"""v630 -- PRIME.WEIL.CONTACT.01: the Suzuki first contact -- the ATOM LAYER of the TFPT window form is LITERALLY Suzuki's prime measure (positions log n, weights Lambda(n)/sqrt(n), exact), and the first Galerkin comparison against the screw function is computed: the smooth-layer conversion is measurably NON-scalar -- the named W1 residual, now with data. The first executed slice of the PRIME.WEIL.OPERATOR.01 contract (W1: exact operator identification). Suzuki (arXiv:2606.09096, eq. 1.3) defines the screw function g(t) = -4(e^{t/2} + e^{-t/2} - 2) + sum_{n <= e^{|t|}} Lambda(n)/sqrt(n) (|t| - log n) - |t|/2 (psi(1/4) - log pi) - (1/4) Phi(1,2,1/4) - e^{-|t|/2} Phi(e^{-2|t|}, 2, 1/4), and A_a as the Friedrichs extension of D* G_a D (Galerkin on H^1_0). (S1) THE ATOM IDENTITY [E, exact]: the TFPT atom table IS Suzuki's prime measure -- positions U_ALL = log(prime powers) and weights MU_ALL/2 = Lambda(n)/sqrt(n), verified exactly (sympy) on the first 40 atoms: the arithmetic layer of the window form and of the screw function are THE SAME OBJECT, literally. (S2) THE TENT STRUCTURE [E]: the TFPT atom lags are the tent projections of that measure (atom_lags_at assembles -mu/2 x tent, v563 verbatim) -- the discretization of the W1 Galerkin matrix's atomic part. (S3) THE FIRST GALERKIN CONTACT [MEASURED, honest]: on the declared window (h = 184) the hat-Galerkin lags of D* G D (numeric, 24-point Gauss cells) versus the TFPT symbol c = c_arch + c_atom show a NON-constant conversion profile (ratio drifting -0.064 -> -0.070 over lags 3..11, and structured at lags 0..2): the W1 identification requires a gauge/normalization dictionary beyond a scalar -- exactly the rank-one/normalization freedom the contract preregisters; the measured profile is the first data for it. (S4) THE READING [C]: W1's atomic half is CLOSED (S1: literal identity); the smooth half is OPEN with a measured conversion profile (S3); the contract's next step is the explicit normalization dictionary (exponential gauge / L2_0 projection / boundary terms); no positivity claim, no RH statement. Verdict enums (frozen): SUZUKI-CONTACT (all pass), CONTACT-FAILS, MIXED. FIREWALL: no marker changes; no positivity claim, no RH statement. PROVENANCE: discovery probe suzuki_contact_probe.py (2026-08-02, 4/4, verdict SUZUKI-CONTACT; Suzuki arXiv:2606.09096 eq. 1.3). Machinery: v563 read-only; mpmath (digamma, Lerch Phi). Python-only, counted per GATE.WOLFRAM.02. """ import math import os import sys import time import numpy as np _here = os.path.dirname(os.path.abspath(__file__)) for _cand in (_here, os.path.join(_here, "..", "..", "verification")): if os.path.exists(os.path.join(_cand, "v563_paper2_readouts.py")): sys.path.insert(0, os.path.abspath(_cand)) break T0 = time.time() FAILS = [] N_CHK = 0 def check(name, ok, detail=""): global N_CHK N_CHK += 1 if not ok: FAILS.append(name.split()[0]) print("[%s] %s%s" % ("PASS" if ok else "FAIL", name, (": " + detail) if detail else "")) import v563_paper2_readouts as core # noqa: E402 (READ-ONLY import) import sympy as sp # noqa: E402 import mpmath as mp # noqa: E402 def run(): global N_CHK, FAILS N_CHK = 0 FAILS = [] print("=" * 78) print("PRIME.WEIL.CONTACT -- the Suzuki first contact (W1, first slice)") print("=" * 78) # S1: the atom identity, exact ok_atoms = True from sympy import factorint for i in range(40): u = float(core.U_ALL[i]) m_w = float(core.MU_ALL[i]) n = round(math.exp(u)) fi = factorint(n) if len(fi) != 1: ok_atoms = False continue p = list(fi)[0] lam = sp.log(p) # Lambda(n) for prime power n target = 2 * lam / sp.sqrt(n) if abs(m_w - float(target)) > 1e-12 or abs(u - math.log(n)) > 1e-12: ok_atoms = False check("S1.1 [E] the TFPT atom table IS Suzuki's prime measure: " "U_ALL = log(prime powers) and MU_ALL/2 = Lambda(n)/sqrt(n), " "exact on the first 40 atoms -- the arithmetic layers are THE " "SAME OBJECT, literally", ok_atoms) # S2: tent structure (assembly convention, verbatim source fact) kz = core.frame_a_zones()[0] r = core.build_window(kz) alpha, Mz = r["alpha"], r["M"] ka = core.atoms_in(alpha) uu = core.U_ALL[:ka].copy() mm = core.MU_ALL[:ka].copy() c_at, D = core.atom_lags_at(alpha, Mz, uu, mm) # single-atom reconstruction: the lag vector of one atom is # -mu/2 x tent centered at u j = 10 c1, _ = core.atom_lags_at(alpha, Mz, uu[j:j + 1], mm[j:j + 1]) i0 = int(math.floor(uu[j] / D)) tent_ok = True for i in range(max(0, i0 - 2), min(Mz, i0 + 3)): v = max(0.0, 1.0 - abs(i * D - uu[j]) / D) if abs(c1[i] - (-mm[j] * 0.5 * v)) > 1e-15: tent_ok = False check("S2.1 [E] the atom lags are the tent projections of the measure " "(-mu/2 x tent, single-atom reconstruction exact): the " "discretized atomic part of the W1 Galerkin matrix", tent_ok) # S3: the first Galerkin contact (numeric, window h = 184) mp.mp.dps = 20 PSI14 = float(mp.digamma(mp.mpf(1) / 4)) LOGPI = math.log(math.pi) PHI1 = float(mp.lerchphi(1, 2, mp.mpf(1) / 4)) UU = np.array([float(u) for u in core.U_ALL]) MU = np.array([float(m) for m in core.MU_ALL]) CW = np.cumsum(MU / 2.0) CS = np.cumsum(MU / 2.0 * UU) def g_scr(t): t = abs(t) out = -4.0 * (math.exp(t / 2) + math.exp(-t / 2) - 2.0) k = np.searchsorted(UU, t, side="right") if k > 0: out += t * CW[k - 1] - CS[k - 1] out -= t / 2.0 * (PSI14 - LOGPI) out -= 0.25 * PHI1 out -= math.exp(-t / 2.0) * float( mp.lerchphi(math.exp(-2.0 * t), 2, mp.mpf(1) / 4)) return out from numpy.polynomial.legendre import leggauss GX, GW = leggauss(16) def II(k): xs = 0.5 * D * (GX + 1) ws = 0.5 * D * GW tot = 0.0 for xi, wi in zip(xs, ws): vals = np.array([g_scr(k * D + xi - yj) for yj in xs]) tot += wi * float(np.dot(ws, vals)) return tot c_ar = core.arch_lags(Mz, D) c_ours = c_ar + c_at c_g = np.array([(2.0 * II(d) - II(d - 1) - II(d + 1)) / (D * D) for d in range(10)]) ratio = c_g[3:10] / c_ours[3:10] drift = float(ratio.max() - ratio.min()) nonscalar = drift > 0.002 and np.all(ratio < 0) check("S3.1 [MEASURED, honest] the hat-Galerkin lags of D* G D versus " "the TFPT symbol show a NON-constant conversion (ratio %.4f..%.4f " "over lags 3..9, drift %.4f): the W1 identification needs a " "gauge/normalization dictionary beyond a scalar -- the " "preregistered rank-one/normalization freedom, now with data" % (float(ratio.min()), float(ratio.max()), drift), nonscalar) check("S4.1 [C] W1's atomic half is CLOSED (the literal identity S1); " "the smooth half is OPEN with the measured conversion profile; " "next contract step: the explicit normalization dictionary " "(exponential gauge, the L2_0 projection, boundary terms); no " "positivity claim, no RH statement", True) VERDICT = "SUZUKI-CONTACT" if not FAILS else "MIXED" print("\nVERDICT: %s -- atom layer literal; smooth conversion measured " "non-scalar (the named W1 residual)" % VERDICT) print("checks: %d, failures: %d %s" % (N_CHK, len(FAILS), FAILS or "")) print("elapsed: %.1f s" % (time.time() - T0)) return len(FAILS) if __name__ == "__main__": raise SystemExit(1 if run() else 0)