"""v586 -- PRIME.LOCKDIR.01: the locking DIRECTION is density-determined: the real locking eigenvector of the pencil B^{-1}S agrees with the prime-free model's locking eigenvector to 0.03 degrees (median; max 0.33, decaying ~ h^{-1.16}) on all 69 regular windows -- while the lock DEPTH differs by a median factor 1350 (v585). Combined statement: THE PRIMES ARE A PURE DEPTH AMPLIFIER ALONG A DENSITY-FIXED DIRECTION. This resolves the v577 puzzle honestly: the null ray (2,-1) was the h = infinity comparator; at finite h the deterministic direction itself drifts (v2/v1: -1.73 at h = 184 to -1.22 at h = 1445), and the real direction tracks IT to fractions of a degree. The drift limit is consistent with -0.5 (the (2,-1) ray) at a 1/log h rate (fit limit -0.551, R^2 0.67) -- typed as consistent, not settled. The deterministic direction moreover CERTIFIES the real defect as a witness: the quadratic form of the real B - S along the prime-free direction reproduces the true minimal eigenvalue within [0.98, 2.9] (median 1.42) on every window -- the arithmetic content of Problem 7.1 is ONE quadratic form along ONE deterministic vector. FIREWALL: v583 model verbatim, no new inputs, no fit; ladder measurements on the declared surface; the deterministic direction's own asymptotics (the classical 1D problem) stays open; no uniformity, no rate claim beyond the surface, NO RH statement. Verdict enums (frozen): DIRECTION-DENSITY-FIXED (median angle < 0.1 deg and scramble must-break fires), DIRECTION-ARITHMETIC, MIXED. PROVENANCE: discovery probe pnt_lock_direction_probe.py (2026-07-31, 7/7, DIRECTION-DENSITY-FIXED); v583 model verbatim, v563 read-only. Python-only, counted per GATE.WOLFRAM.02. """ import math import time import numpy as np T0 = time.time() FAILS = [] N_CHK = 0 ANOMALOUS_H = 1292 GRID_PER_D = 4.0 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 # 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) def pnt_S(r): """The v583 prime-free comb block, verbatim.""" 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:]) s = np.zeros(3) for u_j, l_j in zip(centers, lam): s[0] += l_j * core.spline_project(r["W11"], u_j, D, Mz) s[1] += l_j * core.spline_project(r["W22"], u_j, D, Mz) s[2] += l_j * core.spline_project(r["W12"], u_j, D, Mz) return np.array([[s[0], s[2]], [s[2], s[1]]]) def lock_dir(B, S): """Eigenvector of B^{-1}S with eigenvalue closest to 1.""" ev, V = np.linalg.eig(np.linalg.solve(B, S)) k = np.argmin(abs(ev - 1)) v = V[:, k].real v = v / np.linalg.norm(v) return v if v[0] >= 0 else -v def run(): global N_CHK, FAILS N_CHK = 0 FAILS = [] print("=" * 78) print("PRIME.LOCKDIR -- is the locking direction density-determined?") print("=" * 78) zones = core.frame_a_zones() angles, hs, scal_ratio, drift = [], [], [], [] r_ref = None for kz in zones: r = core.build_window(kz) if r["h"] == ANOMALOUS_H: continue if r["h"] == REF_H: r_ref = (kz, r) B, S = r["B"], r["S"] v_p = lock_dir(B, pnt_S(r)) v_r = lock_dir(B, S) angles.append(math.degrees(math.acos(min(1.0, abs(v_p @ v_r))))) hs.append(r["h"]) drift.append(v_p[1] / v_p[0]) Ah = B - S q = float(v_p @ (Ah @ v_p)) w = np.linalg.eigvalsh(Ah) scal_ratio.append(q / w[np.argmin(abs(w))]) angles = np.array(angles) hs = np.array(hs, float) scal_ratio = np.array(scal_ratio) drift = np.array(drift) check("D1.1 [E, platform] the census runs on all %d regular windows " "(the h = %d anomaly excluded by declaration): locking direction " "= eigenvector of B^{-1}S with eigenvalue closest to 1, real " "comb vs the verbatim v583 prime-free model" % (len(angles), ANOMALOUS_H), len(angles) == 69) sl_ang = float(np.polyfit(np.log(hs), np.log(np.maximum(angles, 1e-9)), 1)[0]) check("D2.1 [MEASURED, THE CENTRAL RESULT -- DIRECTION-DENSITY-FIXED] " "the real locking direction agrees with the PRIME-FREE locking " "direction to max %.3f deg, median %.3f deg, decaying ~ h^%.2f " "across the ladder: the direction is deterministic" % (angles.max(), float(np.median(angles)), sl_ang), angles.max() < 0.5 and np.median(angles) < 0.1 and sl_ang < -0.8) check("D2.2 [MEASURED, the combined statement] with v585 (depth off by " "a median 1350x) this closes the split: THE PRIMES ARE A PURE " "DEPTH AMPLIFIER ALONG A DENSITY-FIXED DIRECTION -- geometry, " "spectator separation (v569) and now the locking direction are " "all density-level; only the depth scalar is arithmetic", angles.max() < 0.5) check("D3.1 [MEASURED, the witness] the deterministic direction " "CERTIFIES the real defect: the quadratic form of the real " "B - S along the prime-free direction reproduces the true " "minimal eigenvalue within [%.3f, %.3f] (median %.3f) on every " "window -- the arithmetic content of Problem 7.1 is ONE " "quadratic form along ONE deterministic vector" % (scal_ratio.min(), scal_ratio.max(), float(np.median(scal_ratio))), scal_ratio.min() > 0.9 and scal_ratio.max() < 3.5) fitx = 1.0 / np.log(hs) A = np.vstack([fitx, np.ones_like(fitx)]).T coef, *_ = np.linalg.lstsq(A, drift, rcond=None) r2 = 1 - (((drift - A @ coef)**2).sum() / ((drift - drift.mean())**2).sum()) check("D4.1 [MEASURED, resolves the v577 puzzle] the deterministic " "direction itself DRIFTS (v2/v1 from %.2f to %.2f across the " "ladder): v577's slow angle decay against the fixed null ray " "(2,-1) was a comparator artifact -- against the correct " "finite-h deterministic direction the alignment is ~0. The " "drift limit under a 1/log h law is %.3f (R^2 %.2f), consistent " "with the null-ray value -0.5 -- typed as consistent, NOT " "settled" % (drift[0], drift[-1], coef[1], r2), abs(coef[1] + 0.5) < 0.2 and r2 > 0.5) kz540, r540 = r_ref v_p540 = lock_dir(r540["B"], pnt_S(r540)) scr = [] for seed in (1, 2, 3): rs = core.build_window(kz540, scramble_seed=seed) v_s = lock_dir(rs["B"], rs["S"]) scr.append(math.degrees(math.acos(min(1.0, abs(v_p540 @ v_s))))) check("D5.1 [E, must-break] the scramble control fires: scrambled " "combs (same masses, uniform positions) misalign from the " "prime-free direction by %.1f--%.1f deg at h = %d (vs %.3f deg " "real): the 0.03-degree alignment is real arithmetic placement " "matching the density profile, not an automatism" % (min(scr), max(scr), REF_H, float(angles[list(hs).index(REF_H)])), min(scr) > 2.0) check("D6.1 [C, the program] the classical route to the geometric " "side of Problem 7.1 is now fully typed: prove the asymptotics " "of the DETERMINISTIC direction and depth (a closed 1D analysis " "of the two-term model -- no primes), then the remaining " "arithmetic content is the depth scalar along that direction " "(zero-oscillation territory, v585). Surface measurements " "only; the deterministic asymptotics stay open; no uniformity, " "no rate, NO RH statement; Problem 7.1 untouched", True) VERDICT = "DIRECTION-DENSITY-FIXED" if not FAILS else "MIXED" print("\nVERDICT: %s -- max angle %.3f deg (median %.3f), decay " "h^%.2f; witness ratio median %.2f; scramble breaks to %.0f+ deg" % (VERDICT, angles.max(), float(np.median(angles)), sl_ang, float(np.median(scal_ratio)), min(scr))) print("checks: %d, failures: %d %s" % (N_CHK, len(FAILS), FAILS or "")) print("elapsed: %.1f s" % (time.time() - T0)) print("FIREWALL: v583 model verbatim; ladder measurements; " "deterministic asymptotics open; no uniformity/rate/RH claim") print("--- PRIME.LOCKDIR.01 density-fixed locking direction: %d passed, %d failed ---" % (N_CHK - len(FAILS), len(FAILS))) return len(FAILS) if __name__ == "__main__": raise SystemExit(1 if run() else 0)