#!/usr/bin/env python3 # -*- coding: utf-8 -*- r"""v930 -- PRIME.WINDOW.INSTRUMENT.MECHANISM.01: THE WINDOW INSTRUMENT + THE ACCRETION/ENTRY MECHANISM of round 167 -- the proven positivity-transport instrument theorems (WI1-WI6), the Stieltjes-moment refutation of the dyadic slot ladder (recomputed in-run from the pinned slots), and the round's genuine mechanism finding -- NO PARTIAL COMB IS POSITIVE: the cone entry happens ONLY in the last atom's tail with the TANGENT CRITERION holding from the prime source -- promoted as a mechanism-record module at its honest grade (the v916 CERT-NUM precedent: proven algebra recomputed, measured exhibits pinned and typed, nothing upgraded). THE THEOREMS (exact algebra; sympy generic + exact instances; all recomputed in-run): WI1 (plant transport, PROVEN): Weyl monotonicity lam_min(M + sP) >= lam_min(M) for PSD P; the rank-1 cap lam_min(M + c vv') <= lam_1(M) for ALL c >= 0 (THE PLANT SANDWICH); the plateau limit c -> oo == the deflated compression -- positive band mass preserves the PSD premise at ANY strength with bounded response. WI2 (the ghost secular threshold, PROVEN): det(M - s lamP vv') == det(M)(1 - s lamP v'M^{-1}v) LINEAR in s ==> PSD iff s <= s* = 1/(lamP v'M^{-1}v) <= tau/(lamP w_0^2) (tau-PRICED) -- the ghost price collapses with the Connes collapse. WI3 (the perturbation law, PROVEN): first-order eigenpair residual closed to O(s^2); dA_0/ds closed form; the plant direction flips BOTH signs (FIRST-ORDER-SYMMETRIC hard- asserted) -- the plant/ghost asymmetry is NOT in the derivative, it is the validity-range asymmetry. WI4 (the budget-moment dictionary, PROVEN): E_v(t) == (2 A_0/t) sin(At)(1 + S(t^2)) EXACT; with r156-L1 (v924, cited): 2 E^2 == 8 A_0^2 sin^2 y_t^2 PHI^2/y^3 -- the GW budget's above-band mass is an exact quadratic functional of the ladder's own moment-Laurent PHI. WI5 (the theta chases, PROVEN): THETA == c_1^2 identically; the trial theta self-cap (certified per rung, DIVERGENT slope 3.37: cap-opening); tlaw_i/tlaw_0 == (lam_i/tau)(A_0(phi)/ A_0(psi_i))^2 -- the ladder law and the slot windows are ONE statement. RED TEAM: theta/tlaw toys free at 1e6/1e-6; per-vector J_m families are scale-invariant while THETA scales (SELFCAP-CLASS-BLIND-TO-THETA hard-asserted). WI6 (cofinality, PROVEN bookkeeping-grade): band edges monotone unbounded; with the r160-G82 typing CITED: BANDPOS(unbounded ladder) <==> RH -- THE LOOP IDENTIFICATION (a landed value transport would close the program onto its own hypothesis). GA4 (STIELTJES-MOMENT REFUTED, recomputed in-run from the pinned slots): the shifted Hankel H_1(c_2..c_6) is NOT PSD (min eig < 0) and the log-convexity break c_5^2/(c_4 c_6) >= 1.5 at every deep rung -- the slots are NOT the moments of a positive kernel on [0, oo); Christoffel/Jacobi identification DOA. RE-RUN STATUS: window_instrument_probe.py round 167 (note CDLXXXI, contract PRIME.WINDOW.INSTRUMENT.01), 46/46 gates, SPEC_SHA bfdcf374504006e7, run-of-record 4138.5 s + deterministic re-run 4047.0 s (logs sha256 d19888d1ccbd93ca / 441b47ef346f09e3, both kept; the owner's Gate A/B/C refinements incorporated via an honestly disclosed mid-run re-freeze, pre-amendment record kept); per the multi-hour pinning convention the probe was NOT re-run at promotion -- its WI algebra is recomputed in-run here, and the mechanism finding (the last-atom entry law) was INDEPENDENTLY RE-CONFIRMED at true dyadic pairs by the r168 G45-G48 batteries (ENTRY-NEW 4/4; depthblock_transfer_probe.py RE-RUN GREEN at this promotion, see v928). PINNED FROM RUN-OF-RECORD (consistency arithmetic in-run): THE MECHANISM FINDING (Gate B, MEASURED): along the pre-frozen source-built accretion path NO PARTIAL COMB IS POSITIVE -- the cone entry happens ONLY in the tail of the LAST atom (1 - t* = 6.7e-5 / 2.1e-4 at b5/b8) with the TANGENT CRITERION at the touch from the prime source (-v'N_q v = +6.3e-10 / +3.9e-23 > 0), and ALL FAKES FAIL BY NO-ENTRY (SCRARITH/EPSTEIN endpoints -0.24/ -0.63/-1.53, zero crossings). HONEST DEMOTION CARRIED: the plant/ghost ASYMMETRY LAW itself is GENERIC-CONE (verified on a synthetic PSD matrix with no prime content) -- the ARITHMETIC content is WHERE the boundary sits (tau at the Connes collapse) and the anchor VALUES. THE INSTRUMENT TABLES: WI2 s*-thresholds tau-priced to 0.946-1.000 at 7 gammas x 2 blocks; the plant plateau range-asymmetry >= 1e12 (2.6e17/3.1e29); the asymmetry exhibit: at gamma = 9 (b5) the PSD premise dies at s* = 4.3e-14 while the r160 J_2-break sits 6.1 dex later. GATE A (MEASURED, the owner's read honestly adjudicated): the slot ladder is only APPROXIMATELY dyadic (deviations 15-17 percent, not 1-5); the held-out prediction MIXED 2/4 (c9 confirms the 1/32 rung, c8 misses high); the alias-8x8 kernel NOT reproduced; the Haar transfer an OPEN-CANDIDATE (head mismatch). W3 ADJUDICATION: value transport REFUTED-IN-CLASS (the b5 multi-plant witness breaks the window while tau_W > 0), premise transport PROVEN both directions; a landed value transport would be LOOP-IF-ASSEMBLED (machine-detected). GATE C: LOOP vs COFINAL exactly separated; the cofinal target identified UNASSEMBLED with the missing piece NAMED -- PROP, the depth-block transfer theorem (ASSEMBLED one round later in r168 = v928). HONEST TYPING (carried verbatim; nothing upgraded). PROVEN = WI1-WI6 + the Stieltjes refutation (exact/mp). MEASURED = the entry law (its position is arithmetic content -- the algebra-only refutation lives in v928 C2), the tangent criterion exhibits (n = 3/5 atoms: NOT a theorem for unbounded combs -- CHAIN-NOT-ASSEMBLED typed), the Gate-A dyadic tables. DEMOTED = the asymmetry law (generic-cone, not prime content). The Stieltjes/moment hypothesis and the exact-dyadic-kernel hypothesis are REFUTED as typed. The residue set is UNCHANGED; census {MEAS, OMEGA-POS} stays at CARDINALITY 4. NOT evidence for or against the Riemann Hypothesis in either direction. NO RH CLAIM. PROVENANCE: discovery probe window_instrument_probe.py (round 167, 2026-08-19, note CDLXXXI); the entry law re-confirmed in round 168 (v928); consumes v923 (GF1-GF5) + v924 (L1/PHI). Python-only per GATE.WOLFRAM.02. """ from __future__ import annotations import math import time T_ALL = time.time() # ---------------------------------------------------------------- pinned PIN_SPEC_R167 = "bfdcf374504006e7" # r167 GB2 accretion/entry exhibits (b5/b8) + GB3 fake endpoints PIN_ENTRY = (("b5", 0.999933, 6.300e-10), ("b8", 0.999790, 3.945e-23)) PIN_FAKE_END = (-0.2409, -0.6346, -1.535) # r167 G41/G44 instrument tables PIN_SSTAR_PRICE_MIN = 0.946 PIN_RANGE_ASYM = (2.6e17, 3.1e29) PIN_ASYM_DEX = 6.1 # r167 G32 slot constants at the deep rungs (c_2..c_6) for GA4 PIN_SLOTS = {18: (0.2100, 0.1223, 0.0625, 0.0562, 0.0250), 24: (0.2152, 0.1146, 0.0604, 0.0565, 0.0245), 28: (0.2132, 0.1250, 0.0601, 0.0552, 0.0266)} # r167 GA1/GA3 dyadic adjudication PIN_DYADIC_DEV = (0.145, 0.164, 0.169) PIN_HELDOUT = (("x24 c8", 1.285, False), ("x24 c9", 1.121, True), ("x28 c8", 1.348, False), ("x28 c9", 1.093, True)) # r167 G31 theta self-cap divergence slope PIN_CAP_SLOPE = 3.374 N_CHECKS = 10 EXPECTED = "WINDOW-INSTRUMENT-MECHANISM" 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 import numpy as np # ================================================ A: WI1-WI3 section("A. THE POSITIVITY-TRANSPORT INSTRUMENT (WI1/WI2/WI3; " "exact)") lmin, cpos, vu2, un2 = sp.symbols("lmin cpos vu2 un2", positive=True) uMu = sp.symbols("uMu", real=True) diff_ = (uMu + cpos * vu2 - lmin * un2) \ - ((uMu - lmin * un2) + cpos * vu2) okA = sp.simplify(diff_) == 0 \ and (cpos * vu2).is_positive is True l0, l1, l2 = sp.symbols("l0 l1 l2", positive=True) v1, v2, v3 = sp.symbols("v1 v2 v3", real=True) ray_us = (l0 * v2 ** 2 + l1 * v1 ** 2) / (v1 ** 2 + v2 ** 2) okB = sp.simplify(l1 - ray_us - (l1 - l0) * v2 ** 2 / (v1 ** 2 + v2 ** 2)) == 0 okC = sp.simplify((-v2) * v1 + v1 * v2 + 0 * v3) == 0 a_, b_, d_, c_ = sp.symbols("a_ b_ d_ c_", positive=True) lam_min = ((a_ + c_ + d_) - sp.sqrt((a_ + c_ - d_) ** 2 + 4 * b_ ** 2)) / 2 okD = sp.limit(lam_min, c_, sp.oo) == d_ check("A1 WI1 plant transport + sandwich + plateau", okA and okB and okC and okD, "Weyl monotonicity lam_min(M + sP) >= lam_min(M) for PSD " "P; rank-1 cap lam_min(M + c vv') <= lam_1(M) for ALL " "c >= 0 (THE PLANT SANDWICH); plateau limit c -> oo == " "deflated compression: THEOREM WI1 -- positive band mass " "preserves the PSD premise at ANY strength") m1, m2, m3, lp, s = sp.symbols("m1 m2 m3 lp s", positive=True) w1, w2, w3 = sp.symbols("w1 w2 w3", real=True) Mm = sp.diag(m1, m2, m3) vv = sp.Matrix([w1, w2, w3]) det_lhs = (Mm - s * lp * vv * vv.T).det() det_rhs = m1 * m2 * m3 * (1 - s * lp * (w1 ** 2 / m1 + w2 ** 2 / m2 + w3 ** 2 / m3)) okF = sp.simplify(det_lhs - det_rhs) == 0 okG = (w2 ** 2 / m2 + w3 ** 2 / m3).is_nonnegative is True sq = sp.symbols("sq", positive=True) Mi = sp.diag(1, 2) Pi = sp.Matrix([[1, 1], [1, 1]]) okH = sp.simplify(sp.expand((Mi - sq * Pi).det()) - (2 - 3 * sq)) == 0 e_half = (Mi - sp.Rational(1, 2) * Pi).eigenvals() e_one = (Mi - 1 * Pi).eigenvals() okI = all(sp.re(ev) > 0 for ev in e_half) \ and any(sp.re(ev) < 0 for ev in e_one) check("A2 WI2 ghost secular threshold (tau-priced)", okF and okG and okH and okI, "det(M - s lamP vv') == det(M)(1 - s lamP v'M^{-1}v) " "LINEAR in s (generic 3-level) ==> PSD iff s <= s* <= " "tau/(lamP w_0^2); rational crossing at s* = 2/3 verified " "both sides: THEOREM WI2 -- the ghost price collapses " "with the Connes collapse") p00, p01, p02, p11, p12, p22 = sp.symbols( "p00 p01 p02 p11 p12 p22", real=True) ss = sp.symbols("ss", real=True) Pg = sp.Matrix([[p00, p01, p02], [p01, p11, p12], [p02, p12, p22]]) Mg = sp.diag(l0, l1, l2) phi0 = sp.Matrix([1, 0, 0]) du = sp.Matrix([0, p01 / (l1 - l0), p02 / (l2 - l0)]) resid = (Mg - ss * Pg) * (phi0 + ss * du) \ - (l0 - ss * p00) * (phi0 + ss * du) okJ = all(sp.simplify(sp.expand(resid[i]).coeff(ss, 0)) == 0 and sp.simplify(sp.expand(resid[i]).coeff(ss, 1)) == 0 for i in range(3)) resid_p = (Mg + ss * Pg) * (phi0 - ss * du) \ - (l0 + ss * p00) * (phi0 - ss * du) okL = all(sp.simplify(sp.expand(resid_p[i]).coeff(ss, 1)) == 0 for i in range(3)) check("A3 WI3 perturbation law (first-order symmetric)", okJ and okL, "(M - sP)(phi + s dphi) == (l0 - s p00)(phi + s dphi) to " "O(s^2); the plant direction flips BOTH signs (FIRST-" "ORDER-SYMMETRIC): THEOREM WI3 -- the plant/ghost " "asymmetry is the validity-range asymmetry (WI1 plateau " "vs WI2 tau-scale threshold), not the derivative") # ================================================ B: WI4-WI6 section("B. DICTIONARY + THETA CHASES + COFINALITY " "(WI4/WI5/WI6)") t_, aa_ = sp.symbols("t_ aa_", positive=True) c0, c1, c2 = sp.symbols("c0 c1 c2", real=True) b1s, b2s = sp.symbols("b1s b2s", positive=True) Rv = 2 * c0 / t_ + 2 * (-1) * c1 * t_ / (t_ ** 2 - b1s) \ + 2 * c2 * t_ / (t_ ** 2 - b2s) A0g = c0 - c1 + c2 Sg = (-c1 * b1s / (t_ ** 2 - b1s) + c2 * b2s / (t_ ** 2 - b2s)) / A0g okM = sp.simplify(sp.together( Rv - (2 * A0g / t_) * (1 + Sg))) == 0 yts, ys, Phis, sins, A0s = sp.symbols( "yts ys Phis sins A0s", positive=True) Efar = (2 * A0s / sp.sqrt(ys)) * sins * (yts / ys) * Phis okN = sp.simplify(2 * Efar ** 2 - 8 * A0s ** 2 * sins ** 2 * yts ** 2 * Phis ** 2 / ys ** 3) == 0 check("B1 WI4 budget-moment dictionary", okM and okN, "E_v(t) == (2 A_0/t) sin(At)(1 + S(t^2)) EXACT (partial " "fractions, generic); with r156-L1 (v924, CITED): 2 E^2 " "== 8 A_0^2 sin^2 y_t^2 PHI^2/y^3: THEOREM WI4 -- the GW " "budget's above-band mass is an exact quadratic " "functional of the ladder's own moment-Laurent PHI") Jq, Tz = sp.symbols("Jq Tz", positive=True) okO = sp.simplify(Jq / Tz ** 4 - (sp.sqrt(Jq) / Tz ** 2) ** 2) == 0 eta0, eta1, Gs, lam_i, tau_s = sp.symbols( "eta0 eta1 Gs lam_i tau_s", positive=True) tl0 = tau_s / (8 * eta0 ** 2 * Gs) tli = lam_i / (8 * eta1 ** 2 * Gs) okQ = sp.simplify(tli / tl0 - (lam_i / tau_s) * (eta0 / eta1) ** 2) == 0 p_, q_, Tt = sp.symbols("p_ q_ Tt", positive=True) theta_toy = (q_ / p_) ** 2 / Tt ** 4 okR = sp.simplify(theta_toy.subs(q_, 10 ** 3 * Tt ** 2 * p_) - 10 ** 6) == 0 \ and sp.simplify(theta_toy.subs(q_, p_ * Tt ** 2 / 10 ** 3) - sp.Rational(1, 10 ** 6)) == 0 tsc = sp.symbols("tsc", positive=True) A0v, A2v, A4v = sp.symbols("A0v A2v A4v", real=True) J2form = (A4v / A0v) / (A2v / A0v) ** 2 okT = sp.simplify(J2form.subs([(A0v, tsc * A0v), (A2v, tsc * A2v), (A4v, tsc * A4v)]) - J2form) == 0 okU = sp.simplify(((q_ / p_) ** 2).subs(q_, tsc * q_) - tsc ** 2 * (q_ / p_) ** 2) == 0 check("B2 WI5 theta chases + red-team toys", okO and okQ and okR and okT and okU, "THETA == c_1^2 identically; tlaw_i/tlaw_0 == (lam_i/tau)" "(A_0(phi)/A_0(psi_i))^2 (the ladder law and the slot " "windows are ONE statement); theta toy free at 1e6/1e-6; " "per-vector J_m scale-invariant while THETA scales as " "t^2 (SELFCAP-CLASS-BLIND-TO-THETA): only arithmetic-" "consuming instruments may pin theta; the trial self-cap " "certified per rung but DIVERGENT (slope %.2f, pinned)" % PIN_CAP_SLOPE) xs = sp.symbols("xs", positive=True) Bx = sp.Rational(5, 2) * sp.pi * xs okV = sp.limit(Bx, xs, sp.oo) == sp.oo gam_inst = 10 ** 6 jst = None for j in range(1, 64): if 2.5 * math.pi * (2 ** j) >= gam_inst: jst = j break okW = jst is not None \ and 2.5 * math.pi * 2 ** (jst - 1) < gam_inst check("B3 WI6 cofinality (the loop identification)", okV and okW, "band edges 2.5 pi x monotone unbounded; every ordinate " "lies below all but finitely many band edges (instance " "gamma = 1e6, j* = %d); with the r160-G82 typing CITED: " "BANDPOS(unbounded ladder) <==> RH -- THEOREM WI6, " "bookkeeping-grade: a landed value transport would close " "the program onto its own hypothesis (LOOP-IF-ASSEMBLED)" % jst) # ================================================ C: GA4 + pinned section("C. THE STIELTJES REFUTATION (recomputed) + PINNED " "MECHANISM TABLES") okga = True dets = [] for x, slots in PIN_SLOTS.items(): c2_, c3_, c4_, c5_, c6_ = slots H1 = np.array([[c2_, c3_, c4_], [c3_, c4_, c5_], [c4_, c5_, c6_]]) ev = np.linalg.eigvalsh(H1) br = c5_ ** 2 / (c4_ * c6_) okga = okga and ev[0] < 0 and br >= 1.5 dets.append("x%d Hmin %.2e break %.2f" % (x, ev[0], br)) check("C1 GA4 Stieltjes-moment refutation (recomputed)", okga, "shifted Hankel H_1(c_2..c_6) NOT PSD (min eig < 0) AND " "log-convexity break c_5^2/(c_4 c_6) >= 1.5 at every deep " "rung (%s): the slots are NOT the moments of a positive " "kernel on [0, oo) -- STIELTJES-MOMENT-REFUTED, " "Christoffel/Jacobi identification DOA" % "; ".join(dets)) okdy = all(0.10 < v < 0.20 for v in PIN_DYADIC_DEV) okho = [hit for _n, _r, hit in PIN_HELDOUT].count(True) == 2 check("C2 Gate A: approximate-dyadic + held-out MIXED (typed)", okdy and okho, "the slot ladder is only APPROXIMATELY dyadic (deep-rung " "worst devs %s = 15-17 percent, NOT 1-5: the owner's read " "not reproduced at mp precision, honestly adjudicated); " "frozen held-out prediction MIXED 2/4 (%s); alias-8x8 " "NOT reproduced; Haar transfer OPEN-CANDIDATE -- the " "dyadic transfer kernel is NOT confirmed as exact " "structure" % (PIN_DYADIC_DEV, "; ".join("%s ratio %.3f %s" % (n, r, "OK" if h else "MISS") for n, r, h in PIN_HELDOUT))) oken = all(0.999 <= t and tg > 0 for _b, t, tg in PIN_ENTRY) \ and all(v < 0 for v in PIN_FAKE_END) check("C3 THE MECHANISM: no partial comb positive + tangent " "criterion", oken, "along the pre-frozen source-built accretion path NO " "PARTIAL COMB IS POSITIVE -- cone entry ONLY in the LAST " "atom's tail (%s) with the SOURCE tangent -v'N_q v > 0 at " "the touch; ALL FAKES FAIL BY NO-ENTRY (endpoints %s, " "zero crossings); INDEPENDENTLY RE-CONFIRMED at true " "dyadic pairs by r168 (ENTRY-NEW 4/4, v928); typed " "MEASURED -- CHAIN-NOT-ASSEMBLED (n = 3/5 atoms, not a " "theorem for unbounded combs)" % (["%s t* %.6f tangent +%.1e" % (b, t, tg) for b, t, tg in PIN_ENTRY], PIN_FAKE_END)) okpr = PIN_SSTAR_PRICE_MIN >= 0.5 \ and all(v >= 1e12 for v in PIN_RANGE_ASYM) \ and PIN_ASYM_DEX > 1 check("C4 instrument tables + honest demotion + W3/GC1", okpr, "WI2 s*-thresholds tau-priced (s*/price %.3f-1.000, 7 " "gammas x 2 blocks); plant plateau range-asymmetry >= " "1e12 (%.1e/%.1e); the premise outlives the window by " "%.1f dex (asymmetry exhibit); HONEST DEMOTION: the " "asymmetry LAW is GENERIC-CONE -- the arithmetic content " "is the boundary LOCATION + anchor VALUES; W3: value " "transport REFUTED-IN-CLASS, premise transport PROVEN; " "GC1: the cofinal target's missing piece NAMED (PROP == " "the depth-block transfer, assembled in r168/v928); " "census cardinality 4 UNCHANGED" % (PIN_SSTAR_PRICE_MIN, PIN_RANGE_ASYM[0], PIN_RANGE_ASYM[1], PIN_ASYM_DEX)) print("\n [TYPED, carried verbatim] PROVEN = WI1-WI6 + the " "Stieltjes refutation.") print(" MEASURED = the entry law + tangent exhibits (typed, " "re-confirmed r168),") print(" the Gate-A dyadic tables. DEMOTED = the asymmetry law " "(generic-cone).") print(" Census cardinality 4 UNCHANGED. NOT RH evidence. NO " "RH claim.") return 0 def run(): global CHECKS, FAILS CHECKS = [] FAILS = [] print("=" * 74) print("v930 -- PRIME.WINDOW.INSTRUMENT.MECHANISM.01 (WI1-WI6 " "exact; Stieltjes") print("refutation recomputed; the no-partial-comb-positive/" "last-atom entry") print("mechanism pinned at its honest grade; round 167; 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("v930: %d/%d checks passed | verdict %s | runtime %.1f s" % (n_run - len(fails), n_run, verdict, time.time() - T_ALL)) print("PINNING: WI1-WI6 + GA4 recomputed in-run; the mechanism/" "instrument tables") print("PINNED from window_instrument_probe.py (SPEC %s, 46/46," % PIN_SPEC_R167) print("record 4138.5 s + re-run 4047.0 s, both logs kept, NOT " "re-run at promotion") print("per the multi-hour convention -- the entry law re-gated " "by the r168") print("batteries re-run green at this promotion, v928). 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.WINDOW.INSTRUMENT.MECHANISM.01 window-" "instrument mechanism record: %d passed, %d failed ---" % (n_run - len(fails), len(fails))) return 0 if ok else 1 if __name__ == "__main__": raise SystemExit(run())