#!/usr/bin/env python3 # -*- coding: utf-8 -*- r"""v913 -- PRIME.NOGO.SIGNED.ONLY.01: THE SIGNED-AND-ALIGNMENT LOCALIZATION OF THE RESIDUAL OBSTRUCTION -- a NO-GO / TYPING module. WHAT KIND OF MODULE THIS IS (read this first). This module prices ARGUMENT CLASSES for the deployed budget shape. It supplies no positivity, no new interval and no new certificate; it closes no gate, narrows no gate and moves NO marker. It is NOT progress toward a proof -- the honest counter-evidence is carried below and printed by the run -- and it must NOT be counted as evidence for or against the Riemann Hypothesis, nor cited in support of any positivity claim. NO RH CLAIM. PROVENANCE: discovery probe signed_only_nogo_probe.py (note CCCLXXIII, 2026-08-13, 87/87, verdict NOGO-COMPOSITE-VERIFIED, 62.3 s; PROBE_SHA d1c42a055d4b17c7349fddef2ac47e7c49067406005c531f20a53fcf8c9a64a9, SPEC_SHA 495308092be1113576a48613c6c2ca98ec96f41f60120c2e67455ae2ac4a07c6). The probe's frozen specification is embedded BYTE-EXACT below and its SHA-256 is re-derived at gate G0.2 before any evaluation. This module promotes the EXACT, CHEAP, SELF-CONTAINED CORE of that probe verbatim and NOTHING ELSE (~30 s). ======================================================================= THE SETTING (finite, explicitly constructed; nothing asymptotic) ======================================================================= For an index kz let n_kz be the kz-th prime power, alpha = log n_kz, gap = log n_{kz+1} - log n_kz, D_k = gap/8, M = 2*ceil(alpha/D_k + 1) rounded up to even, h = M/2, D = alpha/h, N_h = floor(e^{2 alpha+2D}). With tau_D(t) = (1 - |t|/D)_+ the unit tent, K_D(t) = tau_D(t) - (tau_D(t - D) + tau_D(t + D))/2 is the deployed second-difference kernel. The wall assembly produces per rung h and shift theta the bordered Gram Omega_h = [[n, b^T], [b, B]] with B positive definite, its archimedean companion Omega_0 = [[n_0, b_0^T], [b_0, B]], the comb correction b_c = b_0 - b, n_c = n_0 - n, x_0 = B^-1 b_0, q_0 = b_0^T B^-1 b_0, q_c = b_c^T B^-1 b_c, and the source profile w with b_c = -(1/4) A_2 w. The single missing inequality of the programme is, in its best-conditioned form, (L) int_0^1 [-(1/2) - q_c] dtheta > int_0^1 [q_0 - n_0] dtheta on ONE sign-independently predeclared cofinal family of rungs. These data plus the frame family are the DEPLOYED BUDGET SHAPE. ======================================================================= WHAT THIS MODULE CARRIES (probe section ids kept for traceability) ======================================================================= S1.1-S1.7 THE EXACT KERNEL LEDGER, symbolic in D: int K_D = 0, ||K_D||_inf = 1, ||K_D'||_1 = 4 INDEPENDENTLY of D, ||K_D||_1 = 4D/3, D^-1 int K_D^2 = 2/3, and the exponential pairing in both closed forms, T_D = int tau_D e^{-t/2} = (8/D)(cosh(D/2) - 1) and int K_D e^{-t/2} = -(D/8) T_D^2. The third identity is the load-bearing one: the conversion constant of a magnitude hypothesis does not become cheap as the mesh refines. S2.8-S2.10 THE MAGNITUDE CLASS IS UNCONDITIONALLY EMPTY (the comparison, not Littlewood's theorem, which is CITED): the geometric supply is recomputed here by two independent routes, is NEGATIVE and O(1) (|B| <= 2), while the Littlewood floor 4 log log log N_h is >= 2.1654551 at h = 184 and >= 4.1231791 at h = 12632 -- a ratio of 214.7 against this module's OWN conservative slack and 1527.1 against the deployed one (cited). S2.13 SELBERG'S SYMMETRY FORMULA IS PROVABLY NEUTRAL: Lambda = mu * log and mu * log^2 = Lambda.log + Lambda * Lambda are FORMAL identities in the free module on {log p} (n <= 36), so they carry no arithmetic content into the budget. S3.1-S3.4 THE REQUIREMENT IS SIGNED AND CONGRUENCE-INVARIANT, as pure algebra of the bordered Gram: the three-term split s = (n_0 - q_0) - n_c + 2 - q_c is an identity; all three terms are invariant under EVERY congruence T = diag(1, M); under the comb sign flip b_c -> -b_c the outer terms are EVEN and the middle term is ODD; and the F4 kernel K = A_2^T B^-1 A_2 is itself a congruence invariant. Hence no bound that is flip-invariant -- i.e. no unsigned hull, no window choice, no preconditioner, no congruence reformulation -- can lower-bound an odd quantity. S6.3 THE BEURLING-NYMAN TRANSPORT TARGET, exactly in Q: (T3) lam_min(G_N) <= (log 2pi - gamma)/N, so the Baez-Duarte Gram has NO N-uniform floor; (T5) the explicit family G~ = A A^T + (4/5)^2 I meets EVERY hypothesis of the certificate class at every N while d~^2 >= 1/2 forever -- the class is non-implying. ======================================================================= WHAT STAYS IN experiments/ AND WHY (the boundary is deliberate) ======================================================================= The DEPLOYED MATRIX-STAGE READS -- S3.5-S3.12 (the theta-means of s, the measured negativity of the even terms, the certified sign-flipped counter-world, the growth attribution), S4 (SIGNED_h = PRIME(F) and the negative bar) and S5 (the unsigned companion and its class floor) -- are NOT promoted. They inherit the float64 entry-slack premise of the wall generators (X4) and the cited certified ladder (X1), and they belong in experiments/ until that premise is discharged. Everything gated here is exact-symbolic or recomputed from this module's own sieves, and consumes NEITHER of those two premises. ======================================================================= THE LOCALIZATION (the deliverable, stated as what it is) ======================================================================= Any argument that closes (L) for the deployed budget shape must supply information that is (a) SIGNED -- it must orient a quantity that is ODD under the comb sign flip -- and (b) ALIGNMENT-CARRYING -- it must relate the position of the prime atoms to the spectral directions of K, equivalently the ordinates to the sign pattern of Fhat. In particular it can be neither a magnitude bound on psi(x) - x nor a natural-grammar identity, i.e. an identity valid in every arithmetic world and therefore comb-blind. CLASSES PROVEN EMPTY for this budget shape (E1-E10; [here] = gated by this module, [probe] = carried by the frozen probe in experiments/): E1 every magnitude hypothesis |psi(x)-x| <= f(x) sqrt(x) [here] E2 every zero-density input with A > 2; A = 2 and Lindeloef land exactly critical, zero slack [probe] E3 Selberg's symmetry formula (formal identity) [here] E4 every congruence reformulation, window choice, Jacobi/ Cholesky preconditioner, resolvent split [here] E5 every unsigned hull of the comb correction: the parity/ invariance half [here], the measured (cell, theta) negativity [probe] E6 every bound consuming only {w >= 0, ||w||_inf, ||w||_2, supp w} [probe] E7 exact multiplicative transforms (Moebius/Dirichlet): inertia-preserving, so no orientation is created [probe] E8 natural-grammar identities (de Branges/HB, Abel, squares): world-blind, hence non-discriminating [probe] E9 the Beurling-Nyman/Baez-Duarte transport target: no uniform floor (T3), class non-implying (T5) [here] E10 thin/cofinal Li criteria: explicit counterexample, and every subexponential Li envelope is RH-equivalent [probe] MERELY UNEXPLORED, no emptiness claimed: U1 unconditional statements about ordinate POSITIONS; U2 alignment statements; U3 a different budget shape; U4 the four premises of the reverse implication; U5 global constraints on the source profile beyond the cone. WHAT THE STATEMENT DOES NOT SAY (named open scope): (O1) it does not touch arguments that legitimately consume zero-position information -- the remaining statement IS of that kind, which is not the same as false; (O2) it forbids only bounds inside I_cone, so alignment statements are untouched and are exactly what is needed; (O3) every clause is for the deployed kernel, frame family and predeclared rung family -- another budget shape is outside it; (O4) the reverse implication (inequality => RH) rests on four named premises and is NOT established here; (O5) any new global inequality restricting the admissible source profiles beyond {w >= 0, size, support} changes I_cone and escapes S5. ALSO: the statement is FINITE-RUNG (verified at h = 184/388/839 with the ladder to h = 12632); it makes NO all-h claim and says nothing about rungs outside the built cells. And it does NOT say that RH is unprovable, that the wall route fails, or that no proof exists. ======================================================================= PINNING DISCLOSURE (meta-audit CCCXXXVI lesson, applied a priori) ======================================================================= WHAT THE EXECUTED GATES PIN. S1.1-S1.7 pin the kernel ledger SYMBOLICALLY in D (sympy, exact); S3.1-S3.4 pin the split, the congruence invariance, the parity and the kernel invariance as identities in the matrix-stage SYMBOLS, i.e. for every admissible datum and every invertible M, consuming no deployed number; S2.13 pins the Selberg identities exactly in the free module on {log p}; S2.8-S2.10 pin the frames (rebuilt from this module's own prime-power sieve, h PREDICTED not supplied), the geometric supply by two independent routes (closed form and direct quadrature, agreement 1.6e-50), this module's own conservative slack (own von Mangoldt sieve) and the floor/slack ratio; S6.3 pins the two Beurling-Nyman witnesses exactly in Q. WHAT IS CITED, NOT PINNED (a false input here would NOT fail this module): * X3 THE CLASSICAL LITERATURE. Littlewood's Omega-theorem itself -- that S(N) = sup_{x<=N} |psi(x)-x| x^{-1/2} diverges -- is the citation that makes the magnitude class empty; this module gates only the COMPARISON (floor against supply). Likewise Rosser-Schoenfeld, Montgomery-Vaughan/Brun-Titchmarsh, Schoenfeld, Bombieri-Lagarias, Li's criterion, N(1/2,T) >> T log T and CCCLXIII's deployed constants c_B = 0.5523, sigma <= 0.726909. * X2 THE DEPLOYED SLACK 1.5e-05 .. 2.7e-03. The gate uses this module's OWN, strictly larger and hence conservative, slack; the deployed number appears only as a second, cited ratio. * X1 THE CERTIFIED SCHUR INTERVALS AND WITNESS READS at the nine depths. Used ONLY in the counter-evidence print, never inside a gate. * X4 THE FLOAT64 ENTRY SLACK of the wall generators. NOT consumed anywhere here -- which is exactly why the deployed matrix-stage reads stay in experiments/. * X5/X6/X7 CCCLXVIII's demand totals, CCCLXII's chain measurements, and the n0-normalised margin exponents (the flattening counter-evidence). * THE DEPLOYED IDENTIFICATION. That the deployed wall stage IS a bordered Gram of the shape above at the deployed rungs is the probe's measurement (S3.5-S3.12), not gated here; the algebra gated here is what applies TO it. * THE COMPOSITE THEOREM AS A WHOLE (T1-T5 for the deployed budget shape) is the frozen probe run at the SPEC_SHA above; this module carries its exact core. ======================================================================= COUNTER-EVIDENCE, CARRIED AND NOT SMOOTHED (why this is not progress) ======================================================================= NO-WITNESS STANDS: the certified reads are POSITIVE at all NINE depths -- 0 negatives, 0 straddles -- with deepest certified read 2.79579794131272506e-15 at h = 12632 (cited, X1); this module adds none. The normalised decay FLATTENS with depth on the cited normalisation (n0-normalised upper-bound exponents -3.21092511 globally against -2.67942161 on the deepest step 5746 -> 12632, X7), and the re-derived s/D^2 step 1393 -> 2015 is POSITIVE (+0.50302892), so the normalised drift is NOT monotone; no re-derivable exponent comes near the frozen collapse bar -8.0. A no-go about the TYPE of the missing input is not evidence about its EXISTENCE. Registers PRIME.NOGO.SIGNED.ONLY.01 [O] -- a no-go/typing row with a typed scope, NOT an evidence row and NOT a closure. No marker moves. NO RH claim. Python-only per GATE.WOLFRAM.02. """ import ast import hashlib import math import os import sys import time import numpy as np import sympy as sp from mpmath import mp _HERE = os.path.dirname(os.path.abspath(__file__)) if _HERE not in sys.path: sys.path.insert(0, _HERE) EXPECTED = "NOGO-CORE-VERIFIED" N_CHECKS = 24 PROBE_SPEC_SHA = ("495308092be1113576a48613c6c2ca98" "ec96f41f60120c2e67455ae2ac4a07c6") PROBE_SHA = ("d1c42a055d4b17c7349fddef2ac47e7c" "49067406005c531f20a53fcf8c9a64a9") _VERDICTS = {} CHECKS = [] FAILS = [] # ------------------------------------------------- frozen specification MP_DPS = 50 PP_CAP = 200000 RUNTIME_CAP = 1800.0 # (h, kz): h is a PREDICTION of the rebuild from the prime-power list, # never an input to it FRAME_SPECS = ((184, 9), (388, 55), (839, 43), (1393, 88), (2015, 85), (2607, 131), (2854, 222), (5746, 273), (12632, 569)) SUPPLY_RUNGS = (184, 388) FLOOR_RUNGS = (184, 12632) BN_SIZES = (2, 4, 8, 16, 32, 64) SELBERG_N = 36 RATIO_BAR = 100.0 # floor/slack bar for "class empty" # corpus values this module recomputes and must reproduce (regression # wards on its OWN computation, not inputs to it) WARD_FEJER_B = {184: "-1.2648057425013541047", 388: "-1.3147508968858830784"} WARD_FRAME_REL = 2.098e-02 WARD_LITTLEWOOD = {184: 2.1654551, 12632: 4.1231791} WARD_BN_CONST = 1.2606614015 WARD_BN_CB = 0.5523 # (X3) WARD_BN_SIGMA_CAP = 0.726909 # (X3) WARD_BN_SIGMA_RANGE = (0.4106, 0.4260) WARD_SLACK_CITED = 2.7e-03 # (X2) # cited counter-evidence, printed and never gated # (X1/X7) CITED_DEEPEST_READ = "2.79579794131272506e-15" CITED_EXP_U_GLOBAL = -3.21092510922176011 CITED_EXP_U_DEEP = -2.67942161302143633 CITED_EXP_STEP_POS = 0.50302892 CITED_COLLAPSE_BAR = -8.0 # banned CALL names: no zero data, no eigensolver on any deployed form, # no fit anywhere (the probe's firewall, carried verbatim) AST_BANNED = { "zetazero", "zetazeros", "nzeros", "find_zeros", "eigh", "eigvalsh", "eig", "eigvals", "eigs", "eigsh", "tau", "target_sign", "cached_sign", "polyfit", "curve_fit", "lstsq", "least_squares", } # Embedded BYTE-EXACT from the discovery probe: FROZEN_SPEC is the # probe's own frozen docstring, and its SHA-256 must reproduce # PROBE_SPEC_SHA (gate G0.2). FROZEN_SPEC = """\ signed_only_nogo_probe -- PRIME.NOGO.SIGNED.ONLY.01. EXPLORATION ONLY. This probe proves nothing about RH. It is a machine-checked statement ABOUT the remaining obstruction: a composite no-go/localization theorem that says WHICH KINDS of argument cannot close the deployed wall inequality, and names the kinds that are still logically open. No positivity claim, no RH claim, no promotion, no marker moved, nothing written outside experiments/. =========================================================================== THE THEOREM (SIGNED-ONLY LOCALIZATION) =========================================================================== SETTING (all objects finite and explicitly constructed; nothing here is asymptotic). For an index kz let n_kz be the kz-th prime power, alpha := log n_kz, gap := log n_{kz+1} - log n_kz, D_k := gap/8, M := 2*ceil(alpha/D_k + 1) rounded up to even, h := M/2, D := alpha/h, N_h := floor(exp(2 alpha + 2 D)). Let tau_D(t) := max(0, 1 - |t|/D) be the unit tent and K_D(t) := tau_D(t) - (tau_D(t - D) + tau_D(t + D)) / 2 the second-difference kernel. The deployed wall assembly produces, at each rung h and each shift theta in (0,1), a bordered Gram matrix Omega_h(theta) = [[n, b^T], [b, B]] with B positive definite, its archimedean-only companion Omega_0 = [[n_0, b_0^T], [b_0, B]], the comb correction b_c := b_0 - b, n_c := n_0 - n, x_0 := B^-1 b_0, v := A_2^T x_0, q_0 := b_0^T B^-1 b_0, q_c := b_c^T B^-1 b_c, and the source lag profile w with b_c = -(1/4) A_2 w. The single missing inequality of the programme, in its best-conditioned equivalent form, is (L) int_0^1 [ -(1/2) - q_c ] dtheta > int_0^1 [ q_0 - n_0 ] dtheta on ONE sign-independently predeclared cofinal family of rungs. Write need_h for the right-hand side plus the certified slack, SIGNED_h for -(1/2) , and UNSIGNED_h for q_c >= 0. Call this data, together with the frame family above, the DEPLOYED BUDGET SHAPE. INFORMATION CLASSES. For a rung h define I_mag = every statement of the form |psi(x) - x| <= f(x) sqrt(x) for x <= N_h, f arbitrary (including f's that no theorem supplies), consumed through the kernel K_D; I_box = {B, b_0, n_0 known exactly; |n_c| <= R; |b_c,i| <= R} (entrywise size of the comb correction); I_cone = {w >= 0 entrywise; ||w||_inf <= 2 R_c; ||w||_2^2 <= W2; supp w subset J} (size plus positivity of the von Mangoldt weights, no alignment); I_gram = every congruence reformulation T = diag(1, M), M invertible, of the matrix stage, and every window/preconditioner choice that consumes only I_box or I_cone. THEOREM (composite; each clause is machine-checked below in the stated section, at the rungs h in {184, 388, 839} and along the nine-frame ladder h in {184, 388, 839, 1393, 2015, 2607, 2854, 5746, 12632}). (T0) EXACT KERNEL LEDGER. int K_D = 0, ||K_D||_inf = 1, ||K_D'||_1 = 4 independently of D, ||K_D||_1 = 4D/3, D^-1 int K_D^2 = 2/3, int tau_D e^{-t/2} = T_D = (8/D)(cosh(D/2)-1) and int K_D e^{-t/2} = -T_D (cosh(D/2) - 1) = -(D/8) T_D^2. All six are exact symbolic identities in D. [S1] (T1) THE MAGNITUDE CLASS I_mag IS UNCONDITIONALLY EMPTY, NOT MERELY UNPROVEN. Because the kernel constant C_h that converts a magnitude hypothesis into a bound on (L) satisfies C_h >= ||K_D'||_1 = 4 > 0 while the geometric supply B_h stays O(1), any admissible f must beat S(N) := sup_{x<=N} |psi(x)-x| x^{-1/2}, and Littlewood's unconditional Omega-theorem forces S(N) >= c_0 log log log N -> infinity. The resulting floor 4 log log log N_h is >= 2.1654551 at h = 184 and >= 4.1231791 at h = 12632, against a supply-side slack that this probe recomputes from scratch as <= 1.0087e-02 (its own conservative Fejer route) and <= 2.7e-03 (the deployed route, cited). Hence no magnitude hypothesis whatsoever -- proven, conjectural, or merely hypothetical -- can close (L) for this budget shape. In the same typing: the density hypothesis and Lindeloef land exactly critical (the test scale sits ON the square-root barrier, D_h sqrt(N_h) = gap/4 with gap = O(log N_h)), and Selberg's symmetry formula is provably neutral (it is a formal identity in the free module on {log p}). [S2] (T2) THE RESIDUAL REQUIREMENT IS SIGNED AND CONGRUENCE-INVARIANT. The exact three-term split s = (n_0 - q_0) - n_c + 2 - q_c holds identically; each term is invariant under every congruence T = diag(1, M); the outer terms are EVEN and the middle term is ODD under the comb sign flip b_c -> -b_c; the even terms are measured negative at every audited (cell, theta), so positivity is carried exclusively by the odd term. Since every bound built from I_box or I_cone is invariant under that flip, no such bound can produce a positive lower bound for an odd quantity. The flipped datum is admissible in I_box and is certified NEGATIVE by the deployed routine. Therefore no reformulation, window choice, basis change, or preconditioner in I_gram converts the signed requirement into an unsigned one. [S3] (T3) THE SIGNED RESIDUAL IS EXACTLY A WEIL PRIME TERM, AND THE REMAINING STATEMENT IS A ZERO-SUM INEQUALITY AGAINST A NEGATIVE BAR. With Psi_theta the archimedean tent transform of v and the even test function F(u) := -(1/4) Psi_theta(|u|/D), supported in |u| <= S := (h + 2 theta) D, one has SIGNED_h = PRIME(F) := 2 sum_{n <= e^S} Lambda(n) n^{-1/2} F(log n) EXACTLY, so Weil's explicit formula applies verbatim and (L) becomes sum_rho Fhat(gamma_rho) < POLE(F) + ARCH(F) - need_h, whose right-hand side is measured NEGATIVE. This is the opposite of a Weil-positivity statement: it demands a signed evaluation of Fhat on the actual ordinates. It is not self-contradictory -- Fhat takes both signs on a measured 35-38 percent of the grid -- but it is not implied by any upper bound on |PRIME(F)|. [S4] (T4) THE UNSIGNED COMPANION IS UNCONDITIONALLY BOUNDED BUT CERTIFIED NON-CLOSABLE INSIDE I_cone. q_c = (1/16) w^T K w with the congruence-invariant kernel K := A_2^T B^-1 A_2 admits the unconditional bound (1/16) lam_max(K_J) W2 from Brun-Titchmarsh and Chebyshev alone; but the class floor, ATTAINED at an admissible datum of I_cone, still exceeds need_h by a factor > 1 at every audited rung, with a positive growth exponent. So the obstruction is not the sharpness of the bound: the class itself cannot close. Removing q_c requires the ALIGNMENT of w with the spectral directions of K, which is information outside I_cone. [S5] (T5) CONSEQUENCE (the localization). Any argument that closes (L) for the deployed budget shape must supply information that is (a) SIGNED -- it must orient a quantity that is odd under the comb sign flip -- and (b) ALIGNMENT-CARRYING -- it must relate the position of the prime atoms to the spectral directions of K (or, equivalently, the ordinates to the sign pattern of Fhat). In particular it can be neither a magnitude bound on psi(x) - x (T1) nor a natural-grammar identity, i.e. an identity that holds in every arithmetic world and is therefore comb-blind (S6). [S6] WHAT THE THEOREM DOES NOT SAY (named open scope, S8). It does not say RH is unprovable, nor that the wall route fails, nor that no proof exists. Five argument classes are explicitly NOT covered and remain logically open: (O1) arguments that legitimately consume zero-position information (Weil positivity itself, any explicit-formula argument that inputs an unconditional statement about ordinates); (O2) alignment arguments, i.e. any unconditional statement correlating prime positions with the spectral directions of K or with the sign pattern of Fhat; (O3) arguments outside the enumerated classes I_mag, I_box, I_cone, I_gram -- in particular ones that change the budget shape itself (different kernel, different frame family, non-cofinal or differently predeclared rung families); (O4) the reverse implication of the programme (inequality => RH), which rests on four named premises and is NOT established; (O5) any argument using a global inequality that restricts the domain of the source profile beyond I_cone. TYPING AGAINST THE FROZEN GATE RULE. This is a statement ABOUT the problem, not an independent sign source. It supplies no new positivity, closes no gate, narrows no interval, and moves NO marker. Its value is that it prices four failed attack directions at once and names the two properties any future source must have. =========================================================================== DISCIPLINE =========================================================================== RE-DERIVATION, NOT IMPORT. Every load-bearing number below is recomputed here from a generator, not read from a source probe's conclusion. Concretely: the nine frames are rebuilt from an independent prime-power sieve; the Fejer supply B_h is computed by THREE independent routes (closed form, direct quadrature, and the Fourier route int Fhat Theta / 2pi with an explicit tail bound); the kernel ledger and every congruence/inertia statement are re-proved with sympy in this file rather than calling the source probes' symbolic helpers; the Littlewood floors, the criticality of the density hypothesis, the Selberg neutrality, the Li quadruple identity, the Beurling-Nyman T3/T5 witnesses and the cofinal/tail statement are all re-derived from scratch. The deployed matrix stage (B, b_0, b_c, x_0, w, K) is rebuilt through the deployed generators -- that is the object under discussion, so it must be the same object -- and every derived quantity is then recomputed here. CITED, NOT RE-CHECKED (the honest list; each is flagged in S9). (X1) The certified Schur intervals s_h and the certified witness reads U_h at h in {184, 388, 839, 1393, 2015, 2607, 2854, 5746, 12632} come from CCCLX/CCCLXXI. Reproducing the interval arithmetic at the deep rungs costs ~200 s per read and the whole ladder far exceeds this probe's budget. This probe re-derives the decay EXPONENTS from those cited values together with ITS OWN frames, which is where the flattening claim actually lives. (X2) The deployed slack (1.5e-05 .. 2.7e-03) and the CCCXXXI envelope/alignment diagnostics. This probe computes its OWN, strictly larger and hence conservative, supply-side slack from the Fejer route, and reports both. (X3) The classical constants: Rosser-Schoenfeld psi(x) < 1.03883 x, Montgomery-Vaughan Brun-Titchmarsh, Littlewood's Omega-theorem, Schoenfeld's RH bound, Bombieri-Lagarias Cor. 1(c), Li's criterion, N(1/2,T) >> T log T. These are literature; nothing here re-proves them. (X4) The deployed float64 entry slack of the wall generators, typed by CCCXXXIII/CCCLIX, remains the standing premise of every matrix number. (X5) CCCLXVIII's F0..F4 demand values A/G and the geometry budget G_geo. This probe re-derives the three GROWTH FACTORS that carry the attribution claim (beta^-1, R^2, dimension) exactly, but quotes the total 2.9061 orders against which the shares are normalised. (X6) CCCLXII's de Branges chain measurements and CCCLXIII's deployed constants c_B = 0.5523, sigma <= 0.726909. The two decisive Beurling-Nyman witnesses (T3, T5) are re-derived exactly here. (X7) The n0-NORMALISED margin exponents that carry the "flattening" counter-evidence. The normalisation needs n0 at h = 5746 and 12632, i.e. two cells this probe does not build. The RAW witness exponents ARE recomputed here (and are marginally steeper, the direction less favourable to this probe), as is the certified POSITIVE s/D^2 step that makes the normalised drift non-monotone. DISCLOSED DEVIATIONS. Any recomputed value that does not match its cited counterpart within the frozen ward is printed in S9's deviation table. A deviation in a LOAD-BEARING component forces the verdict NOGO-COMPONENT-FAILED; a deviation in a component the source itself types as diagnostic or refused is disclosed and does not. VERDICT ENUM (frozen). NOGO-COMPOSITE-VERIFIED all load-bearing components re-verified and the composite statement stands as written; NOGO-COMPONENT-FAILED a load-bearing component did not re-verify (a bug in a note, HIGH severity); NOGO-SCOPE-NARROWER the honest weaker statement that survives. NO RH CLAIM. No positivity claim. No promotion. No marker moved. """ def check(name, ok, detail=""): CHECKS.append((name, bool(ok))) if not ok: FAILS.append(name.split()[0]) 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 rel(a, b): b = float(b) return abs(float(a) - b) / max(abs(b), 1e-300) def ast_firewall(): with open(os.path.abspath(__file__), encoding="utf-8") as fh: tree = ast.parse(fh.read()) hits = [] for node in ast.walk(tree): if not isinstance(node, ast.Call): continue fn = node.func name = fn.attr if isinstance(fn, ast.Attribute) else ( fn.id if isinstance(fn, ast.Name) else "") if name.lower() in AST_BANNED: hits.append(name) return sorted(set(hits)) # ==================================================== own sieves + frames def prime_power_list(cap): """Ordered prime powers >= 2 up to cap, from an own sieve.""" sieve = np.zeros(cap + 1, dtype=bool) out = [] for p in range(2, cap + 1): if not sieve[p]: sieve[p * p::p] = True q = p while q <= cap: out.append(q) q *= p return sorted(out) def lambda_table(n_max): """von Mangoldt Lambda(n) for n <= n_max, own sieve.""" lam = np.zeros(n_max + 1) sieve = np.zeros(n_max + 1, dtype=bool) for p in range(2, n_max + 1): if not sieve[p]: sieve[p * p::p] = True q, log_p = p, math.log(p) while q <= n_max: lam[q] = log_p q *= p return lam def build_frame(pp, kz): """The frame of index kz, rebuilt from the prime-power list alone.""" n_now, n_next = float(pp[kz]), float(pp[kz + 1]) alpha = math.log(n_now) d_k = 0.5 * (math.log(n_next) - alpha) / 4.0 m_z = int(math.ceil(alpha / d_k - 1.0e-9)) + 1 if m_z % 2: m_z += 1 h = m_z // 2 d_val = alpha / h log_n = 2.0 * alpha + 2.0 * d_val return dict(kz=kz, n=int(pp[kz]), n_next=int(pp[kz + 1]), gap=int(pp[kz + 1]) - int(pp[kz]), alpha=alpha, D=d_val, h=h, log_n=log_n, N=math.floor(math.exp(log_n))) # ========================================= geometric supply, two routes def fejer_closed(s_len): """POLE - P_main + ARCH for the Fejer window F(x) = 1 - x/S, closed.""" s_len = mp.mpf(s_len) q_e = mp.exp(-s_len / 2) pole_minus_pmain = (4 * (1 - q_e) - 8 / s_len + q_e * (4 * s_len + 8) / s_len) psi_q = -mp.euler - 3 * mp.log(2) - mp.pi / 2 z_two = mp.pi ** 2 + 8 * mp.catalan tail = mp.mpf(0) for k in range(0, 80): s_k = 2 * k + mp.mpf(1) / 2 tail += mp.exp(-s_k * s_len) / s_k ** 2 arch = (-mp.log(mp.pi) + psi_q + z_two / (2 * s_len) - 2 * tail / s_len) p_main = -4 + 8 / s_len * (mp.exp(s_len / 2) - 1) v_geo = 4 / s_len * (mp.exp(s_len / 2) - 1) - 1 return dict(B=pole_minus_pmain + arch, pole=pole_minus_pmain + p_main, arch=arch, p_main=p_main, V=v_geo) def fejer_quadrature(s_len): """Independent route: POLE, ARCH, P_main by direct quadrature.""" s_len = mp.mpf(s_len) def f_win(x): return 1 - x / s_len pole = 4 * mp.quad(lambda x: f_win(x) * mp.cosh(x / 2), [0, s_len]) p_main = 2 * mp.quad(lambda x: f_win(x) * mp.exp(x / 2), [0, s_len]) f_zero = mp.mpf(1) def integ(wv): return ((f_zero * mp.exp(-2 * wv) - f_win(wv) * mp.exp(-wv / 2)) / (-mp.expm1(-2 * wv))) body = mp.quad(integ, [0, s_len]) tail = -mp.log1p(-mp.exp(-2 * s_len)) / 2 * f_zero arch = -f_zero * (mp.euler + mp.log(mp.pi)) + 2 * (body + tail) return dict(B=pole - p_main + arch, pole=pole, arch=arch, p_main=p_main) # ================================================== the exact kernel ledger def kernel_ledger(): """S1: the six exact kernel identities, symbolic in the mesh D.""" t_sym, d_sym, q_sym = sp.symbols("t D q", positive=True) def tent_sym(x, d_val): return sp.Max(0, 1 - sp.Abs(x) / d_val) def kd_sym(x, d_val): return (tent_sym(x, d_val) - (tent_sym(x - d_val, d_val) + tent_sym(x + d_val, d_val)) / 2) def piece_lin(a_val, b_val): k_a = sp.simplify(kd_sym(a_val * d_sym, d_sym)) k_b = sp.simplify(kd_sym(b_val * d_sym, d_sym)) return sp.simplify(k_a + (k_b - k_a) * ((t_sym / d_sym - a_val) / (b_val - a_val))) seg = [(-2, -1), (-1, 0), (0, 1), (1, 2)] seg_abs = [(-2, -1), (-1, sp.Rational(-2, 3)), (sp.Rational(-2, 3), 0), (0, sp.Rational(2, 3)), (sp.Rational(2, 3), 1), (1, 2)] node_vals = [sp.simplify(kd_sym(n * d_sym, d_sym)) for n in (-2, -1, sp.Rational(-2, 3), 0, sp.Rational(2, 3), 1, 2)] i_zero = sum(sp.integrate(piece_lin(a, b), (t_sym, a * d_sym, b * d_sym)) for a, b in seg) # each refined piece has constant sign, so |int| = int |.| i_one = sum(sp.Abs(sp.integrate(piece_lin(a, b), (t_sym, a * d_sym, b * d_sym))) for a, b in seg_abs) i_two = sum(sp.integrate(piece_lin(a, b) ** 2, (t_sym, a * d_sym, b * d_sym)) for a, b in seg) tv = sum(sp.Abs(sp.diff(piece_lin(a, b), t_sym)) * (b - a) * d_sym for a, b in seg) i_exp_k = sum(sp.integrate(sp.exp(-t_sym / 2) * piece_lin(a, b), (t_sym, a * d_sym, b * d_sym)) for a, b in seg) t_d_int = (sp.integrate(sp.exp(-t_sym / 2) * (1 + t_sym / d_sym), (t_sym, -d_sym, 0)) + sp.integrate(sp.exp(-t_sym / 2) * (1 - t_sym / d_sym), (t_sym, 0, d_sym))) t_d_target = 8 / d_sym * (sp.cosh(d_sym / 2) - 1) def q_reduce(expr): return sp.simplify(sp.expand(sp.powsimp( expr.rewrite(sp.exp).subs(sp.exp(d_sym / 2), q_sym) .subs(sp.exp(d_sym), q_sym ** 2) .subs(sp.exp(-d_sym / 2), 1 / q_sym) .subs(sp.exp(-d_sym), q_sym ** -2)))) check("S1.1 int K_D = 0 (exact)", sp.simplify(i_zero) == 0, "%s" % sp.simplify(i_zero)) check("S1.2 ||K_D||_inf = 1 (exact)", max(sp.Abs(x) for x in node_vals) == 1 and sp.simplify(kd_sym(0, d_sym)) == 1, "node values %s" % node_vals) check("S1.3 ||K_D'||_1 = 4, independent of D (exact) -- the " "load-bearing identity: the conversion constant of a " "magnitude hypothesis does not get cheap as the mesh refines", sp.simplify(tv - 4) == 0, "%s" % sp.simplify(tv)) check("S1.4 ||K_D||_1 = 4D/3 (exact)", sp.simplify(i_one - 4 * d_sym / 3) == 0, "%s" % sp.simplify(i_one)) check("S1.5 D^-1 int K_D^2 = 2/3 (exact)", sp.simplify(i_two / d_sym - sp.Rational(2, 3)) == 0, "%s" % sp.simplify(i_two / d_sym)) check("S1.6 T_D = int tau_D e^{-t/2} = (8/D)(cosh(D/2)-1) (exact)", q_reduce(t_d_int - t_d_target) == 0) check("S1.7 int K_D e^{-t/2} = -(8/D)(cosh(D/2)-1)^2 = -(D/8) T_D^2 " "(exact)", q_reduce(i_exp_k + 8 / d_sym * (sp.cosh(d_sym / 2) - 1) ** 2) == 0 and q_reduce(i_exp_k + d_sym / 8 * t_d_target ** 2) == 0) # ============================== the split: signed and congruence-invariant def split_algebra(): """S3.1-S3.4: identities in the matrix-stage SYMBOLS, no deployed number consumed.""" h_s = 4 bm_s = sp.Matrix(h_s - 1, h_s - 1, lambda i, j: sp.Symbol("B%d%d" % (min(i, j), max(i, j)))) m_s = sp.Matrix(h_s - 1, h_s - 1, lambda i, j: sp.Symbol("M%d%d" % (i, j))) n0_s, nc_s = sp.symbols("n0 nc") b_s = sp.Matrix(h_s - 1, 1, lambda i, j: sp.Symbol("b%d" % i)) c_s = sp.Matrix(h_s - 1, 1, lambda i, j: sp.Symbol("c%d" % i)) inv_s = bm_s.inv() q0_s = (b_s.T * inv_s * b_s)[0] qc_s = (c_s.T * inv_s * c_s)[0] lin_s = (c_s.T * inv_s * b_s)[0] s_split = sp.expand((n0_s - q0_s) - nc_s + 2 * lin_s - qc_s) s_direct = sp.expand((n0_s - nc_s) - ((b_s - c_s).T * inv_s * (b_s - c_s))[0]) check("S3.1 the three-term split s = (n_0 - q_0) - n_c + 2 " "- q_c is an identity (sympy, exact)", sp.simplify(s_split - s_direct) == 0) bt_s = m_s.T * b_s ct_s = m_s.T * c_s bmt_s = m_s.T * bm_s * m_s inv_t = bmt_s.inv() ok_inv = all(sp.simplify(x) == 0 for x in ( (bt_s.T * inv_t * bt_s)[0] - q0_s, (ct_s.T * inv_t * ct_s)[0] - qc_s, (ct_s.T * inv_t * bt_s)[0] - lin_s)) check("S3.2 all three terms are invariant under EVERY congruence " "T = diag(1, M) (sympy, exact) -- the OBJECT is basis " "independent, the DEMAND is not", ok_inv) flip = {c_s[i, 0]: -c_s[i, 0] for i in range(h_s - 1)} check("S3.3 parity under the comb sign flip b_c -> -b_c: outer terms " "EVEN, middle term ODD (exact) -- so positivity can only be " "carried by the odd term, and no flip-invariant bound can " "lower-bound it", sp.simplify(q0_s.subs(flip) - q0_s) == 0 and sp.simplify(qc_s.subs(flip) - qc_s) == 0 and sp.simplify(lin_s.subs(flip) + lin_s) == 0) a2_sym = sp.Matrix(3, 5, lambda i, j: (1 if j == i else (-2 if j == i + 1 else (1 if j == i + 2 else 0)))) bm5 = sp.Matrix(3, 3, lambda i, j: sp.Symbol("G%d%d" % (min(i, j), max(i, j)))) m3 = sp.Matrix(3, 3, lambda i, j: sp.Symbol("N%d%d" % (i, j))) k_plain = sp.simplify(a2_sym.T * bm5.inv() * a2_sym) k_cong = sp.simplify((m3.T * a2_sym).T * (m3.T * bm5 * m3).inv() * (m3.T * a2_sym)) check("S3.4 the F4 kernel K = A_2^T B^-1 A_2 is itself a congruence " "INVARIANT (sympy, exact)", sp.simplify(k_plain - k_cong) == sp.zeros(5, 5)) def selberg_neutrality(): """S2.13: formal identities in the free module on {log p}.""" primes_sel = [p for p in range(2, SELBERG_N + 1) if all(p % d for d in range(2, int(math.isqrt(p)) + 1))] lp_sym = {p: sp.Symbol("L%d" % p, positive=True) for p in primes_sel} def lam_sym(n): for p in primes_sel: m, k = n, 0 while m % p == 0: m //= p k += 1 if k and m == 1: return lp_sym[p] return sp.Integer(0) def log_sym(n): out, m = sp.Integer(0), n for p in primes_sel: while m % p == 0: m //= p out += lp_sym[p] return out def mu_int(n): m, cnt = n, 0 for p in primes_sel: if m % p == 0: m //= p cnt += 1 if m % p == 0: return 0 return (-1) ** cnt if m == 1 else 0 ok_lam = ok_sel = True for n in range(1, SELBERG_N + 1): divs = [d for d in range(1, n + 1) if n % d == 0] ok_lam = ok_lam and sp.expand( sum(mu_int(d) * log_sym(n // d) for d in divs) - lam_sym(n)) == 0 lhs = sum(mu_int(d) * sp.expand(log_sym(n // d) ** 2) for d in divs) rhs = sp.expand(lam_sym(n) * log_sym(n) + sum(lam_sym(d) * lam_sym(n // d) for d in divs)) ok_sel = ok_sel and sp.expand(lhs - rhs) == 0 check("S2.13 Selberg symmetry is PROVABLY NEUTRAL: Lambda = mu * log " "and mu * log^2 = Lambda.log + Lambda * Lambda are formal " "identities in the free module on {log p} (n <= %d), so they " "add no arithmetic content -- information gain exactly 1.000" % SELBERG_N, ok_lam and ok_sel, "both identities exact") def beurling_nyman(): """S6.3: the two exact Beurling-Nyman/Baez-Duarte witnesses.""" bn_const = mp.log(2 * mp.pi) - mp.euler bn_rows = [] for n_bn in BN_SIZES: a_bn = sp.Matrix(n_bn, 2, lambda i, j: sp.Integer(1) if j == 0 else sp.Rational(1, i + 1)) g_bn = a_bn * a_bn.T + sp.Rational(16, 25) * sp.eye(n_bn) beta_bn = sp.Matrix([sp.Rational(1, 2), sp.Rational(1, 2)]) b_bn = a_bn * beta_bn sig = (b_bn.T * g_bn.inv() * b_bn)[0] floor_ok = all(sp.nsimplify(x) >= 0 for x in (g_bn - sp.Rational(5523, 10000) * sp.eye(n_bn)).eigenvals()) bn_rows.append((n_bn, float(sig), floor_ok)) sig_lo = min(r[1] for r in bn_rows) sig_hi = max(r[1] for r in bn_rows) check("S6.3 Beurling-Nyman exactly in Q: (T3) lam_min(G_N) <= " "(log 2pi - gamma)/N = %.10f/N, so the Baez-Duarte Gram has NO " "N-uniform floor and the deployed hypothesis is FALSE for " "N >= 3; (T5) the explicit family G~ = A A^T + (4/5)^2 I " "satisfies EVERY hypothesis of the certificate class at every " "N while d~^2 >= 1/2 forever -- the class is non-implying" % float(bn_const), rel(bn_const, WARD_BN_CONST) < 1e-9 and float(bn_const) / 3 < WARD_BN_CB and all(r[2] for r in bn_rows) and sig_hi <= WARD_BN_SIGMA_CAP and 1 - sig_hi >= 0.5 - 1e-12 and abs(sig_lo - WARD_BN_SIGMA_RANGE[0]) < 5e-4 and abs(sig_hi - WARD_BN_SIGMA_RANGE[1]) < 5e-4, "cap %.6f < c_B %.4f | sigma~ in [%.4f, %.4f] <= %.6f | " "d~^2 >= 1/2" % (float(bn_const) / 3, WARD_BN_CB, sig_lo, sig_hi, WARD_BN_SIGMA_CAP)) def part(): section("PRIME.NOGO.SIGNED.ONLY.01 -- the localization of the " "residual\nobstruction (a NO-GO / typing module: no " "positivity, no gate, no marker)") # ---------------------------------------------------------- guards print("\n-- G0: firewall, frozen spec, own sieves, frame rebuild") hits = ast_firewall() guards = [check("G0.1 AST firewall clean (no zero data, no " "eigensolver on any deployed form, no fit)", not hits, "hits=%s" % (hits or "none"))] spec_sha = hashlib.sha256(FROZEN_SPEC.encode("utf-8")).hexdigest() guards.append(check("G0.2 the embedded frozen spec reproduces the " "probe SPEC_SHA-256 BEFORE any evaluation", spec_sha == PROBE_SPEC_SHA, "SHA256 %s..." % spec_sha[:16])) pp = prime_power_list(PP_CAP) guards.append(check("G0.3 own prime-power sieve starts 2,3,4,5,7,8,9,11", pp[:8] == [2, 3, 4, 5, 7, 8, 9, 11], "%s" % pp[:8])) frames = {} h_ok = True worst_frame_rel = 0.0 for want_h, kz in FRAME_SPECS: fr = build_frame(pp, kz) frames[want_h] = fr h_ok = h_ok and (fr["h"] == want_h) root = math.exp(fr["alpha"] + fr["D"]) worst_frame_rel = max(worst_frame_rel, rel(fr["D"] * root, fr["gap"] / 4.0)) guards.append(check("G0.4 all nine frames reconstructed from prime " "powers alone (h is PREDICTED, never supplied)", h_ok, "h = %s" % [frames[k]["h"] for k, _ in FRAME_SPECS])) guards.append(check("G0.5 the test scale sits ON the square-root " "barrier: D_h sqrt(N_h) = gap/4", worst_frame_rel < 5e-2, "worst rel %.4e (corpus %.4e)" % (worst_frame_rel, WARD_FRAME_REL))) guards.append(check("G0.6 the integer gap is O(log N) on every frame " "(the family is indexed by kz, not by h)", all(frames[k]["gap"] <= frames[k]["log_n"] for k, _ in FRAME_SPECS), "gaps %s" % [frames[k]["gap"] for k, _ in FRAME_SPECS])) guards_ok = all(guards) # -------------------------------------------------- S1 kernel ledger section("S1 THE EXACT KERNEL LEDGER (sympy, exact in the mesh D)") kernel_ledger() print(" READING: the conversion constant of any magnitude hypothesis " "is bounded\n BELOW by ||K_D'||_1 = 4 uniformly in D -- the " "kernel does not become cheap\n as the frames get finer. " "That exact fact is what makes S2 unconditional.") # ------------------------------------- S2 the magnitude class is empty section("S2 THE MAGNITUDE CLASS IS UNCONDITIONALLY EMPTY " "(Littlewood CITED)") supply = {} ok_supply = True for h_key in SUPPLY_RUNGS: fr = frames[h_key] s_len = mp.mpf(2) * mp.mpf(fr["alpha"]) * (1 + mp.mpf(1) / fr["h"]) closed = fejer_closed(s_len) quadr = fejer_quadrature(s_len) n_cap = int(mp.floor(mp.exp(s_len))) lam = lambda_table(max(n_cap, 2)) nz = np.nonzero(lam > 0)[0] prime = 2.0 * float(sum(mp.mpf(lam[n]) / mp.sqrt(int(n)) * (1 - mp.log(int(n)) / s_len) for n in nz)) delta = prime - float(closed["p_main"]) supply[h_key] = dict(S=s_len, B=closed["B"], N=n_cap, route_dev=abs(closed["B"] - quadr["B"]), delta=delta, slack=0.5 * (float(closed["B"]) - delta)) ok_supply = ok_supply and ( abs(float(closed["B"] - quadr["B"])) < 1e-15 and abs(float(closed["B"] - mp.mpf(WARD_FEJER_B[h_key]))) < 1e-15 and closed["B"] < 0 and abs(float(closed["B"])) <= 2.0) check("S2.8 the geometric supply, recomputed here by TWO independent " "routes (closed form, direct quadrature), is NEGATIVE and O(1) " "(|B| <= 2) at both registered rungs and reproduces the corpus " "value", ok_supply, " | ".join("h=%d B %s (route dev %.1e, corpus %s)" % (k, mp.nstr(supply[k]["B"], 20), float(supply[k]["route_dev"]), WARD_FEJER_B[k]) for k in SUPPLY_RUNGS)) floors = {} for h_key in FLOOR_RUNGS: n_val = mp.exp(mp.mpf(frames[h_key]["log_n"])) floors[h_key] = 4 * mp.log(mp.log(mp.log(n_val))) check("S2.9 h=%d Littlewood floor 4 logloglog N_h >= %.7f " "(Littlewood's Omega-theorem itself is CITED, X3)" % (h_key, WARD_LITTLEWOOD[h_key]), float(floors[h_key]) >= WARD_LITTLEWOOD[h_key] - 5e-7, "floor %.7f (N_h = %d)" % (float(floors[h_key]), frames[h_key]["N"])) own_slack = max(supply[k]["slack"] for k in supply) ratio_own = float(floors[FLOOR_RUNGS[0]]) / own_slack ratio_cited = float(floors[FLOOR_RUNGS[1]]) / WARD_SLACK_CITED check("S2.10 THE CLASS IS EMPTY: floor / slack >> 1 both on this " "module's OWN conservative slack and on the deployed slack " "(X2, cited) -- any admissible f would have to beat the " "unconditionally divergent S(N)", ratio_own > RATIO_BAR and ratio_cited > RATIO_BAR, "own %.1f (slack %.4e) | cited %.1f (slack %.1e)" % (ratio_own, own_slack, ratio_cited, WARD_SLACK_CITED)) selberg_neutrality() # ----------------------------------- S3 signed + congruence-invariant section("S3 THE RESIDUAL REQUIREMENT IS SIGNED AND " "CONGRUENCE-INVARIANT") split_algebra() print(" READING: the signed requirement survives every congruence, " "every window\n and every preconditioner because the split " "terms are invariant and the\n carrying term is ODD; an " "unsigned hull is flip-invariant, hence\n structurally unable " "to bound it below.") # ------------------------------------------------- S6 transport class section("S6 THE BEURLING-NYMAN TRANSPORT TARGET (exact witnesses)") beurling_nyman() # --------------------------------------------------- the localization section("THE LOCALIZATION (the deliverable) AND ITS NAMED SCOPE") print("""\ Any argument that closes (L) for the DEPLOYED BUDGET SHAPE must be (a) SIGNED -- it must orient a quantity that is ODD under the comb sign flip [S3.3], and (b) ALIGNMENT-CARRYING -- it must relate the position of the prime atoms to the spectral directions of K (equivalently the ordinates to the sign pattern of Fhat). In particular it can be NEITHER a magnitude bound on psi(x) - x [S1 + S2] NOR a natural-grammar identity, i.e. one valid in every arithmetic world and therefore comb-blind [S6]. PROVEN EMPTY for this budget shape ([here] = gated by this module, [probe] = carried by the frozen probe in experiments/): E1 every magnitude hypothesis |psi(x)-x| <= f(x) sqrt(x) [here] E2 every zero-density input with A > 2; A = 2 and Lindeloef land exactly critical, with zero slack [probe] E3 Selberg's symmetry formula (formal identity) [here] E4 every congruence reformulation, window choice, Jacobi/ Cholesky preconditioner, resolvent split [here] E5 every unsigned hull of the comb correction: the parity/ invariance half [here], the measured negativity [probe] E6 every bound consuming only {w >= 0, ||w||_inf, ||w||_2, supp w} [probe] E7 exact multiplicative transforms (Moebius/Dirichlet) [probe] E8 natural-grammar identities (de Branges/HB, Abel) [probe] E9 the Beurling-Nyman/Baez-Duarte transport target [here] E10 thin/cofinal Li criteria (explicit counterexample) [probe] MERELY UNEXPLORED (NO emptiness is claimed): U1 unconditional statements about ordinate POSITIONS; U2 alignment statements; U3 a different budget shape; U4 the four premises of the reverse implication; U5 global constraints on the source profile. WHAT THIS DOES NOT SAY (named open scope): (O1) nothing against arguments that legitimately consume zero positions -- the remaining statement IS of that kind, which is not the same as being false; (O2) only bounds INSIDE the cone class are forbidden; alignment statements are untouched and are exactly what is needed; (O3) every clause is for the deployed kernel, frame family and predeclared rung family -- another budget shape is outside; (O4) the reverse implication (inequality => RH) rests on four named premises and is NOT established; (O5) a new global inequality restricting the admissible source profiles changes the cone class and escapes it. ALSO: this is FINITE-RUNG. It is verified at h = 184/388/839 with the ladder to h = 12632; it makes NO all-h claim and says nothing about rungs outside the built cells. It does NOT say that RH is unprovable, that the wall route fails, or that no proof exists.""") # --------------------------------------------- honest counter-evidence section("COUNTER-EVIDENCE, CARRIED AND NOT SMOOTHED (cited, X1/X7)") print("""\ NO-WITNESS STANDS: the certified reads are POSITIVE at all NINE depths -- 0 negatives, 0 straddles -- with deepest certified read %s at h = 12632; this module adds none. The normalised decay FLATTENS with depth on the cited normalisation (n0-normalised upper-bound exponents %.8f globally against %.8f on the deepest step 5746 -> 12632), and the re-derived s/D^2 step 1393 -> 2015 is POSITIVE (+%.8f), so the normalised drift is NOT monotone; no re-derivable exponent comes near the frozen collapse bar %.1f. A no-go about the TYPE of the missing input is not evidence about its EXISTENCE.""" % (CITED_DEEPEST_READ, CITED_EXP_U_GLOBAL, CITED_EXP_U_DEEP, CITED_EXP_STEP_POS, CITED_COLLAPSE_BAR)) # ------------------------------------------------------------- typing section("TYPING AGAINST THE FROZEN GATE RULE") print("""\ This is a statement ABOUT the problem, NOT an independent sign source. It supplies no positivity, no new interval and no new certificate; it closes no gate, narrows no gate and moves NO marker. Its content is the localization of the missing input to a two-property class (signed AND alignment-carrying) plus the enumeration of the classes now provably empty for this budget shape. It must NOT be counted as evidence for or against the Riemann Hypothesis and must not be cited in support of any positivity claim. Registers PRIME.NOGO.SIGNED.ONLY.01 [O] as a no-go/typing row.""") elapsed = time.time() - T_ALL check("V1 runtime below the frozen bar (%.0f s)" % RUNTIME_CAP, elapsed < RUNTIME_CAP, "%.1f s" % elapsed) fails = list(FAILS) if not guards_ok: verdict = "NOGO-CORE-INVALID (guards: %s)" % ",".join( f for f in fails if f.startswith("G0")) elif fails: verdict = "NOGO-CORE-FAILED (%s)" % ",".join(fails) else: verdict = EXPECTED _VERDICTS["v"] = verdict section("VERDICT: %s" % verdict) print("CHECKS %d/%d PASS (%.1f s); fails: %s" % (len(CHECKS) - len(fails), len(CHECKS), time.time() - T_ALL, fails or "none")) return 0 if (guards_ok and not fails) else 1 T_ALL = time.time() def run(): global T_ALL T_ALL = time.time() mp.dps = MP_DPS CHECKS.clear() FAILS.clear() _VERDICTS.clear() print("=" * 74) print("v913 -- PRIME.NOGO.SIGNED.ONLY.01 (the signed-and-alignment " "localization;") print("a NO-GO / typing module: no positivity, no gate, no marker, " "NO RH claim)") print("=" * 74) rc = part() n_run, fails = len(CHECKS), list(FAILS) ok = (rc == 0 and n_run == N_CHECKS and not fails and _VERDICTS.get("v") == EXPECTED) print("\n" + "=" * 74) print("v913: %d/%d checks passed | verdict %s | runtime %.1f s" % (n_run - len(fails), n_run, _VERDICTS.get("v"), time.time() - T_ALL)) print("NO RH claim, no positivity claim. This module prices " "ARGUMENT CLASSES; it") print("closes no gate, narrows no gate and moves no marker, and it " "is not evidence") print("for or against the Riemann Hypothesis. Probe SPEC_SHA " "%s..." % PROBE_SPEC_SHA[:16]) print("PROBE_SHA %s... (experiments/tfpt-discovery/" "signed_only_nogo_probe.py)" % PROBE_SHA[:16]) print("[%s] PATTERN GATE: expected %d checks, zero fails, verdict " "%s (got %d, fails %s, verdict %s)" % ("PASS" if ok else "FAIL", N_CHECKS, EXPECTED, n_run, fails or "none", _VERDICTS.get("v"))) return 0 if ok else 1 if __name__ == "__main__": raise SystemExit(run())