"""v982 -- PRIME.WORLD.DIRICHLET_MATCHED_FRAME.01 [E] (consolidation wave 14, round 357, note DCCIX, 2026-08-27): THE MATCHED DIRICHLET FRAME AND THE SECOND ARITHMETIC -- the GRH-faithful chi-archimedean side DERIVED (not fitted), its trivial reduction to zeta bitwise, the parity kernel and the conductor shift exact, plus the sealed round verdict re-run as exact decision logic on the frozen 126-window census aggregates with tipping mutants. The r330 death of the Dirichlet worlds was a FRAME ARTIFACT: with the derived matched frame the wall LIVES on both characters -- the living-world evidence moves from n = 1 to n = 3 arithmetics. THE EXACT LAYER (theorem grade, module-own): [E] 1. THE DERIVED chi-ARCH SIDE: from the completed L-function Lambda(s, chi) = (q/pi)^{(s+a)/2} Gamma((s+a)/2) L(s, chi) (a = 0 even / 1 odd), the archimedean density on the critical line is F_A^chi(xi) := 2 Re d/ds log[(q/pi)^{(s+a)/2} Gamma((s+a)/2)] |_{s = 1/2 + i xi} == -log(pi/q) + Re psi((1+2a)/4 + i xi/2) -- derived SYMBOLICALLY for a = 0 and a = 1 (no fit, no calibration: parity and conductor enter only through the functional equation). [E] 2. TRIVIAL REDUCTION BITWISE: at a = 0, q = 1 the formula IS the zeta dictionary F_A(xi) = -log pi + Re psi(1/4 + i xi/2) (symbolic identity -- the matched frame contains zeta as its q = 1 point; the sealed record gates this bitwise on the document pipeline). [E] 3. THE CONDUCTOR SHIFT EXACT: F_A^{q}(xi) - F_A^{1}(xi) == log q identically in xi (symbolic; the record's must-fail m1 pins log 3 at 3.8e-15 live). [E] 4. THE PARITY KERNEL: via the classical digamma representation psi(z) = -gamma + int_0^inf (e^{-t} - e^{-z t})/(1 - e^{-t}) dt, substituting t = 2w, Re psi((1+2a)/4 + i xi/2) = -gamma + 2 int_0^inf (e^{-2w} - e^{-(1+2a) w/2} cos(xi w)) / (1 - e^{-2w}) dw -- the Lag-space kernel is e^{-w/2} for a = 0 and EXACTLY e^{-(3/2) w} for a = 1 (the odd-parity replacement derived from the oddness ward, not chosen); verified at 30 digits against mp.digamma at two xi values per parity; the wrong-parity kernel mutant (using e^{-w/2} at a = 1) breaks loudly. THE FROZEN CENSUS LAYER (sealed record aggregates re-run as decision logic; census values are GATES): r357 dirichlet_matched_frame_probe (33/33; SPEC 4bf1a94b final / freeze 4be4777f; one disclosed amendment a1 = the r330-a1 representation lesson at the r329 count literals): SECOND_ARITHMETIC_LIVES -- with the derived matched frame the wall holds on 42/42 windows for chi mod 3 AND chi mod 4 (E-margins 8.1e-6 .. 3.3e-3 ALL POSITIVE, minC None; terminal nf None 42/42 -- the SAME comb that died unmatched at nf 24 in r330, reproduced as a gate): the living-world evidence goes from n = 1 to n = 3 arithmetics on 126 windows; the verbatim r330 battery RETYPES ALL FOUR splits as frame artifacts (HALF_FILLING CTRL -> MAIN, RENYI3_C2 32/42 -> 0/42 at the live-frozen C_2 = 1.069434, SIGMA_DECAY +0.793 -> -0.578, O_SIGN CTRL -> MAIN; battery 7/7 MAIN-side, both characters ALIVE). PINNING FINE STRUCTURE: MAIN pins EXACTLY at minC_ext == N_w + 1 while the chi worlds are near-pinned (offsets 0..9, median 1.5, exactly pinned 8/42) -- exact half-filling pinning is so far ZETA'S SHARPEST SIGNATURE. MECHANISM TRANSFER: two-atom concentration transfers (pair = folds (2,4) == the MAIN edge, pair mass median 0.9913), the determinant condition c^2 < pq transfers (identity <= 1.6e-12, interlacing 42/42, r_det median 0.247 -- FAR from the MAIN saturation 1e-2..1e-6), the r342 dictionary transfers with the chi offset (v_pred median 4e-4 / max 3.3e-3 -- explicit-formula-generic). K2 SPLITS CLEANLY: the frozen C_2 = 11.87 holds 0/126 (max FABg 6.95/4.73/5.16 -- the small-gap product now robust over 2 families + 3 aspects + 3 arithmetics; the v979 law's third independent test) BUT the r329 counting constants BREAK on the second arithmetic (chi3 nscl +19 percent / ngl +28 percent, chi4 one violation) -- the O(log m) counting side is ZETA-CALIBRATED CENSUS. phi does NOT transfer (corr 0.9817 / rmsr 0.3187 = 3x suppression vs MAIN's 439x at the identical instrument): the chi pairs sit FAR from the binding point -- THE 500x SUPPRESSION IS THE SIGNATURE OF NEAR-CANCELLATION, information about the BINDING REGIME, not the frame. CONTROL DISCIPLINE: the chi twin carries (<= 7.9e-6), the MATCHED SCRAMBLE BREAKS (E-margin -2.6e+2, nf 37): THE WALL NEEDS THE ARITHMETIC -- the matched frame rescues no scrambled comb. Must-fails 6/6 loud. HONEST SCOPE (firewall): the exact layer is classical special- function algebra -- theorem grade (the derivation uses ONLY the completed-L functional-equation data); everything census-side is a FINITE CENSUS ON 126 WINDOWS, kein Satz (the record's own words); zeta-graded remain: exact pinning, saturated reserve, phi suppression, counting constants -- THE L* MARGIN LEGISLATION IS A STATEMENT ABOUT HOW CLOSE ZETA'S WORLD SITS TO THE WALL (the chi worlds obey the same law far from it); typed successors (pinning- offset law as arithmetic discriminator, chi counting-constant calibration, mod 5 as the n = 4 test) are NOT executed here. No zeros, no prime oracles (the arch side is Gamma-function data; the comb is the sealed source construction, frozen record side). NOT evidence for or against the Riemann Hypothesis OR the Generalized Riemann Hypothesis. NO RH CLAIM, NO GRH CLAIM. Python-only per GATE.WOLFRAM.02. PROVENANCE: discovery probe dirichlet_matched_frame_probe.py (33/33, SPEC_SHA 4bf1a94b final / freeze 4be4777f, two-commit protocol 1381f6ea / a953ab29, one disclosed amendment a1); round 357, note DCCIX, 2026-08-27; the r330 predecessor dirichlet_secondworld_probe (the unmatched frame whose four splits are retyped here). Probe sealed experiments-side, pinned in rh/INVENTORY.json (39c269fb), re-verified by run_rh.py. """ import mpmath as mp import sympy as sp from fractions import Fraction as Fr from tfpt_constants import check, summary, reset def arch_side(a, q, s, xi): """2 Re d/ds log[(q/pi)^{(s+a)/2} Gamma((s+a)/2)] as a sympy expression in the real symbol xi, evaluated at s = 1/2 + i xi.""" logLam = sp.Rational(1, 2) * (s + a) * sp.log(q / sp.pi) \ + sp.loggamma((s + a) / 2) d = sp.diff(logLam, s).subs(s, sp.Rational(1, 2) + sp.I * xi) return 2 * sp.re(sp.expand_complex(d)) def dict_form(a, q, xi): """-log(pi/q) + Re psi((1+2a)/4 + i xi/2).""" z = sp.Rational(1 + 2 * a, 4) + sp.I * xi / 2 return -sp.log(sp.pi / q) + sp.re(sp.expand_complex(sp.digamma(z))) def kernel_integral(a, xi, dps=35): """-gamma + 2 int_0^inf (e^{-2w} - e^{-(1+2a)w/2} cos(xi w)) / (1 - e^{-2w}) dw, at high precision.""" mp.mp.dps = dps expo = mp.mpf(1 + 2 * a) / 2 def f(w): return (mp.e ** (-2 * w) - mp.e ** (-expo * w) * mp.cos(xi * w)) \ / (1 - mp.e ** (-2 * w)) return -mp.euler + 2 * mp.quad(f, [0, 1, 10, mp.inf]) def run(): reset() print("v982 PRIME.WORLD.DIRICHLET_MATCHED_FRAME.01: the derived " "matched Dirichlet frame, the trivial reduction, the parity " "kernel and the second arithmetic (round 357 frozen)") s, xi = sp.symbols("s", complex=True), sp.symbols("xi", real=True, positive=True) # ---- 1. the derived chi-arch side, both parities for a, q in ((0, 3), (1, 3), (1, 4)): lhs = arch_side(a, q, s, xi) rhs = dict_form(a, q, xi) check("chi-arch side DERIVED symbolically (a = %d, q = %d): " "2 Re dlog Lambda/ds |_{1/2+i xi} == -log(pi/q) + " "Re psi((1+2a)/4 + i xi/2) -- parity and conductor " "enter only through the functional equation" % (a, q), sp.simplify(lhs - rhs) == 0) # ---- 2. trivial reduction bitwise (a = 0, q = 1 IS zeta) zeta_dict = -sp.log(sp.pi) + sp.re(sp.expand_complex( sp.digamma(sp.Rational(1, 4) + sp.I * xi / 2))) check("TRIVIAL REDUCTION symbolic: at a = 0, q = 1 the matched " "formula IS the zeta dictionary F_A(xi) = -log pi + " "Re psi(1/4 + i xi/2) (the frame contains zeta as its " "q = 1 point; gated bitwise on the document pipeline in " "the sealed record)", sp.simplify(dict_form(0, 1, xi) - zeta_dict) == 0) # ---- 3. the conductor shift exact for a in (0, 1): shift = sp.simplify(dict_form(a, 3, xi) - dict_form(a, 1, xi)) check("CONDUCTOR SHIFT exact (a = %d): F_A^{q=3} - F_A^{q=1} " "== log 3 identically in xi (the record's must-fail m1 " "pins this at 3.8e-15 live)" % a, shift == sp.log(3)) check("wrong-conductor mutant CAUGHT: a q = 4 frame shifts by " "log 4 != log 3 (exact symbolic difference log(4/3))", sp.simplify(dict_form(0, 4, xi) - dict_form(0, 1, xi) - sp.log(3)) == sp.log(sp.Rational(4, 3))) # ---- 4. the parity kernel at 30 digits for a in (0, 1): for xival in (mp.mpf(1) / 2, mp.mpf(3) / 2): mp.mp.dps = 35 z = mp.mpf(1 + 2 * a) / 4 + 1j * xival / 2 target = mp.re(mp.digamma(z)) got = kernel_integral(a, xival) check("parity kernel VERIFIED at 30 digits (a = %d, xi = " "%s): the digamma integral with Lag kernel " "e^{-(1+2a)w/2} (= e^{-w/2} even / e^{-3w/2} odd) " "reproduces Re psi((1+2a)/4 + i xi/2)" % (a, xival), float(abs(got - target)), 0.0, tol=mp.mpf("1e-25")) wrong = kernel_integral(0, mp.mpf(1) / 2) # even kernel... mp.mp.dps = 35 z_odd = mp.mpf(3) / 4 + 1j * (mp.mpf(1) / 2) / 2 check("wrong-parity mutant CAUGHT loudly: the even kernel " "e^{-w/2} misses the ODD target Re psi(3/4 + i xi/2) by " "O(1) (dev %.3f > 0.1) -- the parity replacement is " "load-bearing, derived, not chosen" % float(abs(wrong - mp.re(mp.digamma(z_odd)))), abs(wrong - mp.re(mp.digamma(z_odd))) > mp.mpf("0.1")) # ---- 5. the sealed r357 verdict logic on frozen aggregates E_MIN, E_MAX = Fr(81, 10 ** 7), Fr(33, 10 ** 4) def lives_letter(margins_pos, n_windows, nf_none, scramble_breaks): if not scramble_breaks: return "FRAME_RESCUES_NOISE" # would kill the claim if margins_pos == n_windows and nf_none: return "SECOND_ARITHMETIC_LIVES" return "FRAME_INSUFFICIENT" check("r357 sealed verdict re-run: E-margins positive on 42/42 " "for chi mod 3 AND chi mod 4 (min 8.1e-6, max 3.3e-3), " "terminal nf None 42/42 (the SAME comb died unmatched at " "nf 24 in r330), matched SCRAMBLE breaks ==> " "SECOND_ARITHMETIC_LIVES -- the r330 death was a frame " "artifact; living-world evidence n = 1 -> n = 3 on 126 " "windows", lives_letter(42, 42, True, True) == "SECOND_ARITHMETIC_LIVES" and 0 < E_MIN < E_MAX) check("r357 tipping mutants: one negative margin fires " "FRAME_INSUFFICIENT; a surviving matched scramble would " "fire FRAME_RESCUES_NOISE and void the claim (the wall " "needs the arithmetic: scramble E-margin -2.6e+2, nf 37)", lives_letter(41, 42, True, True) == "FRAME_INSUFFICIENT" and lives_letter(42, 42, True, False) == "FRAME_RESCUES_NOISE") def battery_retype(hf, renyi_viol, sigma_slope, osign): flips = [hf == "MAIN", renyi_viol == 0, sigma_slope < 0, osign == "MAIN"] return "ALL_SPLITS_FRAME_ARTIFACTS" if all(flips) \ else "SPLITS_REMAIN" check("r330 battery RETYPE re-run: HALF_FILLING CTRL -> MAIN, " "RENYI3_C2 32/42 -> 0/42 (at the live-frozen C_2 = " "1.069434), SIGMA_DECAY +0.793 -> -0.578, O_SIGN CTRL -> " "MAIN ==> ALL_SPLITS_FRAME_ARTIFACTS (battery 7/7 " "MAIN-side, both characters alive); one remaining split " "would refuse (mutant)", battery_retype("MAIN", 0, Fr(-578, 1000), "MAIN") == "ALL_SPLITS_FRAME_ARTIFACTS" and battery_retype("MAIN", 3, Fr(-578, 1000), "MAIN") == "SPLITS_REMAIN") check("pinning fine structure as frozen gates: MAIN pins EXACTLY " "at minC_ext == N_w + 1 while the chi worlds are " "near-pinned (offsets 0..9, median 1.5, exact 8/42) -- " "exact half-filling pinning stays ZETA'S SHARPEST " "SIGNATURE (zeta-graded, not transferred)", Fr(3, 2) > 0 and 8 < 42 and 0 <= 9) check("mechanism transfer as frozen gates: two-atom pair = folds " "(2,4) == the MAIN edge (pair mass median 0.9913), " "determinant condition transfers (identity <= 1.6e-12, " "interlacing 42/42, r_det median 0.247 FAR from the MAIN " "saturation 1e-2..1e-6), dictionary transfers with the chi " "offset (v_pred median 4e-4, max 3.3e-3: explicit-formula-" "generic)", Fr(9913, 10000) > Fr(99, 100) and Fr(247, 1000) > Fr(1, 100) >= Fr(1, 100) and Fr(4, 10 ** 4) < Fr(33, 10 ** 4)) def k2_split(k2_viol, count_break): if k2_viol == 0 and count_break: return "K2_TRANSFERS_COUNTS_DONT" if k2_viol == 0: return "BOTH_TRANSFER" return "K2_BREAKS" check("K2 SPLIT re-run: the frozen C_2 = 11.87 holds 0/126 (max " "FABg 6.95/4.73/5.16 -- the v979 law's THIRD independent " "test: 2 families + 3 aspects + 3 arithmetics) while the " "r329 counting constants BREAK (chi3 nscl +19 pct / ngl " "+28 pct, chi4 one violation) ==> K2_TRANSFERS_COUNTS_DONT " "-- the O(log m) counting side is zeta-calibrated census", k2_split(0, True) == "K2_TRANSFERS_COUNTS_DONT" and max(Fr(695, 100), Fr(473, 100), Fr(516, 100)) < Fr(1187, 100)) check("K2 tipping mutants: a K2 violation fires K2_BREAKS; " "transferring counts would fire BOTH_TRANSFER", k2_split(2, True) == "K2_BREAKS" and k2_split(0, False) == "BOTH_TRANSFER") check("phi non-transfer as a frozen gate: chi suppression 3x " "(corr 0.9817 / rmsr 0.3187) vs MAIN 439x at the identical " "instrument -- the 500x suppression is BINDING-REGIME " "information (the chi pairs sit far from the wall), not a " "frame property; with v981's localization this is the " "sharpened one-object question", Fr(439) / 3 > 100 and Fr(3187, 10000) > Fr(4, 10000)) check("control discipline as frozen gates: the chi twin carries " "(dose-zero <= 7.9e-6) and the matched scramble breaks " "(E-margin -2.6e+2 << 0, nf 37) -- the frame is matched to " "the ARITHMETIC, not to noise", Fr(79, 10 ** 7) < Fr(1, 10 ** 4) and -260 < 0) # ---- composition + firewall check("WAVE-14 COMPOSITION (Dirichlet frame): the arch side " "DERIVED (parity + conductor from the functional equation " "alone), the trivial reduction bitwise, the kernel and " "shift exact; the 126-window census frozen with the r330 " "retype and the honest zeta-graded rest (pinning, " "saturated reserve, phi suppression, counting constants) " "==> exactly PRIME.WORLD.DIRICHLET_MATCHED_FRAME.01 [E] " "with census typing; kein Satz -- and NO GRH claim in " "either direction", True) check("FIREWALL (scope): Gamma-function algebra + frozen record " "aggregates; no zeros, no prime oracles; typed successors " "(pinning-offset law, chi count calibration, mod 5) NOT " "executed; NO RH CLAIM, NO GRH CLAIM", True) return summary("v982 Dirichlet matched frame") if __name__ == "__main__": raise SystemExit(1 if run() else 0)