#!/usr/bin/env python3 """verify_P317.py -- Verifier for Addendum 317 (complex Hopf-fibered selection). Recomputes the keystone complex-realization probe from scratch and asserts the pre-registered conditions of A317. A316 closed the family-to-mode route in the REAL radial reduction and named the blocker: the families are the COMPLEX omega^k cube-root eigenspaces of the layer-cycle, but the real reduction folds omega and omega^2 together (Re(omega^k) = (1,-1/2,-1/2), so k=1,2 collapse). A317 moves to a complex realization. On the Hopf fibration S^1 -> S^3 -> S^2 the layer-cycle psi -> psi + 2pi/3 acts on the fiber-momentum sector e^{i m psi} by omega^m, so the omega^k eigenspace is { m == k (mod 3) }, lowest rep m=k. The fiber momentum m enters the radial operator as a genuine m-dependent term (P18 S3.3 fiber Laplacian, eigenvalue m^2), tested in two corpus-faithful forms: F-lap : O_m = -d^2/dr^2 - (3/r)d/dr + V_eff(r) + m^2 (flat additive m^2) F-Hopf: O_m = -d^2/dr^2 - (3/r)d/dr + m^2/r^2 + V_eff(r) (cone/Hopf 1/r^2) Canonical map: family k -> ground state of the m=k sector (k=0,1,2). All in r^3 dr; gen_eigh numpy Cholesky reduction (scipy unavailable). Finding: the A316 degeneracy IS broken (m=0,1,2 give three genuinely distinct operators and three distinct entries), the key structural result. But the canonical map OVERSHOOTS: F-lap mu=2.43, F-Hopf mu=2.29, both far above 1.5 with no peak, and neither improves on the A316 real-reduction closest (L2=0.399, mu=1.222). VERDICT NULL: the genuine omega^k phase breaks the degeneracy but does NOT supply the magnitude; the residual is elsewhere. The verifier encodes this honest verdict. S1 degeneracy broken: three distinct entries, both forms - checks 1-3 S2 the canonical map's diagonal/mu/tau/L2/peak, both forms - checks 4-6 S3 NULL verdict honest condition (no hit, no improvement) + clean - checks 7-8 """ import sys import math import numpy as np def gen_eigh(A, Bm): L = np.linalg.cholesky(Bm) Linv = np.linalg.inv(L) Cmat = Linv @ A @ Linv.T Cmat = 0.5 * (Cmat + Cmat.T) w, y = np.linalg.eigh(Cmat) return w, Linv.T @ y PASS = FAIL = 0 def check(n, desc, cond): global PASS, FAIL ok = bool(cond) PASS += ok FAIL += (not ok) print(f" [{'PASS' if ok else 'FAIL'}] {n:>2}. {desc}") PI = math.pi OM = 4 * PI**3 + PI**2 + PI ES = 13.177 C = (2.0, 3.0, 16.0) TGT = [c / C[0] for c in C] N = 1500 CANONICAL_M = [0, 1, 2] A316_L2 = 0.399 A316_MU = 1.222 def rho_p(x): return 48 * PI**3 * x**2 + 6 * PI**2 * x + 2 * PI def Vpot(x): return rho_p(x)**2 / (2 * OM**2) + ES * x**2 * (1 - x)**2 def grid(): dx = 1.0 / N return np.linspace(dx, 1 - dx, N - 1), dx def radial_matrices(x, dx, V_on_grid, weight): n = N - 1 w_mid = 0.5 * (weight[:-1] + weight[1:]) w_mid_left0 = max(0.5 * (0.0 + weight[0]), 0.0) A = np.zeros((n, n)) for i in range(n): wl = w_mid[i - 1] if i > 0 else w_mid_left0 wr = w_mid[i] if i + 1 < n else weight[i] A[i, i] = (wl + wr) / dx**2 if i + 1 < n: A[i, i + 1] = -w_mid[i] / dx**2 A[i + 1, i] = -w_mid[i] / dx**2 A += np.diag(weight * V_on_grid) A = 0.5 * (A + A.T) return A, np.diag(weight) def solve_fiber(m, form, n_modes=2): x, dx = grid() w_arr = x**3 if form == "F_lap": fiber = float(m**2) * np.ones_like(x) elif form == "F_Hopf": fiber = (float(m**2) / x**2) if m > 0 else np.zeros_like(x) else: raise ValueError(form) A, Wm = radial_matrices(x, dx, Vpot(x) + fiber, w_arr) w, v = gen_eigh(A, Wm) o = np.argsort(w) w = w[o] v = v[:, o] vv = v[:, :n_modes].copy() for k in range(n_modes): nrm = math.sqrt(float(np.sum(vv[:, k]**2 * w_arr) * dx)) vv[:, k] = vv[:, k] / nrm return x, dx, w[:n_modes], vv def overlap_row(col, x, dx, w_arr, lam): norm = float(np.sum(col**2 * w_arr) * dx) return [float(np.sum(col**2 * C[k - 1] * PI**k * x**k * w_arr) * dx) / norm / lam for k in (1, 2, 3)] def diag_peaks(M): Z = np.zeros_like(M) for k in range(3): c = M[:, k] Z[:, k] = (c - c.mean()) / c.std() if c.std() > 0 else 0.0 return all(int(np.argmax(Z[:, k])) == k for k in range(3)) def l2(ratio): return math.sqrt(sum((ratio[i] - TGT[i])**2 for i in range(3))) def box_states(x): pb = np.column_stack([math.sqrt(2.0) * np.sin((n + 1) * PI * x) for n in range(3)]) lb = np.array([(n + 1)**2 * PI**2 for n in range(3)]) return pb, lb def eval_form(form, pb, lb): x, dx = grid() w_arr = x**3 states = np.zeros((N - 1, 3)) lams = np.zeros(3) for k in range(3): _, _, w_m, v_m = solve_fiber(CANONICAL_M[k], form, 2) states[:, k] = v_m[:, 0] lams[k] = w_m[0] M = np.array([overlap_row(states[:, n], x, dx, w_arr, lams[n]) for n in range(3)]) Mbox = np.array([overlap_row(pb[:, n], x, dx, w_arr, lb[n]) for n in range(3)]) d = [M[n, n] for n in range(3)] ratio = [v / d[0] for v in d] peak = diag_peaks(M) box_peak = diag_peaks(Mbox) valid_peak = peak and not box_peak within20 = all(abs(ratio[i] - TGT[i]) <= 0.20 * TGT[i] for i in range(3)) degen = any(abs(lams[a] - lams[b]) < 1e-6 and all(abs(M[a, q] - M[b, q]) < 1e-6 for q in range(3)) for a in range(3) for b in range(a + 1, 3)) distinct = (abs(d[0] - d[1]) > 1e-6 and abs(d[0] - d[2]) > 1e-6 and abs(d[1] - d[2]) > 1e-6) return { "mu": ratio[1], "tau": ratio[2], "ratio": ratio, "valid_peak": valid_peak, "within20": within20, "hit": within20 and valid_peak, "L2": l2(ratio), "lams": [float(v) for v in lams], "degen": degen, "distinct": distinct, } # ---- recompute ----------------------------------------------------------- pb, lb = box_states(grid()[0]) R = {f: eval_form(f, pb, lb) for f in ("F_lap", "F_Hopf")} degeneracy_broken = all(R[f]["distinct"] and not R[f]["degen"] for f in R) any_hit = any(R[f]["hit"] for f in R) improves = any(R[f]["L2"] < A316_L2 - 1e-9 for f in R) moves_mu = any(abs(R[f]["mu"] - 1.5) < abs(A316_MU - 1.5) - 1e-9 for f in R) closest = min(R, key=lambda f: R[f]["L2"]) nan_found = any(v != v for f in R for v in R[f]["ratio"] + R[f]["lams"] + [R[f]["mu"], R[f]["tau"]]) verdict = ("FORCED" if any_hit else "NULL" if (not improves and not moves_mu) else "STRENGTHENED_PARTIAL") # the m=0 sector ground state is the pure-radial 4D operator (A316 node-count # ground): its degree-1 diagonal entry is the normalization base; the m=0 # eigenvalue should match the A316 C_4Dself ground eigenvalue ~23.27. m0_eig = R["F_lap"]["lams"][0] # ---- assertions ---------------------------------------------------------- print("S1 the A316 degeneracy is broken: three distinct entries, both forms") check(1, "the complex realization gives m=0,1,2 as three GENUINELY DISTINCT " "operators: F-lap eigenvalues %s differ (m^2 shifts 0,1,4); F-Hopf " "eigenvalues %s differ (m^2/r^2). The key structural fact vs A316." % ([round(v, 3) for v in R["F_lap"]["lams"]], [round(v, 3) for v in R["F_Hopf"]["lams"]]), len(set(round(v, 4) for v in R["F_lap"]["lams"])) == 3 and len(set(round(v, 4) for v in R["F_Hopf"]["lams"])) == 3) check(2, "A316 DEGENERACY BROKEN: both forms give three distinct diagonal " "entries and NO degenerate-selection collapse (F-lap distinct=%s degen=%s; " "F-Hopf distinct=%s degen=%s) -- unlike A316 S_Z3 where k=1,2 collapsed" % (R["F_lap"]["distinct"], R["F_lap"]["degen"], R["F_Hopf"]["distinct"], R["F_Hopf"]["degen"]), degeneracy_broken) check(3, "the m=0 fiber sector reproduces the pure-radial 4D operator (A316 " "node-count ground): m=0 ground eigenvalue %.4f ~ 23.27 (C_4Dself " "ground); both forms agree at m=0 (F-lap %.4f, F-Hopf %.4f)" % (m0_eig, R["F_lap"]["lams"][0], R["F_Hopf"]["lams"][0]), abs(m0_eig - 23.27) < 0.1 and abs(R["F_lap"]["lams"][0] - R["F_Hopf"]["lams"][0]) < 1e-6) print("S2 the canonical complex map's diagonal / mu / tau / L2 / peak") check(4, "F-lap canonical map (family k -> ground of m=k, flat +m^2): diagonal " "(1, %.4f, %.4f), mu=%.4f, tau=%.4f, L2=%.4f -- OVERSHOOTS far above 1.5, " "no valid peak (peak=%s)" % (R["F_lap"]["mu"], R["F_lap"]["tau"], R["F_lap"]["mu"], R["F_lap"]["tau"], R["F_lap"]["L2"], R["F_lap"]["valid_peak"]), abs(R["F_lap"]["mu"] - 2.4274) < 0.01 and abs(R["F_lap"]["tau"] - 21.0403) < 0.05 and R["F_lap"]["mu"] > 1.5 and not R["F_lap"]["valid_peak"]) check(5, "F-Hopf canonical map (family k -> ground of m=k, m^2/r^2): diagonal " "(1, %.4f, %.4f), mu=%.4f, tau=%.4f, L2=%.4f -- also OVERSHOOTS, no valid " "peak (peak=%s)" % (R["F_Hopf"]["mu"], R["F_Hopf"]["tau"], R["F_Hopf"]["mu"], R["F_Hopf"]["tau"], R["F_Hopf"]["L2"], R["F_Hopf"]["valid_peak"]), abs(R["F_Hopf"]["mu"] - 2.2901) < 0.01 and abs(R["F_Hopf"]["tau"] - 20.7261) < 0.05 and R["F_Hopf"]["mu"] > 1.5 and not R["F_Hopf"]["valid_peak"]) check(6, "neither fiber form's canonical map HITS: neither lands within 20%% of " "(1,1.5,8) and neither restores a valid peak (any_hit=%s); the closest " "form is %s (L2=%.4f)" % (any_hit, closest, R[closest]["L2"]), not any_hit and not R["F_lap"]["hit"] and not R["F_Hopf"]["hit"] and not R["F_lap"]["within20"] and not R["F_Hopf"]["within20"]) print("S3 NULL verdict honest condition (no improvement on A316) + clean") check(7, "VERDICT NULL: the complex realization does NOT improve on the A316 " "real-reduction closest (L2=%.3f, mu=%.3f): improves_L2=%s, " "moves_mu_toward_1.5=%s -- both fiber forms overshoot mu (2.43, 2.29) and " "blow up L2 (~13), so the genuine omega^k phase does NOT supply the " "magnitude; OI-287-1 magnitude does NOT close" % (A316_L2, A316_MU, improves, moves_mu), verdict == "NULL" and not improves and not moves_mu and not any_hit) check(8, "clean and honest: no NaN; the CANONICAL map (family k -> m=k omega^k " "eigenspace, declared before computing) was tested for BOTH corpus-faithful " "fiber forms and neither hits -- a genuine NULL, not a fit; the degeneracy " "IS broken (the real result) but the magnitude residual is elsewhere " "(nan=%s, degeneracy_broken=%s)" % (nan_found, degeneracy_broken), (not nan_found) and degeneracy_broken and not any_hit and verdict == "NULL") print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL") sys.exit(0 if FAIL == 0 else 1)