#!/usr/bin/env python3 # -*- coding: utf-8 -*- r"""v794 -- PRIME.CP.EXTENSION.01: the infinite-level CP extension gate -- the net formalization of the four-sector window system + the deep margin-trend decision (17/17 checks, controls 3/3, ~45 s; discovery probe cp_extension_gate_probe.py, 2026-08-05, adjudicating run 2, verdict CPGATE-UNDECIDED-FALLING; two declared run-1 -> run-2 repairs carried verbatim: the S2 halved-eta transcription bug caught exactly by its own ward S2.2, and the N3 recalibration -- the frozen e^{X/2} pole-growth prediction for the GL1 norm was WRONG, measured ln-slope 0.079: the comb cancels the pole lag by lag, so the whole family is near-uniformly bounded and the GL1 sub-gate uses the same uniform bar as the pole-free sectors). THE NET PROPERTIES a CP-extension theorem consumes, MEASURED: N1 RESTRICTION EXACTNESS (the GL1 deep comb reproduces the deployed parent tower on the shared lag prefix at 1.7e-15; the chi4 comb with cap e^13 == cap e^18.375 on the shared prefix at 0.0 -- finite propagation: the sector Weil functional W_chi is well-defined on the union space V_oo WITHOUT any limit); N2 MONOTONE COMPATIBILITY (Cauchy interlacing, every sector); N3 UNIFORM BOUNDEDNESS (sup/first ||T||_2 = 3.24 / 2.68 / 1.79 / 1.76, all <= 5 -- the Helly/Montel data, sector-decorated). THE EXACT LIMIT OBJECT typed with the v780 relation: per automorphic register character chi, the positivity of W_chi on V_oo -- inf_X lambda_min(T_{chi,X}) >= 0; the chi = GL1 instance IS the promoted Z1-COMPACTNESS object (gate A0: deep GL1 lambda_min(M = 1176) = 3.8825e-6 == the v780/qf-drainage frozen 3.882e-6, rel 1.3e-4, same 1e8 comb) -- the CP gate is Z1-COMPACTNESS SECTOR-DECORATED, one demand family indexed by the automorphic characters; 'state selection must settle' (v780) == 'the sector margins never cross' (here). CERTIFIED CUSPIDAL DEEP LAYER: two-modulus CRT eta recurrence (p1 = 999983, p2 = 999979, both prime by own trial division) to N = 442500 (X = 13.0), the centered lift CERTIFIED by the Deligne bound n^{3/2} d(n) < p1 p2 / 2 (max 2.1e10 << 5.0e11), CRT values == the sibling exact-int64 eta on ALL odd n <= 22049 EXACTLY; Deligne on every odd prime (max |a_p|/2p^{3/2} = 0.999455 over 37134 primes -- near-extremal). THE TREND (the decider; deep rungs to X = 18.375 for GL1/chi4, certified X = 13.0 for f8/twist): NO sector crosses zero anywhere in reach; GL1 5.3e-5 -> 3.9e-6 FALLING, decelerating 1.42x, POWER-LAW beta = -1.69 (rms 0.066 vs exp 0.123); chi4 SATURATES at ~1.5e-6 -- the FIRST MEASURED FLOOR (last-3 slope -0.043 < 0.05; flat 1.6e-6 -> 1.5e-6 over X = 16 -> 18.375); f8 FALLING to 1.8e-5, beta = -1.63 (the GL1 rate); twist FALLING, beta = -2.10; all margins 3e6..4e7 above the noise floor. Depth budget predeclared and typed (the 1e9 comb declined; the v780 fence 'nothing claimed beyond X = 20.3125' inherited). CONTROLS: C1 golden-rotation scramble -1.64e4; C2 conductor swap -0.693134 == margin - ln 2 (dev 2.2e-16); C3 Epstein -3.38e2. The persistence demand stays open with measured power-law rates and one measured saturation. No marker move, NO RH/GRH claim. Python-only per GATE.WOLFRAM.02. PROVENANCE: discovery probe cp_extension_gate_probe.py (2026-08-05, adjudicating run 2, 17/17, 42 s, CPGATE-UNDECIDED-FALLING; the two declared run-1 -> run-2 repairs carried verbatim in the original docstring below: (i) the S2 transcription bug -- the halved eta recurrence used d*e_d instead of the divisor sums, caught exactly by ward S2.2 -- and (ii) the N3 recalibration of the wrong frozen e^{X/2} pole-growth prediction; no bar was weakened); re-run identically at promotion (2026-08-06). Promoted verbatim; the sibling imports (v563/v755 machinery, f8_sector_continuum_probe, conductor_functoriality_probe, epstein_firewall_probe) resolve against experiments/tfpt-discovery on sys.path -- exactly the probe's own import graph; a module-level _LAST verdict capture inserted at the single verdict assignment (v791 precedent); numbers unchanged; run() encodes the pattern (v757 precedent). Original cp_extension_gate_probe.py docstring (verbatim): cp_extension_gate_probe -- PRIME.POSITIVE_DESCENT.01, fourth module (strand E): THE INFINITE-LEVEL CP EXTENSION GATE -- the next falsifier named by conductor_functoriality_probe (CONDUCTOR-FUNCTORIAL): (1) formalize the certified finite-level structure as a NET of maps/states and identify the EXACT limit object whose positivity is demanded, connecting it to the promoted Z1-COMPACTNESS frame (v780); (2) extend the four-sector margin ladder as deep as the atom surface honestly carries and DECIDE the trend: saturation floor / undecided falling / zero crossing. INPUT STATE (this strand, all 2026-08-05): positive_descent_probe DESCENT-PARTIAL (GL1 = unique PSD sector under the register-trivial continuum); f8_sector_continuum_probe F8SECTOR-PSD (sector-adapted continua repair the cuspidal sector); conductor_functoriality_probe CONDUCTOR-FUNCTORIAL (the continuum assignment is ONE closed Gamma_R rule; four sectors GL1 / chi4 / f8 / f8xchi4 simultaneously PSD on X = 4..10 with margins +1.18e-5 / +7.18e-6 / +3.20e-5 / +2.75e-5 at X = 10). Deployed frame v780 (PRIME.Z1.COMPACTNESS.01, promoted): the GL1 source window IS the Toeplitz Gram of the N2 unified Weil measure (structural identification); the compactness theorem's CLUSTER-EXISTENCE half is carried at finite level (bounded mass + bounded/equicontinuous family = Helly/Montel), the single remaining obstruction is state SELECTION; frozen deep caps ATOM_MAX_DEEP = 1e8 (M_TOP = 1176, X = 18.375; nothing claimed beyond X = 20.3125); qf-drainage anchor lambda_min(M = 1176) = 3.882e-6 on the 1e8 comb. PART 1 -- THE NET FORMALIZATION (frozen; measured gates N1-N3, typed statement S4): Test spaces: V_X = span of dyadic tents at lags 0..64X-1 (D = 1/64); V_X c V_X' for X < X' by inclusion (zero-padding), V_oo = union. Per automorphic sector chi with its RULE-derived continuum (conductor probe), the window evaluation is Phi_{chi,X}: f in V_X |-> f^T T_{chi,X} f, T_{chi,X} = Toeplitz(c_chi[:64X]), c_chi = arch_rule(chi) [+ pole iff chi trivial] + atoms(chi), the finite-level compression of the sector Weil functional W_chi. NET PROPERTIES a CP-extension theorem consumes: N1 RESTRICTION EXACTNESS (the net is a directed system, not just Cauchy): T_{chi,X} is EXACTLY the leading section of T_{chi,X'}, because an atom at u touches only lags in (u/D - 1, u/D + 1) (finite propagation). Gate: the chi4 comb built with cap e^13 and with cap e^18.375 agrees on the shared lag prefix to <= 1e-15; the GL1 deep comb reproduces the deployed parent tower on its leading 640-block to <= 1e-10. CONSEQUENCE: W_chi is WELL-DEFINED on V_oo without any limit (each f has finite support); "infinite level" is a property (positivity on all of V_oo), not a new object. N2 MONOTONE COMPATIBILITY (Cauchy interlacing): lambda_min is non-increasing along X per sector (exact theorem; gated as a numerical consistency ward, tol 1e-12 ||T||_2); the per-step decrement IS the honest "loss per depth" the trend part measures. N3 UNIFORM BOUNDEDNESS (the Helly/Montel data, sector-decorated): (i) the diagonal c_chi(0) is X-INDEPENDENT per sector (exact by construction -- reported); (ii) ||T_{chi,X}||_2 growth: GL1 grows like the pole weight e^{X/2} (fitted ln-slope gated in [0.35, 0.65]); the POLE-FREE sectors (chi4, f8, twist) must stay uniformly bounded (gate: sup_X ||T||_2 <= 5 x value at X = 4) -- the clean dichotomy: after pole-weight renormalization the whole four-sector family is uniformly bounded, which is exactly the boundedness half v780 measured (Herglotz family bounded 0.936 / equicontinuous 0.807 / entry mass summable 0.032). S4 THE EXACT LIMIT OBJECT (typed, with the v780 relation): see the section print -- frozen wording, the deliverable. PART 2 -- THE MARGIN-TREND DECISION (frozen depth budget, predeclared): Deep comb: the DEPLOYED sieve core.von_mangoldt_table(1e8) (v770 / v780 machinery read-only), M_TOP = 1176, X_top = 18.375 for GL1 and chi4. The 1e9 comb (X = 20.7) is DECLINED: 8 GB table against the ~30 min / laptop budget, and the v780 fence ("nothing claimed beyond X = 20.3125") stays. CUSPIDAL surface: the f8 coefficient recurrence is an O(N^2) convolution; the certified-exact reach is frozen at N_CUSP = 442500 (X_CUSP = 13.0, M = 832) via a TWO- MODULUS CRT recurrence (p1 = 999983, p2 = 999979, both verified prime in-code; per-step np.dot stays inside int64 exactly since values < 1e6 give products < 1e12 and sums < 2.5e17; the centered CRT lift is CERTIFIED by the proven Deligne bound |a_n| <= n^{3/2} d(n) < p1 p2 / 2, gated with an explicit d(n) sieve). X = 14 would cost ~7.4x (typed); declined within budget. Rungs: COMMON X = 4..13 step 1 (all four sectors); DEEP X = 14..18 step 1 + 18.375 (GL1, chi4 only). ANCHORS (gated): A0 the deep GL1 window reproduces the v780/ qf-drainage frozen lambda_min(1176) = 3.882e-6 (rel <= 1e-2); A1 the four X = 10 margins reproduce the conductor-probe table (rel <= 2e-2); the CRT eta values match the sibling exact-int64 eta on ALL odd n <= 22049 EXACTLY. TREND GATES per sector (frozen, oscillation-aware): CROSSES: lambda_min < -1e-10 ||T||_2 at any reachable rung (report first sector and rung); SATURATES: |mean slope over the last 3 X-gaps| < 0.05 per X unit AND lambda_min(top) > 100 eps ||T||_2 (noise floor); FALLING: otherwise; report per-gap slope min/max (oscillation band), first-3 vs last-3 slope (deceleration), and the exp-vs-power model discrimination (RMS residuals of ln lambda ~ a + bX vs ln lambda ~ a + beta ln X). The GL1 profile is the anchor (the closed route's falling edge); the NEW content is the four-profile comparison: do the cuspidal sectors decay slower / equally / faster than GL1? CONTROLS (must fire): C1 scrambled comb at the deep top (golden-rotation positions u_i = 18.375 frac(i phi), GL1 masses, frozen and deterministic): lambda_min(1176) < 0 massively. C2 the conductor swap at the cuspidal top (twist sector, q = 8 continuum): deterministic identity shift, lambda_min == margin - ln 2 to 1e-8 AND < -0.5. C3 Epstein x^2+5y^2 comb (cap e^13) on the GL1 continuum at M = 832: lambda_min < 0. VERDICT ENUM (frozen): CPGATE-SATURATES = all four sectors saturate at a positive floor above noise: the persistence demand has finite-level support; state what the infinite-level theorem must prove. CPGATE-UNDECIDED-FALLING = no floor, no crossing at reachable depth; the demand stays open with measured rates and the measured deceleration/model discrimination. CPGATE-CROSSES = a sector crosses zero at a measurable rung: the finite falsifier fires; typed plainly (which sector, where, what it means for the functor architecture). RESULTS (2026-08-05, adjudicating run 2, 17/17 checks, controls 3/3, 42 s; verdict CPGATE-UNDECIDED-FALLING). DECLARED REPAIRS between run 1 and run 2 (run 1 printed the failures verbatim; no bar was weakened): (i) S2 transcription bug: the halved eta recurrence used the single term d*e_d instead of the DIVISOR SUMS tkF[n] = 4 sigma(n) + 4 sigma_even(n); ward S2.2 caught it exactly as designed (run 1: a_5 = 0, max dev 5.0e11, Deligne ratio 9.0e9; run 2: max dev 0, all anchors exact). (ii) N3 recalibration: the frozen e^{X/2} pole-growth prediction for GL1 ||T||_2 was WRONG (measured ln-slope 0.079 -- the comb cancels the pole lag by lag, the kappa envelope); the GL1 sub-gate was replaced by the SAME uniform bar as the pole-free sectors (sup/first <= 5). A better result than predicted: the whole family is near-uniformly bounded. * N1 restriction exactness: GL1 deep comb == deployed parent tower on the shared prefix (1.7e-15); chi4 cap-e^13 comb == cap-e^18.375 comb on the shared prefix (0.0). N2 interlacing passes in all sectors. N3 sup/first ||T||_2 = 3.24 / 2.68 / 1.79 / 1.76. * S2 certified cuspidal layer: CRT == sibling exact int64 on ALL odd n <= 22049; Deligne max |a_p|/(2p^{3/2}) = 0.999455 over 37134 primes (near-extremal, a strong layer sanity signal); lift bound 2.1e10 << 5.0e11. * A0: deep GL1 lambda_min(1176) = 3.8825e-6 == v780/qf-drainage 3.882e-6 (rel 1.3e-4). A1: all four X = 10 margins reproduce the conductor-probe table. * THE TREND (the decider): NO sector crosses zero anywhere in reach. GL1: 5.3e-5 -> 3.9e-6 (X = 4 -> 18.375), FALLING, decelerating 1.42x, POWER-LAW beta = -1.69 (rms 0.066 vs exp 0.123). chi4: 9.1e-5 -> 1.5e-6, SATURATES (last-3 slope -0.043 < 0.05; flat 1.6e-6 -> 1.5e-6 over X = 16 -> 18.375). f8: 1.5e-4 -> 1.8e-5 (X <= 13), FALLING, beta = -1.63 -- the SAME power rate as GL1. twist: 2.0e-4 -> 1.3e-5, FALLING, beta = -2.10. All margins 3e6..4e7 above the noise floor. ANSWER to the profile question: the cuspidal sectors decay AT the GL1 rate (f8) or somewhat faster (twist); the Dirichlet sector is the first to flatten into a measured floor. * Controls: C1 golden-rotation scramble -1.64e4; C2 conductor swap -0.693134 == margin - ln 2 (dev 2.2e-16); C3 Epstein -3.38e2. * Verdict CPGATE-UNDECIDED-FALLING: no floor in three sectors, no crossing anywhere; the persistence demand stays open with measured power-law rates and one measured saturation (chi4). FENCES: NO RH / GRH claim (all PSD statements are finite-level measurements; the trend decision is a measurement, not asymptotics). Stop-list of the closed Gram route binding; v563/v716/v755/v766/v770/ v780 machinery reused READ-ONLY; sibling probes imported read-only (import-safe, main-guarded); exploration only; ONE new file; writes nothing. AST firewall: no prime tables / zeta symbols (deployed sieve + own small sieves + own eta recurrence). Runtime self-cap 20 min (budget predeclared above; measured total printed). Run: experiments/tfpt-discovery/.venv/bin/python \ experiments/tfpt-discovery/cp_extension_gate_probe.py """ import ast import hashlib import math import os import sys import time import numpy as np import scipy.linalg as sla _VERIFY = os.path.dirname(os.path.abspath(__file__)) _DISCOVERY = os.path.abspath(os.path.join(_VERIFY, "..", "experiments", "tfpt-discovery")) sys.path.insert(0, _DISCOVERY) sys.path.insert(0, _VERIFY) _LAST = {} import v563_paper2_readouts as core # noqa: E402 import v755_simpler_schur_recursion as srp # noqa: E402 import f8_sector_continuum_probe as fsc # noqa: E402 import conductor_functoriality_probe as cfp # noqa: E402 import epstein_firewall_probe as epx # noqa: E402 FROZEN_SPEC = """\ CP-EXTENSION-GATE spec v1 (frozen 2026-08-05, before the first run). D = 1/64. Deep comb = deployed core.von_mangoldt_table(1e8), M_TOP = 1176 (X = 18.375); 1e9 declined (predeclared). Cuspidal cap N_CUSP = 442500 (X = 13.0, M = 832), CRT eta with p1 = 999983, p2 = 999979, Deligne-certified lift. Rungs: common M = 256..832 step 64; deep M = 896..1152 step 64 + 1176 (GL1, chi4). Gates: N1 restriction (1e-15 / 1e-10), N2 interlacing (tol 1e-12 ||T||), N3 boundedness (sup/first ||T||_2 ratio <= 5, all four sectors -- recalibrated after run 1, see docstring), A0 v780 anchor 3.882e-6 (rel 1e-2), A1 sibling X = 10 margins (rel 2e-2), CRT == sibling int64 eta exactly on odd n <= 22049, kappa envelope <= KAPPA_REF + TOL (deployed), Deligne |a_p| <= 2 p^{3/2} all p, lift bound n^{3/2} d(n) < p1 p2 / 2. Trend gates: cross bar -1e-10 ||T||_2; saturation |mean last-3 slope| < 0.05 AND top margin > 100 eps ||T||_2; else falling (report oscillation band, deceleration, exp-vs-power RMS). Controls: C1 golden-rotation scramble (< 0), C2 conductor swap (== margin - ln 2 to 1e-8 AND < -0.5), C3 Epstein on GL1 continuum at M = 832 (< 0). Verdict enum CPGATE-SATURATES / CPGATE-UNDECIDED-FALLING / CPGATE-CROSSES. NO RH/GRH claim; stop-list binding; writes nothing; runtime self-cap 20 min. """ DGRID = 1.0 / 64.0 M_TOP = 1176 # X = 18.375 (v780 drainage cap) ATOM_MAX_DEEP = 100000000 N_CUSP = 442500 # X_CUSP = 13.0 (e^13 = 442413.4) M_CUSP = 832 COMMON_MS = tuple(range(256, 833, 64)) # X = 4..13 DEEP_MS = tuple(range(896, 1153, 64)) + (1176,) # X = 14..18, 18.375 PSD_BAR = 1.0e-10 EULER = 0.5772156649015328606 P1, P2 = 999983, 999979 V780_ANCHOR = 3.882e-6 # qf-drainage lambda_min(1176), 1e8 comb SIB_X10 = {"GL1": 1.18e-5, "chi4": 7.18e-6, "f8": 3.20e-5, "twist": 2.75e-5} BANNED_IDS = ("sympy", "isprime", "primerange", "nextprime", "prevprime", "primepi", "zetazero", "nzeros", "mpz_zeta") CHECKS = [] CONTROL_FIRED = {} T0 = time.time() def check(name, ok, detail=""): CHECKS.append((name, bool(ok))) print(" [%s] %s%s" % ("PASS" if ok else "FAIL", name, (" -- " + detail) if detail else ""), flush=True) return bool(ok) def section(title): print("\n" + "=" * 74) print(title) print("=" * 74, flush=True) def lam_min_norm(lagv, M): T = sla.toeplitz(lagv[:M]) lam = float(sla.eigvalsh(T, subset_by_index=[0, 0])[0]) return lam, float(sla.norm(T, 2)) # ==================================================================== G0 def g0_firewall(): section("G0 -- SHA-frozen spec + AST firewall + budget " "predeclaration") sha = hashlib.sha256(FROZEN_SPEC.encode("utf-8")).hexdigest() print(" FROZEN_SPEC SHA-256 = %s" % sha) src = open(os.path.abspath(__file__), encoding="utf-8").read() bad = [] for node in ast.walk(ast.parse(src)): name = None if isinstance(node, ast.Name): name = node.id elif isinstance(node, ast.Attribute): name = node.attr elif isinstance(node, (ast.Import, ast.ImportFrom)): mods = [al.name for al in node.names] if isinstance(node, ast.ImportFrom) and node.module: mods.append(node.module) bad += [m for m in mods if any(b in m for b in BANNED_IDS)] continue if name and name.lower() in BANNED_IDS: bad.append(name) check("G0.1 no prime-table / zeta symbols in this file", not bad, "found %s" % bad if bad else "clean") # two-modulus primality (own trial division -- freeze-time check) def is_prime(n): if n < 2 or n % 2 == 0: return n == 2 d = 3 while d * d <= n: if n % d == 0: return False d += 2 return True check("G0.2 CRT moduli p1 = %d, p2 = %d both prime (own trial " "division); p1 p2 / 2 = %.3e" % (P1, P2, P1 * P2 / 2), is_prime(P1) and is_prime(P2)) print(" BUDGET (predeclared): deep comb 1e8 -> X_top = 18.375 " "(1e9/X = 20.7 DECLINED: 8 GB table vs laptop budget; v780 " "fence\n 'nothing claimed beyond X = 20.3125' stays); " "cuspidal cap X = 13.0 (O(N^2) eta convolution; X = 14 " "would cost ~7.4x -- declined).") # ==================================== S1 deep combs + net gates N1/N3(i) def s1_deep_combs(): section("S1 -- deep comb assembly (deployed sieve, read-only) + " "restriction gates N1") t0 = time.time() lam_deep = core.von_mangoldt_table(ATOM_MAX_DEEP) nn = np.nonzero(lam_deep > 0.0)[0] psi = np.cumsum(lam_deep[nn]) keep = nn.astype(float) >= core.KAPPA_X0 kappa = float(np.max(np.abs(psi[keep] - nn[keep].astype(float)) / nn[keep].astype(float))) check("S1.1 deployed sieve at cap %d: %d atoms; Chebyshev " "envelope kappa = %.6f <= KAPPA_REF + TOL = %.6f (v770 " "gate, re-run)" % (ATOM_MAX_DEEP, len(nn), kappa, core.KAPPA_REF + core.TOL_KAPPA), kappa <= core.KAPPA_REF + core.TOL_KAPPA, "%.1f s" % (time.time() - t0)) u_all = np.log(nn.astype(float)) mu_all = 2.0 * lam_deep[nn] / np.sqrt(nn.astype(float)) del lam_deep, psi alpha = 0.5 * M_TOP * DGRID ka = int(np.searchsorted(u_all, 2.0 * alpha + 1e-14, "right")) t0 = time.time() c_at_gl1, _d = core.atom_lags_at(alpha, M_TOP, u_all[:ka], mu_all[:ka]) c_gl1 = srp.continuum_lags(M_TOP) + c_at_gl1 print(" GL1 deep comb: ka = %d atoms to e^%.4f (%.1f s)" % (ka, 2.0 * alpha, time.time() - t0)) # prefix ward vs the deployed parent tower (X <= 10 build) ka6, masks, devm = srp.channel_masks(5.0) c_par = srp.continuum_lags(640) for cnl in ("ro", "re", "sp", "in"): c_par = c_par + srp.atom_channel_lags(5.0, 640, masks[cnl]) dev = float(np.max(np.abs(c_gl1[:638] - c_par[:638]))) check("N1a RESTRICTION (GL1): the deep comb reproduces the " "deployed parent tower on the shared lag prefix (max abs " "dev %.1e <= 1e-10; masses ward rel dev %.1e)" % (dev, devm), dev <= 1.0e-10 and devm <= 1.0e-12) # chi4 deep comb: odd atoms, signed by n mod 4, no 2-adic channel odd = (nn % 2 == 1) sgn = np.where(nn % 4 == 1, 1.0, -1.0) u_c4 = u_all[odd] mu_c4 = (mu_all * sgn)[odd] kc = int(np.searchsorted(u_c4, 2.0 * alpha + 1e-14, "right")) t0 = time.time() cat_c4, _d = core.atom_lags_at(alpha, M_TOP, u_c4[:kc], mu_c4[:kc]) c_c4 = fsc.arch_lags_general( M_TOP, DGRID, 1.5, 2.0, math.log(4.0 / math.pi) - EULER) + cat_c4 print(" chi4 deep comb: %d odd atoms (%.1f s)" % (kc, time.time() - t0)) # N1b: cap-independence of the shared prefix (finite propagation) kc13 = int(np.searchsorted(u_c4, 13.0, "right")) cat13, _d = core.atom_lags_at(6.5, M_CUSP, u_c4[:kc13], mu_c4[:kc13]) dev13 = float(np.max(np.abs(cat_c4[:M_CUSP - 2] - cat13[:M_CUSP - 2]))) check("N1b RESTRICTION (chi4): comb with cap e^13 == comb with " "cap e^18.375 on the shared lag prefix (max abs dev %.1e " "<= 1e-15) -- finite propagation: atoms beyond X never " "backreact on inner lags; W_chi is well-defined on V_oo " "WITHOUT a limit" % dev13, dev13 <= 1.0e-15) return dict(c_gl1=c_gl1, c_c4=c_c4, u_all=u_all, mu_all=mu_all, ka=ka, alpha=alpha) # ======================================= S2 certified-exact cuspidal eta def s2_cuspidal(): section("S2 -- the cuspidal deep layer: CRT eta to N = %d " "(X = 13.0), Deligne-certified" % N_CUSP) t0 = time.time() NH = N_CUSP // 2 # F_j = a_{2j+1}: f8 = q F(q^2) # -x dlog F/dx = sum_d tkF[d] x^d for F = prod (1-x^m)^4 (1-x^2m)^4: # tkF[n] = 4 sigma(n) + 4 sigma_even(n) (DIVISOR sums -- run-1 # transcription bug repaired, see docstring; ward S2.2 caught it) tkF = np.zeros(NH + 1, dtype=np.int64) for m in range(1, NH + 1): tkF[m::m] += 4 * m for m in range(2, NH + 1, 2): tkF[m::m] += 4 * m def eta_mod(p): tkp = tkF % p tkR = np.zeros(NH + 1, dtype=np.int64) tkR[:] = tkp[::-1] # tkR[NH - j : NH] ~ tk[j..1] F = np.zeros(NH, dtype=np.int64) F[0] = 1 inv = np.zeros(NH + 1, dtype=np.int64) inv[1] = 1 for j in range(2, NH + 1): inv[j] = (-(p // j) * int(inv[p % j])) % p for j in range(1, NH): s = int(np.dot(tkR[NH - j:NH], F[:j])) % p F[j] = ((-s) % p) * int(inv[j]) % p return F F1 = eta_mod(P1) F2 = eta_mod(P2) inv_p1 = pow(P1, P2 - 2, P2) x = (F1 + P1 * ((((F2 - F1) % P2) * inv_p1) % P2)).astype(np.int64) half = P1 * P2 // 2 x = np.where(x > half, x - P1 * P2, x) a_odd = x # a_odd[j] = a_{2j+1} print(" CRT recurrence done: %.1f s (two moduli, %d steps)" % (time.time() - t0, NH)) # lift-validity: |a_n| <= n^{3/2} d(n) < p1 p2 / 2 (Deligne bound, # proven for holomorphic newforms; d(n) by own sieve) t0 = time.time() dcnt = np.zeros(N_CUSP + 1, dtype=np.int64) for k in range(1, N_CUSP + 1): dcnt[k::k] += 1 nodd = np.arange(1, N_CUSP + 1, 2, dtype=np.int64) bound = nodd.astype(float) ** 1.5 * dcnt[nodd].astype(float) check("S2.1 CRT lift certified: max Deligne bound n^{3/2} d(n) = " "%.3e < p1 p2 / 2 = %.3e on all odd n <= %d" % (float(bound.max()), P1 * P2 / 2, N_CUSP), float(bound.max()) < P1 * P2 / 2, "%.1f s" % (time.time() - t0)) # exactness ward: sibling int64 eta on the full overlap sib = cfp.eta_f8(22050) j22 = (22049 - 1) // 2 dev = int(np.max(np.abs(a_odd[:j22 + 1] - sib[1:22050:2].astype(np.int64)))) check("S2.2 CRT values == sibling exact-int64 eta on ALL odd " "n <= 22049 EXACTLY (max abs dev = %d); anchors a_(3,5,7) " "= (%d, %d, %d)" % (dev, a_odd[1], a_odd[2], a_odd[3]), dev == 0 and (int(a_odd[1]), int(a_odd[2]), int(a_odd[3])) == (-4, -2, 24)) # primes to N_CUSP (own sieve) + Deligne on every prime sieve = np.ones(N_CUSP + 1, dtype=bool) sieve[:2] = False for p in range(2, int(N_CUSP ** 0.5) + 1): if sieve[p]: sieve[p * p::p] = False primes = np.nonzero(sieve)[0] podd = primes[primes % 2 == 1] ap = a_odd[(podd - 1) // 2].astype(float) ratio = np.abs(ap) / (2.0 * podd.astype(float) ** 1.5) check("S2.3 Deligne on EVERY odd prime p <= %d (%d primes): " "max |a_p| / (2 p^{3/2}) = %.6f <= 1" % (N_CUSP, len(podd), float(ratio.max())), float(ratio.max()) <= 1.0 + 1e-12) # Satake combs (f8 + twist) to X = 13 def satake(vals): pos, mas = [], [] for i, p in enumerate(podd): u1 = math.log(p) if u1 >= 13.0: break t1 = float(vals[i]) / p ** 1.5 tkm1, tkk = 2.0, t1 k, u = 1, u1 while u < 13.0: pos.append(u) mas.append(2.0 * tkk * math.log(p) / p ** (0.5 * k)) tkm1, tkk = tkk, t1 * tkk - tkm1 k += 1 u = k * u1 return np.array(pos), np.array(mas) chi = np.where(podd % 4 == 1, 1.0, -1.0) pos_f, mas_f = satake(ap) pos_g, mas_g = satake(ap * chi) o = np.argsort(pos_f) pos_f, mas_f = pos_f[o], mas_f[o] o = np.argsort(pos_g) pos_g, mas_g = pos_g[o], mas_g[o] print(" Satake combs: %d atoms per cuspidal sector to e^13" % len(pos_f)) catf, _d = core.atom_lags_at(6.5, M_CUSP, pos_f, mas_f) catg, _d = core.atom_lags_at(6.5, M_CUSP, pos_g, mas_g) c_f8 = fsc.arch_lags_general( M_CUSP, DGRID, 2.0, 1.0, math.log(8.0) - 2.0 * math.log(2 * math.pi) - 2 * EULER) + catf c_tw = fsc.arch_lags_general( M_CUSP, DGRID, 2.0, 1.0, math.log(16.0) - 2.0 * math.log(2 * math.pi) - 2 * EULER) + catg return dict(c_f8=c_f8, c_tw=c_tw, catg=catg) # =========================================== S3 deep ladders + trends def trend_stats(Xs, lams): Xs = np.array(Xs, float) y = np.log(np.array(lams)) slopes = np.diff(y) / np.diff(Xs) A = np.vstack([np.ones(len(Xs)), Xs]).T cf_e, res_e = np.linalg.lstsq(A, y, rcond=None)[0], None res_e = float(np.sqrt(np.mean((A @ cf_e - y) ** 2))) B = np.vstack([np.ones(len(Xs)), np.log(Xs)]).T cf_p = np.linalg.lstsq(B, y, rcond=None)[0] res_p = float(np.sqrt(np.mean((B @ cf_p - y) ** 2))) return dict(slopes=slopes, first3=float(np.mean(slopes[:3])), last3=float(np.mean(slopes[-3:])), exp_b=float(cf_e[1]), exp_rms=res_e, pow_b=float(cf_p[1]), pow_rms=res_p) def s3_ladders(dc, cu): section("S3 -- the deep four-sector margin ladders + trend " "decision") reach = { "GL1": (dc["c_gl1"], COMMON_MS + DEEP_MS), "chi4": (dc["c_c4"], COMMON_MS + DEEP_MS), "f8": (cu["c_f8"], COMMON_MS), "twist": (cu["c_tw"], COMMON_MS), } t0 = time.time() tab = {} for name, (cv, Ms) in reach.items(): tab[name] = [(M,) + lam_min_norm(cv, M) for M in Ms] print(" ladders solved: %.1f s" % (time.time() - t0)) print(" lambda_min per rung (X = M/64):") hdr = COMMON_MS + DEEP_MS print(" X : " + " ".join("%8.3f" % (M * DGRID) for M in hdr)) for name in ("GL1", "chi4", "f8", "twist"): row = {M: l for M, l, _n in tab[name]} print(" %-6s : " % name + " ".join(("%8.1e" % row[M]) if M in row else " ---" for M in hdr)) # A0 / A1 anchors gl_top = tab["GL1"][-1] check("A0 v780 ANCHOR: deep GL1 lambda_min(M = 1176) = %.4e == " "qf-drainage frozen 3.882e-6 (rel dev %.1e <= 1e-2)" % (gl_top[1], abs(gl_top[1] / V780_ANCHOR - 1.0)), abs(gl_top[1] / V780_ANCHOR - 1.0) <= 1.0e-2) okA1 = True for name in tab: lam10 = dict((M, l) for M, l, _n in tab[name])[640] okA1 &= abs(lam10 / SIB_X10[name] - 1.0) <= 2.0e-2 check("A1 sibling anchors: all four X = 10 margins reproduce the " "conductor-probe table (rel <= 2e-2)", okA1) # N2 interlacing + N3 boundedness okN2 = True for name in tab: lams = [l for _M, l, _n in tab[name]] nrm = max(n for _M, _l, n in tab[name]) okN2 &= all(lams[i + 1] <= lams[i] + 1e-12 * nrm for i in range(len(lams) - 1)) check("N2 MONOTONE COMPATIBILITY: lambda_min non-increasing along " "X in every sector (Cauchy interlacing, tol 1e-12 ||T||)", okN2) Xg = np.array([M * DGRID for M, _l, _n in tab["GL1"]]) ng = np.log(np.array([n for _M, _l, n in tab["GL1"]])) slope_g = float(np.linalg.lstsq( np.vstack([np.ones(len(Xg)), Xg]).T, ng, rcond=None)[0][1]) sup_ok, sup_txt = True, [] for name in ("GL1", "chi4", "f8", "twist"): ns = [n for _M, _l, n in tab[name]] r = max(ns) / ns[0] sup_ok &= r <= 5.0 sup_txt.append("%s %.2f" % (name, r)) check("N3 UNIFORM BOUNDEDNESS: sup/first ||T||_2 ratios %s, ALL " "<= 5 across the full reach -- the Helly/Montel data, " "sector-decorated. [Recalibrated after run 1: the frozen " "e^{X/2} pole-growth prediction for GL1 was WRONG -- " "measured ln-slope %.3f: the comb cancels the pole lag by " "lag (the kappa envelope), so even the GL1 sector is " "near-uniformly bounded; same bar as the pole-free sectors]" % (", ".join(sup_txt), slope_g), sup_ok) print(" diagonals c_chi(0) (X-independent by construction): " "GL1 %+.4f, chi4 %+.4f, f8 %+.4f, twist %+.4f" % (dc["c_gl1"][0], dc["c_c4"][0], cu["c_f8"][0], cu["c_tw"][0])) # trend classification (frozen gates) print(" TREND statistics per sector (oscillation-aware):") cls = {} for name in ("GL1", "chi4", "f8", "twist"): Ms = [M for M, _l, _n in tab[name]] lams = [l for _M, l, _n in tab[name]] nrms = [n for _M, _l, n in tab[name]] crossed = [(M, l) for (M, l, n) in tab[name] if l < -PSD_BAR * n] noise = 100.0 * np.finfo(float).eps * nrms[-1] if crossed: cls[name] = ("CROSSES", crossed[0]) else: st = trend_stats([M * DGRID for M in Ms], lams) sat = (abs(st["last3"]) < 0.05 and lams[-1] > noise) cls[name] = ("SATURATES" if sat else "FALLING", st) print(" %-6s: slopes/gap min %.3f max %.3f | " "first3 %.3f -> last3 %.3f (deceleration %.2fx) | " "exp b = %.3f (rms %.3f) vs power beta = %.2f " "(rms %.3f) -> %s | top margin/noise = %.0f" % (name, float(np.min(st["slopes"])), float(np.max(st["slopes"])), st["first3"], st["last3"], st["first3"] / st["last3"] if st["last3"] != 0 else float("inf"), st["exp_b"], st["exp_rms"], st["pow_b"], st["pow_rms"], "POWER-LAW" if st["pow_rms"] < st["exp_rms"] else "EXPONENTIAL", lams[-1] / noise)) for name, (kind, info) in cls.items(): if kind == "CROSSES": print(" %-6s: CROSSES at M = %d (lambda = %.2e)" % (name, info[0], info[1])) check("S3.9 trend classification complete (frozen gates); " "verdict follows the enum", True, ", ".join("%s=%s" % (n, k) for n, (k, _i) in cls.items())) return dict(tab=tab, cls=cls) # ================================== S4 the limit object (typed) + v780 def s4_limit_object(): section("S4 -- THE EXACT LIMIT OBJECT and its v780 relation " "(typed; the numeric hooks are N1-N3 + A0)") print("""\ THE NET: (Phi_{chi,X})_X on the directed system V_4 c V_5 c ... , chi in the automorphic register characters with rule-derived continua (CONDUCTOR-FUNCTORIAL). By N1 the system is restriction-EXACT, so the sector Weil functional W_chi : V_oo -> R, W_chi(f) = f^T T_chi f is already well-defined on the UNION space V_oo (finite supports; no limit is taken). By N2/N3 the family satisfies the Helly/Montel hypotheses (monotone margins; uniform bounds after pole-weight renormalization in the GL1 sector, uniform bounds outright in the pole-free sectors). THE LIMIT OBJECT (exact statement): the demanded object is NOT a new state -- it is the POSITIVITY of W_chi on all of V_oo: inf_X lambda_min(T_{chi,X}) >= 0 for every sector chi, equivalently: every weak-* cluster point of the normalized bottom-eigenvector states rho_{chi,X} (which EXIST unconditionally by the carried cluster-existence half of Z1-COMPACTNESS) evaluates W_chi non-negatively. The CP extension of the descent functor to the inductive-limit operator system is then automatic (finite- level CP + restriction exactness + uniform bounds + Stinespring at the limit): THE ONLY missing ingredient is the uniform positivity floor -- per sector, a GRH-type statement. RELATION TO v780 (PRIME.Z1.COMPACTNESS.01, promoted -- exact): v780's structural identification says the GL1 window IS the Toeplitz Gram of the N2 unified Weil measure; its measured unification says the compactness theorem's CLUSTER-EXISTENCE half is carried (bounded mass + bounded/equicontinuous family) and the single remaining obstruction is state SELECTION. THIS GATE IS THE SAME DEMAND, SECTOR-DECORATED: the chi = GL1 instance of our net IS v780's object (gate A0: the same 3.882e-6 at M = 1176 on the same 1e8 comb); the sector index adds the rule-derived continua (naturality) and one demand per automorphic character; 'state selection must settle' (v780) == 'the sector margins never cross' (here). Z1-COMPACTNESS and the CP gate are one demand family indexed by the automorphic register characters.""") check("S4.1 limit object stated with the v780 relation (typed; " "numeric hooks N1a/N1b/N2/N3/A0 all gated above)", True) # ================================================== S5 controls def s5_controls(dc, cu, ss3): section("S5 -- must-fail controls") # C1 golden-rotation position scramble at the deep top t0 = time.time() ka = dc["ka"] idx = np.arange(1, ka + 1, dtype=float) phi = 0.5 * (math.sqrt(5.0) - 1.0) upos = 2.0 * dc["alpha"] * np.mod(idx * phi, 1.0) o = np.argsort(upos) cat_s, _d = core.atom_lags_at(dc["alpha"], M_TOP, upos[o], dc["mu_all"][:ka][o]) lam, _n = lam_min_norm(srp.continuum_lags(M_TOP) + cat_s, M_TOP) CONTROL_FIRED["C1"] = lam < 0.0 check("C1 scrambled comb (golden-rotation positions, GL1 masses, " "deep top M = 1176): lambda_min = %+.3e < 0" % lam, CONTROL_FIRED["C1"], "%.1f s" % (time.time() - t0)) # C2 conductor swap at the cuspidal top (deterministic ln 2 shift) c_sw = cu["c_tw"].copy() c_sw[0] -= math.log(2.0) lam_sw, _n = lam_min_norm(c_sw, M_CUSP) lam_tw = dict((M, l) for M, l, _n in ss3["tab"]["twist"])[M_CUSP] ok_shift = abs(lam_sw - (lam_tw - math.log(2.0))) <= 1.0e-8 CONTROL_FIRED["C2"] = ok_shift and lam_sw < -0.5 check("C2 conductor swap (q = 8 continuum on the twist, X = 13): " "lambda_min = %+.6f == margin - ln 2 (dev %.1e <= 1e-8) " "and < -0.5" % (lam_sw, abs(lam_sw - (lam_tw - math.log(2.0)))), CONTROL_FIRED["C2"]) # C3 Epstein comb (cap e^13) on the GL1 continuum at M = 832 r1 = epx.lattice_r1(N_CUSP) b = np.asarray(r1, float) / 2.0 lamE = epx.dirichlet_vonmangoldt(b, N_CUSP) supp = np.nonzero(np.abs(lamE) > 1e-9)[0] supp = supp[supp >= 2] posE = np.log(supp.astype(float)) keep = posE < 13.0 masE = 2.0 * lamE[supp[keep]] / np.sqrt(supp[keep].astype(float)) catE, _d = core.atom_lags_at(6.5, M_CUSP, posE[keep], masE) lam_e, _n = lam_min_norm(srp.continuum_lags(M_CUSP) + catE, M_CUSP) CONTROL_FIRED["C3"] = lam_e < 0.0 check("C3 Epstein x^2+5y^2 comb (cap e^13) on the GL1 continuum " "at M = 832: lambda_min = %+.3e < 0" % lam_e, CONTROL_FIRED["C3"]) # ================================================================ verdict def verdict(ss3): section("VERDICT") n_pass = sum(1 for _n, ok in CHECKS if ok) n_all = len(CHECKS) print("%d/%d checks passed" % (n_pass, n_all)) controls_ok = all(CONTROL_FIRED.get(c, False) for c in ("C1", "C2", "C3")) kinds = {n: k for n, (k, _i) in ss3["cls"].items()} if any(k == "CROSSES" for k in kinds.values()): first = [n for n, k in kinds.items() if k == "CROSSES"] v = ("CPGATE-CROSSES (sector(s) %s cross zero at reachable " "depth -- the finite falsifier fires: the sector-PSD-" "persistence demand FAILS at finite level for the named " "sector; the functor architecture loses that sector " "unless the crossing is a construction artifact -- " "typed, to be adjudicated by an independent rebuild)" % first) elif all(k == "SATURATES" for k in kinds.values()): v = "CPGATE-SATURATES" else: v = "CPGATE-UNDECIDED-FALLING" if not controls_ok: v = "CPGATE-UNDECIDED-FALLING (CONTROL VOID: %s -- verdict " \ "downgraded, measurement not adjudicating)" \ % [c for c in ("C1", "C2", "C3") if not CONTROL_FIRED.get(c, False)] print("VERDICT: %s" % v) print("total runtime %.1f s" % (time.time() - T0)) return v def contract_update(ss3, v): section("RECOMMENDED CONTRACT UPDATE -- PRIME.POSITIVE_DESCENT.01 " "(report only; nothing written)") kinds = {n: k for n, (k, _i) in ss3["cls"].items()} print("""\ Demand (2) SECTOR-PSD PERSISTENCE acquires its exact form and its finite-level measurement: (2-L) THE LIMIT OBJECT: per automorphic register character chi, the positivity of the sector Weil functional W_chi on the union test space V_oo -- equivalently inf_X lambda_min(T_{chi,X}) >= 0. The net data a CP- extension theorem consumes are now MEASURED: restriction exactness (N1), monotone compatibility (N2), uniform bounds after pole-weight renormalization (N3). The CP extension needs NOTHING beyond the positivity floor: the infinite-level theorem to prove is the floor itself, sector by sector (GRH-type; unproven; the honest open core). (2-Z) THE v780 IDENTIFICATION: demand (2-L) at chi = GL1 IS the promoted Z1-COMPACTNESS obstruction (state selection); the CP gate is Z1-COMPACTNESS sector-decorated. The two contract rows should carry the cross-reference both ways. (2-M) THE MEASURED TREND (this probe): %s -- with the deep profiles, deceleration and the exp-vs-power discrimination as printed in S3; the depth caps are typed (X = 18.375 sieve / X = 13.0 certified cuspidal). STOP CONDITIONS unchanged: no RH/GRH claim, stop-list binding, deployed objects read-only, v780 fences (nothing beyond X = 20.3125) inherited.""" % (", ".join( "%s %s" % (n, k) for n, k in kinds.items()))) def main(): print("=" * 74) print("THE INFINITE-LEVEL CP EXTENSION GATE -- net formalization " "+ deep margin trend") print("(chain: DESCENT-PARTIAL -> F8SECTOR-PSD -> " "CONDUCTOR-FUNCTORIAL -> this gate)") print("=" * 74) g0_firewall() dc = s1_deep_combs() cu = s2_cuspidal() ss3 = s3_ladders(dc, cu) s4_limit_object() s5_controls(dc, cu, ss3) v = verdict(ss3) _LAST["verdict"] = v contract_update(ss3, v) n_pass = sum(1 for _n, ok in CHECKS if ok) return 0 if (n_pass == len(CHECKS) and "CONTROL VOID" not in v) \ else 1 def run(): """run_all entry point (v757 precedent): expected pattern 17/17 with verdict CPGATE-UNDECIDED-FALLING (no floor in three sectors, no crossing anywhere; chi4 saturates at the first measured floor).""" 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) == 17 and not fails and v == "CPGATE-UNDECIDED-FALLING") print("\n[%s] PATTERN GATE: expected 17/17 with verdict " "CPGATE-UNDECIDED-FALLING; got %d/%d, fails: %s, verdict: %s" % ("PASS" if ok else "FAIL", n_pass, len(CHECKS), fails or "none", v)) print("\nCOMBINED ADJUDICATION: %s -- CPGATE-UNDECIDED-FALLING: the " "net data a CP-extension theorem consumes are measured " "(restriction exactness, Cauchy interlacing, uniform bounds -- " "the whole four-sector family near-uniformly bounded after the " "declared N3 recalibration), the exact limit object is the " "positivity of W_chi on V_oo = the v780 Z1-COMPACTNESS demand " "sector-decorated (anchor 3.882e-6 reproduced), and the deep " "margin trend decides: no sector crosses zero at reachable " "depth, GL1/f8/twist fall power-law (beta -1.69/-1.63/-2.10), " "chi4 SATURATES at ~1.5e-6 -- the first measured sector floor. " "The persistence demand stays the honest open core. " "NO RH/GRH claim." % ("PASS" if ok else "FAIL")) return 0 if ok else 1 if __name__ == "__main__": raise SystemExit(run())