"""v537 -- HECKE.GEOM.HALFINT.01: the half-integral bridge. Shimura preimage + Hecke equivariance + Kohnen scope fence + Waldspurger/Baruch-Mao constancy. Consolidated from discovery T38/T41/T44/T45; builds on the weight-4 Hecke/Eichler package (v535/v536) without duplicating those checks. [E] (A) UNIQUE SHIMURA PREIMAGE AT SCALE -8 (T38). In the 70-element compiler theta-monoid of weight 5/2 (theta2/3/4 plain and q^2-scaled, total exponent sum 5, integer-q only) there is exactly ONE object with the signed identity Sh_{t=2, psi=1}(g) = -8 * f8 coefficientwise (screen n <= 40; witness extended to n <= 120). Witness: g = theta2(q^2)^2 * theta3(q^2) * theta4 * theta4(q^2) (key (0,2,0,1,1,1)). Three further monoid elements lift to other scales (+8, -16, +16)*f8 (documented; not the -8 witness). Shimura convention (Shimura 1973 / Cipra): A_t(n) = sum_{d|n} psi(d) (t/d)_Kronecker d^{k-1} b(t n^2/d^2) with k=2 (weight 5/2 -> 4), psi=1, t=2. [E] (B) HECKE EQUIVARIANCE (T41). g is an exact T(p^2)-eigenform of weight 5/2, trivial nebentypus, with eigenvalues a_p(f8) for p = 3,5,7,11,13 (= -4,-2,24,-44,22). Shimura 1973 formula (T(p^2)f)(n) = a(p^2 n) + chi(p)(n/p) p^{k-1} a(n) + chi(p^2) p^{2k-1} a(n/p^2) (k=2 => p and p^3). Trial FE of the 5/2-side around centre 5/4 with N=512, eps=+1 as a numeric anchor (AFE↔direct). [E] (C) KOHNEN SCOPE FENCE (T44) -- NEGATIVE RESULT PART OF CLAIM. U4(g)=0 exactly (mass mod 4 = {0:0, 1:..., 2:..., 3:0}); the spectral Kohnen pr+_4 sees no plus component (H=W4U4 vanishes on ker(U4)). Level hypothesis 32=4*8 with M=8 even lies outside Kohnen 1982 (needs M odd squarefree; 4a-Mbc=1 unsolvable). Plus-route structurally closed. [C] (D) WALDSPURGER / BARUCH-MAO CONSTANCY (T45). With AFE-based twist L-values (AFE vs direct at s=3.5, rel <= 1e-8; eps_d = chi_d(8); eps=-1 => L(2)=0) the quotient R(d) = b(|d|)^2 / (|d|^{3/2} L(f8 x chi_d, 2)) is CONSTANT over 10 fundamental discriminants d ≡ 1 mod 8 (R = 23.1873585645..., spread <= 1e-12; includes d=89). The class d ≡ 5 mod 8 vanishes structurally (b=0, root number -1). HONEST FENCES (load-bearing typing): * Shimura 1973, Kohnen 1980/82, Waldspurger 1981 / Baruch-Mao are CLASSICAL -- the claim is: (i) the preimage sits IN the compiler monoid and is UNIQUE there; (ii) the full chain is in-suite machine-checked; (iii) the constant R is MEASURED, not derived. * GL(2) centre s=2 (weight-4 normalisation), NOT the xi-line. * NO RH statement; ZETA.HP.CARRIER untouched. * Trial FE of the 5/2-side is a numeric anchor, not a derivation of the half-integral functional equation package. * NO marker upgrades of any pre-existing contract. Status: [E] exact integer Shimura uniqueness / Hecke equivariance / Kohnen scope fence; [C] AFE-numeric Waldspurger constancy (measured R). Python; Wolfram-mirrored (exact a_p / Shimura scale / Kohnen scope / root-number arithmetic -- q-series builds, AFE/L-values, and R-constancy stay Python-only), counted per GATE.WOLFRAM.02. Discovery provenance: experiments/tfpt-discovery/cuspidal_bridge_probe.py (T38) experiments/tfpt-discovery/functor_wiring_halfintegral_probe.py (T41) experiments/tfpt-discovery/kohnen_operator_projection_probe.py (T44) experiments/tfpt-discovery/baruch_mao_metaplectic_probe.py (T45) (+ f8 scaffolding from e8_glue_chi4_signed_theta_probe.py, T11) """ from __future__ import annotations import itertools import math import time from fractions import Fraction import mpmath import numpy as np import sympy as sp from tfpt_constants import check, summary, reset # ---------------------------------------------------------------- budgets A_P = {3: -4, 5: -2, 7: 24, 11: -44, 13: 22} PRIMES_HECKE = (3, 5, 7, 11, 13) WITNESS_KEY = (0, 2, 0, 1, 1, 1) # th2(q2)^2 th3(q2) th4 th4(q2) K_HALF = 2 # weight = k+1/2 = 5/2 Q_CAND = 3200 # uniqueness screen (t=2 => n_cap ~40) N_SHIMURA_SEARCH = 40 N_SHIMURA_EXT = 120 Q_WITNESS = max(2 * (N_SHIMURA_EXT ** 2) + 16, 13 * 13 * 100 + 16) N_HECKE = 100 N_F8 = 100_000 # AFE twist L-values (T45) Q_G_SMALL = 200 # g coeffs for Waldspurger (d <= 120) DMAX = 120 SMALL_D_VALIDATE = (5, 13, 17, 29, 37, 41, 89) D_LITMUS = 89 AFE_DIRECT_TOL = mpmath.mpf("1e-8") FE_EPS_RATIO = 10.0 STABLE_REL_TOL = mpmath.mpf("1e-5") L_VANISH_TOL = mpmath.mpf("1e-20") R_SPREAD_TOL = mpmath.mpf("1e-12") R_TARGET = mpmath.mpf("23.1873585645") LEVEL_FE_TRIALS = (16, 32, 64, 128, 256, 512) FE_THRESH = 5e-3 AFE_SAFETY = 50.0 # nterms ~ safety * sqrt(N)/(2pi) # ---------------------------------------------------------------- helpers def chi4(n: int) -> int: r = n % 4 if r == 1: return 1 if r == 3: return -1 return 0 def kronecker(d: int, n: int) -> int: return int(sp.kronecker_symbol(d, n)) def legendre(a: int, p: int) -> int: if a % p == 0: return 0 return int(sp.legendre_symbol(a % p, p)) def eta_pass(d, e, order): s = np.zeros(order + 1, dtype=np.int64) s[0] = 1 for k in range(d, order + 1, d): for _ in range(e): s[k:] = s[k:] - s[:-k] return s def conv_i64(a, b, order): return np.convolve(a, b)[: order + 1].astype(np.int64) def is_fundamental_discriminant(d: int) -> bool: if d == 0: return False if d % 4 == 1: return abs(sp.mobius(abs(d))) == 1 if d % 4 != 0: return False m = d // 4 if m % 4 not in (2, 3): return False return abs(sp.mobius(abs(m))) == 1 def twist_root_number(d: int, eps_f: int = 1, N_f: int = 8) -> int: """eps_d = eps_f * chi_d(N_f) for weight-4 trivial nebentypus. With eps_f=+1, N_f=8: eps_d = chi_d(8) = +1 (d≡1 mod 8), -1 (d≡5). """ assert d % 2 != 0 return int(eps_f * kronecker(d, N_f)) # ---------------------------------------------------------------- thetas _TH_CACHE: dict = {} def theta2_t(order_t, scale_q=1): s = np.zeros(order_t + 1, dtype=np.int64) o = 1 while True: exp = scale_q * o * o if exp > order_t: break s[exp] = 2 o += 2 return s def theta3_t(order_t, scale_q=1): s = np.zeros(order_t + 1, dtype=np.int64) s[0] = 1 n = 1 while True: exp = 4 * scale_q * n * n if exp > order_t: break s[exp] = 2 n += 1 return s def theta4_t(order_t, scale_q=1): s = np.zeros(order_t + 1, dtype=np.int64) s[0] = 1 n = 1 while True: exp = 4 * scale_q * n * n if exp > order_t: break s[exp] = 2 * ((-1) ** n) n += 1 return s def _th(kind, scale_q, order_t): key = (kind, scale_q, order_t) if key not in _TH_CACHE: if kind == 2: _TH_CACHE[key] = theta2_t(order_t, scale_q) elif kind == 3: _TH_CACHE[key] = theta3_t(order_t, scale_q) else: _TH_CACHE[key] = theta4_t(order_t, scale_q) return _TH_CACHE[key] def monomial_t(a0, a2, b0, b2, c0, c2, order_t): s = np.zeros(order_t + 1, dtype=np.int64) s[0] = 1 for _ in range(a0): s = conv_i64(s, _th(2, 1, order_t), order_t) for _ in range(a2): s = conv_i64(s, _th(2, 2, order_t), order_t) for _ in range(b0): s = conv_i64(s, _th(3, 1, order_t), order_t) for _ in range(b2): s = conv_i64(s, _th(3, 2, order_t), order_t) for _ in range(c0): s = conv_i64(s, _th(4, 1, order_t), order_t) for _ in range(c2): s = conv_i64(s, _th(4, 2, order_t), order_t) return s def to_q_series(s_t, qmax): for r in range(1, 4): if np.any(s_t[r::4] != 0): return None out = [0] * (qmax + 1) lim = min(qmax, (len(s_t) - 1) // 4) for n in range(lim + 1): out[n] = int(s_t[4 * n]) return out def build_candidates(qmax): """Integer-q weight-5/2 theta monoid (plain + q^2-scaled).""" tmax = 4 * qmax cands = [] def add(label, a0, a2, b0, b2, c0, c2): if a0 + a2 + b0 + b2 + c0 + c2 != 5: return if (a0 + 2 * a2) % 4 != 0: return st = monomial_t(a0, a2, b0, b2, c0, c2, tmax) bq = to_q_series(st, qmax) if bq is None or all(x == 0 for x in bq): return cands.append((label, bq, (a0, a2, b0, b2, c0, c2))) for b in range(6): add(f"V0:th3^{b}*th4^{5 - b}", 0, 0, b, 0, 5 - b, 0) for b in range(2): add(f"V0:th2^4*th3^{b}*th4^{1 - b}", 4, 0, b, 0, 1 - b, 0) for a0, a2, b0, b2, c0 in itertools.product(range(6), repeat=5): c2 = 5 - (a0 + a2 + b0 + b2 + c0) if c2 < 0: continue if a2 == b2 == c2 == 0: continue add(f"V1:{a0},{a2},{b0},{b2},{c0},{c2}", a0, a2, b0, b2, c0, c2) uniq = {} for lab, bq, key in cands: tup = tuple(bq) if tup not in uniq: uniq[tup] = (lab, bq, key) return list(uniq.values()) # ---------------------------------------------------------------- Shimura def shimura_A(bq, n, t, psi): total = 0 for d in sp.divisors(n): d = int(d) idx = t * (n // d) * (n // d) if idx >= len(bq): return None total += psi(d) * kronecker(t, d) * (d ** (K_HALF - 1)) * bq[idx] return int(total) def shimura_lift(bq, t, psi, nmax): out = [0] * (nmax + 1) for n in range(1, nmax + 1): val = shimura_A(bq, n, t, psi) if val is None: return None out[n] = val return out def max_lift_n(bq_len, t): return int(((bq_len - 1) // t) ** 0.5) # ---------------------------------------------------------------- Hecke def T_p2_half(a, p, chi_p, chi_p2, order, n_check): assert p * p * n_check <= order p1 = p ** (K_HALF - 1) p3 = p ** (2 * K_HALF - 1) out = [Fraction(0) for _ in range(n_check + 1)] for n in range(n_check + 1): np2 = n * p * p term = Fraction(a[np2] if np2 <= order else 0) if n >= 1: term += (Fraction(chi_p) * Fraction(legendre(n, p)) * Fraction(p1) * Fraction(a[n])) if n % (p * p) == 0: term += Fraction(chi_p2) * Fraction(p3) * Fraction(a[n // (p * p)]) out[n] = term return out # ---------------------------------------------------------------- FE / AFE def L_g_direct(g, s, terms): s = mpmath.mpf(s) tot = mpmath.mpf(0) for n in range(1, min(terms, len(g) - 1) + 1): bn = g[n] if bn: tot += mpmath.mpf(bn) * mpmath.power(n, -s) return tot def L_g_afe(g, s, Nlev, eps, terms): s = mpmath.mpf(s) wt = mpmath.mpf("2.5") sqN = mpmath.sqrt(Nlev) lam = mpmath.mpf(0) for n in range(1, min(terms, len(g) - 1) + 1): bn = g[n] if not bn: continue xx = 2 * mpmath.pi * n / sqN pref = sqN / (2 * mpmath.pi * n) c = mpmath.mpf(bn) lam += c * (pref ** s * mpmath.gammainc(s, xx) + eps * pref ** (wt - s) * mpmath.gammainc(wt - s, xx)) return lam / ((sqN / (2 * mpmath.pi)) ** s * mpmath.gamma(s)) def nterms_for(Nlev: int, safety: float = AFE_SAFETY) -> int: sq = math.sqrt(Nlev) need = int(math.ceil(safety * sq / (2 * math.pi))) + 80 return min(N_F8, max(1200, need)) def L_twist_direct(a_f8, d, s, terms): s = mpmath.mpf(s) tot = mpmath.mpf(0) for n in range(1, min(terms, len(a_f8) - 1) + 1): an = a_f8[n] if not an: continue ch = kronecker(d, n) if not ch: continue tot += mpmath.mpf(an * ch) * mpmath.power(n, -s) return tot def L_twist_afe(a_f8, d, s, Nlev, eps, terms): s = mpmath.mpf(s) sqN = mpmath.sqrt(Nlev) two_pi = 2 * mpmath.pi lam = mpmath.mpf(0) kms = mpmath.mpf(4) - s for n in range(1, min(terms, len(a_f8) - 1) + 1): an = a_f8[n] if not an: continue ch = kronecker(d, n) if not ch: continue xx = two_pi * n / sqN pref = sqN / (two_pi * n) c = mpmath.mpf(an * ch) lam += c * (pref ** s * mpmath.gammainc(s, xx) + eps * pref ** kms * mpmath.gammainc(kms, xx)) return lam / ((sqN / (2 * mpmath.pi)) ** s * mpmath.gamma(s)) # ====================================================================== def run(): reset() mpmath.mp.dps = 25 t0 = time.time() print("v537 HECKE.GEOM.HALFINT.01 -- half-integral bridge " "(Shimura preimage + Hecke + Kohnen fence + Waldspurger)") # ============================================================ S0 print("S0 -- scaffold: f8 = eta(2t)^4 eta(4t)^4") f8 = np.roll(conv_i64(eta_pass(2, 4, N_F8), eta_pass(4, 4, N_F8), N_F8), 1) f8[0] = 0 a_f8 = [int(f8[n]) for n in range(N_F8 + 1)] check("S0.f8: eta(2t)^4 eta(4t)^4 matches T11 a_p head " "{3:-4,5:-2,7:24,11:-44,13:22}; a_1=1; a_even=0 on n<=200", a_f8[1] == 1 and all(a_f8[p] == v for p, v in A_P.items()) and all(a_f8[n] == 0 for n in range(2, 201, 2))) # ============================================================ A print("A -- unique Shimura preimage in the compiler theta-monoid (T38)") t1 = time.time() cands = build_candidates(Q_CAND) print(f" built {len(cands)} integer-q candidates to O(q^{Q_CAND}) " f"in {time.time() - t1:.1f}s") check(f"A.space: compiler weight-5/2 theta monoid has " f"{len(cands)} distinct integer-q series " f"(~70 expected; V0 + q^2-scaled V1)", 60 <= len(cands) <= 80) n_cap = min(N_SHIMURA_SEARCH, max_lift_n(Q_CAND + 1, 2)) matches = [] # any nonzero scale * f8 for lab, bq, key in cands: sh = shimura_lift(bq, 2, lambda d: 1, n_cap) if sh is None or sh[1] == 0: continue scale = Fraction(int(sh[1]), int(a_f8[1])) if all(scale * a_f8[n] == sh[n] for n in range(1, n_cap + 1)): matches.append((lab, key, scale)) matches_m8 = [m for m in matches if m[2] == Fraction(-8)] print(f" Shimura t=2,psi=1 scale-multiples of f8 on " f"n=1..{n_cap}: {len(matches)}") for lab, key, scale in matches: print(f" key={key} scale={scale} {lab}") check(f"A.unique: exactly ONE candidate with Sh_{{t=2,psi=1}} = " f"-8*f8 on n=1..{n_cap} (signed scale fixed); " f"witness key={WITNESS_KEY}; " f"{len(matches)} monoid elements lift to other scales " f"(+8,-16,+16 documented)", len(matches_m8) == 1 and matches_m8[0][1] == WITNESS_KEY and len(matches) == 4 and {m[2] for m in matches} == {Fraction(-8), Fraction(8), Fraction(-16), Fraction(16)}) # Extended witness verify to n=120 t1 = time.time() g_big = to_q_series(monomial_t(*WITNESS_KEY, 4 * Q_WITNESS), Q_WITNESS) assert g_big is not None sh_ext = shimura_lift(g_big, 2, lambda d: 1, N_SHIMURA_EXT) ext_ok = (sh_ext is not None and all( sh_ext[n] == -8 * a_f8[n] for n in range(1, N_SHIMURA_EXT + 1))) print(f" extended Sh verify n=1..{N_SHIMURA_EXT} on O(q^{Q_WITNESS}): " f"{ext_ok} ({time.time() - t1:.1f}s)") print(f" g head: {g_big[:16]}") print(f" Sh head: {sh_ext[:16] if sh_ext else None}") check(f"A.witness: g=theta2(q^2)^2*theta3(q^2)*theta4*theta4(q^2); " f"Sh_{{t=2,psi=1}}(g)=-8*f8 exactly on n=1..{N_SHIMURA_EXT}; " f"g[0]=0; NOT Kohnen-plus", ext_ok and g_big[0] == 0 and any(g_big[n] != 0 and n % 4 in (2, 3) for n in range(1, 200))) check("A.convention: Shimura formula documented " "(A_t(n)=sum_{d|n} psi(d)(t/d) d^{k-1} b(t n^2/d^2); " "k=2, psi=1, t=2; weight 5/2 -> 4)", K_HALF == 2 and 2 * K_HALF == 4 and N_SHIMURA_EXT >= 120) # ============================================================ B print("B -- T(p^2) Hecke equivariance + trial FE (T41)") g = g_big # reuse extended table for Hecke hecke_ok = True for p in PRIMES_HECKE: b = T_p2_half(g, p, 1, 1, Q_WITNESS, N_HECKE) ap = A_P[p] if any(b[n] != Fraction(ap) * Fraction(g[n]) for n in range(N_HECKE + 1)): hecke_ok = False print(f" p={p}: FAIL") else: print(f" p={p}: T(p^2)g = {ap} g on n=0..{N_HECKE}") check("B.hecke: g is exact T(p^2)-eigenform (trivial chi) with " f"lam=a_p(f8) for p={PRIMES_HECKE} on n<= {N_HECKE} " f"(values {-4,-2,24,-44,22})", hecke_ok) # Trial FE around centre 5/4 S_AFE = [mpmath.mpf("2.8"), mpmath.mpf("3.2"), mpmath.mpf("3.6")] AFE_TERMS = 4000 DIR_TERMS = min(Q_WITNESS, 18000) Ldir_cache = {float(s): L_g_direct(g, s, DIR_TERMS) for s in S_AFE} fe_best = None for Nlev in LEVEL_FE_TRIALS: for eps in (1, -1): max_gap = mpmath.mpf(0) for s in S_AFE: Ld = Ldir_cache[float(s)] La = L_g_afe(g, s, Nlev, eps, AFE_TERMS) gap = (abs(La / Ld - 1) if Ld != 0 else mpmath.mpf(1)) max_gap = max(max_gap, gap) if fe_best is None or max_gap < fe_best[2]: fe_best = (Nlev, eps, max_gap) print(f" best FE: N={fe_best[0]}, eps={fe_best[1]:+d}, " f"max|AFE/dir-1|={float(fe_best[2]):.3e}") check("B.FE: trial half-integral FE Lambda(s)=eps Lambda(5/2-s) " f"anchors at N=512, eps=+1 (centre 5/4); " f"max|AFE/direct-1|={float(fe_best[2]):.3e} < {FE_THRESH}", fe_best[0] == 512 and fe_best[1] == 1 and float(fe_best[2]) < FE_THRESH) check("B.fence: half-integral L-series generally have NO Euler " "product (multiplicativity of g-coeffs fails on coprime pairs)", any(g[n] * g[m] != g[n * m] for n in range(1, 40) for m in range(1, 40) if sp.gcd(n, m) == 1 and n * m <= 200 and g[n] and g[m])) # ============================================================ C print("C -- Kohnen scope fence / U4(g)=0 (T44; NEGATIVE in claim)") g_small = g[: max(Q_G_SMALL, 801) + 1] mass_mod4 = { r: sum(abs(g_small[n]) for n in range(1, 801) if n % 4 == r) for r in range(4) } print(f" |g| mass by n mod 4 (n=1..800): {mass_mod4}") u4_zero = all(g[4 * n] == 0 for n in range(0, Q_WITNESS // 4 + 1) if 4 * n <= Q_WITNESS) check("C.U4: U4(g)=0 exactly (no Fourier mass on n≡0 mod 4); " f"mass mod 4 = {{0:0, 1:{mass_mod4[1]}, 2:{mass_mod4[2]}, " f"3:0}}", u4_zero and mass_mod4[0] == 0 and mass_mod4[3] == 0 and mass_mod4[1] > 0 and mass_mod4[2] > 0) check("C.spectral: on ker(U4), H=W4∘U4 vanishes so raw " "Kohnen pr⁺_4=(1/3)Id is NOT a plus projector " "(no plus component of g)", u4_zero) # structural: H(g)=0 ⇒ pr4(g)=g/3 not plus-supported M_HYP = 8 level_hyp = 4 * M_HYP M_odd = (M_HYP % 2 == 1) M_sqfree = (abs(sp.mobius(M_HYP)) == 1) w4_solvable = None for a in range(0, 64): for b in range(1, 64, 4): if (4 * a - 1) % (M_HYP * b) == 0: c = (4 * a - 1) // (M_HYP * b) w4_solvable = (a, b, c) break if w4_solvable is not None: break outside = (not (M_odd and M_sqfree)) and (w4_solvable is None) print(f" level {level_hyp}=4*{M_HYP}: M odd={M_odd}, " f"sqfree={M_sqfree}; W(4) solvable={w4_solvable}") check("C.scope: level 32=4·8 has M=8 even (not odd squarefree); " "W(4) condition 4a−Mbc=1 unsolvable; OUTSIDE Kohnen 1982 " "plus-new isomorphism scope (plus-route structurally closed)", outside and level_hyp == 32 and w4_solvable is None) # ============================================================ D print("D -- Waldspurger / Baruch-Mao constancy (T45)") g_w = to_q_series(monomial_t(*WITNESS_KEY, 4 * Q_G_SMALL), Q_G_SMALL) assert g_w is not None # Untwisted Fricke of f8 gaps_f8 = {} s45 = mpmath.mpf("4.5") Ld_f8 = sum(mpmath.mpf(a_f8[n]) * mpmath.power(n, -s45) for n in range(1, 8001) if a_f8[n]) for eps in (1, -1): sqN = mpmath.sqrt(8) lam = mpmath.mpf(0) for n in range(1, 2001): an = a_f8[n] if not an: continue xx = 2 * mpmath.pi * n / sqN pref = sqN / (2 * mpmath.pi * n) c = mpmath.mpf(an) lam += c * (pref ** s45 * mpmath.gammainc(s45, xx) + eps * pref ** (4 - s45) * mpmath.gammainc(4 - s45, xx)) La = lam / ((sqN / (2 * mpmath.pi)) ** s45 * mpmath.gamma(s45)) gaps_f8[eps] = abs(La - Ld_f8) eps_f8_num = 1 if gaps_f8[1] < gaps_f8[-1] else -1 check("D.f8-Fricke: numeric Fricke of f8 is +1 (T12); " "gap(+1)/gap(-1) ratio ≥ 10", eps_f8_num == 1 and max(gaps_f8.values()) / min(gaps_f8.values()) > 10) def evaluate_twist(d: int): Nlev = 8 * d * d eps_th = twist_root_number(d, 1, 8) nterm = nterms_for(Nlev) s_hi = mpmath.mpf("3.5") nterm_afe = min(N_F8, max(nterm, 8000)) nterm_dir = min(N_F8, max(40000, 8 * nterm_afe)) L_dir_hi = L_twist_direct(a_f8, d, s_hi, terms=nterm_dir) gap_th = abs(L_twist_afe(a_f8, d, s_hi, Nlev, eps_th, nterm_afe) - L_dir_hi) gap_wrong = abs(L_twist_afe(a_f8, d, s_hi, Nlev, -eps_th, nterm_afe) - L_dir_hi) eps_hi_ok = (gap_th < gap_wrong and (gap_wrong / gap_th > FE_EPS_RATIO if gap_th > 0 else True)) rel_gap_hi = (gap_th / abs(L_dir_hi) if L_dir_hi != 0 else mpmath.mpf(1)) L_afe = L_twist_afe(a_f8, d, 2, Nlev, eps_th, nterm_afe) L_dir_c = L_twist_direct(a_f8, d, 2, terms=nterm_afe) n_lo = max(400, nterm_afe // 2) n_hi = min(N_F8, max(nterm_afe, n_lo + 200)) L_lo = L_twist_afe(a_f8, d, 2, Nlev, eps_th, n_lo) L_hi = L_twist_afe(a_f8, d, 2, Nlev, eps_th, n_hi) stab_rel = (abs(L_hi - L_lo) / abs(L_hi) if abs(L_hi) > L_VANISH_TOL else mpmath.mpf(0)) return { "d": d, "eps": eps_th, "L_afe": L_afe, "L_dir_c": L_dir_c, "rel_gap_hi": rel_gap_hi, "eps_hi_ok": eps_hi_ok, "stab_rel": stab_rel, "centre_vanish_forced": eps_th == -1, "centre_is_zero": abs(L_afe) < L_VANISH_TOL, } val_rows = [evaluate_twist(d) for d in SMALL_D_VALIDATE if is_fundamental_discriminant(d)] eps_ok_all = all(r["eps_hi_ok"] for r in val_rows) afe_dir_ok = all(r["rel_gap_hi"] < AFE_DIRECT_TOL for r in val_rows) vanish_ok = all(r["centre_is_zero"] for r in val_rows if r["centre_vanish_forced"]) for r in val_rows: print(f" d={r['d']}: eps={r['eps']:+d}, " f"L_afe(2)={r['L_afe']}, " f"rel@3.5={float(r['rel_gap_hi']):.3e}") check("D.AFE(i): theoretical eps_d=chi_d(8) preferred over wrong " f"sign at s=3.5 (gap ratio ≥ {FE_EPS_RATIO}) for " f"d in {SMALL_D_VALIDATE}", eps_ok_all and len(val_rows) >= 6) check("D.AFE(ii): AFE matches direct sums at s=3.5 " f"(rel ≤ {float(AFE_DIRECT_TOL):.0e}); eps=-1 ⇒ L(2)=0", afe_dir_ok and vanish_ok) row89 = next(r for r in val_rows if r["d"] == D_LITMUS) d89_ok = (row89["stab_rel"] < STABLE_REL_TOL and row89["eps_hi_ok"] and row89["rel_gap_hi"] < AFE_DIRECT_TOL and not row89["centre_is_zero"]) d89_dir_bad = (abs(row89["L_afe"] - row89["L_dir_c"]) / abs(row89["L_afe"]) > mpmath.mpf("1e-2")) print(f" litmus d=89: L_afe(2)={row89['L_afe']}, " f"stab={float(row89['stab_rel']):.3e}") check("D.AFE(iii): d=89 AFE stabilises (term-doubling < 1e-5, " "s=3.5 match); nonvanishing; direct-at-s=2 still disagrees " "(T44 trap)", d89_ok and d89_dir_bad) fund_odd = [d for d in range(1, DMAX + 1, 2) if is_fundamental_discriminant(d)] vanish_class = [d for d in fund_odd if d % 8 == 5 and d <= Q_G_SMALL] vanish_b_ok = all(g_w[d] == 0 for d in vanish_class) check(f"D.vanish: all {len(vanish_class)} fund. d≡5 mod 8 have " "b(|d|)=0 (root number −1 class)", vanish_b_ok and len(vanish_class) >= 10) twist_cache = {r["d"]: r for r in val_rows} def get_twist(d): if d not in twist_cache: twist_cache[d] = evaluate_twist(d) return twist_cache[d] live_rows = [] for d in fund_odd: if d > Q_G_SMALL or d % 8 != 1: continue bn = g_w[d] Lw = get_twist(d) Lval = Lw["L_afe"] if abs(Lval) < L_VANISH_TOL or bn == 0: continue denom = mpmath.power(d, mpmath.mpf("1.5")) * Lval R = (mpmath.mpf(bn) ** 2) / denom live_rows.append({"d": d, "b": bn, "L": Lval, "R": R, "eps": Lw["eps"]}) print(f" live R(d) rows (d≡1 mod 8): {len(live_rows)}") for r in live_rows: print(f" d={r['d']}: b={r['b']}, L={float(r['L']):.12g}, " f"R={float(r['R']):.12g}") Rs = [r["R"] for r in live_rows] R_med = sorted(Rs, key=float)[len(Rs) // 2] if Rs else None spread = (max(abs(r - R_med) / abs(R_med) for r in Rs) if Rs and R_med else mpmath.mpf(1)) print(f" R_med={R_med}, spread={float(spread):.3e}") check(f"D.sample: ≥10 live fund. d≡1 mod 8 with b≠0 and L≠0 " f"(got {len(live_rows)}; includes d=89)", len(live_rows) >= 10 and any(r["d"] == 89 for r in live_rows)) check("D.constancy: R(d)=b(|d|)^2/(|d|^{3/2} L(f8×χ_d,2)) CONSTANT " f"on live class (R≈{float(R_TARGET)}, spread≤1e-12; " f"measured spread={float(spread):.3e})", R_med is not None and abs(R_med - R_TARGET) / R_TARGET < mpmath.mpf("1e-8") and spread <= R_SPREAD_TOL) check("D.eps-class: eps_d=+1 on all live d≡1 mod 8 rows", all(r["eps"] == 1 for r in live_rows)) # ============================================================ FENCES print("F -- honesty fences (named, not failures)") check("FENCE: Shimura/Kohnen/Waldspurger/Baruch-Mao named classical; " "claim = unique monoid preimage + in-suite chain + measured R; " "GL(2) centre s=2 NOT xi-line; NO RH; ZETA.HP.CARRIER untouched; " "no marker moves", len(matches_m8) == 1 and matches_m8[0][2] == Fraction(-8) and hecke_ok and outside and spread <= R_SPREAD_TOL and float(fe_best[2]) < FE_THRESH) elapsed = time.time() - t0 print(f" walltime {elapsed:.1f}s") return summary("HECKE.GEOM.HALFINT.01 half-integral bridge") if __name__ == "__main__": raise SystemExit(run())