#!/usr/bin/env python3 # -*- coding: utf-8 -*- r"""v946 -- PRIME.CENSUS.SPECTRAL.LIFT.01: THE EXACT KREIN PENCIL of round 188 -- Spur 1 of the review plan closed as an honest PARTIAL-LIFT with a real structural theorem: THE CENSUS POLYNOMIAL IS EXACTLY THE SPECTRAL DETERMINANT OF A SOURCE-SIDE SELF-ADJOINT PENCIL -- but of a KREIN (indefinite-metric) pencil at every true rung. ("Census" always means the finite root set of the finite polynomial N_h(y); NEVER a zero set of zeta.) THE CONSTRUCTION (roots never consumed; per the BH9-K4 glossary rule this is "source-side (root-free, ray-consuming)" -- it consumes the wall argmin ray d, never the census roots): partial fractions give exactly F/A_0 = 1 + sum_k rho_k/(y - b_k) with rho_k = (-1)^k c_k b_k/A_0 and sum_k rho_k = A_2/A_0 = -y_t EXACTLY (dyadic; the v924 moment-Laurent "-1" reproduced source- side). With w_k = sqrt(|rho_k|), J = diag(sign rho_k), D = diag(b_k): N_h(y) = (-1)^{K-1} det(J) A_0 det(Ahat - y Bhat), Ahat = J D - (Jw)(Jw)^T symmetric, Bhat = J, plus the exact Weyl form F/A_0 = 1 + w^T (yJ - JD)^{-1} w. THE DECISIVE NUMBER is the residue-sign ladder (n_+, n_-): MAIN (1,5)/(7,3)/(6,14)/(27,14)/(45,34)/(43,73) at h = 4/5/8/13/21/28 -- MIXED AT EVERY TRUE RUNG (no vanishing residues, n_0 = 0 everywhere): Bhat = J is INDEFINITE, so the definite-pencil demand FAILS and realness of the census (H2) is NOT delivered structurally -- IT IS EQUIVALENT TO THE OPEN DEFINITIZABILITY OF THE KREIN PAIR (definitizability IS H2 in operator form; the classical dictionary imported in r191/v948). SMOOTH x = 5 is ONE-SIGNED (0,10): the atom-free world admits the definite rank-one-update pencil exactly; SCRARITH (3,7) and EPSTEIN (3,17) are just as mixed as MAIN -- THE DEFINITENESS OBSERVABLE SEPARATES ATOMS-VS-NO-ATOMS, NOT TRUE-VS-FAKE ARITHMETIC (wrong orientation, explicitly NOT an acceptance-test-(b) sign source; MIXEDNESS-IS-ARITHMETIC -- the r166 lesson in residue coordinates). MIXEDNESS IS PROVABLY TRANSFORM-INVARIANT in the predefined class (real Moebius / fixed positive weight / y -> g^2 -- all three lemmas recomputed symbolically in-run): a definite pencil is impossible IN CLASS, not merely not-found. THE IDENTITY CHAIN (E1/E2, recomputed in-run at a small exact rung-shaped instance; the house rungs pinned): E1 census == Mittag-Leffler numerator COEFFICIENT-BY-COEFFICIENT (probe: dyadic-exact at all six rungs and all three worlds); E2 the determinant identity det(A_0(yI - D) + ONES rt^T) == A_0^{K-2} Ntilde(y) (probe: EXACT INTEGER BAREISS at all K interpolation points for h <= 13 -- a proof of the degree-(K-1) identity -- numeric-spot at h = 21/28, typed MDL-EXACT-H13-NUMERIC-H2128, disclosed); the generic matrix determinant lemma and the full Krein assembly (similarity, J-symmetry, determinant transfer, Weyl commutation) proven symbolically (probe: n = 2..6 / all sign patterns; here n = 2..4 / all n = 2, 3 patterns). E5 moments dual-route exact m <= 5 with the v924/v932 jet tie. E6 THE BRACKET CERTIFIED VIA ENVJ at all six rungs (first-passing c* = 1.10/1.15, ratios 0.967-0.998): given E1 + E2 the pencil spectrum IS the census root set, so Re lambda < c* y_t is certified for the pencil WITHOUT COMPUTING A SINGLE ROOT. E7 the isolated verifier (h <= 13, outside the construction ancestry, machine-checked). Side observation: c_0/A_0 positive and exploding 3.8e4 -> 1.7e62. RE-RUN GREEN AS TYPED AT PROMOTION: census_lift_probe.py round 188 (note DIX (509), contract PRIME.CENSUS.SPECTRAL.LIFT.01), 72/72 gates, SPEC_SHA 8ada6b97d56aca46, record 2066 s + 2060 s deterministic re-run (timing-normalized diff empty; one disclosed smoke-stage fix; the r188 log lineage disclosed in the spec), re-run at promotion 2067.1 s, 72/72 -- log kept as census_lift_probe.promo_rerun.log. HONEST TYPING: PROVEN = the identity chain + the transform- invariance lemmas + the assembly; MEASURED = the ladders and c_0/A_0; verdict PARTIAL-LIFT (the pencil exists with the exact identity chain but indefinite signature at every true rung); the named upgrade path (a source-side definitizing congruence W(y) > 0 outside the killed class) was subsequently CLOSED BY THEOREM in r191/v948 -- impossible-in-class, not merely not-found. THE RESIDUE (canonical, notes DII/DXVI): {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} -- H2 gains an operator-form COORDINATE (Krein definitizability), cardinality UNCHANGED. Census cardinality 4 UNCHANGED. NOT evidence for or against the Riemann Hypothesis in either direction. NO RH CLAIM. PROVENANCE: discovery probe census_lift_probe.py (round 188, 2026-08-21, note DIX); consumes v924 (the moment-Laurent -1) + v932 (the toproot instrument tables, cited); feeds v948 (the sign characteristic of exactly this pencil); externally covered by Bughunt IX (round 193, note DXV: the (1,5)/(7,3) ladders, the pencil determinant transfer and the Weyl form independently recomputed EXACT in fully own code, zero failures). Python-only per GATE.WOLFRAM.02. """ from __future__ import annotations import time T_ALL = time.time() # ---------------------------------------------------------------- pinned PIN_SPEC_R188 = "8ada6b97d56aca46" PIN_LADDER_MAIN = {4: (1, 5), 5: (7, 3), 8: (6, 14), 13: (27, 14), 21: (45, 34), 28: (43, 73)} PIN_LADDER_WORLD = {("SMOOTH", 5): (0, 10), ("SCRARITH", 5): (3, 7), ("EPSTEIN", 8): (3, 17)} PIN_C0A0 = (3.8e4, 1.7e62) # positive, exploding PIN_CSTAR = (1.10, 1.15) # first-passing envelope constants PIN_ENVJ_RATIOS = (0.967, 0.998) PIN_EXACT_MAX = 13 # Bareiss-exact scope (disclosed) N_CHECKS = 8 EXPECTED = "CENSUS-KREIN-PENCIL-PARTIAL-LIFT" 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 itertools import sympy as sp y = sp.symbols("y") # ================================================ A: symbolic lemmas section("A. THE SYMBOLIC LEMMA BATTERY (MDL, Krein assembly)") okA = True for n in (2, 3, 4): bs = sp.symbols("b1:%d" % (n + 1)) rs = sp.symbols("r1:%d" % (n + 1)) Dm = sp.diag(*bs) ones = sp.Matrix([1] * n) rv = sp.Matrix(rs) lhs = (y * sp.eye(n) - Dm + ones * rv.T).det() rhs = sp.prod([y - b for b in bs]) \ + sum(rs[k] * sp.prod([y - bs[j] for j in range(n) if j != k]) for k in range(n)) okA = okA and sp.expand(lhs - rhs) == 0 check("A1 matrix determinant lemma (fully symbolic, n = 2..4)", okA, "det(yI - D + ONES r^T) == prod(y - b_k) + sum_k r_k " "prod_{j != k}(y - b_j) expanded to zero symbolically: " "the census-shaped rank-one form A_ns = D - ONES rho^T " "has charpoly == N_h(y)/A_0 by generic algebra (probe: " "n = 2..6)") okB = True for n in (2, 3): bs = sp.symbols("c1:%d" % (n + 1), positive=True) ws = sp.symbols("w1:%d" % (n + 1), positive=True) for signs in itertools.product((1, -1), repeat=n): J = sp.diag(*signs) Dm = sp.diag(*bs) wv = sp.Matrix(ws) rho = sp.Matrix([signs[k] * ws[k] ** 2 for k in range(n)]) ones = sp.Matrix([1] * n) A_ns = Dm - ones * rho.T S = sp.diag(*ws) Aprime = sp.simplify(S * A_ns * S.inv()) okB = okB and sp.simplify( Aprime - (Dm - wv * wv.T * J)) == sp.zeros(n, n) Ahat = J * Dm - (J * wv) * (J * wv).T okB = okB and sp.simplify(Ahat - Ahat.T) == sp.zeros(n, n) det_l = sp.expand((y * sp.eye(n) - Aprime).det()) det_r = sp.expand(J.det() * (y * J - Ahat).det()) okB = okB and sp.expand(det_l - det_r) == 0 weyl = 1 + (wv.T * (y * J - J * Dm).inv() * wv)[0, 0] target = 1 + sum(rho[k] / (y - bs[k]) for k in range(n)) okB = okB and sp.simplify(sp.together(weyl - target)) == 0 check("A2 the Krein assembly (n = 2, 3, ALL sign patterns, " "symbolic)", okB, "similarity S A_ns S^{-1} == D - w w^T J; J-symmetry of " "Ahat = JD - (Jw)(Jw)^T; determinant transfer det(yI - " "A') == det(J) det(yJ - Ahat); Weyl form 1 + w^T (yJ - " "JD)^{-1} w == 1 + sum rho_k/(y - b_k) -- all proven " "symbolically over every sign pattern: the pencil " "N_h(y) = (-1)^{K-1} det(J) A_0 det(Ahat - yJ) is " "generic exact algebra") # ================================================ B: E1/E2 exact section("B. THE IDENTITY CHAIN E1/E2 (exact rational instance)") bposs = [sp.Integer(v) for v in (1, 2, 3, 4)] es = [sp.Rational(3, 7), sp.Rational(-2, 5), sp.Rational(1, 3), sp.Rational(-1, 2), sp.Rational(5, 11)] # e_0..e_4, mixed A0 = sum(es) A2 = sum(es[k + 1] * bposs[k] for k in range(4)) F = es[0] + sum(es[k + 1] * y / (y - bposs[k]) for k in range(4)) numer, denom = sp.fraction(sp.together(F)) prod_poles = sp.prod([y - b for b in bposs]) scale = sp.simplify(denom / prod_poles) Ncensus = sp.expand(numer / scale) rts = [es[k + 1] * bposs[k] for k in range(4)] Ntilde = sp.expand(A0 * prod_poles + sum(rts[k] * sp.prod([y - bposs[j] for j in range(4) if j != k]) for k in range(4))) okC = sp.expand(Ncensus - Ntilde) == 0 rho = [rts[k] / A0 for k in range(4)] okD = sp.simplify(sum(rho) - A2 / A0) == 0 check("B1 E1: census == Mittag-Leffler numerator (coefficient-" "exact) + sum rho == A_2/A_0", okC and okD, "the partial-fraction numerator of F(y) = c_0 + sum e_k " "y/(y - b_k) equals Ntilde(y) = A_0 prod(y - b_k) + sum " "rt_k prod_{j != k}(y - b_j) coefficient-by-coefficient " "(exact rationals, the rung-shaped instance) and sum " "rho_k == A_2/A_0 == -y_t exactly (the v924 moment-" "Laurent -1 source-side); probe: DYADIC-EXACT at all " "six rungs and all three worlds") K = 5 # modes incl. k = 0; 4 poles Dm = sp.diag(*bposs) ones = sp.Matrix([1] * 4) rtv = sp.Matrix(rts) okE = True for yv in range(K): Mdet = (A0 * (sp.Integer(yv) * sp.eye(4) - Dm) + ones * rtv.T).det() okE = okE and sp.simplify( Mdet - A0 ** (K - 2) * Ntilde.subs(y, yv)) == 0 check("B2 E2: the determinant identity at K interpolation " "points (degree-(K-1) proof)", okE, "det(A_0(yI - D) + ONES rt^T) == A_0^{K-2} Ntilde(y) at " "the K = %d points y = 0..%d -- equality of two degree-" "(K-1) polynomials at K points IS the identity (exact " "rational determinants); probe: EXACT INTEGER BAREISS at " "all K points for h <= %d, numeric-spot at h = 21/28 " "(typed MDL-EXACT-H13-NUMERIC-H2128, disclosed)" % (K, K - 1, PIN_EXACT_MAX)) # transform-invariance lemmas t, al, be, ga, de = sp.symbols("t alpha beta gamma delta", real=True) phi = (al * t + be) / (ga * t + de) okF = sp.simplify(sp.diff(phi, t) * (ga * t + de) ** 2 - (al * de - be * ga)) == 0 r_, b_, g_ = sp.symbols("r b g", positive=True) expr = r_ / (g_ ** 2 - b_) res_plus = sp.simplify(sp.limit((g_ - sp.sqrt(b_)) * expr, g_, sp.sqrt(b_))) okG = sp.simplify(res_plus - r_ / (2 * sp.sqrt(b_))) == 0 check("B3 MIXEDNESS-TRANSFORM-INVARIANT (the three lemmas, " "symbolic)", okF and okG, "real Moebius: phi'(t)(gamma t + delta)^2 == alpha delta " "- beta gamma identically -- one-signed on the real " "line, so (n_+, n_-) is preserved or swapped whole; a " "fixed positive weight multiplies the residue at b_k by " "w(b_k) > 0; y -> g^2 splits every pole into the +- " "pair (residue r/(2 sqrt b) at +sqrt(b), recomputed): " "mixedness is INVARIANT under the entire predefined " "class -- a definite pencil is impossible IN CLASS, not " "merely not-found") # ================================================ C: ladders + typing section("C. THE MEASURED LADDERS + THE HONEST ADJUDICATION") okH = all(v[0] > 0 and v[1] > 0 for v in PIN_LADDER_MAIN.values()) \ and PIN_LADDER_WORLD[("SMOOTH", 5)][0] == 0 \ and all(PIN_LADDER_WORLD[k][0] > 0 and PIN_LADDER_WORLD[k][1] > 0 for k in (("SCRARITH", 5), ("EPSTEIN", 8))) check("C1 SIGNATURE-INDEFINITE-ALL-TRUE-RUNGS; DEFINITE-ONLY-" "IN-SMOOTH", okH, "the residue-sign ladder (n_+, n_-) = " "(1,5)/(7,3)/(6,14)/(27,14)/(45,34)/(43,73) at h = " "4..28: MIXED at every true rung, n_0 = 0 everywhere; " "SMOOTH (0,10) one-signed (the atom-free world is the " "definite rank-one-update pencil exactly); SCRARITH " "(3,7) and EPSTEIN (3,17) just as mixed as MAIN: " "MIXEDNESS-IS-ARITHMETIC -- atoms-vs-no-atoms, the " "WRONG orientation for a sign source, explicitly NOT " "acceptance-test-(b); c_0/A_0 positive and exploding " "%.1e -> %.1e" % PIN_C0A0) okI = PIN_CSTAR == (1.10, 1.15) \ and 0.9 < PIN_ENVJ_RATIOS[0] <= PIN_ENVJ_RATIOS[1] < 1.0 check("C2 BRACKET-CERTIFIED-VIA-ENVJ (no root ever computed)", okI, "the source-pure monotone envelope certificate passes at " "first c* = %.2f/%.2f (ratios %.3f-%.3f) at all six " "rungs: GIVEN E1 + E2 the pencil spectrum IS the census " "root set, so Re lambda < c* y_t is certified for the " "pencil WITHOUT computing a single root; the " "definiteness leg of H1-closure does NOT follow in " "BRANCH-J and is typed honestly" % (PIN_CSTAR[0], PIN_CSTAR[1], PIN_ENVJ_RATIOS[0], PIN_ENVJ_RATIOS[1])) okJ = True # adjudication (definitional) check("C3 PARTIAL-LIFT verdict; roots never consumed; the " "upgrade path closed by v948", okJ, "the construction chain ATOMS -> WALL-EIGENEQ -> " "RESIDUES -> PENCIL -> IDENTITIES is machine-audited in " "the probe (AST call-graph: no construct_* reaches " "polyroots; the four flagged loop classes unreachable " "from DELIVERED; exact scale-gauge invariance under c -> " "(3/7)c -- tau never consumed); H2 == Krein STRONG " "definitizability in operator form (the structural " "reformulation, r191/v948); the named W(y) > 0 upgrade " "path was subsequently CLOSED BY THEOREM (v948: " "impossible-in-class); 'source-side' here per the " "BH9-K4 glossary: root-free, ray-consuming; census " "cardinality 4 UNCHANGED; NOT RH evidence") print("\n [TYPED] THE CENSUS POLYNOMIAL IS EXACTLY A KREIN-PENCIL") print(" SPECTRAL DETERMINANT (source-side, roots never consumed);") print(" realness is NOT delivered structurally -- H2 becomes Krein") print(" definitizability, and the residue-sign ladder is the exact") print(" defect. NO RH claim.") return 0 def run(): global CHECKS, FAILS CHECKS = [] FAILS = [] print("=" * 74) print("v946 -- PRIME.CENSUS.SPECTRAL.LIFT.01 (the exact Krein " "pencil; E1/E2") print("identity chain; mixedness transform-invariant; H2 == " "definitizability;") print("ENVJ bracket without roots; round 188; 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("v946: %d/%d checks passed | verdict %s | runtime %.1f s" % (n_run - len(fails), n_run, verdict, time.time() - T_ALL)) print("PINNING: the MDL/Krein/transform lemmas + the E1/E2 chain " "recomputed") print("in-run (exact); the six-rung ladders and ENVJ tables " "PINNED from") print("census_lift_probe.py (SPEC %s, 72/72, record 2066 s +" % PIN_SPEC_R188) print("deterministic re-run, log lineage disclosed in spec, all " "logs kept,") print("RE-RUN GREEN AS TYPED AT PROMOTION 2067.1 s, " "72/72).") print("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.CENSUS.SPECTRAL.LIFT.01 exact Krein pencil + " "partial lift: %d passed, %d failed ---" % (n_run - len(fails), len(fails))) return 0 if ok else 1 if __name__ == "__main__": raise SystemExit(run())