#!/usr/bin/env python3 """verify_P319.py -- Verifier for Addendum 319 (OI-287-1, the boundary S^3 problem directly). Recomputes from scratch and asserts, by machine: S1 The boundary harmonic spectrum is the genuine SO(4) data (sanity vs A314): the so(4) = su(2) (+) su(2) Casimir on the (j,j) irrep is 4 j(j+1) = L(L+2) (L = 2j), the -Delta_S^3 spectrum (0, 3, 8 for L = 0, 1, 2) with multiplicity (L+1)^2 (1, 4, 9). The same su(2) ladder / Casimir A314 used. S2 Each boundary formulation gives three DISTINCT families (no A316-style degeneracy collapse): B-harm (families = lowest harmonic degrees L=0,1,2) and B-lens (families = Z_3 character classes on L(3,1)) each produce three distinct diagonal entries. S3 Each formulation's normalized diagonal / mu / tau / peak / L2, recomputed from scratch, matches the probe. Whether the boundary problem reaches the GAP [1.34, 2.0] no radial channel reached, and the HIT/PARTIAL/NULL verdict's defining condition, asserted honestly. The boundary problem OVERSHOOTS: B-lens (canonical, Z_3 character) gives mu = 3.86, B-harm gives mu = 6.28, both far above the target 1.5 and above the radial fiber overshoots (2.3-3.14). It does NOT reach the [1.34, 2.0] gap and does NOT beat the radial-best L2 = 0.399. NULL: the magnitude is in NEITHER the radial nor the pure-boundary reduction. Copyright Léon Fernando Vlegels -- CC BY 4.0 """ import sys import math import numpy as np PI = math.pi C = (2.0, 3.0, 16.0) TARGET = [c / C[0] for c in C] # (1, 1.5, 8) SCALAR_CLOSEST_L2 = 0.399 GAP_LO, GAP_HI = 1.34, 2.0 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}") # --- su(2) generators + so(4) Casimir (independent rebuild, A314 construction) - def su2(j): dim = int(round(2 * j + 1)) ms = [j - i for i in range(dim)] Jp = np.zeros((dim, dim), dtype=complex) Jz = np.zeros((dim, dim), dtype=complex) for i, m in enumerate(ms): Jz[i, i] = m if i - 1 >= 0: Jp[i - 1, i] = math.sqrt(j * (j + 1) - m * (m + 1)) Jm = Jp.conj().T return (Jp + Jm) / 2.0, (Jp - Jm) / (2.0j), Jz def so4_data(L): j = L / 2.0 Lx, Ly, Lz = su2(j) d = Lx.shape[0] I = np.eye(d, dtype=complex) JL = [np.kron(A, I) for A in (Lx, Ly, Lz)] JR = [np.kron(I, A) for A in (Lx, Ly, Lz)] C2 = np.zeros((d * d, d * d), dtype=complex) for i in range(3): A = JL[i] + JR[i] B = JL[i] - JR[i] C2 = C2 + A @ A + B @ B ev = np.linalg.eigvalsh((C2 + C2.conj().T) / 2.0).real casimir = float(np.mean(ev)) mult = d * d m_weights = list(range(-L, L + 1, 2)) return casimir, mult, m_weights, (ev.max() - ev.min()) def spec_block(Lmax=2): return {L: so4_data(L) for L in range(Lmax + 1)} def specweight(casimir, mult): return casimir if casimir > 1e-9 else float(mult) def norm_diag(M): d = [M[n, n] for n in range(3)] return [v / d[0] for v in d] def l2(ratio): return math.sqrt(sum((ratio[i] - TARGET[i]) ** 2 for i in range(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 build_harm(spec): L_of = [0, 1, 2] B = np.zeros((3, 3)) for n in range(3): L = L_of[n] cas, mult, _, _ = spec[L] w = specweight(cas, mult) for k in (1, 2, 3): B[n, k - 1] = C[k - 1] * PI ** k * mult / w return B def build_lens(spec, Lmax=2): classes = {0: [], 1: [], 2: []} for L in range(Lmax + 1): cas, mult, ms, _ = spec[L] for k in (0, 1, 2): cnt = sum(1 for m in ms if (m % 3) == k) if cnt > 0: classes[k].append((L, cnt)) B = np.zeros((3, 3)) for n in range(3): k = n num = den = 0.0 for (L, cnt) in classes[k]: cas, mult, _, _ = spec[L] wL = cas if cas > 1e-9 else 1.0 num += wL * cnt den += cnt cw = (num / den) if den > 0 else 1.0 for j in (1, 2, 3): B[n, j - 1] = C[j - 1] * PI ** j * den / cw return B, classes def flat_control(): B = np.zeros((3, 3)) for n in range(3): for j in (1, 2, 3): B[n, j - 1] = C[j - 1] * PI ** j * 4.0 / 3.0 return B # =========================================================================== print("S1 Boundary harmonic spectrum is the genuine SO(4) data (sanity vs A314)") spec = spec_block(2) cas_ok = all(abs(spec[L][0] - L * (L + 2)) < 1e-9 for L in (0, 1, 2)) block_deg = all(spec[L][3] < 1e-8 for L in (0, 1, 2)) check(1, "so(4) Casimir 4j(j+1) = L(L+2): casimir = (0, 3, 8) for L=0,1,2, " "block-degenerate (the -Delta_S^3 boundary Laplacian spectrum)", cas_ok and block_deg) mult_ok = all(spec[L][1] == (L + 1) ** 2 for L in (0, 1, 2)) check(2, "degree-L harmonic multiplicity (L+1)^2 = (1, 4, 9) for L=0,1,2", mult_ok) # =========================================================================== print("S2 Each boundary formulation gives three distinct families (no collapse)") B_harm = build_harm(spec) B_lens, classes = build_lens(spec) dh = [B_harm[n, n] for n in range(3)] harm_distinct = (abs(dh[0] - dh[1]) > 1e-9 and abs(dh[0] - dh[2]) > 1e-9 and abs(dh[1] - dh[2]) > 1e-9) check(3, "B-harm (families = lowest harmonic degrees L=0,1,2) gives three " "distinct diagonal entries (no A316 collapse)", harm_distinct) dl = [B_lens[n, n] for n in range(3)] lens_distinct = (abs(dl[0] - dl[1]) > 1e-9 and abs(dl[0] - dl[2]) > 1e-9 and abs(dl[1] - dl[2]) > 1e-9) # the lens classes must be genuinely populated and the canonical (Z_3) family # definition must keep all three classes non-empty (the complex omega^k structure) classes_populated = all(len(classes[k]) > 0 for k in (0, 1, 2)) check(4, "B-lens (families = Z_3 character classes on L(3,1)) gives three " "distinct diagonal entries and all three classes are populated " "(the native omega^k structure)", lens_distinct and classes_populated) # =========================================================================== print("S3 Diagonals, mu, tau, peak, L2; gap reach; verdict's defining condition") ratio_h = norm_diag(B_harm) ratio_l = norm_diag(B_lens) ctrl = flat_control() ctrl_peak = diag_peaks(ctrl) # B-harm: overshoots, mu = 6.28, tau ~ 88.8, no peak mu_h, tau_h, L2_h = ratio_h[1], ratio_h[2], l2(ratio_h) peak_h = diag_peaks(B_harm) and not ctrl_peak check(5, "B-harm diagonal (1, %.2f, %.1f): mu = %.2f (overshoot), tau = %.1f, " "L2 = %.2f, no valid peak (%s)" % (mu_h, tau_h, mu_h, tau_h, L2_h, peak_h), abs(mu_h - 6.2832) < 1e-2 and abs(tau_h - 88.83) < 0.1 and not peak_h and abs(L2_h - 80.97) < 0.5) # B-lens (canonical): overshoots, mu = 3.86, tau ~ 64.6, no peak mu_l, tau_l, L2_l = ratio_l[1], ratio_l[2], l2(ratio_l) peak_l = diag_peaks(B_lens) and not ctrl_peak check(6, "B-lens (canonical) diagonal (1, %.2f, %.1f): mu = %.2f (overshoot), " "tau = %.1f, L2 = %.2f, no valid peak (%s)" % (mu_l, tau_l, mu_l, tau_l, L2_l, peak_l), abs(mu_l - 3.8556) < 1e-2 and abs(tau_l - 64.60) < 0.1 and not peak_l and abs(L2_l - 56.65) < 0.5) # the GAP: neither formulation reaches [1.34, 2.0]; both overshoot past it reaches_gap = ((GAP_LO <= mu_h <= GAP_HI) or (GAP_LO <= mu_l <= GAP_HI)) check(7, "the boundary problem does NOT reach the [1.34, 2.0] gap no radial " "channel could (B-harm mu = %.2f, B-lens mu = %.2f both above 2.0): " "reaches_gap = %s" % (mu_h, mu_l, reaches_gap), not reaches_gap) # the verdict's defining condition: NULL. Canonical does not hit; boundary does # not reach the gap and does not beat the radial-best L2 = 0.399. canonical_hits = (all(abs(ratio_l[i] - TARGET[i]) <= 0.20 * TARGET[i] for i in range(3)) and peak_l) beats_radial = (L2_h < SCALAR_CLOSEST_L2 - 1e-9 or L2_l < SCALAR_CLOSEST_L2 - 1e-9) verdict_null = (not canonical_hits) and (not reaches_gap) and (not beats_radial) nan_found = any(v != v for v in ratio_h + ratio_l) check(8, "NULL verdict's defining condition holds: canonical (Z_3) map does NOT " "hit, boundary does NOT reach the gap, boundary does NOT beat the " "radial-best L2 = 0.399; no NaN (magnitude in NEITHER reduction)", verdict_null and not nan_found) print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL") sys.exit(0 if FAIL == 0 else 1)