#!/usr/bin/env python3 # -*- coding: utf-8 -*- r"""v947 -- PRIME.LOEWNER.PICK.01: THE EXACT LOEWNER-PICK DICTIONARY + THE PICK-CLASS REFUTATION of round 189 -- Loewner's theorem adjudicated as a currency for wall positivity: the Pick reading is REFUTED with an exact new structural law in its place. THE EXACT DICTIONARY (the round's positive delivery): the wall is NOT the canonical Loewner matrix of its potential but a DIAGONALLY-SHIFTED one -- M_h PSD <=> Raw_h PSD by exact Sylvester congruence, and Raw_h = L_f + diag(Delta) with the off-diagonal divided-difference form upgraded from r186's float64 to mp at every rung (dev <= 1.5e-61). THE NEW EXACT LAW (probe: sympy per-atom + mp <= 6.0e-61; BH9 own independent recompute <= 6.3e-61 numeric + per-atom symbolic): THE COSINE-QUADRATURE DIAGONAL LAW -- the diagonal shift on the prime block is Raw[k,k] - f_prime'(b_k) = 2a pc_k, THE COSINE QUADRATURE pc_k = sum_q w_q cos(om_k u_q) OF THE SAME SIEVE ATOMS whose SINE quadrature pj feeds the potential: THE WALL CARRIES BOTH QUADRATURES OF THE SAME ATOMS IN DIFFERENT SLOTS (the k = 0 doubling mirrors the mode norm nrm_0^2 = 2a vs a). The POLE block is fully canonical (Delta_pole == 0 identically -- recomputed symbolically in-run: divided difference == rank-1 Cauchy AND diagonal == f_pole') and f_pole is a GENUINE PICK FUNCTION with exact Herglotz representation = the point mass 2 sinh^2(a/2) delta_{t = -1/4} (weight = the builder constant, location = the completed square of s(1-s) -- recomputed symbolically in-run, including Im f_pole > 0 on the upper half plane). The ARCH excess is measured (its 1/(2w) singularity's cos-transform is log-divergent, the builder's gamma + ln pi counterterm; closed form honestly NOT claimed). Not a sum kernel (anticommutator s3/s1 = 0.372..0.622, Bhatia class excluded), not a two-function pair (g = ones exact). Loewner 1934 / Donoghue 1974 / Simon 2019 apply to the canonical L_f only. THE PICK-CLASS REFUTATION (the decisive numbers, all pinned): lambda_min(L_f) < 0 at ALL 14 rungs (log10(|lambda_min|/||F||) = -0.84..-2.00, roughly constant, zero refusals) -- the Pick costume misses wall positivity by 9.4 (h=4) to 87.9 (h=20) ORDERS; f_h is non-monotone on every hull (descent counts 2-34: order-1 already dead); ALL grid sign changes are carried by the prime leg (10/18/35/68 at the PICK rungs; pole + arch: zero); refinement margins DEEPEN (-0.044 -> -0.206: no hidden interval-wide margin); bonus exhibit: a 17-node window below the first f'-zero is INDEFINITE while f' > 0 on it -- the classical scalar-vs-matrix order separation live in the wall potential. HERGLOTZ per leg: pole EXACT source-expressible point mass; arch measured-monotone representation open; prime leg non-Pick and its formal density IS the signed atom data -- a positive rewrite of that leg would be the wall positivity itself: the dream statement dies honestly at the prime leg in both directions (relabeling barrier named, not crossed). WORLDS + SCREENS: the dictionary is world-blind (typed), the VALUES are mixed: MAIN indefinite 14/14 but THE SMOOTH CELL IS PSD WITH 0 DESCENTS -- its 2a-periodic oscillation vanishes exactly on the node lattice (sin(2a om_k) == 0 at om_k = k pi/a, COMMENSURATE SAMPLING -- recomputed symbolically in-run) which incommensurate prime atoms cannot do; SCRARITH/EPSTEIN mixed; the r172 witness is matrix-side-invariant by construction (typed, not sold); ATOMJET detected exactly linearly (devs 4.2e-61/3.0e-61); tau-screens BOTH FLAT (-0.011/-0.011) -- not a relabel and not a sign source either; the r45 LOEWNER-DEAD prior art distinguished: THIS kills a THIRD reading (positivity-as-function-class), not r45's, and the IIKS/RHP lane is untouched. RE-RUN GREEN AS TYPED AT PROMOTION: loewner_pick_probe.py round 189 (note DX (510), contract PRIME.LOEWNER.PICK.01), 28/28 gates, SPEC_SHA a547448468899af9, record 1046 s + 1058 s deterministic re-run (timing-normalized diff empty, all logs kept, three pre-freeze amendments disclosed: A1 the commensurate jet atom q = 5 -> q = 2; A2 window gate -> resolve-and-record; A3 world enum -> ternary), re-run at promotion 1082.9 s, 28/28 -- log kept as loewner_pick_probe.promo_rerun.log. HONEST TYPING: PROVEN = the dictionary algebra + the pole-block canonicity + the Herglotz point mass + the commensurate-sampling mechanism; MEASURED = the lambda_min/descent/grid/refinement/ window/world ladders; the K1 correction of record (r186) is CARRIED here by construction -- this round's record already states the corrected dictionary ("NOT literally the canonical L_f") and needed NOTHING per BH9; the named next question (WHICH source functional makes L_f + diag(Delta) PSD when neither summand is) stays recorded, NOT claimed, experiments-side. 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}. Census cardinality 4 UNCHANGED. NOT evidence for or against the Riemann Hypothesis in either direction. NO RH CLAIM. PROVENANCE: discovery probe loewner_pick_probe.py (round 189, 2026-08-21, note DX); consumes v944 (the r186 off-diagonal law it upgraded and diagonally completed); externally covered by Bughunt IX (round 193, note DXV: the cosine-quadrature diagonal law independently recomputed <= 6.3e-61 numeric + per-atom symbolic proof with the pole block == rank-1 Cauchy == canonical Loewner of f_pole, zero failures; X1 adjudication: r189's record correct as written). Python-only per GATE.WOLFRAM.02. """ from __future__ import annotations import time T_ALL = time.time() # ---------------------------------------------------------------- pinned PIN_SPEC_R189 = "a547448468899af9" PIN_OFFDIAG_MP = 1.5e-61 # off-diagonal Loewner form, mp PIN_COSLAW_DEV = 6.0e-61 # prime cos-quadrature law, mp PIN_COSLAW_BH9 = 6.3e-61 # BH9 own independent recompute PIN_LM_RANGE = (-2.00, -0.84) # log10(|lambda_min(L_f)|/||F||) PIN_PICK_MISS_ORDERS = (9.4, 87.9) # gap to wall positivity PIN_DESC_RANGE = (2, 34) PIN_GRID_NSC = (10, 18, 35, 68) # ALL carried by the prime leg PIN_REFINE_H4 = (-0.0442, -0.1198, -0.1498, -0.2063) # R0..R3 PIN_WIN_LM = (-4.6e-2, -4.3e-2) # 17-node window indefinite PIN_WIN_FP_MIN = 0.26 # while f' > 0 on it PIN_SUMK_S3S1 = (0.372, 0.622) # not a sum kernel PIN_JET_DEVS = (4.2e-61, 3.0e-61) # ATOMJET linear detection PIN_TAU_SLOPES = (-0.011, -0.011) # both flat PIN_TAU_FLAT_BAR = 0.30 N_CHECKS = 9 EXPECTED = "LOEWNER-PICK-DICTIONARY-MISMATCH" 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: exact dictionary section("A. THE EXACT DICTIONARY (pole canonical, Pick, " "quadrature slots)") a_, bi, bj, b_ = sp.symbols("a b_i b_j b", positive=True) s2 = 2 * sp.sinh(a_ / 2) ** 2 fpole = lambda t: -s2 / (sp.Rational(1, 4) + t) # noqa: E731 dd = sp.simplify((fpole(bi) - fpole(bj)) / (bi - bj) - s2 / ((sp.Rational(1, 4) + bi) * (sp.Rational(1, 4) + bj))) ddiag = sp.simplify(sp.diff(fpole(b_), b_) - s2 / (sp.Rational(1, 4) + b_) ** 2) okA = dd == 0 and ddiag == 0 check("A1 the pole block is FULLY CANONICAL (Delta_pole == 0 " "identically)", okA, "divided difference of f_pole(b) = -2 sinh^2(a/2)/(1/4 + " "b) == the rank-1 Cauchy kernel 2 sinh^2(a/2)/((1/4 + " "b_i)(1/4 + b_j)) AND the diagonal == f_pole'(b) exactly " "(sympy): the pole block is the ONE fully canonical " "Loewner block of the wall -- BH9 independently verified " "pole block == rank-1 Cauchy == canonical Loewner of " "f_pole") x_, y_ = sp.symbols("x y", real=True) zc = x_ + sp.I * y_ imf = sp.simplify(sp.im(-s2 / (sp.Rational(1, 4) + zc))) # Im f_pole = s2 * y / |1/4 + z|^2 > 0 for y > 0 target = s2 * y_ / ((sp.Rational(1, 4) + x_) ** 2 + y_ ** 2) okB = sp.simplify(imf - target) == 0 cauchy = sp.integrate(sp.DiracDelta(x_ + sp.Rational(1, 4)) / (x_ - b_), (x_, -sp.oo, sp.oo)) okC = sp.simplify(s2 * cauchy - fpole(b_)) == 0 check("A2 f_pole is a GENUINE PICK function with exact Herglotz " "point mass", okB and okC, "Im f_pole(z) == 2 sinh^2(a/2) Im z / |1/4 + z|^2 > 0 on " "the upper half plane (sympy exact) and the Cauchy " "transform of the point mass 2 sinh^2(a/2) " "delta_{t = -1/4} == f_pole exactly: weight = the " "builder constant, location = the completed square of " "s(1 - s) -- the pole leg's Herglotz representation is " "EXACT and source-expressible") u, w_ = sp.symbols("u w", positive=True) om = sp.symbols("omega", positive=True) fprime_atom = 2 * sp.sqrt(b_) * w_ * sp.sin(sp.sqrt(b_) * u) deriv = sp.simplify(sp.diff(fprime_atom, b_) - (w_ * sp.sin(sp.sqrt(b_) * u) / sp.sqrt(b_) + w_ * u * sp.cos(sp.sqrt(b_) * u))) okD = deriv == 0 okE = sp.simplify(sp.diff(w_ * sp.sin(om * u), u) - w_ * om * sp.cos(om * u)) == 0 check("A3 THE COSINE-QUADRATURE DIAGONAL LAW (per-atom " "structure; probe sympy-exact)", okD and okE, "per atom (u, w): f_prime'(b) == w sin(om u)/om + " "w u cos(om u) at om = sqrt(b) (exact chain rule, " "recomputed) -- the canonical diagonal; the probe proves " "per-atom (sympy) and gates at every rung (mp <= %.1e; " "BH9 own recompute <= %.1e) that the wall diagonal " "EXCEEDS it by exactly 2a pc_k with pc_k = sum w cos(om_k " "u): THE SINE QUADRATURE SITS IN THE POTENTIAL, THE " "COSINE QUADRATURE IN THE DIAGONAL SHIFT -- both " "quadratures of the same atoms in different slots " "(k = 0 doubled, mirroring nrm_0^2 = 2a)" % (PIN_COSLAW_DEV, PIN_COSLAW_BH9)) k = sp.symbols("k", positive=True, integer=True) omk = k * sp.pi / a_ okF = sp.simplify(sp.sin(2 * a_ * omk)) == 0 \ and sp.simplify(sp.cos(2 * a_ * omk)) == 1 check("A4 the commensurate-sampling mechanism (SMOOTH-PSD-AT-" "NODES, exact)", okF, "at the node lattice om_k = k pi/a: sin(2a om_k) == " "sin(2 pi k) == 0 and cos(2a om_k) == 1 exactly (sympy, " "k integer): the SMOOTH world's whole 2a-periodic " "oscillation vanishes exactly on the nodes -- which " "incommensurate prime atoms (u = log q) cannot do: the " "mechanism behind WORLD-MIXED-SMOOTH-PSD-AT-NODES, typed " "as lattice geometry, not sold as a prime fingerprint") # ================================================ B: the refutation section("B. THE PICK-CLASS REFUTATION (pinned, typed MEASURED)") okG = PIN_LM_RANGE[0] < PIN_LM_RANGE[1] < 0 \ and PIN_PICK_MISS_ORDERS == (9.4, 87.9) check("B1 CANONICAL-COMPLETION-INDEFINITE at all 14 rungs " "(misses by 9.4-87.9 orders)", okG, "lambda_min(L_f) < 0 at every rung with log10(|lambda_" "min|/||F||) in [%.2f, %.2f] (roughly constant, zero " "refusals): the Pick costume misses wall positivity by " "%.1f (h=4) to %.1f (h=20) ORDERS -- L_f is INDEFINITE " "while the wall is PD: the wall is NOT positivity-of-a-" "matrix-monotone-function; Loewner 1934/Donoghue 1974/" "Simon 2019 apply to the canonical L_f only" % (PIN_LM_RANGE[0], PIN_LM_RANGE[1], PIN_PICK_MISS_ORDERS[0], PIN_PICK_MISS_ORDERS[1])) okH = PIN_DESC_RANGE == (2, 34) \ and all(PIN_REFINE_H4[i + 1] < PIN_REFINE_H4[i] for i in (0, 1)) and PIN_REFINE_H4[3] < PIN_REFINE_H4[0] check("B2 order-1 dead + refinement DEEPENS (no hidden " "margin)", okH, "descent counts %d..%d of K-1 on every hull (f_h non-" "monotone: matrix monotonicity of ANY order requires f " "nondecreasing -- order-1 already dead); the refinement " "battery R0 -> R3 at h = 4: %s (recomputed monotone " "deepening): no interval-wide margin hides between the " "nodes; ALL grid sign changes %s carried by the PRIME " "leg (pole + arch: zero)" % (PIN_DESC_RANGE[0], PIN_DESC_RANGE[1], str(PIN_REFINE_H4), str(PIN_GRID_NSC))) okI = PIN_WIN_LM[1] < 0 and PIN_WIN_FP_MIN > 0 check("B3 the scalar-vs-matrix order separation (live " "exhibit)", okI, "the 17-node window below the first f'-zero is " "INDEFINITE (lambda_min/||F|| ~ %.1e..%.1e) while f' > 0 " "on it (min f' %.2f..1.32): the classical separation " "between scalar monotonicity and matrix monotonicity is " "LIVE in the wall potential -- the refutation is not a " "boundary artifact" % (PIN_WIN_LM[0], PIN_WIN_LM[1], PIN_WIN_FP_MIN)) okJ = PIN_SUMK_S3S1[0] > 1e-6 \ and abs(PIN_TAU_SLOPES[0]) <= PIN_TAU_FLAT_BAR \ and abs(PIN_TAU_SLOPES[1]) <= PIN_TAU_FLAT_BAR \ and PIN_JET_DEVS[0] < 1e-28 and PIN_JET_DEVS[1] < 1e-28 check("B4 not-a-sum-kernel; ATOMJET linear; tau-screens BOTH " "FLAT", okJ, "the anticommutator ratio s3/s1 = %.3f..%.3f excludes " "the Bhatia sum-kernel class; the ATOMJET (double the " "q = 2 atom weight) shifts potential and diagonal law " "EXACTLY linearly (devs %.1e/%.1e): the dictionary " "detects source jets linearly; tau-screen slopes " "%.3f/%.3f both inside the flat bar %.2f -- the Pick-" "defect coordinates do NOT ride the tau currency (and " "need not, being indefiniteness certificates); the r45 " "LOEWNER-DEAD distinction carried: THIS kills the " "positivity-as-function-class reading, the IIKS/RHP " "lane untouched" % (PIN_SUMK_S3S1[0], PIN_SUMK_S3S1[1], PIN_JET_DEVS[0], PIN_JET_DEVS[1], PIN_TAU_SLOPES[0], PIN_TAU_SLOPES[1], PIN_TAU_FLAT_BAR)) # ================================================ C: adjudication section("C. HERGLOTZ PER LEG + THE HONEST BARRIER") okK = True # definitional/adjudication check("C1 HERGLOTZ-POLE-EXACT-ARCH-OPEN-PRIME-BARRIER " "(the dream dies honestly)", okK, "pole leg: exact source-expressible point mass (A2); " "arch leg: measured-monotone, representation OPEN (the " "1/(2w) singularity's cos-transform is log-divergent, " "the builder's gamma + ln pi counterterm -- closed form " "honestly NOT claimed); prime leg: non-Pick BOTH " "directions, and its formal density IS the signed atom " "data -- a positive rewrite of that leg would BE the " "wall positivity: the relabeling barrier is NAMED, not " "crossed; the exactly-posed cancellation question " "(which source functional makes L_f + diag(Delta) PSD " "when neither summand is) stays RECORDED, not claimed, " "experiments-side; census cardinality 4 UNCHANGED; NOT " "RH evidence") print("\n [TYPED] THE WALL IS EXACTLY L_f + diag(Delta) -- a") print(" diagonally-shifted one-function Loewner matrix, NOT the") print(" canonical object of Loewner's theorem; the Pick reading is") print(" dead by 9-88 orders; the cosine-quadrature diagonal law is") print(" the new exact structure. NO RH claim.") return 0 def run(): global CHECKS, FAILS CHECKS = [] FAILS = [] print("=" * 74) print("v947 -- PRIME.LOEWNER.PICK.01 (the exact dictionary Raw = " "L_f +") print("diag(Delta); the cosine-quadrature diagonal law; the pole-" "leg Herglotz") print("point mass; the Pick-class refutation; SMOOTH-PSD-AT-NODES; " "round 189;") 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("v947: %d/%d checks passed | verdict %s | runtime %.1f s" % (n_run - len(fails), n_run, verdict, time.time() - T_ALL)) print("PINNING: the pole-canonicity/Pick/Herglotz/chain-rule/" "commensurability") print("algebra recomputed in-run (exact); the lambda_min/descent/" "grid/window/") print("world tables PINNED from loewner_pick_probe.py (SPEC %s," % PIN_SPEC_R189) print("28/28, record 1046 s + 1058 s deterministic re-run, three " "pre-freeze") print("amendments disclosed, all logs kept, RE-RUN GREEN AS TYPED " "AT PROMOTION") print("1082.9 s, 28/28). 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.LOEWNER.PICK.01 exact dictionary + Pick " "refutation: %d passed, %d failed ---" % (n_run - len(fails), len(fails))) return 0 if ok else 1 if __name__ == "__main__": raise SystemExit(run())