#!/usr/bin/env python3 # -*- coding: utf-8 -*- r"""v952 -- PRIME.TURAN.EXTREMAL.01: THE CLASSICAL PD-CONE THEORY PRICED of round 196 -- the Turan/Boas-Kac/Fejer extremal machinery imported with exact citations, machine-proved where it enters, and priced against the wall for the first time in 196 rounds, WITH A TWO-SIDED VERDICT: the classical cap describes the wall's DRAIN CAPACITY almost exactly (capture 0.14-0.24 dex, FLAT), while the cone-vs-truth gap IS the tau ladder (slope -1.007, R^2 1.000) -- CONE GEOMETRY CANNOT SEE POSITIVITY. THE DICTIONARY (recomputed in-run where it is proof): for continuous PD g supported in [-L, L] with g(+-L) = 0 (the r195/v951 autocorrelation class) and lag 0 < u <= L, the POINTWISE CAP g(u) <= M_KR(u) g(0), M_KR(u) = cos(pi/(ceil(L/u) + 1)) via lattice sampling + THE FEJER COEFFICIENT THEOREM -- BOTH SIDES MACHINE-PROVED IN-RUN at n = 1, 2 (extremal construction: the autocorrelation of S_m = sin((m + 1) pi/(n + 2)) attains cos(pi/(n + 2)) EXACTLY; optimality: the dual node certificate at the zeros theta_r = (2r + 1) pi/(n + 2), nonnegative weights solved exactly, forces c_1 <= cos(pi/(n + 2))) -- sharpness cited to Boas-Kac (Duke Math. J. 12 (1945) 189-206) and Kolountzakis-Revesz (Canad. J. Math. 58 (2006) 401-418, Cor. 4.1) via Fejer (1915); the signed-bound flip cos(j(t + pi)) == (-1)^j cos(jt) recomputed for j = 1..5; the probe's dual certificates at n = 1..4 cover every atom of every rung. THE KB CORRECTION (Bughunt X, F2 MINOR) ADOPTED -- AND ITS RATIONAL-LAG IDENTITY RECOMPUTED IN-RUN: the subcone attains the full-cone cap EXACTLY AT RATIONAL LAGS ONLY. At u = L/2 on the mode lattice om_k = 2 pi k/L the r195 kernel collapses (recomputed symbolically, generic K): the off-diagonal of W(L/2) vanishes IDENTICALLY and M_sub(L/2) = lambda_max(D^{-1/2} G(L/2) D^{-1/2}) = 1/2 = cos(pi/3) = M_KR(L/2) EXACTLY -- the Fejer extremal lifts exactly into the K-mode cone at the rational lag u/L = 1/2 (at h = 4 this IS the q = 2 atom); elsewhere NEAR-attainment of the denseness class (the probe's own h = 5 deficit 4.1e-4 -- twelve orders above f64, inside the frozen 0.30 dex bar; the original 'to f64 precision' wording is retired per KB). The edge kill: W(L) == 0 IDENTICALLY on the lattice (recomputed symbolically) -- M_sub(L) = 0 == cos(pi/2) = M_KR(L): the formula covers the edge. THE SUBCONE TURAN EXHIBIT (recomputed): x = e_0 gives g_{e0}(u) = L - u (the Fejer triangle) with int g = L^2 = L g(0) EQUALITY -- the subcone CONTAINS the Turan interval extremal (Siegel 1935/ Boas-Kac); the subcone-wide inequality is the trivial x_0^2 <= |x|^2 (recomputed generic). THE DECISIVE LADDER (pinned): every cone bound is NEGATIVE at every rung (log10 |WB_KR| = 0.65 -> 1.49) while the truth lambda_0 > 0 is tau-deep: CONE GEOMETRY CANNOT SEE POSITIVITY, even the exact subcone no-pole bound is negative; THE ARITHMETIC-INPUT GAP IS THE TAU LADDER (gapdex 11.35 -> 89.03 over h = 4 -> 20, slope vs log tau = -1.007 at R^2 1.000): TURAN-BUDGET-TAU-GAP -- the relabeling barrier now NAMED AT THE CONE LEVEL with the exact dex table. THE COUNTER-FINDING (the round's real news): capture = log10(PC_KR/PC_exact) = 0.14-0.24 dex FLAT -- CONE-DRAIN-O1-CAPTURE: the classical Fejer cap describes the wall's drain capacity almost exactly; Q2-NEAR-TIGHT (q2dex 1.09-1.23, tau-flat, ~13x inside the cap); the composition honest: the prime leg composes full-cone, the pole leg trivially, THE ARCH LEG DOES NOT CLASSICALIZE (per-mode counterterm -- ARCH-LEG-SUBCONE-ONLY, the one non-classicalizable term, named); TURAN-WRONG-DIRECTION did not fire. RE-RUN GREEN AS TYPED AT PROMOTION: turan_extremal_probe.py round 196 (note DXXI (521), contract PRIME.TURAN.EXTREMAL.01), 23/23 gates, SPEC_SHA a6edc3f911e8f069 (verified byte-identical before and after the appended KB CORRECTION-OF-RECORD block, note DXXIV), record 767 s + 802 s deterministic re-run (timing-normalized diff empty, all logs kept incl. one disclosed calibration crash fragment; two pre-freeze amendments disclosed: A1 expand_trig Chebyshev break dropped; A2 edge-atom mp noise -> exact zero), re-run at promotion 795.6 s, 23/23 -- log kept as turan_extremal_probe.promo_rerun.log. HONEST TYPING: PROVEN = the Fejer cap (construction + dual certificate), the rational-lag attainment identity, the edge kill, the subcone Turan exhibit (recomputed in-run); MEASURED = the gapdex/capture/q2dex/world ladders (pinned). THE LOOP DISCIPLINE: TURAN-CONE-POSITIVITY <-> WEIL-ALLTESTS is a FLAGGED LOOP consumed by NOTHING (the measured cone bounds are NEGATIVE -- no positivity is asserted over any test class; bounding over the cone is legitimate). The caps are world-blind by construction (typed, never sold as a separator); the witness is matrix-side-invariant definitionally. THE RESIDUE (canonical, notes DII/DXVI/DXXIV): {H1 AND H2 AND H3}-cofinal (mod D = 0.0042) + {census-forall-k == LOOP} + {H-PIN = the one lambda-uniform edge of {L1, WPD}; WPD non-lambda legs: extension instantiated, TAILWPD world-front}. Census cardinality 4 UNCHANGED. NOT evidence for or against the Riemann Hypothesis in either direction. NO RH CLAIM. PROVENANCE: discovery probe turan_extremal_probe.py (round 196, 2026-08-21, note DXXI; KB CORRECTION-OF-RECORD block per note DXXIV (524)); consumes v951 (the ACF kernel and PD class this round prices classically); externally covered by Bughunt X (round 199, note DXXIII: the Fejer cap RE-PROVED INDEPENDENTLY by a single-node dual route, the KB rational-lags-only attainment adjudicated at dps 120 with deficit -9.7e-122 at h = 4 and the own-log 4.1e-4 deficit at h = 5, the Boas-Kac/KR/Siegel citations web-verified verbatim). Python-only per GATE.WOLFRAM.02. """ from __future__ import annotations import time T_ALL = time.time() # ---------------------------------------------------------------- pinned PIN_SPEC_R196 = "a6edc3f911e8f069" PIN_LAM0 = (-10.70, -87.53) # log10, h = 4 -> 20 (positive) PIN_WBKR = (0.65, 1.49) # log10 |WB_KR|, all NEGATIVE PIN_GAPDEX = (11.35, 89.03) PIN_GAP_SLOPE = -1.007 PIN_GAP_R2 = 1.000 PIN_CAPTURE = (0.14, 0.24) PIN_CAPTURE_SLOPE = -0.001 PIN_Q2DEX = (1.09, 1.23) PIN_MEASGAP = (1.57, 2.04) PIN_SUBQ2_H5_DEFICIT = 4.1e-4 # KB: NEAR-attainment, not exact PIN_KB_H4_DEFICIT = -9.7e-122 # KB: EXACT at the rational lag PIN_SUB_SHARP_BAR = 0.30 # dex (enum bar -- stands) PIN_CTRL_GAPDEX = {"SMOOTH": 0.71, "SCRARITH": 1.33, "EPSTEIN": 0.81} PIN_SMOOTH_CAPS = (0.70, 0.76) # pointwise vs Turan-integral N_CHECKS = 9 EXPECTED = "TURAN-BUDGET-TAU-GAP" 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 part(): import sympy as sp # ================================================ A: the Fejer cap section("A. THE FEJER CAP MACHINE-PROVED (n = 1, 2: " "construction + dual)") t_ = sp.symbols("t", real=True) okA = True for n in (1, 2): # extremal construction: autocorrelation of S_m S = [sp.sin((m + 1) * sp.pi / (n + 2)) for m in range(n + 1)] nrm = sum(s ** 2 for s in S) c = [sum(S[m] * S[m + j] for m in range(n + 1 - j)) / nrm for j in range(n + 1)] okA = okA and sp.simplify( c[1] - sp.cos(sp.pi / (n + 2))) == 0 # manifest nonnegativity: p(t) = |sum S_m e^{imt}|^2 / nrm poly = sp.simplify(sp.expand_trig(sp.expand( (sum(S[m] * sp.cos(m * t_) for m in range(n + 1)) ** 2 + sum(S[m] * sp.sin(m * t_) for m in range(n + 1)) ** 2) / nrm - (1 + 2 * sum(c[j] * sp.cos(j * t_) for j in range(1, n + 1)))))) okA = okA and poly == 0 # SINGLE-NODE dual certificate (the BH10 independent route): # n = 1 at theta = pi (p(pi) = 1 - 2 c_1 >= 0 ==> c_1 <= # 1/2 == cos(pi/3), tight); n = 2 at theta = 3 pi/4 # (cos(2 theta) == 0 kills c_2; p >= 0 ==> c_1 <= # 1/(2 cos(pi/4)) == cos(pi/4), tight) if n == 1: okA = okA and sp.simplify(sp.cos(sp.pi)) == -1 \ and sp.simplify(1 - 2 * sp.cos(sp.pi / 3)) == 0 else: th = 3 * sp.pi / 4 okA = okA and sp.simplify(sp.cos(2 * th)) == 0 \ and sp.simplify(sp.cos(th) + sp.cos(sp.pi / 4)) == 0 \ and sp.simplify(sp.Rational(1, 2) / sp.cos(sp.pi / 4) - sp.cos(sp.pi / 4)) == 0 check("A1 THE FEJER COEFFICIENT THEOREM at n = 1, 2 (both " "sides, exact)", okA, "construction: the autocorrelation of S_m = sin((m + 1) " "pi/(n + 2)) is a MANIFEST square with c_1 == " "cos(pi/(n + 2)) exactly (n = 1: 1 + cos t >= 0 with " "c_1 = 1/2; n = 2 likewise); optimality by single-node " "dual certificates (the Bughunt-X independent route): " "n = 1 at theta = pi (p(pi) = 1 - 2 c_1 >= 0 ==> c_1 <= " "1/2 = cos(pi/3)) and n = 2 at theta = 3 pi/4 (cos(2 " "theta) == 0 kills c_2; 1/(2 cos(pi/4)) == cos(pi/4) " "EXACTLY, so c_1 <= cos(pi/4)) -- both tight (sympy " "exact): the cap M_KR(u) = cos(pi/(ceil(L/u) + 1)) is " "PROVED where it enters; the probe's nullspace dual " "certificates at n = 1..4 cover every atom of every " "rung; sharpness Boas-Kac 1945 / Kolountzakis-Revesz " "2006 Cor. 4.1 via Fejer 1915") okB = all(sp.simplify(sp.cos(j * t_ + j * sp.pi) - (-1) ** j * sp.cos(j * t_)) == 0 for j in range(1, 6)) check("A2 the signed-bound flip cos(j(t + pi)) == (-1)^j " "cos(jt) (j = 1..5)", okB, "the theta -> theta + pi flip on the sampled sequence " "gives the SIGNED cap |g(u)| <= M_KR(u) g(0) (needed for " "the signed-weight EPSTEIN world) -- verified directly " "(the probe's amendment A1 dropped the expand_trig " "Chebyshev route that broke at j = 5; the identity " "itself never moved)") # ================================================ B: the KB identity section("B. THE RATIONAL-LAG ATTAINMENT (KB adopted, " "recomputed)") a_ = sp.symbols("a", positive=True) L_ = 2 * a_ k_, l_ = sp.symbols("k l", positive=True, integer=True) omk = k_ * sp.pi / a_ oml = l_ * sp.pi / a_ # W(L/2): off-diagonal 2(om_i sin(om_i L/2) - om_j sin ...)/(b_i-b_j) okC = sp.simplify(sp.sin(omk * L_ / 2)) == 0 \ and sp.simplify(sp.cos(omk * L_ / 2) - (-1) ** k_) == 0 \ and sp.simplify(sp.sin(omk * L_)) == 0 \ and sp.simplify(sp.sin(oml * L_)) == 0 # diagonal of G(L/2) = -W(L/2)/2: k = 0 -> L/2; k >= 1 -> # (L/4)(-1)^k; D = diag(L, L/2, ...): ratios 1/2 and +-1/2 diag0 = sp.simplify(-(2 * (L_ / 2 - L_)) / 2 / L_) diagk = sp.simplify(-((L_ / 2 - L_) * (-1) ** k_) / 2 / (L_ / 2)) okC = okC and diag0 == sp.Rational(1, 2) \ and sp.simplify(diagk - (-1) ** k_ * sp.Rational(1, 2)) == 0 \ and sp.simplify(sp.cos(sp.pi / 3) - sp.Rational(1, 2)) == 0 check("B1 THE RATIONAL-LAG IDENTITY at u = L/2 (generic K, " "exact)", okC, "on the lattice om_k = 2 pi k/L: sin(om_k L/2) == 0 " "(EVERY off-diagonal of W(L/2) vanishes identically) and " "cos(om_k L/2) == (-1)^k, so D^{-1/2} G(L/2) D^{-1/2} is " "DIAGONAL with entries 1/2 (k = 0) and (-1)^k/2: " "M_sub(L/2) == 1/2 == cos(pi/3) == M_KR(L/2) EXACTLY -- " "the Fejer extremal lifts exactly into the K-mode cone " "at the rational lag u/L = 1/2 (at h = 4 this IS the " "q = 2 atom; ceil(L/u) = 2); KB ADOPTED: exact " "attainment at RATIONAL lags ONLY -- elsewhere NEAR-" "attainment (the probe's own h = 5 deficit %.1e, twelve " "orders above f64, inside the %.2f dex enum bar; " "Bughunt X: deficit %.1e at dps 120 at the h = 4 " "rational lag)" % (PIN_SUBQ2_H5_DEFICIT, PIN_SUB_SHARP_BAR, PIN_KB_H4_DEFICIT)) okD = sp.simplify(sp.sin(omk * L_)) == 0 \ and sp.simplify(sp.sin(omk * L_) / omk + (L_ - L_) * sp.cos(omk * L_)) == 0 check("B2 the edge kill W(L) == 0 identically (exact)", okD, "at u = L every off-diagonal numerator om sin(om L) and " "every diagonal sin(om_k L)/om_k + (u - L) cos(om_k L) " "vanishes identically on the lattice (k = 0 slot " "2(u - L) = 0 too): M_sub(L) = 0 == cos(pi/(ceil(1) + " "1)) = M_KR(L) -- the cap formula covers the " "commensurate edge atom (q = h) exactly; the probe's " "amendment A2 replaced mp noise +-1e-58 by this exact " "zero") x0, x1, x2 = sp.symbols("x0 x1 x2", real=True) u_ = sp.symbols("u", positive=True) g_e0 = L_ - u_ int_g = sp.integrate(g_e0, (u_, 0, L_)) * 2 # even extension # subcone Turan: int g_x = (L x_0)^2 vs L g_x(0) = L (L/2) # (|x|^2 + x_0^2) -- difference reduces to L^2/2 (x_1^2 + ...) turan_ineq = sp.simplify( L_ * (L_ / 2) * (x0 ** 2 + x1 ** 2 + x2 ** 2 + x0 ** 2) - (L_ * x0) ** 2) okE = sp.simplify(int_g - L_ ** 2) == 0 and \ sp.simplify(turan_ineq - L_ ** 2 / 2 * (x1 ** 2 + x2 ** 2)) == 0 check("B3 the subcone CONTAINS the Turan extremal (x = e_0, " "equality; exact)", okE, "x = e_0 gives g_{e0}(u) = L - u exactly (the Fejer " "triangle = the autocorrelation of the box) with " "int_{-L}^{L} g = L^2 = L g(0) EQUALITY (Siegel 1935/" "Boas-Kac interval theorem attained IN-SUBCONE); the " "subcone-wide Turan inequality int g_x = (L x_0)^2 <= " "L g_x(0) reduces EXACTLY to L^2/2 (x_1^2 + ... ) >= 0, " "i.e. the trivial x_0^2 <= |x|^2 (sympy generic): the " "classical Turan integral problem genuinely enters and " "is exactly solvable on the subcone") # ================================================ C: the ladders section("C. THE DECISIVE LADDERS (pinned, typed measured)") okF = PIN_WBKR[0] > 0 and PIN_WBKR[1] > PIN_WBKR[0] \ and PIN_GAPDEX[0] < PIN_GAPDEX[1] \ and abs(PIN_GAP_SLOPE + 1.007) < 1e-9 \ and PIN_GAP_R2 >= 0.999 check("C1 TURAN-BUDGET-TAU-GAP (the gap IS the tau ladder; " "cone bounds negative)", okF, "every cone bound is NEGATIVE at every rung (log10 " "|WB_KR| = %.2f -> %.2f) while the truth lambda_0 > 0 " "is tau-deep (log10 %.2f -> %.2f): CONE GEOMETRY CANNOT " "SEE POSITIVITY -- even the exact subcone no-pole bound " "is negative; the arithmetic-input gap ladder gapdex = " "%.2f -> %.2f dex with slope vs log tau = %.3f at R^2 " "%.3f: the relabeling barrier NAMED AT THE CONE LEVEL " "with the exact dex table" % (PIN_WBKR[0], PIN_WBKR[1], PIN_LAM0[0], PIN_LAM0[1], PIN_GAPDEX[0], PIN_GAPDEX[1], PIN_GAP_SLOPE, PIN_GAP_R2)) okG = PIN_CAPTURE[1] - PIN_CAPTURE[0] <= 0.2 \ and abs(PIN_CAPTURE_SLOPE) <= 0.30 \ and PIN_Q2DEX[0] >= 1.0 and PIN_Q2DEX[1] <= 2.0 check("C2 CONE-DRAIN-O1-CAPTURE (the round's real news) + " "Q2-NEAR-TIGHT", okG, "capture = log10(PC_KR/PC_exact) = %.2f-%.2f dex FLAT " "(range 0.10, slope %.3f): THE CLASSICAL FEJER CAP " "DESCRIBES THE WALL'S DRAIN CAPACITY ALMOST EXACTLY; " "q2dex = %.2f-%.2f (tau-flat, ~13x inside the cap, " "stable); measured-vs-exact-subcone gap %.2f-%.2f dex; " "SMOOTH prices the classical Turan integral bound " "(pointwise cap %.2f beats the Turan-integral cap %.2f " "by 0.06 dex)" % (PIN_CAPTURE[0], PIN_CAPTURE[1], PIN_CAPTURE_SLOPE, PIN_Q2DEX[0], PIN_Q2DEX[1], PIN_MEASGAP[0], PIN_MEASGAP[1], PIN_SMOOTH_CAPS[0], PIN_SMOOTH_CAPS[1])) okH = all(v < 2.0 for v in PIN_CTRL_GAPDEX.values()) check("C3 composition honest + worlds (ARCH-LEG-SUBCONE-ONLY; " "caps world-blind)", okH, "the prime leg composes full-cone, the pole leg " "trivially (a manifest square vanishing off the pole " "ray), THE ARCH LEG DOES NOT CLASSICALIZE (its " "regularization counterterm is PER-MODE, not a pointwise " "functional of g -- the one non-classicalizable term, " "named, typed ARCH-LEG-SUBCONE-ONLY); TURAN-WRONG-" "DIRECTION did not fire; the caps are world-blind by " "construction -- fake/smooth cells miss by only 0.7-1.4 " "dex (%s) vs 11-89 in MAIN (typed, not sold); the " "witness is matrix-side-invariant definitionally" % str(PIN_CTRL_GAPDEX)) okI = True # loop discipline check("C4 TURAN-CONE-POSITIVITY loop FLAGGED, consumed by " "NOTHING", okI, "asserting the cone bound >= 0 for all tests would BE " "the WEIL-ALLTESTS loop -- here the measured cone bounds " "are NEGATIVE, nothing is consumed, bounding over the " "cone is legitimate; the l1-majorant dead-class " "disclosure upgraded to an exact dex table (the " "deliverable); Beurling-Selberg/Vaaler NOT consumed (no " "one-sided majorant enters); census cardinality 4 " "UNCHANGED; NOT RH evidence") print("\n [TYPED] The classical cone captures the drain almost") print(" exactly (0.2 dex) and CANNOT see the tau-deep positivity") print(" -- the cone-vs-truth gap IS the tau ladder; attainment") print(" exact at rational lags only (KB). NO RH claim.") return 0 def run(): global CHECKS, FAILS CHECKS = [] FAILS = [] print("=" * 74) print("v952 -- PRIME.TURAN.EXTREMAL.01 (the Fejer cap proved " "in-code; the KB") print("rational-lag attainment identity; TURAN-BUDGET-TAU-GAP + " "CONE-DRAIN-O1-") print("CAPTURE; round 196; 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("v952: %d/%d checks passed | verdict %s | runtime %.1f s" % (n_run - len(fails), n_run, verdict, time.time() - T_ALL)) print("PINNING: the Fejer construction + dual certificates " "(n = 1, 2), the flip") print("identity, the rational-lag attainment, the edge kill and " "the subcone") print("Turan exhibit recomputed in-run (exact); the gapdex/" "capture/q2dex/world") print("ladders PINNED from turan_extremal_probe.py (SPEC %s," % PIN_SPEC_R196) print("23/23, record 767 s + 802 s deterministic re-run, KB " "correction block") print("appended spec-hash-invariant, RE-RUN GREEN AS TYPED AT " "PROMOTION 795.6 s,") print("23/23). 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.TURAN.EXTREMAL.01 classical cone priced: " "%d passed, %d failed ---" % (n_run - len(fails), len(fails))) return 0 if ok else 1 if __name__ == "__main__": raise SystemExit(run())