"""Complete second-order spectrum of the actual rotating spatial count projector.""" from datetime import datetime, timezone from hashlib import sha256 from importlib.util import module_from_spec, spec_from_file_location from math import comb, pi, sqrt from pathlib import Path import json import numpy as np from scipy.integrate import solve_ivp from scipy.linalg import eig, eigvals, eigvalsh from scipy.optimize import brentq, root ROOT = Path(__file__).resolve().parent HELPER = ROOT/'moving-contact-checks.py' spec = spec_from_file_location('spatial_rotation_weights', HELPER) spatial = module_from_spec(spec) spec.loader.exec_module(spatial) DEPTHS, ASSIGNMENTS = spatial.DEPTHS, spatial.ASSIGNMENTS CONDITIONS, ERRORS = {}, {} GRID = (0., .25, .5, .75, .99, 1.1, 1.3, 1.5, 1.6, 1.75, 1.9, 2.1, 2.4, 3., 4., 6.) def check(value, label): if not bool(value): raise AssertionError(label) CONDITIONS[label] = CONDITIONS.get(label, 0)+1 def close(actual, expected, label, tolerance=3e-9): actual, expected = np.asarray(actual), np.asarray(expected) error = float(np.max(np.abs(actual-expected))/max(1.0, float(np.max(np.abs(expected))))) ERRORS[label] = max(ERRORS.get(label, 0.0), error) check(error < tolerance, label) def data(depth, cutoff, sigma=1): rotations, weights = spatial.compressed(*spatial.joint_weights(depth, cutoff, sigma)[1:]) b = np.sqrt(weights) potential = 4*np.eye(len(b))-3*np.outer(b, b) return rotations.astype(float), weights, b, potential def frequency_matrix(rotations, potential, speed): transport = np.diag(speed*rotations) return np.block([[transport, np.eye(len(rotations))], [potential, transport]]) def energy_threshold(rotations, weights, cutoff): if not cutoff: return None def deficit(g): return float(1-3*np.sum(weights/(4-(g*rotations/cutoff)**2))) threshold = brentq(deficit, 1, 2-1e-11, xtol=3e-13, rtol=3e-14) check(1 < threshold < 2, 'positive rotating energy threshold lies between universal bounds') _, _, _, potential = data_for_energy(rotations, weights) minimum = eigvalsh(potential-np.diag((threshold*rotations/cutoff)**2))[0] close(minimum, 0, 'secular energy threshold matches complete stiffness matrix') return threshold def data_for_energy(rotations, weights): b = np.sqrt(weights) return rotations, weights, b, 4*np.eye(len(b))-3*np.outer(b, b) def bright_state(rotations, weights, potential, speed): def secular(z): return float(1-3*np.sum(weights/(4-(z-speed*rotations)**2))) z = brentq(secular, .5, 1.5, xtol=2e-14, rtol=1e-14) vector = np.sqrt(weights)/(4-(z-speed*rotations)**2) vector /= np.linalg.norm(vector) covariant_frequency = z-speed*rotations residual = potential@vector-covariant_frequency**2*vector close(residual, np.zeros_like(vector), 'bright frequency solves complete quadratic pencil') charge = 2*float(np.dot(covariant_frequency, vector**2)) angular_charge = -2*float(np.dot(rotations*covariant_frequency, vector**2)) energy = 2*float(np.dot(covariant_frequency**2, vector**2)) close(energy-speed*angular_charge, z*charge, 'second-order rotating energy retains signed phase and rotation charges') check(charge > 0, 'continued slow bright branch has positive signed phase charge') return z, vector, charge, angular_charge, energy def slow_checks(depth, cutoff, rotations, weights, potential): if not cutoff: z, _, charge, _, energy = bright_state(rotations, weights, potential, 1) close(z, 1, 'ground clock is unchanged by any source angle speed') close(energy/charge, 1, 'ground field energy per positive charge equals its clock') return h = .001/cutoff def coefficient(step): plus = bright_state(rotations, weights, potential, step) minus = bright_state(rotations, weights, potential, -step) zp, zm = plus[0], minus[0] field = (plus[4]/plus[2]+minus[4]/minus[2]-2)/(2*step*step) return (zp+zm-2)/(2*step*step), (zp-zm)/(2*step), field full, first, full_field = coefficient(h) half, _, half_field = coefficient(h/2) estimate = (4*half-full)/3 metric = float(spatial.geometry(depth, cutoff)) close(estimate, -7*metric/6, 'finite-inertia second-order shift reuses the spatial metric', 7e-5) close(first, cutoff/(depth+1), 'slow bright linear shift is the geometric rotation mean', 5e-6) close((4*half_field-full_field)/3, 7*metric/6, 'field energy per signed charge retains the opposite quadratic shift', 7e-5) def augmented_energy_checks(rotations, weights, potential, speed, count): z, u, phase, _, _ = bright_state(rotations, weights, potential, speed) shifted = z-speed*rotations stiffness = potential-np.diag(shifted**2) spectrum = eigvalsh(stiffness) close(spectrum[0], 0, 'relative-equilibrium augmented field form has its exact bright zero mode') gap = float(spectrum[1]) if len(spectrum) > 1 else None if gap is not None: check(gap > 0, 'continued relative equilibrium minimizes complete augmented energy transverse to symmetry') rng = np.random.default_rng(671) q = rng.normal(size=len(u))+1j*rng.normal(size=len(u)) momentum = rng.normal(size=len(u))+1j*rng.normal(size=len(u)) source_p = .83 charge = -2*np.vdot(q, momentum).imag angular = source_p+2*np.vdot(q, rotations*momentum).imag energy = source_p**2/(2*count)+np.vdot(momentum, momentum).real+np.vdot(q, potential@q).real augmented = energy-speed*angular-z*charge+count*speed**2/2 squares = (source_p-count*speed)**2/(2*count) squares += np.vdot(momentum+1j*shifted*q, momentum+1j*shifted*q).real squares += np.vdot(q, stiffness@q).real close(augmented, squares, 'complete autonomous augmented energy is the sum of rotor momentum and field forms') q = u/sqrt(phase) momentum = -1j*shifted*q source_p = count*speed close(-2*np.vdot(q, momentum).imag, 1, 'relative-equilibrium action normalizer is its signed phase charge') check(z+abs(speed)*max(abs(rotations)) < 2, 'continued branch has a positive diagonal secular factor at every retained rotation') return {'transverse_augmented_stiffness_gap': gap, 'positive_diagonal_margin': float(2-max(abs(shifted))), 'conditional_autonomous_orbital_stability': True} def onset_checks(): cases = [] for depth in DEPTHS: cutoff = 2 rotations, weights, b, potential = data(depth, cutoff) low, high = 1.2, 1.6 def growth(g): return float(np.max(eigvals(frequency_matrix(rotations, potential, g/cutoff)).imag)) check(growth(low) < 1e-8 and growth(high) > .1, 'all depths have a resolved stable/unstable control bracket') for _ in range(34): middle = (low+high)/2 if growth(middle) > 1e-6: high = middle else: low = middle zlist = eigvals(frequency_matrix(rotations, potential, high/cutoff)) starting_z = float(zlist[np.argmax(zlist.imag)].real) def equations(x): z, g = x shifted = z-g*rotations/cutoff den = 4-shifted**2 return [1-3*np.sum(weights/den), -6*np.sum(weights*shifted/den**2)] solution = root(equations, [starting_z, high], tol=1e-11) close(equations(solution.x), [0, 0], 'instability onset solves secular equation and its frequency derivative') z, g = map(float, solution.x) check(abs(g-high) < 2e-7, 'independent double-root onset agrees with spectral bisection') vector = b/(4-(z-g*rotations/cutoff)**2) vector /= np.linalg.norm(vector) signed_charge = 2*float(np.dot(z-g*rotations/cutoff, vector**2)) close(signed_charge, 0, 'colliding frequency branches have zero limiting signed phase charge') threshold = energy_threshold(rotations, weights, cutoff) check(g > threshold+1e-4, 'loss of positive rotating energy precedes this instability onset') cases.append({'depth': depth, 'cutoff': cutoff, 'energy_threshold_g': threshold, 'first_control_instability_g': g, 'double_root_frequency_over_lambda': z, 'bisection_bracket': [low, high], 'double_root_signed_phase_charge': signed_charge}) return cases def growth_checks(): cases = [] for depth in DEPTHS: rotations, weights, b, potential = data(depth, 2) speed = .8 values, vectors = eig(frequency_matrix(rotations, potential, speed)) selected = int(np.argmax(values.imag)) z = values[selected] q = vectors[:len(b), selected] q /= np.linalg.norm(q) momentum = -1j*(z-speed*rotations)*q check(z.imag > .1, 'direct growth test uses a resolved unstable frequency') def rhs(time, y): position, velocity = y[:len(b)], y[len(b):2*len(b)] moving = b*np.exp(1j*rotations*speed*time) overlap = np.vdot(moving, position) mixed = np.vdot(moving, rotations*position) force = -4*position+3*moving*overlap work = -6*speed*np.imag(np.conj(overlap)*mixed) return np.concatenate((velocity, force, [work])) endpoint = 2/float(z.imag) times = np.linspace(0, endpoint, 25) initial = np.concatenate((q, momentum, [0j])) solution = solve_ivp(rhs, [0, endpoint], initial, t_eval=times, method='DOP853', rtol=2e-11, atol=2e-12) check(solution.success, 'direct second-order moving-source integration completed') expected = np.array([np.exp(1j*rotations*speed*t)*q*np.exp(-1j*z*t) for t in times]).T close(solution.y[:len(b)], expected, 'direct laboratory growth agrees with complete quadratic pencil', 5e-9) energies = [] for time, y in zip(times, solution.y.T): position, velocity = y[:len(b)], y[len(b):2*len(b)] moving = b*np.exp(1j*rotations*speed*time) energies.append(float(np.vdot(velocity, velocity).real+4*np.vdot(position, position).real- 3*abs(np.vdot(moving, position))**2)) close(np.array(energies)/energies[0], np.exp(2*z.imag*times), 'unstable mode energy grows at twice its amplitude growth rate', 5e-9) close(np.array(energies)-energies[0], solution.y[-1].real, 'prescribed source work accounts for the growing field energy', 5e-9) cases.append({'depth': depth, 'cutoff': 2, 'rotation_g': 1.6, 'amplitude_growth_over_lambda': float(z.imag), 'amplitude_efolding_in_target_clock_cycles': 1/(2*pi*float(z.imag)), 'time_in_lambda_units': endpoint, 'integration_evaluations': solution.nfev}) return cases def reciprocal_checks(): cases = [] for name, depth, cutoff in [('control', d, 2) for d in DEPTHS]+[('electron', 6, 12), ('muon', 6, 34)]: rotations, weights, b, potential = data(depth, cutoff) count = comb(cutoff+depth, depth) q = b+(.07+.03j)*(rotations-np.dot(rotations, weights))*b/(cutoff+1) q /= sqrt(2)*np.linalg.norm(q) momentum = -1j*q n = len(b) initial = np.concatenate(([.17, .83], q.real, q.imag, momentum.real, momentum.imag)) def unpack(y): return y[0], y[1], y[2:2+n]+1j*y[2+n:2+2*n], y[2+2*n:2+3*n]+1j*y[2+3*n:] def rhs(time, y): theta, source_p, position, velocity = unpack(y) moving = b*np.exp(1j*rotations*theta) overlap = np.vdot(moving, position) mixed = np.vdot(moving, rotations*position) force = -4*position+3*moving*overlap torque = 6*np.imag(np.conj(overlap)*mixed) return np.concatenate(([source_p/count, torque], velocity.real, velocity.imag, force.real, force.imag)) solution = solve_ivp(rhs, [0, 6], initial, t_eval=np.linspace(0, 6, 31), method='DOP853', rtol=2e-11, atol=2e-12) check(solution.success, 'autonomous second-order source integration completed') energies, phases, angular, norms, source_momenta = [], [], [], [], [] for y in solution.y.T: theta, source_p, position, velocity = unpack(y) moving = b*np.exp(1j*rotations*theta) energy = source_p**2/(2*count)+np.vdot(velocity, velocity).real energy += 4*np.vdot(position, position).real-3*abs(np.vdot(moving, position))**2 energies.append(float(energy)) phases.append(-2*np.vdot(position, velocity).imag) angular.append(source_p+2*np.vdot(position, rotations*velocity).imag) norms.append(float(np.vdot(position, position).real)) source_momenta.append(source_p) close(energies, np.full(len(energies), energies[0]), 'complete second-order field/source energy conserved', 5e-9) close(phases, np.ones(len(phases)), 'signed phase charge is conserved with both oscillator branches', 5e-9) close(angular, np.full(len(angular), angular[0]), 'complete second-order source/field rotation charge conserved', 5e-9) check(np.ptp(norms) > 1e-7, 'configuration norm is not substituted for the signed phase charge') check(np.ptp(source_momenta) > 1e-7, 'second-order source backreaction is retained') cases.append({'name': name, 'depth': depth, 'cutoff': cutoff, 'simplex_count': count, 'interval_in_lambda_units': 6, 'source_momentum_range': float(np.ptp(source_momenta)), 'configuration_norm_range': float(np.ptp(norms)), 'relative_energy_drift': float(np.max(np.abs(np.array(energies)/energies[0]-1))), 'signed_phase_charge_drift': float(np.max(np.abs(np.array(phases)-1))), 'combined_rotation_charge_drift': float(np.max(np.abs(np.array(angular)-angular[0])))}) return cases def convective_checks(): for depth in DEPTHS: rotations, weights, b, potential = data(depth, 2) speed, theta, inertia = .73, .37, 1.3 phase = np.exp(1j*rotations*theta) q = b+(.07+.03j)*(rotations-np.dot(rotations, weights))*b qdot = (.11-.31j)*q+.07*rotations*q velocity = qdot-1j*speed*rotations*q chi = np.conj(phase)*q chidot = np.conj(phase)*velocity moving = b*phase energy = 4*np.vdot(q, q).real-3*abs(np.vdot(moving, q))**2 original_l = np.vdot(velocity, velocity).real-energy+inertia*speed**2/2 reduced_l = np.vdot(chidot, chidot).real-np.vdot(chi, potential@chi).real+inertia*speed**2/2 close(original_l, reduced_l, 'full convective second-order kinetic action rotates without omitted terms') j = 2*np.vdot(q, rotations*velocity).imag canonical_p = inertia*speed-j complete_h = (canonical_p+j)**2/(2*inertia)+np.vdot(velocity, velocity).real+energy close(complete_h, inertia*speed**2/2+np.vdot(chidot, chidot).real+ np.vdot(chi, potential@chi).real, 'convective source momentum shift retains positive complete Hamiltonian') close(potential@b, b, 'inverse bright frequency square remains one in the full convective action') def main(): assignments, scans = [], [] for name, depth, index in ASSIGNMENTS: cutoff, count = index-1, comb(index-1+depth, depth) rotations, weights, b, potential = data(depth, cutoff) threshold = energy_threshold(rotations, weights, cutoff) slow_checks(depth, cutoff, rotations, weights, potential) for ratio in (.1, 1., 10.): speed = ratio/count matrix = frequency_matrix(rotations, potential, speed) values = eigvals(matrix) growth = float(np.max(values.imag)) if cutoff == 0 or ratio*cutoff/count < threshold: check(growth < 2e-8, 'maintained state is spectrally stable whenever rotating energy is positive') record = {'name': name, 'depth': depth, 'mode_index': index, 'cutoff': cutoff, 'simplex_count': count, 'source_omega_over_sector_mu': ratio, 'source_omega_over_target_lambda': speed, 'rotation_g': speed*cutoff, 'energy_threshold_g': threshold, 'amplitude_growth_over_lambda': growth, 'amplitude_efolding_in_target_clock_cycles': 1/(2*pi*growth) if growth > 2e-8 else None, 'resolved_growing_mode': growth > 2e-8, 'oscillator_frequency_count': len(values)} if speed*cutoff < .25: z, vector, charge, j, energy = bright_state(rotations, weights, potential, speed) record.update(rotating_bright_frequency_over_lambda=z, field_energy_per_signed_charge_over_lambda=energy/charge, field_rotation_charge_per_signed_charge=j/charge, complete_energy_per_unit_signed_charge_over_lambda=energy/charge+ratio*ratio/(2*count)) record.update(augmented_energy_checks(rotations, weights, potential, speed, count)) assignments.append(record) for g in GRID: speed = g/max(1, cutoff) values = eigvals(frequency_matrix(rotations, potential, speed)) growth = float(np.max(values.imag)) if not cutoff or g < threshold: check(growth < 2e-8, 'all sampled positive-energy spectra retain both real frequency branches') scans.append({'name': name, 'depth': depth, 'cutoff': cutoff, 'rotation_g': g, 'source_omega_over_target_lambda': speed, 'amplitude_growth_over_lambda': growth, 'amplitude_efolding_in_target_clock_cycles': 1/(2*pi*growth) if growth > 2e-8 else None}) other = data(depth, cutoff, -1)[3] reversal = np.eye(len(b))[::-1] transform = np.block([[reversal, np.zeros_like(reversal)], [np.zeros_like(reversal), -reversal]]) plus = frequency_matrix(rotations, potential, .37/max(1, cutoff)) minus = frequency_matrix(rotations, other, .37/max(1, cutoff)) close(transform@plus@transform, -minus, 'complete opposite-polarization frequency spectra are exact mirrors') for depth in DEPTHS: rotations, weights, _, _ = data(depth, 1) threshold = energy_threshold(rotations, weights, 1) close(threshold, 2*sqrt((depth+1)/(4*depth+1)), 'first-cutoff positive energy threshold has the exact common formula') onset = onset_checks() growth = growth_checks() reciprocal = reciprocal_checks() convective_checks() report = {'experiment_key': 'actual-spatial-second-order-moving-contact', 'created_utc': datetime.now(timezone.utc).isoformat(), 'sources': {'experiments/second-order-contact-checks.py': sha256(Path(__file__).read_bytes()).hexdigest(), 'experiments/moving-contact-checks.py': sha256(HELPER.read_bytes()).hexdigest()}, 'classification': 'conditional full finite-inertia count-projector and collective-source comparison', 'assumptions': ['same positive sphere-band spatial evaluation kernel and actual rotation weights', 'inverse restoring square lambda^2(4I-3P), lambda=mu_d S_N is stipulated', 'positive complex second-order canonical kinetic action with both oscillator branches', 'bare spatial Dirac dispersion is not included in this isolated inverse-restoring control', 'source path is a prescribed great circle or a new positive reciprocal collective rotor', 'comparison source inertia I mu_d=1, never inferred as observed particle inertia'], 'limits': ['finite scans are not an exhaustive stability chart', 'a growing internal mode is not an established outgoing decay channel or observed lifetime', 'electron persistence is assessed separately; short-lived particles need not be infinitely stable', 'no local master source, selected indices, full spatial support or relativistic particle is derived', 'no special depth-five selection or new favorable sector-assignment evidence'], 'condition_count': len(CONDITIONS), 'evaluation_count': sum(CONDITIONS.values()), 'conditions': CONDITIONS, 'maximum_errors': ERRORS, 'assignments': assignments, 'rotation_scans': scans, 'low_cutoff_instability_onsets': onset, 'direct_growth': growth, 'reciprocal_source': reciprocal} destination = ROOT/('second-order-contact-results-'+datetime.now(timezone.utc).strftime('%Y%m%dT%H%M%S.%fZ')+'.json') destination.write_text(json.dumps(report, indent=2)+'\n') print(json.dumps({'report': str(destination), 'conditions': report['condition_count'], 'evaluations': report['evaluation_count'], 'assignment_cases': len(assignments), 'scan_cases': len(scans), 'maximum_errors': ERRORS}, indent=2), flush=True) if __name__ == '__main__': main()