None, PSD all K = 2..40, support F = 4, Levinson rank-saturates at depth 4 (|k| -> 1: finite unitary rep); prism violator kstar = 15. Epstein (N = 3721): genus identity exact, Lambda_E has 67 negative sites (first n = 36), window form lambda_min = -1.05e+3 / -4.08e+3 vs floor ~4e-12 (~14 orders), Levinson BREAKDOWN at depth 49/62 -- NOT a state. Both controls calibrate. * E3.4: VERDICT MOONSHOT-STAGE3-STATE-ON-TRUNCATIONS. PROVENANCE: discovery probe moonshot_state_probe.py (2026-08-03, 21/21 PASS, verdict MOONSHOT-STAGE3-STATE-ON-TRUNCATIONS: the Weil functional is a STATE on every truncation -- constructive GNS vector state via Levinson PD (min headroom 0.76); the naive delta_e tower reading fails by |Delta|/W3-margin = 104..1563 (the trace-formula boundary = the localized RH substance); KMS beta = 1 detailed balance measured (half-density exponent -1/2 exact); Ihara control is an exact state, Epstein control breaks (Levinson breakdown depth 49/62)). Promoted verbatim (sibling imports now point at the promoted v716/v668 modules); numbers unchanged. """ import ast import math import os import sys import time import numpy as np _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 (declared surfaces) import v716_moonshot_arch_glue as stage2 # noqa: E402 import v668_ground_truth as ihara # noqa: E402 (control lab, promoted) import v668_ground_truth as eps # noqa: E402 (control lab, promoted) T0 = time.time() CHECKS = [] EPS = float(np.finfo(float).eps) FLOOR_SAFETY = 20.0 N_RAND = 12 # random battery vectors per window SEED = 3 N_THETA = 20000 # symbol grid on (0, pi) E32A_WINS = 3 # tower comparison on the 3 smallest windows BAR_BIND = 1.0e-10 # extension identity: two float summation # orders at M = 2866 (calib 1.0e-12) BAR_KEY = 1.0e-10 # keystone matrix identity (calib 3.6e-14) BAR_RANK1 = 1.0e-10 # P rank-1: lam2/lam1 (calib 2.7e-19) BAR_POLSQ = 1.0e-9 # closed pole square vs P (calib 2.7e-13) BAR_FEJ = -1.0e-9 # Fejer symbol floor BAR_GIBBS = 1.0e-12 # Gibbs sector identities (calib 2.2e-16) BAR_BETA = 1.0e-10 # half-density residual (calib 8.9e-16) BETA_MISFIT = 1.0 # wrong-exponent misfit floor BAR_ATOMBIND = 1.0e-12 # deployed atoms vs counting side BAR_EF = 1.0e-6 # Ihara explicit formula (integer) LEV_HEADROOM = 1.0e-3 # zeta Levinson: min(1 - |k|) floor EPSTEIN_FACTOR = 1.0e3 # |lambda_min| must exceed floor by this BANNED_IDS = ("sympy", "isprime", "primerange", "nextprime", "prevprime", "primepi", "zetazero", "zetas", "nzeros") RNG = np.random.default_rng(SEED) def check(name, ok, detail=""): CHECKS.append((name, bool(ok))) print(" [%s] %s%s" % ("PASS" if ok else "FAIL", name, (" -- " + detail) if detail else "")) return bool(ok) # ------------------------------------------------------------ small helpers def odd_extend(V, M): """Full odd lattice vector u (length M) from the free half V (length h = M/2): u_j = V_j (j < h), u_{M-1-j} = -V_j.""" u = np.zeros(M) h = M // 2 u[:h] = V u[h:] = -V[::-1] return u def toep_quad(c, u): """u^T Toeplitz(c) u via the autocorrelation (discrete convolution algebra route): sum_m r_m c_{|m|}.""" M = len(u) r = np.correlate(u, u, "full") # lags -(M-1) .. (M-1) idx = np.abs(np.arange(-(M - 1), M)) return float(np.dot(r, np.asarray(c)[idx])) def levinson(p): """Levinson-Durbin on the moment vector p (p[0] > 0). Returns (ks, ok_pd, depth): ok_pd iff all |k| < 1 with positive prediction error through full depth; depth = first failure index (or len).""" r = np.asarray(p, float) / float(p[0]) M = len(r) a = np.zeros(M) a[0] = 1.0 e = 1.0 ks = [] for m in range(1, M): acc = r[m] + float(np.dot(a[1:m], r[m - 1:0:-1])) k = -acc / e ks.append(k) if not (abs(k) < 1.0) or e <= 0.0: return np.array(ks), False, m a_new = a.copy() a_new[1:m + 1] += k * a[:m][::-1] a = a_new e *= (1.0 - k * k) return np.array(ks), True, M def symbol_scan(p, fejer): """min over theta in (0, pi) of the (raw | Fejer) window symbol.""" p = np.asarray(p, float) M = len(p) th = np.linspace(1.0e-4, math.pi - 1.0e-4, N_THETA) w = np.ones(M) if fejer: w = 1.0 - np.arange(M) / M vals = (p[0] * w[0] + 2.0 * np.cos(np.outer(th, np.arange(1, M))) @ (p[1:] * w[1:])) i = int(np.argmin(vals)) return float(vals[i]), float(th[i]) def interp_odd(u, M, D, pts): """Phi_V(t) = sum_j u_j tent((t - q_j)/D), q_j = (j-(M-1)/2) D.""" pts = np.asarray(pts, float) idx = pts / D + 0.5 * (M - 1) i0 = np.floor(idx).astype(int) fr = idx - i0 out = np.zeros_like(pts) ok0 = (i0 >= 0) & (i0 <= M - 1) out[ok0] += u[i0[ok0]] * (1.0 - fr[ok0]) ok1 = (i0 + 1 >= 0) & (i0 + 1 <= M - 1) out[ok1] += u[i0[ok1] + 1] * fr[ok1] return out def pole_square_closed(V, M, D): """The closed rank-1 pole square on the odd sector: cp_d = K_D 2cosh(dD/2), K_D = (4/D)(2cosh(D/2) - 2), and with the odd extension u: -u^T Toep(cp) u = 2 K_D e^{-(M-1)D/2} (a+ . u)^2.""" u = odd_extend(V, M) K = (4.0 / D) * (2.0 * math.cosh(0.5 * D) - 2.0) ap = np.exp(0.5 * D * np.arange(M)) A = float(ap @ u) return 2.0 * K * math.exp(-0.5 * (M - 1) * D) * A * A # ------------------------------------------------------------- G0 firewall def g0_firewall(wins): print("\nG0 -- Firewall + deklarierte Flaechen") with open(os.path.abspath(__file__), "r", encoding="utf-8") as fh: tree = ast.parse(fh.read()) names = set() for node in ast.walk(tree): if isinstance(node, ast.Name): names.add(node.id) elif isinstance(node, ast.Attribute): names.add(node.attr) elif isinstance(node, (ast.Import, ast.ImportFrom)): for al in node.names: names.add(al.name.split(".")[0]) hits = sorted(n for n in names for b in BANNED_IDS if b in n.lower()) check("G0.1 AST-Firewall: keine Primtabellen-/Zeta-Ladung im eigenen " "Konstruktionspfad (Kontroll-Labore ihara/epstein deklariert, " "Runde-10-Flaechen unberuehrt)", not hits, str(hits)) fam = ", ".join("h=%d (X=%.0f)" % (w["M"] // 2, math.exp(2.0 * w["alpha"])) for w in wins) check("G0.2 deklarierte 5-Fenster-Familie reproduziert: %s" % fam, len(wins) == 5 and all(w["M"] % 2 == 0 for w in wins)) # stage-2 glue spot-check: own quadrature/closed lags vs deployed w0 = wins[0] dev_ar = max(abs(stage2.tent_read(stage2.rho, d, w0["D"]) + w0["car"][d]) / abs(w0["car"][d]) for d in (5, 50, w0["M"] // 2)) cp0 = stage2.pole_lags_closed(w0["M"], w0["D"]) K = (4.0 / w0["D"]) * (2.0 * math.cosh(0.5 * w0["D"]) - 2.0) dev_cp = float(np.max(np.abs( cp0 - K * 2.0 * np.cosh(0.5 * w0["D"] * np.arange(w0["M"]))) / np.max(np.abs(cp0)))) check("G0.3 Stage-2-Verklebung (Stichprobe): eigene rho-Quadratur == " "deployte Arch-Lags (%.1e), Pol-Lags == K_D 2cosh(dD/2) " "geschlossen (%.1e)" % (dev_ar, dev_cp), dev_ar <= 1.0e-10 and dev_cp <= 1.0e-9) # ------------------------------------------------------------------- E3.1 def e31(wins): print("\nE3.1 -- Faltungs-Positivitaet, exakt gestellt " "(Bindung an die W1-Form)") dev_bind = 0.0 n_batt = 0 for w in wins: M, h = w["M"], w["M"] // 2 c = w["car"] + w["cat"] B = core.odd_toeplitz(c, M) w["B"] = B g = np.zeros(M) g[0], g[1] = 2.0 * w["D"] / 3.0, w["D"] / 6.0 w["Gm"] = core.odd_toeplitz(g, M) batt = [np.eye(h)[j] for j in (0, h // 2, h - 1)] batt += list(RNG.standard_normal((N_RAND, h))) lmg, vmin, _rad = eps.gen_min_eig2(B, w["Gm"]) w["lmin_gen"], w["vmin"] = lmg, vmin batt.append(vmin) w["batt"] = batt for V in batt: u = odd_extend(V, M) t_conv = toep_quad(c, u) # convolution-algebra route t_form = 2.0 * float(V @ (B @ V)) # deployed W1 window form dev_bind = max(dev_bind, abs(t_conv - t_form) / max(1.0, abs(t_form))) n_batt += 1 check("E3.1a KONSISTENZ-IDENTITAET: T(f*f~) via diskreter Faltungs-" "Algebra (Autokorrelation x unifizierte Lags) == deployte " "W1-Fensterform 2 V^T B V, %d Batterie-Vektoren x 5 Fenster" % n_batt, dev_bind <= BAR_BIND, "max rel dev %.1e" % dev_bind) # keystone v701: B = odd(p) + P, P rank-1 PSD; closed pole square dev_key = dev_r1 = dev_sq = 0.0 lam_ratio = 0.0 for w in wins: M = w["M"] p = w["car"] + w["cat"] + w["cp"] w["p"] = p P = -core.odd_toeplitz(w["cp"], M) rec = core.odd_toeplitz(p, M) + P dev_key = max(dev_key, float(np.max(np.abs(rec - w["B"])) / np.max(np.abs(w["B"])))) ev = np.linalg.eigvalsh(P) lam_ratio = max(lam_ratio, abs(ev[-2]) / ev[-1]) dev_r1 = max(dev_r1, -min(0.0, ev[0]) / ev[-1]) for V in w["batt"][:4]: q1 = 2.0 * float(V @ (P @ V)) q2 = pole_square_closed(V, M, w["D"]) dev_sq = max(dev_sq, abs(q1 - q2) / max(1.0, abs(q1))) check("E3.1b KEYSTONE (v701): B == odd_toeplitz(p) + P exakt (dev " "%.1e); P = -odd(pole) ist Rang-1 PSD (lam2/lam1 %.1e)" % (dev_key, lam_ratio), dev_key <= BAR_KEY and lam_ratio <= BAR_RANK1 and dev_r1 <= BAR_RANK1) check("E3.1c Pol-Quadrat GESCHLOSSEN: P-Form == 2 K_D e^{-(M-1)D/2} " "(a+ . u)^2 -- der Trivialmoden-Sektor ist ein Rang-1-" "VEKTORZUSTAND in geschlossener Form", dev_sq <= BAR_POLSQ, "max rel dev %.1e" % dev_sq) # measured PD margins (the W3 numbers) ok_pd = True rows = [] for w in wins: lp = float(np.linalg.eigvalsh(w["B"])[0]) w["lmin_plain"] = lp floor = FLOOR_SAFETY * EPS * (w["M"] // 2) \ * float(np.max(np.abs(w["car"] + w["cat"]))) ok_pd &= (lp > floor) and (w["lmin_gen"] > 0.0) rows.append("h=%d: %.3e/%.3e" % (w["M"] // 2, lp, w["lmin_gen"])) check("E3.1d PD GEMESSEN (W3-Margen, plain/generalisiert): %s" % ", ".join(rows), ok_pd) # Levinson positive-feasibility on p (full depth) ok_lev = True head = 1.0 kmax = 0.0 for w in wins: ks, ok, depth = levinson(w["p"]) ok_lev &= ok if len(ks): kmax = max(kmax, float(np.max(np.abs(ks)))) head = min(head, float(np.min(1.0 - np.abs(ks)))) w["lev_ok"] = ok w["lev_depth"] = depth check("E3.1e Levinson auf p in VOLLER Tiefe PD auf allen 5 Fenstern " "(max |k| = %.4f, min Headroom 1-|k| = %.1e)" % (kmax, head), ok_lev and head >= LEV_HEADROOM) return dev_bind # ------------------------------------------------------------------ E3.2a def e32a(wins, comb): print("\nE3.2a -- die natuerliche Darstellung (Kern)") # (i) constructive GNS/CMV state + symbol profile ok_fej = True rows = [] for w in wins: mn_f, _th_f = symbol_scan(w["p"], fejer=True) mn_r, th_r = symbol_scan(w["p"], fejer=False) w["sym"] = (mn_f, mn_r, th_r) ok_fej &= mn_f >= BAR_FEJ rows.append("h=%d: Fejer %.3e, raw %.3e @ tau %.2f" % (w["M"] // 2, mn_f, mn_r, th_r / w["D"])) check("E3.2a-1 KONSTRUKTIVER ZUSTAND: Levinson-PD (E3.1e) + Fejer-" "Symbol >= 0 auf allen Fenstern -- T ist Vektorzustand " " der GNS/Shift-Darstellung des trunkierten Flusses " "(Skalen-Limes = L1-Operator)", ok_fej, "; ".join(rows[:2]) + " ...") neg_raw = all(w["sym"][1] < 0.0 for w in wins) taus = [w["sym"][2] / w["D"] for w in wins] check("E3.2a-2 ROHES Symbol sigma_p dippt NEGATIV auf jedem Fenster " "(tau* = %.1f .. %.1f, fensterabhaengig im Nullstellen-" "Ordinaten-Band) -- die Trunkierungs-Korrektur ist BAND-" "LOKALISIERT, PD haelt trotzdem (Fejer >= 0): Struktur des " "Differenz-Terms, spektral gelesen" % (min(taus), max(taus)), neg_raw, "raw mins: " + ", ".join("%.2f" % w["sym"][1] for w in wins)) # (ii) tower delta_e comparison on the smallest windows us = np.array(sorted(comb)) mu = np.array([2.0 * comb[int(n)] / math.sqrt(float(n)) for n in us]) lg = np.log(us.astype(float)) stats = [] n_neg = n_tot = 0 ok_struct = True dev_split = 0.0 far_lo, far_hi = 0.0, 0.0 ok_min = True for w in wins[:E32A_WINS]: M, D = w["M"], w["D"] reach = math.exp(2.0 * w["alpha"]) sel = us <= reach + 0.5 half = 0.5 * (M - 1) * D + D seli = sel & (lg <= half) rows_w = [] for V in w["batt"]: nrm = math.sqrt(float(V @ (w["Gm"] @ V))) Vn = V / nrm u = odd_extend(Vn, M) T = 2.0 * float(Vn @ (w["B"] @ Vn)) ppole = pole_square_closed(Vn, M, D) phi = interp_odd(u, M, D, lg[seli]) porb = float(np.dot(mu[seli], phi * phi)) t_ar = toep_quad(w["car"], u) t_at = toep_quad(w["cat"], u) # exact split identities: geometric sectors + keystone dev_split = max( dev_split, abs(T - (t_ar + t_at)) / max(1.0, abs(T)), abs(T - (toep_quad(w["p"], u) + ppole)) / max(1.0, abs(T))) # arch far part (origin Pf/delta_0 cell separated) far = t_ar - w["car"][0] * float(u @ u) far_lo, far_hi = min(far_lo, far), max(far_hi, far) delta = T - ppole - porb ok_struct &= (ppole >= -1.0e-15) and (porb >= -1.0e-15) n_tot += 1 n_neg += int(delta < 0.0) rows_w.append((T, ppole, porb, delta, t_ar, t_at)) w["cmp"] = rows_w Tm, pp, po, dl, ta, tt = rows_w[-1] # generalized minimizer ok_min &= (dl < 0.0) and (abs(dl) / Tm >= 10.0) stats.append("h=%d @min: T=%.3e, phi_pole=%.3f, phi_orb=%.3f, " "Delta=%.3f (|Delta|/T=%.0f, /Marge=%.0f)" % (M // 2, Tm, pp, po, dl, abs(dl) / Tm, abs(dl) / w["lmin_gen"])) check("E3.2a-3 STRUKTUR der delta_e-Zerlegung: exakte Sektor-" "Additivitaet T = T_ar + T_at = odd(p)-Form + phi_pole (max " "dev %.1e); phi_pole, phi_orb >= 0 auf allen %d Batterie-" "Vektoren; Arch-Ursprungszelle car[0] = %.4f > 0 traegt die " "positive Pf/delta_0-Masse (Ferm-Anteil in [%.2f, %.2f])" % (dev_split, n_tot, wins[0]["car"][0], far_lo, far_hi), dev_split <= BAR_BIND and ok_struct) check("E3.2a-4 DIFFERENZ-TERM (die RH-Substanz): der naive " "delta_e-Turm-Zustand liefert T NICHT -- Delta = T - " "phi_pole - phi_orb hat GEMISCHTES Vorzeichen auf der " "Batterie (Delta < 0 auf %d/%d) und am generalisierten " "Minimierer traegt die Konspiration: %s" % (n_neg, n_tot, " | ".join(stats)), ok_min) print(" Typ: Delta = (Arch-Fluss-Spur inkl. Pf/delta_0-" "Ursprungsblock) + (Orbit-INTERFERENZ - Diagonale) -- exakt " "der Rand der fehlenden Spurformel; der delta_e-Vektorzustand " "der Turm-Darstellung liefert T NICHT; der GNS/Shift-Zustand " "liefert T EXAKT (E3.2a-1).") # ------------------------------------------------------------------ E3.2b def e32b(wins, comb): print("\nE3.2b -- KMS-Variante (Bost-Connes-Graduierung)") tg = np.linspace(0.25, 6.0, 231) d1 = float(np.max(np.abs(stage2.rho(tg) * np.exp(0.5 * tg) * (1.0 - np.exp(-2.0 * tg)) - 1.0))) d2 = float(np.max(np.abs(stage2.pole_density(tg) * np.exp(0.5 * tg) - (np.exp(tg) + 1.0)))) check("E3.2b-1 GIBBS-FAKTORISIERUNG, kontinuierliche Sektoren: " "rho(u) e^{u/2} (1-e^{-2u}) == 1 (dev %.1e) und " "2cosh(u/2) e^{u/2} == e^u + 1 (dev %.1e) -- EINE globale " "Halbdichte e^{-u/2}" % (d1, d2), d1 <= BAR_GIBBS and d2 <= BAR_GIBBS) reach2 = math.exp(2.0 * wins[E32A_WINS - 1]["alpha"]) ka = core.atoms_in(wins[E32A_WINS - 1]["alpha"]) nn = np.round(np.exp(np.asarray(core.U_ALL[:ka]))).astype(int) dev_at = max(abs(float(core.MU_ALL[k]) * math.sqrt(float(n)) / 2.0 - comb[int(n)]) for k, n in enumerate(nn)) check("E3.2b-2 GIBBS-FAKTORISIERUNG, Atom-Sektor: mu_n sqrt(n)/2 == " "Lambda_geo(n) (Orbit-Zaehlseite, %d Atome bis X = %.0f)" % (ka, reach2), dev_at <= BAR_ATOMBIND, "max dev %.1e" % dev_at) lgn = np.log(nn.astype(float)) y = np.log(np.asarray(core.MU_ALL[:ka], float) / 2.0) \ - np.log(np.array([comb[int(n)] for n in nn])) mis = {} for s in (-0.5, -0.25, -1.0): mis[s] = float(np.max(np.abs(y - s * lgn))) check("E3.2b-3 SCHAERFE der inversen Temperatur: Halbdichte-Exponent " "GEMESSEN -1/2 (Residuum %.1e); Fehlanpassung bei -1/4: %.3f, " "bei -1: %.3f -- beta = 1/2 in Laengeneinheiten ist Messwert, " "nicht Setzung" % (mis[-0.5], mis[-0.25], mis[-1.0]), mis[-0.5] <= BAR_BETA and mis[-0.25] >= BETA_MISFIT and mis[-1.0] >= BETA_MISFIT) u0 = math.log(2.0) m_p = 2.0 * comb[2] / math.sqrt(2.0) * math.exp(0.5 * u0) m_m = 2.0 * comb[2] / math.sqrt(2.0) * math.exp(-0.5 * u0) db = abs(m_m - math.exp(-u0) * m_p) check("E3.2b-4 KMS-SPIEGELUNG: W gerade (Funktionalgleichung, " "deployte Spiegel-Konvention) == detaillierte Balance " "M(-u) = e^{-u} M(u) der Zaehlseite M = W e^{u/2}: " "KMS-beta = 1 (BC-kritisch), s = 1/2-Achse = symmetrische " "GNS-Aufspaltung (Instanz u = log 2, dev %.1e)" % db, db <= 1.0e-14) print(" Typ: volle KMS-Bedingung braucht das nichtkommutative " "BC-Kreuzprodukt (Zeitentwicklung sigma_t(u_n) = n^{it} u_n) " "-- fehlender Satz, hier nur die Gibbs-Struktur gemessen.") # ------------------------------------------------------------------- E3.3 def e33(wins): print("\nE3.3 -- Kontrollen (Ihara MUSS Zustand sein, Epstein DARF " "keiner sein)") # ---- Ihara lab (v668 conventions, verbatim import) nv_p, ee_p = ihara.petersen_edges() pet = ihara.analyze("PETERSEN", nv_p, ee_p, False, ihara.closed_spectrum("PETERSEN")) ok_psd = all(r["lmin"] >= -r["floor"] for r in pet["rows"]) rel_ef = pet["dev_ef"] / float(np.max(pet["N_path"])) check("E3.3a IHARA/Petersen: Spurformel EXAKT (Pfad- == Spektral-" "Seite, rel dev %.1e ueber m <= 39, Zaehlungen bis %d), " "Toeplitz-Formen PSD fuer alle K = 2..40 (kstar = %s), " "Traeger F = %d" % (rel_ef, int(np.max(pet["N_path"])), pet["kstar"], pet["F"]), rel_ef <= BAR_EF and pet["kstar"] is None and ok_psd) ks, ok, depth = levinson(pet["t"]) check("E3.3b IHARA/Petersen: Levinson RANG-SATURIERT bei Tiefe %d " "(F = %d, |k|->1, e->0): der Zustand kommt aus der ENDLICHEN " "unitaeren Darstellung des Graphen -- so sieht Ausgang (i) " "aus, wenn die Spurformel ein SATZ ist (Margen -> 0 ist das " "erwartete Profil echter Positivitaet, v668)" % (depth, pet["F"]), (not ok) and pet["F"] <= depth <= pet["F"] + 2) nv_r, ee_r = ihara.prism_edges(16) pri = ihara.analyze("PRISM16", nv_r, ee_r, True, ihara.closed_spectrum("PRISM", 16)) check("E3.3c IHARA/Violator (Prisma C16 x K2, nicht-Ramanujan): " "Fensterform bricht bei K* = %s -- das Labor detektiert " "'RH falsch'" % pri["kstar"], pri["kstar"] is not None) # ---- Epstein side (x^2 + 5y^2, disc -20, h = 2) picks = wins[:2] N_E = int(math.exp(2.0 * picks[-1]["alpha"]) + 0.5) chi4, chi5, chi20 = eps.chi_arrays(N_E) b = (eps.divisor_transform(chi20, N_E) + eps.convolution_45(chi4, chi5, N_E)) r1 = eps.lattice_r1(N_E) # genus identity (v668 B-block): r_Q1(n) = sum_{d|n} chi_-20(d) # + (chi_-4 * chi_5)(n), exact ints genus_ok = int(np.max(np.abs(r1[1:] - b[1:]))) == 0 aE = np.zeros(N_E + 1) aE[1:] = r1[1:].astype(float) / float(r1[1]) lam_E = eps.dirichlet_vonmangoldt(aE, N_E) neg = np.where(lam_E < -1.0e-9)[0] check("E3.3d EPSTEIN: Genus-Identitaet r_Q1 vs zeta L_-20 + L_-4 L_5 " "koeffizientenweise exakt bis %d; Lambda_E hat %d NEGATIVE " "Stellen (erste: n = %d) -- das Orbit-Mass selbst ist NICHT " "positiv: kein Gibbs-Zustands-Kandidat" % (N_E, len(neg), int(neg[0]) if len(neg) else -1), genus_ok and len(neg) > 0 and (len(neg) == 0 or neg[0] == 36)) ok_break = True rows = [] for w in picks: reach = math.exp(2.0 * w["alpha"]) idx = np.where(np.abs(lam_E[:int(reach) + 1]) > 1.0e-9)[0] pos = np.log(idx.astype(float)) ms = 2.0 * lam_E[idx] / np.sqrt(idx.astype(float)) A_E, Gm_E, cat_E = eps.window_form(w["alpha"], w["M"], pos, ms, w["car"]) lmE, _v, _rad = eps.gen_min_eig2(A_E, Gm_E) floor = FLOOR_SAFETY * EPS * (w["M"] // 2) \ * float(np.max(np.abs(w["car"] + cat_E))) p_E = w["car"] + cat_E + w["cp"] _ks, okE, depthE = levinson(p_E) ok_break &= (lmE < -EPSTEIN_FACTOR * floor) and (not okE) rows.append("h=%d: lambda_min = %.3e (floor %.1e), Levinson-" "BRUCH bei Tiefe %d" % (w["M"] // 2, lmE, floor, depthE)) zrow = "h=%d zeta-Zwilling: lambda_min = %.3e > 0, Levinson " \ "voll PD" % (w["M"] // 2, w["lmin_gen"]) rows.append(zrow) check("E3.3e EPSTEIN: identische Fenster-Maschinerie -- KEIN Zustand " "auf denselben Trunkierungen (%s) -- die Konstruktion misst " "etwas" % " | ".join(rows), ok_break) # ------------------------------------------------------------------- E3.4 def e34(): print("\nE3.4 -- Verdict + Vertragsnotiz + fehlende Saetze") n_pass = sum(1 for _n, ok in CHECKS if ok) n_tot = len(CHECKS) if all(ok for _n, ok in CHECKS): verdict = "MOONSHOT-STAGE3-STATE-ON-TRUNCATIONS" elif any(ok for n, ok in CHECKS if n.startswith("E3.1d")): verdict = "MOONSHOT-STAGE3-STATE-PLUS-CORRECTION" else: verdict = "MOONSHOT-STAGE3-NOT-A-STATE" print(" Checks: %d/%d PASS" % (n_pass, n_tot)) print(" VERDICT: %s" % verdict) print(""" PRIME.Z1.MOONSHOT.03 (Vertragsnotiz-Update, Etappe 3): T ist auf JEDER Trunkierung des verklebten Objekts ein Zustand -- konstruktiv (Levinson-PD auf p => GNS/Shift-Darstellung des trunkierten Flusses, Zyklenvektor e0; Pol = geschlossener Rang-1- Trivialmoden-Vektorzustand). Positivitaet dort AUTOMATISCH; die RH-Frage ist damit die KONVERGENZ dieser Zustands-Familie im L1-Skalenlimes (Quantor im Limes -- KEIN RH-Claim). Der naive delta_e-Vektorzustand der Turm-Darstellung liefert T NICHT: der Differenz-Term Delta (Arch-Fluss-Spur inkl. positiver Pf/delta_0- Ursprungsmasse + Orbit-Interferenz minus Diagonale) ist die exakte Spurformel-Grenze; am Minimierer |Delta|/T = 52..781, |Delta|/W3-Marge = 104..1563 -- die Konspiration ist die RH-Substanz. KMS: beta = 1-detaillierte-Balance (BC-kritisch) gemessen, s = 1/2 = symmetrische GNS-Aufspaltung; volle KMS braucht das BC- Kreuzprodukt. Kontrollen kalibrieren beide Ausgaenge. FEHLENDE SAETZE nach Etappe 3 (Reihenfolge): (1) SPURFORMEL fuer das verklebte Gruppoid-Objekt (geometrische == spektrale Seite; im Ihara-Labor exakt, hier offen) -- verschiebt Delta auf die Spektralseite. (2) SKALENLIMES-KONVERGENZ der GNS-Zustandsfamilie (Mosco/Norm- Resolvente des CMV/Jacobi-Turms; L1-Raten N^-2 gemessen) -- der Quantor von Ausgang (i). (3) GLEICHMAESSIGE Symbol-Positivitaet sigma_p >= 0 (v701- Kompression) == Grenzwert-Positivitaet == RH-Stufe; auf Trunkierungen nur Fejer >= 0 (gemessen), roh band-negativ. (4) KMS-CHARAKTERISIERUNG auf dem BC-Kreuzprodukt (beta = 1 detaillierte Balance gemessen; modulare Theorie fehlt). (5) E8-ARCH-EINDEUTIGKEIT (welcher unimodulare Traeger verklebt; Stage-2-Kontrafaktuale gemessen, Eindeutigkeits-Satz fehlt). RADIKALE EHRLICHKEIT: kein RH-Claim in keinem Ausgang; auch STATE-ON-TRUNCATIONS verschiebt nur den Quantor in den Limes.""") return verdict def run(): print("MOONSHOT ETAPPE 3 -- Positivitaets-Struktur des verklebten " "Objekts (Zustands-Typung)") wins = stage2.declared_family() g0_firewall(wins) e31(wins) x_geo = int(math.exp(2.0 * wins[E32A_WINS - 1]["alpha"]) + 0.5) comb, _meta = stage2.geo_comb(x_geo) e32a(wins, comb) e32b(wins, comb) e33(wins) verdict = e34() print("\n[%s] %d checks, %.1f s" % (verdict, len(CHECKS), time.time() - T0)) return 0 if all(ok for _n, ok in CHECKS) else 1 if __name__ == "__main__": sys.exit(run())