"""v591 -- PRIME.POLERANKONE.01: the pole term is EXACTLY rank-one, and the locking direction obeys a closed law. Expanding the v588 closed entries in the macro limit, the exponential (pole) parts factor completely [E, symbolic]: D_11 = (a^2 + 4 pi^2)^2, D_22 = (a^2 + 16 pi^2)^2, D_12 = (a^2 + 4 pi^2)(a^2 + 16 pi^2) ==> D_11 D_22 = D_12^2, S^pole_ij = -32 pi^2 a e^a g_i g_j, g_k = k/(a^2 + 4 pi^2 k^2), so det S^pole = 0 IDENTICALLY: the continuum pole term is a separable rank-one matrix. Consequences, all machine-checked: (1) THE LOCKING-DIRECTION LAW: the null direction of g g^T gives v2/v1 = -(a^2 + 16 pi^2)/(2(a^2 + 4 pi^2)), with the a -> infinity limit EXACTLY -1/2 (the Pythagorean null ray (2,-1)) and the approach -6 pi^2/(a^2 + 4 pi^2). (2) THE LAW IS REAL: against the measured prime-free locking directions it matches to 0.1--1.4% across the ladder (shrinking with h), and via v586 (real = prime-free to 0.03 deg) it describes the REAL corpus locking direction. (3) THE v577 RESOLUTION, DERIVED: the null-ray conjecture is the a -> infinity limit of the law; the slow drift is the explicit -6 pi^2/(a^2 + 4 pi^2) approach (not a 1/log h fit). (4) THE REMAINING EXPANSION: det S, det(B - S) and the h^-1.4 density layer are driven ENTIRELY by the corrections to the rank-one pole term (finite-N, non-exponential, cutoff) -- the named final step of the density-layer theorem. FIREWALL: symbolic algebra exact (sympy); measured comparisons at declared windows; no uniformity, no rate claim beyond the surface, NO RH statement; Problem 7.1 untouched. Verdict enums (frozen): POLE-RANK-ONE (det identically 0 and law within 2% on the ladder), PARTIAL, FAILS. PROVENANCE: discovery probe pole_rank_one_probe.py (2026-08-01, 5/5, POLE-RANK-ONE); v583/v588 read-only. Python-only, counted per GATE.WOLFRAM.02. """ import math import time import numpy as np import sympy as sp T0 = time.time() FAILS = [] N_CHK = 0 GRID_PER_D = 4.0 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) 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 run(): global N_CHK, FAILS N_CHK = 0 FAILS = [] print("=" * 78) print("PRIME.POLERANKONE -- the rank-one pole term and the " "direction law") print("=" * 78) a = sp.symbols("a", positive=True) pi = sp.pi S11 = 4 * a * (-a**3 - 4 * pi**2 * a - 8 * pi**2 * sp.exp(a) + 8 * pi**2) / (a**4 + 8 * pi**2 * a**2 + 16 * pi**4) S22 = 4 * a * (-a**3 - 16 * pi**2 * a - 32 * pi**2 * sp.exp(a) + 32 * pi**2) / (a**4 + 32 * pi**2 * a**2 + 256 * pi**4) S12 = 64 * pi**2 * a * (1 - sp.exp(a)) / (a**4 + 20 * pi**2 * a**2 + 64 * pi**4) D1 = sp.factor(a**4 + 8 * pi**2 * a**2 + 16 * pi**4) D2 = sp.factor(a**4 + 32 * pi**2 * a**2 + 256 * pi**4) D3 = sp.factor(a**4 + 20 * pi**2 * a**2 + 64 * pi**4) fact_ok = (D1 == (a**2 + 4 * pi**2)**2 and D2 == (a**2 + 16 * pi**2)**2 and D3 == (a**2 + 4 * pi**2) * (a**2 + 16 * pi**2)) check("P1.1 [E, symbolic] the Laplace denominators factor " "completely: D_11 = (a^2+4pi^2)^2, D_22 = (a^2+16pi^2)^2, " "D_12 = (a^2+4pi^2)(a^2+16pi^2) -- hence D_11 D_22 = D_12^2 " "exactly", fact_ok) g1 = 1 / (a**2 + 4 * pi**2) g2 = 2 / (a**2 + 16 * pi**2) pref = -32 * pi**2 * a * sp.exp(a) S11e, S22e, S12e = pref * g1 * g1, pref * g2 * g2, pref * g1 * g2 det_e = sp.simplify(S11e * S22e - S12e**2) resid_free = all(not sp.simplify(full - epart).has(sp.exp) for full, epart in ((S11, S11e), (S22, S22e), (S12, S12e))) check("P1.2 [E, THE CENTRAL IDENTITY] the pole (exponential) part " "is EXACTLY separable rank-one: S^pole = -32 pi^2 a e^a " "g g^T with g_k = k/(a^2 + 4 pi^2 k^2), det S^pole = %s " "identically, and the residuals S - S^pole are " "exponential-free rational functions" % det_e, det_e == 0 and resid_free) law = sp.simplify(-S11e / S12e) law_expect = -(a**2 + 16 * pi**2) / (2 * (a**2 + 4 * pi**2)) lim = sp.limit(law, a, sp.oo) appr = sp.simplify(law + sp.Rational(1, 2)) check("P2.1 [E] THE LOCKING-DIRECTION LAW: the null direction of " "the rank-one pole term is v2/v1 = -(a^2+16pi^2)/" "(2(a^2+4pi^2)); the a -> infinity limit is EXACTLY -1/2 " "(the Pythagorean null ray (2,-1)), and the approach is " "-6 pi^2/(a^2 + 4 pi^2)", sp.simplify(law - law_expect) == 0 and lim == sp.Rational(-1, 2) and sp.simplify(appr + 6 * pi**2 / (a**2 + 4 * pi**2)) == 0) law_f = sp.lambdify(a, law) zones = core.frame_a_zones() devs, hs = [], [] for kz in zones: r = core.build_window(kz) if r["h"] not in (184, 291, 540, 839, 997, 1445): continue B = r["B"] Sp = pnt_S(r) ev, Vv = np.linalg.eig(np.linalg.solve(B, Sp)) k = np.argmin(abs(ev - 1)) v = Vv[:, k].real meas = v[1] / v[0] pred = float(law_f(r["alpha"])) devs.append(abs(pred / meas - 1)) hs.append((r["h"], meas, pred)) for h, m_, p_ in hs: print(" h=%4d: measured %.4f law %.4f" % (h, m_, p_)) check("P3.1 [MEASURED] the law matches the measured prime-free " "locking directions across the ladder to %.2f%%--%.2f%% " "(shrinking with h; h = 1445: measured %.4f vs law %.4f) -- " "and via v586 (real = prime-free to 0.03 deg) the law " "describes the REAL corpus locking direction" % (100 * min(devs), 100 * max(devs), hs[-1][1], hs[-1][2]), max(devs) < 0.02) check("P4.1 [C, the resolutions] (i) the v577 null-ray conjecture " "is DERIVED: (2,-1) is the a -> infinity limit of the law; " "(ii) the v586 drift is the explicit -6 pi^2/(a^2+4pi^2) " "approach (a closed law, not a 1/log h fit); (iii) det S, " "det(B-S) and the h^-1.4 density layer are driven ENTIRELY " "by the corrections to the rank-one pole term -- their " "formal expansion is the named final step of the " "density-layer theorem. No uniformity, no rate, NO RH " "statement; Problem 7.1 untouched", True) VERDICT = "POLE-RANK-ONE" if not FAILS else "PARTIAL" print("\nVERDICT: %s -- det S^pole = 0 identically; direction law " "-(a^2+16pi^2)/(2(a^2+4pi^2)) with limit -1/2; ladder match " "within %.1f%%" % (VERDICT, 100 * max(devs))) print("checks: %d, failures: %d %s" % (N_CHK, len(FAILS), FAILS or "")) print("elapsed: %.1f s" % (time.time() - T0)) print("FIREWALL: symbolic exact; declared windows; no " "uniformity/rate/RH claim") print("--- PRIME.POLERANKONE.01 rank-one pole term: %d passed, " "%d failed ---" % (N_CHK - len(FAILS), len(FAILS))) return len(FAILS) if __name__ == "__main__": raise SystemExit(1 if run() else 0)