"""Actual Clifford and spatial-jet checks for the coexact current family.""" from datetime import datetime, timezone from hashlib import sha256 from importlib.util import spec_from_file_location, module_from_spec from pathlib import Path from itertools import combinations import json import numpy as np ROOT = Path(__file__).resolve().parent spec = spec_from_file_location('radial', ROOT/'inverse-current-radial.py') radial = module_from_spec(spec) spec.loader.exec_module(radial) COUNTS, ERRORS = {}, {} def close(actual, expected, label, tolerance=2e-11): error = float(np.linalg.norm(np.asarray(actual)-np.asarray(expected))/max(1., np.linalg.norm(expected))) ERRORS[label] = max(ERRORS.get(label, 0.), error) if error >= tolerance: raise AssertionError((label, error)) COUNTS[label] = COUNTS.get(label, 0)+1 def main(): rng = np.random.default_rng(202610120530) cases = [] for depth in radial.DEPTHS: matrices = radial.gammas(depth+1) alpha, beta = matrices[:depth], matrices[depth] rank = len(beta) eye = np.eye(rank) clifford = [] for grade in range(depth+1): for indices in combinations(range(depth), grade): matrix = eye.copy() for index in indices: matrix = matrix@alpha[index] clifford.append(matrix) tvalue = (-1)**grade*(depth-2*grade)/2 actual = sum(a@matrix@a for a in alpha)/2 close(actual, tvalue*matrix, 'all exterior grades have the exact Killing-frame connection term') close(sum(a@beta@matrix@a for a in alpha)/2, -tvalue*beta@matrix, 'parallel beta reverses the complete Killing-frame connection term') close(np.trace(matrix), rank if grade == 0 else 0, 'all nontrivial Clifford characters have zero trace in the retained carrier') if depth % 2: clifford += [beta@m for m in clifford] close(len(clifford), rank*rank, 'Clifford forms with the stated odd-depth doubling span the complete matrix carrier') flat = np.array(clifford).reshape(len(clifford), -1)/np.sqrt(rank) # For this monomial Clifford basis the product of distinct monomials # is a phase times a nonidentity monomial; the trace check above # proves its Gram orthogonality without a dense cubic Gram product. close(np.sum(abs(flat)**2, axis=1), np.ones(rank*rank), 'complete Clifford matrix basis has unit diagonal Gram entries') if depth % 2: close([np.trace(m) for m in clifford[2**depth:]], np.zeros(2**depth), 'odd-depth beta monomials remain trace orthogonal to the original Clifford algebra') for level in [1, 2]: eigenvalue = (level+1)*(level+depth-2) for sample in range(3): ambient = rng.normal(size=depth+1) ambient /= np.linalg.norm(ambient) orthogonal, _ = np.linalg.qr(np.column_stack([ambient, rng.normal(size=(depth+1, depth))])) tangent = orthogonal[:, 1:] rotation = np.zeros((depth+1, depth+1)) rotation[0, 1], rotation[1, 0] = 1., -1. linear = np.zeros(depth+1) if level == 2: linear[2] = 1. f = np.dot(linear, ambient) if level == 2 else 1. vector = f*rotation@ambient jacobian = f*rotation+np.outer(rotation@ambient, linear) values = tangent.T@vector derivative = tangent.T@jacobian@tangent derivative = derivative.T # derivative index first, component index second second = np.zeros((depth, depth, depth)) for k in range(depth): for i in range(depth): ambient_second = (np.dot(linear, tangent[:, k])*rotation@tangent[:, i] +np.dot(linear, tangent[:, i])*rotation@tangent[:, k]) second[k, i, :] = tangent.T@ambient_second if k == i: second[k, i, :] -= level*values second[k, i, k] += values[i] close(np.dot(ambient, vector), 0., 'ambient polynomial vector is exactly tangent to the ordinary sphere') close(np.trace(derivative), 0., 'the spatial vector harmonic is divergence free') rough = np.einsum('iij->j', second) close(-rough+(depth-1)*values, eigenvalue*values, 'independent ambient second jets verify the coexact Hodge eigenvalue') m1 = -1j*sum(v*a for v, a in zip(values, alpha)) dm1 = [-1j*sum(derivative[k, j]*alpha[j] for j in range(depth)) for k in range(depth)] m2 = -sum(alpha[i]@alpha[j]*derivative[i, j] for i in range(depth) for j in range(depth))/np.sqrt(eigenvalue) dm2 = [-sum(alpha[i]@alpha[j]*second[k, i, j] for i in range(depth) for j in range(depth))/np.sqrt(eigenvalue) for k in range(depth)] d1 = -1j*sum(a@dm for a, dm in zip(alpha, dm1))+sum(a@m1@a for a in alpha)/2 d2 = -1j*sum(a@dm for a, dm in zip(alpha, dm2))+sum(a@m2@a for a in alpha)/2 close(d1, (1-depth/2)*m1+np.sqrt(eigenvalue)*m2, 'actual full spatial Dirac derivative closes the coexact vector into its two-form') close(d2, np.sqrt(eigenvalue)*m1+(depth/2-2)*m2, 'actual full spatial Dirac derivative closes the two-form back into its vector') if level == 1: positive = m1+np.sqrt((depth-1)/2)*m2 close(positive.conj().T, -positive, 'first coexact positive Killing endomorphism is anti-Hermitian') for k in range(depth): derivative_positive = dm1[k]+np.sqrt((depth-1)/2)*dm2[k] close(derivative_positive, .5j*(alpha[k]@positive-positive@alpha[k]), 'zero coexact endomorphism is a constant right generator in the Killing frame') base = (2*eye+beta)/np.sqrt(5*rank) variation = (2*eye+beta)@positive/np.sqrt(5*rank) close(variation@base.conj().T+base@variation.conj().T, np.zeros_like(base), 'particle right-unitary zero mode changes no left density matrix to first order') close(positive+positive.conj().T, np.zeros_like(positive), 'background right-unitary zero mode changes no left density matrix to first order') if depth >= 3: # The two-form and its beta partner carry no physical # vector trace in this full retained Clifford module. close([np.trace(a@m2) for a in alpha], np.zeros(depth), 'coexact two-form carries no direct vector current') close([np.trace(a@beta@m2) for a in alpha], np.zeros(depth), 'beta coexact two-form carries no direct vector current') close([np.trace(a@beta@m1) for a in alpha], np.zeros(depth), 'unmixed beta vector has no independent vector trace above depth two') close([np.trace(a@m1)/rank for a in alpha], -1j*values, 'actual Clifford vector trace derives the current-source normalization') cases.append(dict(depth=depth, spinor_rank=rank, coexact_degree=level, sample=sample, hodge_eigenvalue=eigenvalue, classification='actual tangent vector, covariant second jets and full matrix Dirac closure')) stamp = datetime.now(timezone.utc).strftime('%Y%m%dT%H%M%S.%fZ') report = dict(experiment_key='covariant-clock-spatial-geometry', created_utc=datetime.now(timezone.utc).isoformat(), sources={'experiments/'+p.name: sha256(p.read_bytes()).hexdigest() for p in [Path(__file__), ROOT/'inverse-current-radial.py']}, evaluations=sum(COUNTS.values()), conditions=COUNTS, maximum_errors=ERRORS, cases=cases, limits=['ordinary unit sphere and full balanced matrix carrier are supplied premises', 'first two coexact harmonics receive explicit ambient polynomial jet checks', 'global Hodge completeness requires the accompanying analytical argument']) output = ROOT/('covariant-clock-geometry-results-'+stamp+'.json') output.write_text(json.dumps(report, indent=2)+'\n') print(json.dumps(dict(report=str(output), evaluations=report['evaluations'], maximum_errors=ERRORS)), flush=True) if __name__ == '__main__': main()