#!/usr/bin/env python3 """Check squared-linear-curvature symbols and a retained-orientation comparison.""" from pathlib import Path import argparse import hashlib import importlib.util import json import math import numpy as np from scipy.integrate import quad from scipy.linalg import eigh CHECKS = 0 CASES = [] def check(condition, label): global CHECKS CHECKS += 1 if not condition: raise AssertionError(label) def close(a, b, label, tolerance=3e-8): check(np.allclose(a, b, atol=tolerance, rtol=tolerance), label) def symbols(second, depth, direction): q = second.shape[0] matrices = [] for i in range(depth): vector = second[:, i, :]@direction matrices.append(np.stack([(np.outer(np.eye(q)[a], vector)-np.outer(vector, np.eye(q)[a])).ravel() for a in range(q)], axis=1)) kinetic = sum(x.T@x for x in matrices) along = sum(direction[i]*matrices[i] for i in range(depth)) defect = along.T@along magnetic = kinetic-defect return kinetic, magnetic, defect, matrices def bump(x): return math.exp(1-1/(1-x*x)) if abs(x) < 1 else 0. def main(): parser = argparse.ArgumentParser() parser.add_argument('--geometry-source', type=Path, required=True) parser.add_argument('--report', type=Path, required=True) args = parser.parse_args() spec = importlib.util.spec_from_file_location('nested_geometry', args.geometry_source) geometry = importlib.util.module_from_spec(spec) spec.loader.exec_module(geometry) hessian = geometry.full_hessian() rng = np.random.default_rng(2026101052) expected = {2: ([4, 4, 2, 2, 4, 4, 4, 4], [2, 2, 2, 0, 2, 2, 2, 2]), 3: ([4, 4, 4, 6, 6, 6], [4, 2, 2, 4, 4, 4]), 4: ([6, 6, 6, 6, 8, 8], [6, 4, 4, 4, 6, 6]), 5: ([2, 2, 4, 4], [2, 2, 4, 4]), 6: ([2, 2, 4, 4], [2, 2, 4, 4])} for d in geometry.DEPTHS: even = d+d % 2 q = 10-even m = even//2 h = int(m % 2 == 0) if not q: CASES.append({'case': 'depth-10-no-normal-connection', 'real_depth': d, 'hodge_active': False, 'normal_curvature_response_present': False}) continue second = hessian[even:, :even, :even] direction = np.zeros(even); direction[0] = 1 kinetic, magnetic, defect, _ = symbols(second, d, direction) close(kinetic, np.diag(expected[d][0]), 'actual graph axial kinetic symbol') close(magnetic, np.diag(expected[d][1]), 'actual graph axial magnetic symbol') for _ in range(4): direction[:d] = rng.normal(size=d); direction /= np.linalg.norm(direction) a, b, c, matrices = symbols(second, d, direction) direct = sum((direction[j]*matrices[i]-direction[i]*matrices[j]).T@ (direction[j]*matrices[i]-direction[i]*matrices[j]) for i in range(d) for j in range(i+1, d)) close(direct, b, 'exterior spatial curvature identity gives magnetic symbol') close(a-b, c, 'difference is a positive longitudinal curvature square') check(np.linalg.eigvalsh(b).min() > -1e-9, 'magnetic symbol is nonnegative') check(np.linalg.eigvalsh(c).min() > -1e-9, 'electric symbol bounds magnetic symbol') for k in (.1, 1., 10.): frequencies = eigh(k*k*np.eye(q)+h*k**4*b, np.eye(q)+h*k*k*a, eigvals_only=True) check(frequencies.min() > 0 and frequencies.max() <= k*k*(1+1e-9), 'positive frozen squared frequencies have phase speeds at most the base speed') CASES.append({'case': f'induced-curvature-depth-{d}', 'real_depth': d, 'hodge_active': bool(h), 'axial_electric_symbol_diagonal': expected[d][0], 'axial_magnetic_symbol_diagonal': expected[d][1], 'independent_gauss_source_constraint_survives_induced_reduction': False, 'frozen_symbol_is_complete_variable_geometry_solution': False, 'background_curvature_second_variation_included': False, 'arbitrary_real_graph_waves_preserve_unitary_normal_carrier': False}) if not h: continue for a, b in sorted(set(zip(expected[d][0], expected[d][1]))): length = math.sqrt(a) tail = quad(lambda y: math.exp(-(2-y)/length)*bump(y)/(2*length), -1., 1., epsabs=1e-13)[0] check(tail > 0, 'a compact local probe has an exterior frozen acceleration response') unforced_tail = (1-b/a)*tail/a check((unforced_tail > 0) == (a > b), 'compact unforced displacement also has an exterior acceleration tail when a exceeds b') # A retained tilt mode replaces the exact derivative locking relation. lock = 1/3 bare_speed_squared = 1-lock for k in (0., .01, .1, 1., 10., 100.): stiffness = np.array([[(bare_speed_squared+lock)*k*k, 1j*lock*k], [-1j*lock*k, lock+b*k*k]]) inertia = np.diag([1., a]) eigenvalues = eigh(stiffness, inertia, eigvals_only=True) check(eigenvalues.min() > -1e-9, 'finite-lock retained-mode energy is positive') if k: for z in eigenvalues: determinant = ((bare_speed_squared+lock)*k*k-z)*(lock+b*k*k-a*z)-lock*lock*k*k close(determinant/(1+k**4+z*z), 0, 'full two-mode poles obey the exact Schur determinant') else: close(eigenvalues, [0., lock/a], 'retained orientation has its own deformation clock') check(0 <= b/a <= 1 and bare_speed_squared+lock == 1, 'retained system has finite principal speeds bounded by the common unit speed') CASES.append({'case': f'active-depth-{d}-axial-symbol-{a}-{b}', 'real_depth': d, 'electric_symbol_a': a, 'magnetic_symbol_b': b, 'high_frequency_locked_speed_squared': b/a, 'compact_probe_acceleration_at_x_2': tail, 'compact_unforced_displacement_acceleration_at_x_2': unforced_tail, 'retained_lock_coefficient': lock, 'retained_bare_speed_squared': bare_speed_squared, 'retained_orientation_gap_squared': lock/a, 'independent_orientation_is_exact_integrable_graph': False}) report = {'title': 'Induced Normal Curvature Gives Mixed Spatial Inertia and a Retained-Orientation Comparison', 'sources': {'experiments/induced-curvature-response-checks.py': hashlib.sha256(Path(__file__).read_bytes()).hexdigest(), 'experiments/nested-normal-curvature-checks.py': hashlib.sha256(args.geometry_source.read_bytes()).hexdigest()}, 'checks_passed': CHECKS, 'case_count': len(CASES), 'occupied_depths': list(geometry.DEPTHS), 'scope': 'actual squared-first-curvature-variation symbols, not the complete nonzero-background curvature Hessian; frozen real-normal response and finite-lock retained-orientation comparison', 'cases': CASES} args.report.write_text(json.dumps(report, indent=2)+'\n') print(f'{CHECKS} induced-response checks passed across {len(CASES)} cases') if __name__ == '__main__': main()