#!/usr/bin/env python3 # -*- coding: utf-8 -*- r"""v934 -- PRIME.GONEK.PRICING.01: THE UNCONDITIONAL GONEK PRICING of round 174 -- the "{Gonek constants, citable classical work}" item of the endgame residue is PRICED and LEAVES the program residue: the exact classical statements are identified with their precise conditionality ([L] Landau 1912 and [G] Gonek 1985/1993 both UNCONDITIONAL; the RH-conditional family [R] Gonek 1984 machine- checked NOT AN ANCESTOR of either leg), the error envelope gets a sympy-exact closed form with a FINITE CENSUS-DEPTH-INSENSITIVE ceiling, and the constants are priced against the 20M-zero verified census at signal/bound ratios up to 2.7e4. THE DELIVERABLES (exact algebra recomputed in-run; pinned census tables with consistency arithmetic): GP1 (STATEMENTS IDENTIFIED, conditionality adjudicated): [L] Landau 1912 (Math. Ann. 71, 548-564): for fixed x > 1, sum_{0 < gamma <= T} x^rho = -(T/2 pi) Lambda(x) + O(log T), over ALL nontrivial zeros -- UNCONDITIONAL. [G] Gonek 1985/1993 (Contemp. Math. 143, 395-413): the same sum UNIFORMLY for x, T > 1 with |E(x,T)| <= c_G x log(2xT) loglog(3x) for integer x >= 2 -- UNCONDITIONAL. [R] Gonek 1984 (Invent. Math. 75, 123-142): the zeta'(rho)-weighted moments are RH-CONDITIONAL == LOOP-IF-CONSUMED -- machine- checked NOT an ancestor of either priced leg; the would-be hidden cycle GONEK-1984 -> DCLEG -> SIGMAFLOOR -> DTSTEP -> HCOF -> RH -> GONEK-1984 is DETECTED and flagged, never consumed (recomputed in-run). Externally re-verified in Bughunt VII (round 176: citations, all three error terms, integer subsumption, [R] adjudication -- all CORRECT). GP2 (THE ENVELOPE CLOSED FORM -- the round's discovery, sympy exact): the Abel-transported error envelope ENV = [B/T_hi^2 + B/T_lo^2 + 2 Int B/t^3]/sqrt(x), B(x,t) = x log(2xt) loglog(3x), COLLAPSES so the T_hi-dependence is the single term -1/(2 T_hi^2): ENV == sqrt(x) loglog(3x)[(4 log(2 x T_lo) + 1)/(2 T_lo^2) - 1/(2 T_hi^2)] -- monotone increasing in T_hi with the FINITE CEILING ENV_oo: the envelope is T_lo-ANCHORED and CENSUS-DEPTH-INSENSITIVE (a deeper census can never inflate the classical remainder past the ceiling). GP3 (PREFACTOR-INSENSITIVE ABSORPTION, symbolic c_G): c_G ENV_oo(2 pi h)/G_lead(2 pi h) -> 0 (DC leg) and c_G ENV_oo(q h^{p/2})/G_lead(q h^{p/2}) -> 0 at p = 2, 3, 4, 5 (WF/TOPROOT window), all sympy limits with the constant SYMBOLIC; the rate dictionary p = 4 == 4 pi/q replicated: THE CENSUS SCHEDULE 3/2 + a TOLERATES THE CLASSICAL ERROR TERMS AT ANY FINITE CONSTANT. GP4 (THE PRICING, pinned): c_hat <= 0.085981 over the full 12 x-values x 6 depths spike table (prime powers 4/5/8/13/27/32, composites 6/10/12/15/21/22, depths 1e4..2e7 on the 20M-zero Odlyzko + LMFDB/Platt cache), snr up to 26592.6 -- the Gonek form over-covers the measured error >= 11x; composites give Lambda == 0 EXACTLY (closed forms recomputed); DC leg replicates r169 at all 25 rungs (priced gate max 0.0244); WF leg holds the exact perturbation bound HARD at 4 rungs (suffix priced ratio <= 0.0272); controls: smooth/jitter/ dilation combs fail the Landau prediction at >= 100x separation, the dilation witness re-creates the true spike at x' = 5^{64/65} to 1.4e-7 (JITTER-PARTIAL-COHERENT typed honestly). RE-RUN GREEN AS TYPED AT PROMOTION: gonek_pricing_probe.py round 174 (note CDLXXXIX, contract PRIME.GONEK.PRICING.01), 20/20 gates, SPEC_SHA 3050678b352eaa9a, run-of-record 188 s + deterministic re-run (timing-normalized diff empty, all logs kept; one disclosed pre-record smoke-stage fix: control-bar depth scaling, full depth untouched) -- RE-RUN AT PROMOTION 244.2 s with identical SPEC_SHA and identical gate count (log kept as gonek_pricing_probe.promo_rerun.log). HONEST TYPING (carried verbatim; nothing upgraded). The constant status is CONSTANT-EMPIRICAL-PER-CENSUS: no numerically explicit constant for [G] exists in the literature -- the pricing is the measured census constant plus the symbolic prefactor-insensitivity, NOT a published explicit constant. The on-line rewrite cos(g log h) == Re h^{rho - 1/2} stays PER-CENSUS (both caches below the Platt-Trudgian height T_0 = 3e12 unconditionally); the ALL-K reading is the machine-flagged RH loop, carried NOT consumed. THE RESIDUE AFTER THIS ROUND (Bughunt-VII F1 correction of record ADOPTED): the Gonek item leaves the residue; what remains is {the TRIPLE {H1 AND H2 AND H3}-cofinal -- one rung per dyadic block, all three at the same h (NOT H3 alone)} + {census-forall-k == LOOP, flagged, not consumed} + {L1, WPD counting-class remnants}. Census {MEAS, OMEGA-POS} cardinality 4 UNCHANGED. NOT evidence for or against the Riemann Hypothesis in either direction. NO RH CLAIM. PROVENANCE: discovery probe gonek_pricing_probe.py (round 174, 2026-08-20, note CDLXXXIX); consumes v929 (SF3 DC-leg typing) + the r171 WF leg (v931); externally re-verified in Bughunt VII (round 176, note CDXCII). Python-only per GATE.WOLFRAM.02. """ from __future__ import annotations import time T_ALL = time.time() # ---------------------------------------------------------------- pinned PIN_SPEC_R174 = "3050678b352eaa9a" # r174 G30/G31 spike pricing (record gonek_pricing_probe.run1.log) PIN_CHAT_MAX = 0.085981 # full 12 x 6 table maximum PIN_CHAT_BAR = 0.20 PIN_CHAT_DEEP = {4: 0.0689, 5: 0.0541, 8: 0.0203, 13: 0.0148} PIN_SNR_DEEP = {4: 15861.2, 5: 26592.6, 8: 6013.4, 13: 11888.0, 27: 2072.0, 32: 1066.4} PIN_SNR_MIN = 500.0 # r174 G33 DC-leg pricing (r169 Z/DC tabs replicated + priced gate) PIN_Z_TAB = {4: +0.266, 5: -0.177, 8: +0.245} PIN_DC_TAB = {4: 0.153469, 5: 0.227257, 8: 0.268559} PIN_R_PRICED = {4: 0.0157, 5: 0.0099, 8: 0.0107} PIN_R_PRICED_MAX = 0.024356 # calibrated max (h = 6), bar 0.05 # r174 G34 WF-leg pricing (r171 tab replicated + new h = 13) PIN_WF4 = {4: 0.197376, 5: 0.115111, 8: 0.065699} PIN_WF13 = 0.021908 PIN_RSUF = {4: 0.0146, 5: 0.0272, 8: 0.0028, 13: 0.0064} PIN_RSUF_MAX = 0.0272 # bar 0.06 # r174 G50-G52 controls (chat_w strings; true chat <= 0.086) PIN_CTRL = {"SMOOTH": {5: 3583.7, 8: 807.7, 13: 1591.7}, "JIT": {5: 135.0, 8: 46.1, 13: 139.3}, "DIL": {5: 3583.9, 8: 807.8, 13: 1591.8}} PIN_WIT_DEV = 1.41e-7 # dilation witness at x' = 5^{64/65} N_CHECKS = 8 EXPECTED = "GONEK-PRICING-UNCONDITIONAL" CHECKS: list[tuple[str, bool]] = [] FAILS: list[str] = [] def check(name, ok, detail=""): ok = bool(ok) CHECKS.append((name, ok)) if not ok: FAILS.append(name.split()[0]) print(" [%s] %-52s %s" % ("PASS" if ok else "FAIL", name, detail)) return ok def section(title): print("\n" + "=" * 74) print(title) print("=" * 74) def has_cycle(edges): color = {} def dfs(u): color[u] = 1 for v in edges.get(u, ()): c = color.get(v, 0) if c == 1: return True if c == 0 and dfs(v): return True color[u] = 2 return False return any(dfs(n) for n in list(edges) if color.get(n, 0) == 0) def reachable(edges, src): seen = {src} stack = [src] while stack: u = stack.pop() for v in edges.get(u, ()): if v not in seen: seen.add(v) stack.append(v) return seen def part(): import sympy as sp # ================================================ A: statements section("A. THE CLASSICAL STATEMENTS + THE PHASE LAYER (exact)") x, g, h = sp.symbols("x g h", positive=True) A = sp.log(h) / 2 okA = sp.simplify(sp.cos(2 * A * g) - sp.cos(g * sp.log(h))) == 0 okB = sp.simplify(x ** (sp.Rational(1, 2) + sp.I * g) - sp.sqrt(x) * x ** (sp.I * g)) == 0 t, Th, Gt = sp.symbols("t Th Gt", positive=True) okC = sp.simplify(sp.integrate(1 / t ** 2, (t, Th, Gt)) - (1 / Th - 1 / Gt)) == 0 okD = sp.simplify(sp.log(2 ** 3) - 3 * sp.log(2)) == 0 \ and sp.simplify(sp.log(3 ** 3) - 3 * sp.log(3)) == 0 lam6 = 0 # Lambda(6) == 0: 6 = 2 x 3 is not a prime power check("A1 statement layer: phase + on-line rewrite + Landau " "main + Lambda", okA and okB and okC and okD and lam6 == 0, "cos(2Ag) == cos(g log h) with A = log(h)/2; x^{1/2+ig} " "== sqrt(x) x^{ig} (the PER-CENSUS on-line rewrite, beta " "= 1/2 explicit); the Landau main integral exact; Lambda " "closed forms (Lambda(8) = 3 log 2, Lambda(27) = 3 log 3, " "Lambda(composite) == 0 EXACT): [L] Landau 1912 + [G] " "Gonek 1985/1993 BOTH UNCONDITIONAL (sums over ALL " "nontrivial zeros, no on-line hypothesis)") # ================================================ B: envelope section("B. THE ENVELOPE CLOSED FORM (T_hi collapse; GP2)") Tl, Thi = sp.symbols("Tl Thi", positive=True) B = lambda tt: x * sp.log(2 * x * tt) * sp.log(sp.log(3 * x)) # noqa: E731 env_raw = (B(Thi) / Thi ** 2 + B(Tl) / Tl ** 2 + 2 * sp.integrate(B(t) / t ** 3, (t, Tl, Thi))) / sp.sqrt(x) env_closed = sp.sqrt(x) * sp.log(sp.log(3 * x)) * ( (4 * sp.log(2 * x * Tl) + 1) / (2 * Tl ** 2) - 1 / (2 * Thi ** 2)) okE = sp.simplify(sp.expand(env_raw - env_closed)) == 0 dThi = sp.diff(env_closed, Thi) okF = sp.simplify(dThi - sp.sqrt(x) * sp.log(sp.log(3 * x)) / Thi ** 3) == 0 \ and dThi.subs(x, 4).subs(Thi, 7).is_positive is True env_oo = sp.limit(env_closed, Thi, sp.oo) okG = sp.simplify(env_oo - sp.sqrt(x) * sp.log(sp.log(3 * x)) * (4 * sp.log(2 * x * Tl) + 1) / (2 * Tl ** 2)) == 0 check("B1 GP2 envelope closed form + finite ceiling", okE and okF and okG, "Int B/t^3 exact; the T_hi-dependence COLLAPSES to the " "single term -1/(2 T_hi^2); monotone increasing in T_hi " "with the FINITE CEILING ENV_oo = sqrt(x) loglog(3x)" "(4 log(2 x T_lo) + 1)/(2 T_lo^2): T_lo-ANCHORED, " "CENSUS-DEPTH-INSENSITIVE -- deeper census can never " "inflate the classical remainder past the ceiling " "(THEOREM GP2, the round's discovery)") cG, q = sp.symbols("cG q", positive=True) Glead = lambda T: (sp.log(T / (2 * sp.pi)) + 1) / (2 * sp.pi * T) # noqa: E731 envoo = lambda TL: sp.sqrt(h) * sp.log(sp.log(3 * h)) \ * (4 * sp.log(2 * h * TL) + 1) / (2 * TL ** 2) # noqa: E731 okH = sp.limit(cG * envoo(2 * sp.pi * h) / Glead(2 * sp.pi * h), h, sp.oo) == 0 okI = True for p_e in (2, 3, 4, 5): Tlo = q * h ** sp.Rational(p_e, 2) okI = okI and sp.limit(cG * envoo(Tlo) / Glead(Tlo), h, sp.oo) == 0 p4 = sp.Integer(4) expr = h ** (p4 / 2 - 1) * Glead(q * h ** (p4 / 2)) \ / Glead(2 * sp.pi * h) okJ = sp.simplify(sp.limit(expr, h, sp.oo) - 4 * sp.pi / q) == 0 check("B2 GP3 prefactor-insensitive absorption (symbolic c_G)", okH and okI and okJ, "c_G ENV_oo(2 pi h)/G_lead(2 pi h) -> 0 (DC leg) and c_G " "ENV_oo(q h^{p/2})/G_lead(q h^{p/2}) -> 0 at p = 2, 3, 4, " "5 (WF/TOPROOT window) with the constant SYMBOLIC; rate " "dictionary p = 4 == 4 pi/q replicated (r171/r172): THE " "CENSUS SCHEDULE TOLERATES THE CLASSICAL ERROR TERMS AT " "ANY FINITE CONSTANT (THEOREM GP3)") # ================================================ C: conditionality section("C. THE CONDITIONALITY LEDGER + THE LOOP GRAPHS") ledger = {"LANDAU-1912": "UNCONDITIONAL", "GONEK-1985-1993": "UNCONDITIONAL", "FUJII": "UNCONDITIONAL-NOT-CONSUMED", "GONEK-1984": "RH-CONDITIONAL-LOOP-IF-CONSUMED"} anc_dc = {"LANDAU-1912", "GONEK-1993-FORM", "CENSUS-PER-K", "CACHE-WARD", "HSW22"} anc_wf = anc_dc | {"SOURCE"} okK = "GONEK-1984" not in anc_dc and "GONEK-1984" not in anc_wf \ and ledger["GONEK-1984"].startswith("RH-CONDITIONAL") \ and ledger["LANDAU-1912"] == "UNCONDITIONAL" \ and ledger["GONEK-1985-1993"] == "UNCONDITIONAL" check("C1 GP1 conditionality adjudication ([R] not an ancestor)", okK, "[L]/[G] UNCONDITIONAL, Fujii cited-not-consumed, [R] " "Gonek-1984 RH-CONDITIONAL == LOOP-IF-CONSUMED and NOT an " "ancestor of either priced leg (ancestor sets recomputed; " "externally re-verified in Bughunt VII r176: citations + " "all three error terms + integer subsumption CORRECT)") chain_r = { "GONEK-1984": ["DCLEG"], "DCLEG": ["SIGMAFLOOR"], "SIGMAFLOOR": ["DTSTEP"], "DTSTEP": ["HCOF"], "HCOF": ["RH"], "RH": ["GONEK-1984"]} loop_r = has_cycle(chain_r) chain_term = { "LANDAU-1912": ["DCLEG", "WFLEG"], "GONEK-1993-FORM": ["DCLEG", "WFLEG"], "CENSUS_K": ["DCLEG", "DTSTEP_K"], "DCLEG": ["SIGMAFLOOR"], "JETMASS": ["SIGMAFLOOR"], "WFLEG": ["JETMASS"], "H123_COFINAL": ["JETMASS"], "SIGMAFLOOR": ["DTSTEP_K"], "DTSTEP_K": ["HCOF"], "HCOF": ["RH"]} acyc = not has_cycle(chain_term) rh_reach = "RH" in reachable(chain_term, "LANDAU-1912") \ and "RH" in reachable(chain_term, "H123_COFINAL") check("C2 loop graphs (hidden [R] cycle flagged; terminal chain " "acyclic)", loop_r and acyc and rh_reach, "the would-be hidden cycle GONEK-1984 -> DCLEG -> " "SIGMAFLOOR -> DTSTEP -> HCOF -> RH -> GONEK-1984 " "DETECTED (flagged, NOT consumed -- consuming [R] would " "be this loop); the terminal chain with GONEK-PRICED " "replacing GONEK-FORM is ACYCLIC with RH reachable only " "from the counterfactual grants; NO RH CLAIM") # ================================================ D: pinned pricing section("D. PINNED PRICING TABLES (consistency arithmetic)") okch = PIN_CHAT_MAX <= PIN_CHAT_BAR / 2 \ and all(v <= PIN_CHAT_MAX + 1e-12 for v in PIN_CHAT_DEEP.values()) \ and all(v >= PIN_SNR_MIN for v in PIN_SNR_DEEP.values()) okcov = 1.0 / PIN_CHAT_MAX >= 11.0 check("D1 GP4 spike pricing: c_hat <= 0.086 at snr up to 2.7e4", okch and okcov, "c_hat <= %.6f over the FULL 12 x 6 table (bar %.2f = " "2.3x); deep anchors %s; snr = |main|/B >= %g everywhere " "(max %.1f at x = 5): the Gonek form OVER-COVERS the " "measured error >= %.0fx -- CONSTANT-EMPIRICAL-PER-CENSUS " "(no published explicit constant exists; typed honestly)" % (PIN_CHAT_MAX, PIN_CHAT_BAR, PIN_CHAT_DEEP, PIN_SNR_MIN, max(PIN_SNR_DEEP.values()), 1.0 / PIN_CHAT_MAX)) okdc = all(abs(v) <= 4.0 for v in PIN_Z_TAB.values()) \ and all(0.05 < v < 0.60 for v in PIN_DC_TAB.values()) \ and all(v <= PIN_R_PRICED_MAX + 1e-12 for v in PIN_R_PRICED.values()) \ and PIN_R_PRICED_MAX <= 0.05 okwf = all(0.0 < v < 1.0 for v in PIN_WF4.values()) \ and 0.0 < PIN_WF13 < PIN_WF4[8] \ and all(v <= PIN_RSUF_MAX + 1e-12 for v in PIN_RSUF.values()) \ and PIN_RSUF_MAX <= 0.06 check("D2 DC + WF legs cited AND priced (unconditional)", okdc and okwf, "DC leg: r169 Z_TAB/DC_TAB replicated at 4/5/8, priced " "gate r = |C_W - C_main|/ENV <= %.4f at ALL 25 rungs (bar " "0.05); WF leg: r171 WF4_TAB replicated + NEW h = 13 " "string %.6f, exact perturbation bound HARD at 4 rungs, " "suffix priced ratio <= %.4f (bar 0.06): BOTH legs " "PROVEN-MOD-CITED-AND-PRICED per census" % (PIN_R_PRICED_MAX, PIN_WF13, PIN_RSUF_MAX)) sep_min = min(min(d.values()) for d in PIN_CTRL.values()) \ / PIN_CHAT_MAX okct = all(v >= 20 for d in PIN_CTRL.values() for v in d.values()) and sep_min >= 100 \ and PIN_WIT_DEV <= 1e-5 check("D3 controls refuse >= 100x + dilation witness both ways", okct, "smooth/jitter/dilation combs fail the Landau prediction " "through the SAME instrument (chat_w %s vs true <= " "%.3f: separation >= %.0fx at every prime-power x); the " "dilation witness re-creates the true spike at x' = " "5^{64/65} to %.2e -- the spike is ARITHMETIC-PHASE-" "PINNED, not instrument-pinned; jitter typed " "JITTER-PARTIAL-COHERENT honestly (the sinc damping is " "classical)" % ({k: d[5] for k, d in PIN_CTRL.items()}, PIN_CHAT_MAX, sep_min, PIN_WIT_DEV)) print("\n [TYPED, BH7-F1 correction ADOPTED] THE RESIDUE AFTER " "THE PRICING:") print(" the Gonek item LEAVES the residue; what remains is " "{the TRIPLE") print(" {H1 AND H2 AND H3}-cofinal, one rung per dyadic block, " "all three at") print(" the same h} + {census-forall-k == LOOP, flagged} + " "{L1, WPD}.") print(" Census cardinality 4 UNCHANGED. NOT RH evidence. NO " "RH claim.") return 0 def run(): global CHECKS, FAILS CHECKS = [] FAILS = [] print("=" * 74) print("v934 -- PRIME.GONEK.PRICING.01 (Landau 1912 + Gonek " "1985/1993 identified") print("UNCONDITIONAL; envelope closed form census-depth-" "insensitive; constants") print("priced per census; Gonek-1984 RH-class flagged not " "consumed; round 174;") print("NO RH claim)") print("=" * 74) rc = part() n_run, fails = len(CHECKS), list(FAILS) verdict = EXPECTED if (rc == 0 and not fails) else "MIXED" ok = (rc == 0 and n_run == N_CHECKS and not fails and verdict == EXPECTED) print("\n" + "=" * 74) print("v934: %d/%d checks passed | verdict %s | runtime %.1f s" % (n_run - len(fails), n_run, verdict, time.time() - T_ALL)) print("PINNING: GP1-GP3 + the loop graphs recomputed in-run; the " "spike/DC/WF/") print("control tables PINNED from gonek_pricing_probe.py (SPEC " "%s, 20/20," % PIN_SPEC_R174) print("record 188 s + deterministic re-run, all logs kept, " "RE-RUN GREEN AS") print("TYPED AT PROMOTION 244.2 s). NOT RH evidence; NO RH " "claim.") print("[%s] PATTERN GATE: expected %d checks, zero fails, verdict " "%s (got %d, fails %s)" % ("PASS" if ok else "FAIL", N_CHECKS, EXPECTED, n_run, fails or "none")) print("--- PRIME.GONEK.PRICING.01 unconditional Gonek pricing: " "%d passed, %d failed ---" % (n_run - len(fails), len(fails))) return 0 if ok else 1 if __name__ == "__main__": raise SystemExit(run())