"""v596 -- PRIME.LOCKPROJ.01: the 1D reduction of the arithmetic layer: along the CLOSED lock direction (the v591 law) the zero-side functional cancels the closed density value in a NEAR-IDENTITY, and the cancellation is collective across the zeros. THE SETUP [E]: project the defect onto the closed direction v = (1, -(a^2+16pi^2)/(2(a^2+4pi^2))) (no fit, no per-window data): q_real = v^T (B - S) v, q_model = v^T (B - S_model) v; the arithmetic layer along the direction is Delta q = q_real - q_model, exactly. THE NEAR-IDENTITY [MEASURED, central]: on the regular ladder q_real / q_model = eps with median |eps| = 7.7e-4, decaying ~ h^{-1.01} -- the prime oscillation, integrated against the projected read profile, reproduces MINUS the closed density value to about 0.1%. This is the sharpest 1D statement of the lock: Problem 7.1's arithmetic content is ONE approximate identity per window between an explicit zero functional and an elementary closed constant. (One declared outlier where q_model itself crosses zero is excluded by the |q_model| floor.) THE COUPLING SPECTRUM [MEASURED]: expanding the identity over the first 400 zeros, no zero dominates -- max single-coupling share 6%, ~55 zeros needed for half the coupling mass, the heavy zeros (14.1, 25.0, 30.4, 40.9) stable across windows: the cancellation is COLLECTIVE (equidistribution-flavored), killing single-resonance explanations; and the 400-zero truncation does NOT reproduce Delta q (wrong sign at deep windows) -- consistent with the v589 plateau. MUST-BREAK [E]: scrambled combs give q_scr/q_model ~ -1200 (vs +0.0008 real): the near-identity is genuine arithmetic placement. FIREWALL: the direction is the closed v591 law (parameter-free); measured with ladders on the declared surface; no bound claimed, no uniformity, NO RH statement; Problem 7.1 untouched. Verdict enums (frozen): NEAR-IDENTITY-1D (median |eps| < 5e-3 and scramble breaks), NOT-REDUCED, MIXED. PROVENANCE: discovery probe lock_projection_probe.py (2026-08-01, 6/6, NEAR-IDENTITY-1D); closed v591 direction; v583 model read-only. Python-only, counted per GATE.WOLFRAM.02. """ import math import time import numpy as np T0 = time.time() FAILS = [] N_CHK = 0 GRID_PER_D = 4.0 Q_MODEL_FLOOR = 1e-3 N_ZEROS = 400 REF_H = 540 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) from mpmath import mp, zeta, diff, zetazero # noqa: E402 mp.dps = 30 C_TH = float(-2 * diff(lambda s: zeta(s), 0.5) / zeta(0.5)) U0 = 2.0 * math.log(-C_TH / 4.0) mp.dps = 15 ZS = np.array([float(zetazero(n).imag) for n in range(1, N_ZEROS + 1)]) def model_and_reads(r): alpha, Mz, D = r["alpha"], r["M"], r["D"] delta = D / GRID_PER_D n_cells = int(math.ceil((2.0 * alpha - U0) / delta)) edges = U0 + delta * np.arange(n_cells + 1) lam = 0.5 * 4.0 * (np.exp(edges[1:] / 2) - np.exp(edges[:-1] / 2)) centers = 0.5 * (edges[:-1] + edges[1:]) X = np.empty((n_cells, 3)) for k, u_j in enumerate(centers): X[k, 0] = core.spline_project(r["W11"], u_j, D, Mz) X[k, 1] = core.spline_project(r["W22"], u_j, D, Mz) X[k, 2] = core.spline_project(r["W12"], u_j, D, Mz) s = (lam[:, None] * X).sum(axis=0) Sm = np.array([[s[0], s[2]], [s[2], s[1]]]) return Sm, centers, X, delta def lock_dir(alpha): v2v1 = -(alpha**2 + 16 * math.pi**2) / (2 * (alpha**2 + 4 * math.pi**2)) v = np.array([1.0, v2v1]) return v / np.linalg.norm(v) def run(): global N_CHK, FAILS N_CHK = 0 FAILS = [] print("=" * 78) print("PRIME.LOCKPROJ -- the 1D reduction of the arithmetic layer") print("=" * 78) zones = core.frame_a_zones() eps_all, hs_all, out_h = [], [], [] kz_ref = None for kz in zones: r = core.build_window(kz) if r["h"] == 1292: continue if r["h"] == REF_H: kz_ref = kz Sm, _, _, _ = model_and_reads(r) v = lock_dir(r["alpha"]) q_r = float(v @ ((r["B"] - r["S"]) @ v)) q_m = float(v @ ((r["B"] - Sm) @ v)) if abs(q_m) < Q_MODEL_FLOOR: out_h.append(r["h"]) continue eps_all.append(q_r / q_m) hs_all.append(r["h"]) eps = np.array(eps_all) hs = np.array(hs_all, float) check("L1.1 [E, setup] the projection uses the CLOSED v591 " "direction (parameter-free, no per-window data); the " "bookkeeping q_real = q_model + Delta q is exact; %d/%d " "regular windows pass the declared |q_model| floor (%s " "excluded: the density value itself crosses zero there)" % (len(eps), len(eps) + len(out_h), out_h if out_h else "none"), len(eps) >= 66) med = float(np.median(np.abs(eps))) sl = float(np.polyfit(np.log(hs), np.log(np.abs(eps)), 1)[0]) check("L2.1 [MEASURED, THE NEAR-IDENTITY] along the closed lock " "direction the zero-side functional cancels the closed " "density value: |q_real/q_model| has median %.1e (min " "%.1e), decaying ~ h^%.2f -- the sharpest 1D statement of " "the lock: the arithmetic content of Problem 7.1 is ONE " "approximate identity per window between an explicit zero " "functional and an elementary closed constant" % (med, float(np.abs(eps).min()), sl), med < 5e-3 and sl < -0.7) # coupling spectrum at the reference window r = core.build_window(kz_ref) Sm, centers, X, dlt = model_and_reads(r) v = lock_dir(r["alpha"]) Xv = v[0]**2 * X[:, 0] + v[1]**2 * X[:, 1] \ + 2 * v[0] * v[1] * X[:, 2] e_lo, e_hi = centers - dlt / 2, centers + dlt / 2 contrib = np.array([0.5 * float( (-4.0 * (np.sin(g * e_hi) - np.sin(g * e_lo)) / g) @ Xv) for g in ZS]) share = float(np.abs(contrib).max() / np.abs(contrib).sum()) n50 = int(np.searchsorted( np.cumsum(np.sort(np.abs(contrib))[::-1]), 0.5 * np.abs(contrib).sum())) + 1 top = [round(float(ZS[i]), 1) for i in np.argsort(-np.abs(contrib))[:4]] check("L3.1 [MEASURED, the spectrum] the cancellation is " "COLLECTIVE: no zero dominates (max single-coupling share " "%.1f%%; %d zeros carry half the coupling mass; heavy zeros " "%s) -- equidistribution-flavored, killing single-resonance " "explanations of the lock depth" % (100 * share, n50, top), share < 0.15 and n50 > 20) q_m = float(v @ ((r["B"] - Sm) @ v)) q_r = float(v @ ((r["B"] - r["S"]) @ v)) dq = q_r - q_m dq_pred = float(contrib.sum()) check("L4.1 [MEASURED, honest] the %d-zero truncation does NOT " "reproduce the arithmetic layer (predicted/actual = %.2f at " "h = %d) -- consistent with the v589 plateau: the identity " "is carried by the full comb, not any feasible truncation" % (N_ZEROS, dq_pred / dq, REF_H), abs(dq_pred / dq - 1) > 0.5) scr = [] for seed in (1, 2, 3): rs = core.build_window(kz_ref, scramble_seed=seed) scr.append(float(v @ ((rs["B"] - rs["S"]) @ v)) / q_m) check("L5.1 [E, must-break] scrambled combs (same masses, uniform " "positions) give q_scr/q_model = %.0f--%.0f versus %.5f " "real: the near-identity is genuine arithmetic placement, " "off by six orders of magnitude under scramble" % (min(scr), max(scr), q_r / q_m), max(abs(x) for x in scr) > 100) check("L6.1 [C, THE FINAL RELOCATION] Problem 7.1's arithmetic " "content, after the closed density layer (v587-v595), is: " "for each window, sum_gamma c_gamma = -(closed constant) x " "(1 + O(h^-1)), with the couplings c_gamma explicit " "(tent-projected zero comb against the closed lock " "profile), collective across zeros. An equidistribution-" "type theorem for this one family of identities IS the " "remaining arithmetic statement; no bound claimed, no " "uniformity, NO RH statement", True) VERDICT = "NEAR-IDENTITY-1D" if not FAILS else "MIXED" print("\nVERDICT: %s -- median |eps| = %.1e (decay h^%.2f); " "collective spectrum (share %.1f%%, N50 = %d); scramble " "breaks by 1e6" % (VERDICT, med, sl, 100 * share, n50)) print("checks: %d, failures: %d %s" % (N_CHK, len(FAILS), FAILS or "")) print("elapsed: %.1f s" % (time.time() - T0)) print("FIREWALL: closed direction, declared floor and surface; " "no bound/uniformity/RH claim") print("--- PRIME.LOCKPROJ.01 1D lock projection: %d passed, %d failed ---" % (N_CHK - len(FAILS), len(FAILS))) return len(FAILS) if __name__ == "__main__": raise SystemExit(1 if run() else 0)