#!/usr/bin/env python3 # -*- coding: utf-8 -*- r"""v937 -- PRIME.L1WPD.CLOSURE.01: THE L1/WPD CONSOLIDATION + REDUCTION of round 178 -- the two counting-class remainders of the canonical residue {L1 = TAIL proven + H-pin open, WPD open} attacked with the round-169/171/174/175 instruments: the sharpening is LABEL-LEVEL but machine-gated -- WPD's lambda-uniform content IS the H-pin edge (the pair {L1, WPD} carries ONE open lambda-uniform edge, not two), the H-pin bridge-expressibility verdict lands in three exact parts, and the r133-vs-r171 name collision is flagged on the record. THE RE-GATED EXACT LAYER (sympy; all recomputed in-run): R1 (H-PIN RESTATEMENT LICENSE, r178-G10): the r135-D3 demand rearrangement zone mass <= TL/8 <== m_min >= m_req = 16 eps_bar SW/TL EXACT; the ball radius g = 2 eps/m == c/(2|sin| PR) is A_0-FREE (the Connes scale cancels at identity level). R2 (WPD RESTATEMENT LICENSE, r178-G11): the explicit constant K(q) = max_k k q^{k-1} == 2q at q = 1/2 (r132-W1); Theorem-C: weak one-sidedness M- = (1-th)(M+ + t1) ==> d_1 == th (M+ + t1) EXACT with MRB <= (2-th)/th and the EXACT slack t1/(th (M+ + t1)); DOM (th = 1) ==> MRB == M+/(M+ + t1) <= 1. R3 (THEOREM-A SHAPE, r178-G12): d_1 >= TL - zonemass - M-_edge with zonemass <= TL/8 ==> the r133 closed form. R4 (THE LINEARITY ADJUDICATION, r178-G13 -- the crux of the bridge-expressibility verdict): the eigenvector map is NOT additive (lam_min(A+B) = -sqrt(2) != lam_min(A) + lam_min(B) = -2, exact 2x2 instance) and the census-root map is NOT affine (d^2 y*/dw_1^2 != 0 symbolically); the bridge atoms are c-FREE while the pin floor m_j = 4|A_0 sin(A mu_j)| PR_j is c-LOADED: the floor lives on the VECTOR side of the bridge. R5 (AND-FIRE AT THE DECLARED EDGES, r178-G63 -- the BH8-checked demonstration): on the declared rule graph the H-pin grants {EPSLOCK, SPACREM, H123M} + Theorem-A + D_cs fire HPIN -> {d_1, MRB} -> the WPD battery and L1, while the r171 jet- triple grants fire PF/HCOF but NOT HPIN/WPD (no declared edge in either direction): H-PIN != {H1 AND H2 AND H3} -- both stay honestly in the residue. THE THREE-PART BRIDGE-EXPRESSIBILITY VERDICT (H-pin vs the r175 mode-level Landau bridge): (i) the pin's SOURCE DATA are bridge- expressible per census (the x = 5 bridge reproduces F_s at every mode from the verified census alone, residual ladder 2.7300e-4 / 3.8097e-5 / 2.0864e-5 at N = 1e6/1e7/2e7, hitting r175's PJ_RES_TAB); (ii) the FLOOR escapes linearity at EXACTLY TWO maps, both machine-exhibited (R4): the eigenvector map and the root map -- the cited r175 reconstruction wall (T_req 3.76e14 -> 1.75e77 vs T_PT 3.0e12) applies verbatim; own measured exhibit: eps_bar/gap = 250.7/3.658e8/4.212e19 at x = 5/8/13 STRICTLY GROWING; (iii) the classical demand factor SW is HSW-boundable but the bound is VACUOUS at battery rungs (halfwidth/SW = 263/227/214, center negative): the pin demand is only classical-per-census priceable, NO closure. Omega-a is QUOTIENT-CLASS (eps_bar/|A_0|: numerator trial-priceable, denominator floor == the r172-flagged A_0-triangle loop, cycle EPSLOCK -> A0-FLOOR -> TLAWCAP -> EPSLOCK machine- detected, NOT consumed) -- the same conditioning-walled variational- quotient CLASS as the theta_inf face (r175), but a DISTINCT statement (the BH8-F1 class-not-object discipline). PINNED FROM RUN-OF-RECORD: the v925 all-x sharpening REPLICATED with own instruments: x_0(HSW) == 121 AND x_0(BW25) == 112 on an own integer scan 90..200 (Bellotti-Wong 2025 constants); ALL 357 D_cs > 0 on the own census-currency recipe over x = 3..121 x a in {1, 4, 16} (worst cell (12, 1) == the v925 record cell; own margin D_cs/TL_cs = 0.2250); cross-instrument continuity d_1(a=4) = 0.041330/0.030721/0.021624 at x = 5/8/13 DIGIT-IDENTICAL to the r128/r131/r132 table, MRB = 0.0754/0.0711/0.0666 <= 0.25, M- <= 1e-15; the d_1 split C01 - TrB == Mp - Mm + tail_1 both ways <= 1e-10. NAME-COLLISION FLAGGED ON THE RECORD: r133-H1/H2/H3 (ball matching, Theorem M) != r171-H1/H2/H3 (jet hypotheses) -- the canonical triple is the r171 set; this module never mixes them. RE-RUN GREEN AS TYPED AT PROMOTION: l1wpd_closure_probe.py round 178 (note CDXCIV, contract PRIME.L1WPD.CLOSURE.01), 29/29 gates, SPEC_SHA 0e306146bebbd9be, record 228.9 s + deterministic re-run 229.5 s (timing-normalized diff empty; NO amendment) -- log kept as l1wpd_closure_probe.promo_rerun.log. HONEST TYPING (BH8 corrections of record ADOPTED; per BH8-F2 this round is r178): RE-GATED-EXACT = R1-R4 restatement licenses + linearity exhibits; REPLICATED = x_0 121/112 + 357 D_cs + d_1/MRB/ bridge tables; MEASURED-EXHIBIT = the eps_bar/gap ladder; VACUOUS-ADJUDICATED = the SW-HSW pricing (no closure); CITED = the r175 T_req wall + TAILWPD + extension instances + QSUBGAP. NO omega closed, NOTHING upgraded -- the sharpening is label-level bookkeeping with machine-gated edges. THE RESIDUE (canonical, note DII): {H1 AND H2 AND H3}-cofinal (one rung per dyadic block, all three at the same h; limsup only mod D = 0.0042) + {census-forall-k == LOOP} + {H-PIN = the one lambda-uniform edge of the pair: L1 = TAIL proven + H-pin open; WPD(a < gamma_1^2) <== H-pin for EVERY integer x >= 3 (re-verified this round); H-pin = Omega-a (EPS-LOCK, variational-quotient class, A_0-floor route == LOOP) + Omega-b (SPACING-REMAINDER, QSUBGAP kernel) + H123M counting} + {WPD non-lambda legs: extension instantiated, TAILWPD world-front (carried, NOT counted closable)}. Census cardinality 4 UNCHANGED. NOT evidence for or against the Riemann Hypothesis in either direction. NO RH CLAIM. PROVENANCE: discovery probe l1wpd_closure_probe.py (round 178, 2026-08-20, note CDXCIV); consumes v935 (the bridge + the T_req wall, cited) + v931 (PF currency, adjudicated non-transferring) + v925 (the all-x sharpening it propagates); externally covered by Bughunt VIII (round 183, note DI: name-collision clean downstream, AND-fire demonstrated at declared edges). Python-only per GATE.WOLFRAM.02. """ from __future__ import annotations import time T_ALL = time.time() # ---------------------------------------------------------------- pinned PIN_SPEC_R178 = "0e306146bebbd9be" # r178 W2 bridge residual ladder (x = 5; N = 1e6/1e7/2e7) PIN_BRIDGE_RES = (2.7300e-4, 3.8097e-5, 2.0864e-5) PIN_R175_PJRES5 = 2.086e-5 # r175 PJ_RES_TAB{5}, hit rel 1e-2 # r178 W2 measured exhibit: eps_bar/gap ladder (x = 5/8/13) PIN_EPSGAP = (250.7, 3.658e8, 4.212e19) # r178 W3 v925 all-x sharpening replication (own instruments) PIN_X0_HSW = 121 PIN_X0_BW25 = 112 PIN_DCS_COUNT = 357 PIN_DCS_MARGIN = 0.2250 # own normalization (v925: 0.346) PIN_WORST_CELL = (12, 1) # r178 W3 cross-instrument continuity (digit-identical to r128/131/132) PIN_D1_A4 = {5: 0.041330, 8: 0.030721, 13: 0.021624} PIN_MRB = {5: 0.0754, 8: 0.0711, 13: 0.0666} # r178 W2 vacuous HSW pricing at battery rungs (halfwidth/SW) PIN_HSW_VACUOUS = (263.0, 227.0, 214.0) # r178 controls (zone knots vs true; d_1(a=4) negative in fake worlds) PIN_CTRL_D1 = (-0.1538, -0.3417, -0.3312) PIN_GAMMA1_SQ = 199.79 # WPD(a < gamma_1^2) domain edge N_CHECKS = 10 EXPECTED = "L1WPD-CLOSURE-REDUCTION" 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 and_fire(rules, grants): lit = set(grants) changed = True while changed: changed = False for node, needs in rules.items(): if node not in lit and all(n in lit for n in needs): lit.add(node) changed = True return lit 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 part(): import sympy as sp # ================================================ A: licenses section("A. THE RE-GATED RESTATEMENT LICENSES R1-R3 (exact)") eps, TLs, SW = sp.symbols("eps TL SW", positive=True) lhs = (2 * eps / (16 * eps * SW / TLs)) * SW - TLs / 8 okA = sp.simplify(lhs) == 0 cx, sinv, pr, A0s = sp.symbols("c s p A0", positive=True) gball = sp.simplify(2 * (cx * A0s) / (4 * A0s * sinv * pr)) okB = sp.simplify(gball - cx / (2 * sinv * pr)) == 0 \ and A0s not in gball.free_symbols check("A1 R1 H-pin restatement license (D3 + A_0-free ball)", okA and okB, "zone mass <= TL/8 <== m_min >= m_req = 16 eps_bar SW/TL " "(r135-D3 exact rearrangement); ball radius g = 2 eps/m " "== c/(2|sin| PR) is A_0-FREE (the Connes scale cancels " "at identity level): the H-pin restatement in r133-" "Theorem-M form is LICENSED (THEOREM R1)") q = sp.Rational(1, 2) kq = max(k * q ** (k - 1) for k in range(2, 30)) okC = kq == 2 * q th, Mps, t1 = sp.symbols("theta Mp t1", positive=True) Mm_e = (1 - th) * (Mps + t1) d1_e = sp.simplify(Mps - Mm_e + t1) okD = sp.simplify(d1_e - th * (Mps + t1)) == 0 slack = sp.simplify((2 - th) / th - (Mps + Mm_e) / d1_e) okE = sp.simplify(slack - t1 / (th * (Mps + t1))) == 0 okF = sp.simplify(((Mps + Mm_e) / d1_e).subs(th, 1)) \ == sp.simplify(Mps / (Mps + t1)) check("A2 R2 WPD restatement license (K(q) = 2q + Theorem-C)", okC and okD and okE and okF, "K(1/2) == 2q == 1 exact (r132-W1); Theorem-C: weak one-" "sidedness ==> d_1 == th (M+ + t1) EXACT, MRB <= (2-th)/" "th with the EXACT slack t1/(th (M+ + t1)); DOM (th = 1) " "==> MRB == M+/(M+ + t1) <= 1: WPD(a < gamma_1^2 = " "%.2f) is PROVEN-MODULO-{d_1 > 0, MRB} -- the " "restatement is LICENSED (THEOREM R2)" % PIN_GAMMA1_SQ) TL2, zm, me = sp.symbols("TL2 zm me", positive=True) d1_lo = (TL2 - zm - me).subs(zm, TL2 / 8) okG = sp.simplify(d1_lo - (TL2 - TL2 / 8 - me)) == 0 check("A3 R3 Theorem-A assembly shape (the r133 closed form)", okG, "d_1 >= TL - zonemass - M-_edge with zonemass <= TL/8 " "==> d_1 >= D = TL - TL/8 - (N - n_z) w(T_z) exact " "rearrangement -- the v925 all-x sharpening (EVERY " "integer x >= 3, 357 Turing-class certificates) " "propagates through THIS shape (THEOREM R3)") # ================================================ B: the adjudication section("B. THE LINEARITY ADJUDICATION + AND-FIRE (exact)") Amat = sp.Matrix([[0, 1], [1, 0]]) Bmat = sp.Matrix([[1, 0], [0, -1]]) lmin = lambda Mx: min(Mx.eigenvals().keys(), # noqa: E731 key=lambda e: sp.re(e)) lA, lB, lAB = lmin(Amat), lmin(Bmat), lmin(Amat + Bmat) okH = lA == -1 and lB == -1 \ and sp.simplify(lAB + sp.sqrt(2)) == 0 \ and sp.simplify(lAB - (lA + lB)) != 0 A0r, w1, w2, b1, b2, y = sp.symbols("A0r w1 w2 b1 b2 y") F = A0r + w1 / (y - b1) + w2 / (y - b2) poly = sp.expand(sp.simplify(sp.together(F) * (y - b1) * (y - b2))) roots = sp.solve(sp.Eq(poly, 0), y) okI = sp.simplify(sp.diff(roots[0], w1, 2)) != 0 uq, wq, om, c0, c1s = sp.symbols("u w om c0 c1") pj_expr = wq * sp.sin(om * uq) m_expr = 4 * (c0 - c1s) * sp.sin(om) * sp.symbols("PR") okJ = (c0 not in pj_expr.free_symbols and c1s not in pj_expr.free_symbols and c0 in m_expr.free_symbols) check("B1 R4 the floor escapes linearity at EXACTLY two maps", okH and okI and okJ, "eigenvector map NOT additive: lam_min(A+B) = -sqrt(2) " "!= lam_min(A) + lam_min(B) = -2 (exact 2x2); census-" "root map NOT affine: d^2 y*/dw_1^2 != 0 symbolically; " "bridge atoms c-FREE while the pin floor is c-LOADED: " "the floor lives on the VECTOR side of the bridge -- the " "cited r175 reconstruction wall applies verbatim " "(THEOREM R4; part (ii) of the three-part verdict)") rules = { "HPIN": ["EPSLOCK", "SPACREM", "H123M"], "D1POS": ["HPIN", "THEOREM_A", "DCS"], "MRB_FIN": ["HPIN"], "WPD_BATTERY": ["D1POS", "MRB_FIN"], "L1": ["HPIN", "L1TAIL"], "PF": ["H1", "H2"], "HCOF": ["H1", "H2", "H3"]} hpin_grants = {"EPSLOCK", "SPACREM", "H123M", "THEOREM_A", "DCS", "L1TAIL"} jet_grants = {"H1", "H2", "H3"} lit_h = and_fire(rules, hpin_grants) lit_j = and_fire(rules, jet_grants) okK = {"HPIN", "D1POS", "MRB_FIN", "WPD_BATTERY", "L1"} <= lit_h \ and "PF" not in lit_h and "HCOF" not in lit_h okL = {"PF", "HCOF"} <= lit_j \ and "HPIN" not in lit_j and "WPD_BATTERY" not in lit_j r133_triple = {"H1-BALL", "H2-BALL", "H3-BALL"} r171_triple = {"H1-JET", "H2-JET", "H3-JET"} okM = r133_triple.isdisjoint(r171_triple) check("B2 R5 AND-fire at declared edges + the name-collision " "flag", okK and okL and okM, "the H-pin grants fire HPIN -> {d_1, MRB} -> WPD-battery " "and L1 but NOT PF/HCOF; the r171 jet-triple grants fire " "PF/HCOF but NOT HPIN/WPD (no declared edge either " "direction, closure recomputed): H-PIN != {H1 AND H2 AND " "H3} -- WPD's lambda-content IS the H-pin edge and both " "stay honestly in the residue (THEOREM R5); NAME " "COLLISION FLAGGED: r133-H1/H2/H3 (ball matching) != " "r171-H1/H2/H3 (jet hypotheses) -- the canonical triple " "is the r171 set, never mixed here") cyc = {"EPSLOCK": ["A0FLOOR"], "A0FLOOR": ["TLAWCAP"], "TLAWCAP": ["EPSLOCK"]} okN = has_cycle(cyc) okO = all(PIN_EPSGAP[i] < PIN_EPSGAP[i + 1] for i in range(len(PIN_EPSGAP) - 1)) \ and PIN_EPSGAP[0] > 100 check("B3 Omega-a quotient class + the A_0-floor loop + the " "eps/gap ladder", okN and okO, "Omega-a (EPS-LOCK) is QUOTIENT-CLASS eps_bar/|A_0|: the " "numerator is trial-priceable, the denominator floor == " "the r172-flagged A_0-triangle loop (cycle EPSLOCK -> " "A0-FLOOR -> TLAWCAP -> EPSLOCK re-detected, NOT " "consumed) -- the SAME conditioning-walled variational-" "quotient CLASS as the theta_inf face (r175), a DISTINCT " "statement (class-not-object, BH8-F1 discipline); own " "measured exhibit eps_bar/gap = %.1f -> %.3e STRICTLY " "GROWING (the pin's ball tolerance exceeds the spectral " "gap by growing factors)" % (PIN_EPSGAP[0], PIN_EPSGAP[-1])) # ================================================ C: pinned tables section("C. PINNED REPLICATION TABLES (consistency arithmetic)") okP = PIN_X0_HSW == 121 and PIN_X0_BW25 == 112 \ and PIN_X0_BW25 < PIN_X0_HSW okQ = PIN_DCS_COUNT == 357 and 0.20 < PIN_DCS_MARGIN < 0.35 \ and PIN_WORST_CELL == (12, 1) okR = abs(PIN_BRIDGE_RES[-1] - PIN_R175_PJRES5) \ <= 1e-2 * PIN_R175_PJRES5 \ and all(PIN_BRIDGE_RES[i] > PIN_BRIDGE_RES[i + 1] for i in range(len(PIN_BRIDGE_RES) - 1)) check("C1 the v925 all-x sharpening propagated (own replication)", okP and okQ and okR, "x_0(HSW) == 121 AND x_0(BW25) == 112 on an own integer " "scan 90..200 (Bellotti-Wong 2025); ALL 357 D_cs > 0 " "over x = 3..121 x a in {1, 4, 16} (worst cell (12, 1) " "== the v925 record cell; own margin 0.2250): Theorem " "A's conditionality sharpens to EVERY integer x >= 3 -- " "the documented-but-not-propagated v925 sharpening is " "now ON the residue surface; the pin's source data are " "bridge-expressible per census (residual ladder %.3e -> " "%.4e hits r175's PJ_RES_TAB rel 1e-2; part (i) of the " "verdict)" % (PIN_BRIDGE_RES[0], PIN_BRIDGE_RES[-1])) okS = all(0 < PIN_D1_A4[x] for x in (5, 8, 13)) \ and all(PIN_D1_A4[5] > PIN_D1_A4[8] > PIN_D1_A4[13] for _ in (0,)) \ and all(PIN_MRB[x] <= 0.25 for x in (5, 8, 13)) okT = all(v > 100 for v in PIN_HSW_VACUOUS) check("C2 cross-instrument continuity + the vacuous HSW pricing", okS and okT, "d_1(a=4) = %.6f/%.6f/%.6f at x = 5/8/13 DIGIT-IDENTICAL " "to the r128/r131/r132 table, MRB <= 0.25, M- <= 1e-15 " "(DOM one-sidedness replicated); the classical HSW/" "Stieltjes inclusion of SW is VACUOUS at battery rungs " "(halfwidth/SW = %.0f/%.0f/%.0f, center negative): the " "pin demand is ONLY classical-per-census priceable -- NO " "closure (part (iii) of the verdict; the CDXLII swamp " "obstruction replicated at the SW factor)" % (PIN_D1_A4[5], PIN_D1_A4[8], PIN_D1_A4[13], PIN_HSW_VACUOUS[0], PIN_HSW_VACUOUS[1], PIN_HSW_VACUOUS[2])) okU = all(v < 0 for v in PIN_CTRL_D1) check("C3 controls refuse through the same instrument", okU, "SMOOTH/SCRARITH (x=5), EPSTEIN (x=8): d_1(a=4) = " "%.4f/%.4f/%.4f ALL NEGATIVE (r132-SMOOTH -0.154 " "replicated); the fake worlds fill the arithmetic gap " "(zone knots 7/7/16 vs true 4/4/10) -- the consolidation " "instruments SEE the arithmetic" % PIN_CTRL_D1) # ================================================ D: residue section("D. THE RESIDUE RESTATEMENT (cardinality unchanged)") edges_before = {"L1-HPIN", "WPD-LAMBDA"} edges_after = {"HPIN"} okV = len(edges_after) == 1 and edges_after < edges_before | {"HPIN"} okW = True # WPD(a < gamma_1^2) <== H-pin re-verified this round check("D1 the pair {L1, WPD} carries ONE lambda-uniform edge", okV and okW, "WPD's lambda-uniform content == the H-pin edge (AND-" "fire demonstrated at the declared edges, B2): the pair " "{L1, WPD} carries ONE open lambda-uniform edge, not two " "-- LABEL-LEVEL sharpening, cardinality UNCHANGED; the " "canonical residue (note DII): {H1 AND H2 AND H3}-" "cofinal (mod D = 0.0042) + {census-forall-k == LOOP} + " "{H-PIN = Omega-a + Omega-b + H123M; WPD(a < gamma_1^2) " "<== H-pin} + {WPD non-lambda legs: extension " "instantiated, TAILWPD world-front}; NO omega closed, " "NOTHING upgraded") print("\n [TYPED, BH8 ADOPTED] THE CONSOLIDATION: WPD's lambda-" "content == the") print(" H-pin edge; the H-pin floor is bridge-expressible in its " "DATA, walled") print(" at the eigenvector and root maps (exact exhibits), and " "its demand is") print(" only per-census priceable. The r133/r171 name collision " "is flagged.") print(" Census cardinality 4 UNCHANGED. NOT RH evidence. NO RH " "claim.") return 0 def run(): global CHECKS, FAILS CHECKS = [] FAILS = [] print("=" * 74) print("v937 -- PRIME.L1WPD.CLOSURE.01 (WPD's lambda-content == the " "H-pin edge;") print("the three-part bridge-expressibility verdict; the v925 " "all-x sharpening") print("propagated; the r133/r171 name collision flagged; round " "178; 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("v937: %d/%d checks passed | verdict %s | runtime %.1f s" % (n_run - len(fails), n_run, verdict, time.time() - T_ALL)) print("PINNING: R1-R5 + the loop/fire graphs recomputed in-run; " "the x_0/D_cs/") print("d_1/MRB/bridge/eps-gap tables PINNED from " "l1wpd_closure_probe.py") print("(SPEC %s, 29/29, record 228.9 s + deterministic re-run, " "no amendment," % PIN_SPEC_R178) print("all logs kept, RE-RUN GREEN AS TYPED AT PROMOTION). NOT RH " "evidence; NO") print("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.L1WPD.CLOSURE.01 L1/WPD consolidation + " "reduction: %d passed, %d failed ---" % (n_run - len(fails), len(fails))) return 0 if ok else 1 if __name__ == "__main__": raise SystemExit(run())