""" Demo: KK Tower Hamiltonian -- Genuine 11D Navigation Uses a physically motivated Kaluza-Klein tower Hamiltonian with 64-level Hilbert space to demonstrate genuine per-dimension navigation. Unlike the toy 4-level system, this Hamiltonian encodes actual KK mass physics: E_d(n_d) = n_d^2 / (2 R_d^2) Key improvements over the toy system: - 64 eigenvalues populate multiple spectral bands -> genuine curvature - Navigation coordinates are quantum observables: Tr(P_d @ rho) * R_d - Energy bands derived from physical KK mass thresholds - Coupling between modes creates non-trivial spectral structure Answers the same 5 core questions as demo_11d_navigation.py, now with physical KK structure backing every calculation. """ import sys import os sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..')) import numpy as np from fce.kk_hamiltonian import KKTowerBuilder from fce.engine import EngineConfig from fce.navigation import ( NavigationEngine, NavigationConfig, SpectralDimensionMapper, DimensionalScaling, build_dimension_info, ) from fce.multi_clock import MultiClockAnalyzer, MultiClockConfig from fce.frenet_serret import QuantumFrenetSerret def main(): print("=" * 70) print(" Fractal Correction Engine -- KK Tower 11D Navigation Demo") print("=" * 70) # ---- Build the KK Hamiltonian ---- print("\n Building KK tower Hamiltonian (dim_q=64, 7 compact dims)...") builder = KKTowerBuilder(dim_q=64, coupling=0.02) H = builder.build_hamiltonian() dim_q = H.shape[0] # Print spectrum summary summary = builder.spectrum_summary() print(f"\n Hilbert space dimension: {summary['dim_q']}") print(f" Energy range: [{summary['E_min']:.4f}, {summary['E_max']:.4f}]") print(f" Off-diagonal elements: {summary['n_off_diagonal']}") print(f"\n Compactification radii and first-mode energies:") print(f" {'Dim':>4s} {'Radius (nat.)':>14s} {'E1 (1st mode)':>14s}") print(" " + "-" * 36) for d in range(7): print(f" {d+4:>4d} {builder.radii[d]:>14.6f} " f"{builder.first_mode_energies[d]:>14.4f}") print(f"\n Band populations (eigenvalues per band):") print(f" {'Band':>5s} {'Type':>10s} {'Count':>6s}") print(" " + "-" * 25) band_labels = ['time', 'x', 'y', 'z'] + [f'compact_{d}' for d in range(4, 11)] for b, count in enumerate(summary['band_populations']): btype = 'spacetime' if b < 4 else 'compact' print(f" {b:>5d} {btype:>10s} {count:>6d}") # ---- Initial state ---- print("\n Initial state: superposition of 8 lowest eigenstates") rho_init = builder.build_initial_state('superposition') purity = float(np.real(np.trace(rho_init @ rho_init))) E_init = float(np.real(np.trace(H @ rho_init))) print(f" Purity: {purity:.6f}") print(f" Energy: {E_init:.4f}") # Initial KK momenta P_ops = builder.build_momentum_operators() print(f"\n Initial KK momentum observables:") print(f" {'Dim':>4s} {'':>12s} {'coord (R*

)':>14s}") print(" " + "-" * 34) for d in range(7): p_exp = float(np.real(np.trace(P_ops[d] @ rho_init))) coord = p_exp * builder.radii[d] print(f" {d+4:>4d} {p_exp:>12.6f} {coord:>14.6e}") # ---- System parameters ---- system_params = {'T1': 1e-3, 'T2': 5e-4, 'gate_fidelity': 0.999} engine_config = EngineConfig( feedback_gain=0.1, correction_threshold=0.001, qec_enabled=True, enable_thermal_monitoring=False, ) nav_config = NavigationConfig( kk_activation_sharpness=10.0, enable_boundary_effects=True, navigation_temperature=1.0, ) dt = 1e-6 n_steps = 100 # ================================================================ # QUESTION 1: KK Activation Thresholds # ================================================================ print("\n" + "=" * 70) print(" Q1: KK Activation Thresholds (from physical radii)") print("=" * 70) scaler = DimensionalScaling() dim_info = build_dimension_info() mapper = SpectralDimensionMapper(H, scaler, dim_info, kk_builder=builder) print(f"\n System eigenvalue range: [{mapper.E_min:.4f}, {mapper.E_max:.4f}]") print() print(f" {'Dim':>4s} {'Name':>12s} {'Type':>10s} " f"{'Threshold':>12s} {'Active?':>8s}") print(" " + "-" * 52) for d in range(11): di = dim_info[d] thresh = mapper.activation_energies[d] act = mapper.activation_function(E_init, d) active_str = "YES" if act > 0.5 else f"{act:.3f}" thresh_str = f"{thresh:.4f}" if np.isfinite(thresh) else "-inf" print(f" {d:>4d} {di.name:>12s} {di.dim_type:>10s} " f"{thresh_str:>12s} {active_str:>8s}") # ================================================================ # QUESTION 2: Per-Dimension Curvature # ================================================================ print("\n" + "=" * 70) print(" Q2: Per-Dimension Curvature (Spectral Decomposition)") print("=" * 70) fs = QuantumFrenetSerret() dim_curv = mapper.compute_dimensional_curvature( rho_init, fs, E_init, sharpness=nav_config.kk_activation_sharpness ) total_kappa = fs.compute_curvature(rho_init, H) total_tau = fs.compute_torsion(rho_init, H) print(f"\n Total scalar curvature: kappa = {total_kappa:.6f}") print(f" Total scalar torsion: tau = {total_tau:.6f}") print() genuine_count = 0 print(f" {'Dim':>4s} {'kappa_d':>10s} {'tau_d':>10s} " f"{'activation':>12s} {'weight':>10s} {'source':>14s}") print(" " + "-" * 68) for d in range(11): k = dim_curv.curvatures[d] t = dim_curv.torsions[d] a = dim_curv.activation[d] w = dim_curv.subspace_weights[d] source = "spectral" if w > 1e-12 else "extrapolated" if source == "spectral": genuine_count += 1 print(f" {d:>4d} {k:>10.6f} {t:>10.6f} " f"{a:>12.6f} {w:>10.6f} {source:>14s}") print(f"\n Genuine (spectral) curvature in {genuine_count}/11 dimensions") # ================================================================ # QUESTION 3: Multi-Clock Dimensional Resolution # ================================================================ print("\n" + "=" * 70) print(" Q3: Multi-Clock KK Probing (64-dim projectors)") print("=" * 70) mc_config = MultiClockConfig(n_clocks=5, offset_multiplier=1.0) mc = MultiClockAnalyzer( hamiltonian=H, system_params=system_params, config=mc_config ) kk_result = mc.evolve_kk_probing_clocks( rho_init, dt, min(n_steps, 50), mapper.projectors ) print(f"\n Occupied energy bands: {kk_result['occupied_bands']}") print() print(f" {'Band':>5s} {'Divergence':>12s} {'Sensitivity':>12s} {'Status':>10s}") print(" " + "-" * 45) for b in range(min(11, len(kk_result['divergence_per_band']))): div = kk_result['divergence_per_band'][b] sens = kk_result['sensitivity_per_dim'][b] status = "ACTIVE" if b in kk_result['occupied_bands'] else "empty" print(f" {b:>5d} {div:>12.6f} {sens:>12.6f} {status:>10s}") n_resolved = len(kk_result['occupied_bands']) print(f"\n Dimensions resolved by multi-clock: {n_resolved}/11") # ================================================================ # QUESTION 4: QFT Boundary Effects # ================================================================ print("\n" + "=" * 70) print(" Q4: QFT Boundary Effects") print("=" * 70) # Run navigation print("\n Running 11D navigation (100 steps, dim_q=64)...") nav = NavigationEngine( hamiltonian=H, system_params=system_params, engine_config=engine_config, nav_config=nav_config, kk_builder=builder, ) nav_result = nav.navigate(rho_init, dt, n_steps) print(f" Navigation complete. Fidelity = {nav_result.total_fidelity:.6f}") print() if nav_result.qft_boundary_effects: print(f" {'Dim':>4s} {'Casimir (J)':>14s} {'Hawking T':>14s} " f"{'Radius (m)':>14s} {'KK mass (GeV)':>14s}") print(" " + "-" * 66) for d, effects in sorted(nav_result.qft_boundary_effects.items()): print(f" {d:>4d} {effects['casimir_energy']:>14.4e} " f"{effects['hawking_temperature']:>14.4e} " f"{effects['compactification_radius']:>14.4e} " f"{effects['kk_mass_GeV']:>14.4e}") else: print(" No boundary transitions. Static boundary analysis:") print() print(f" {'Dim':>4s} {'Casimir (J)':>14s} {'Hawking T':>14s} " f"{'Radius (m)':>14s} {'KK mass (GeV)':>14s}") print(" " + "-" * 66) for d in range(4, 11): effects = nav._qft_boundary_analysis(d) print(f" {d:>4d} {effects['casimir_energy']:>14.4e} " f"{effects['hawking_temperature']:>14.4e} " f"{effects['compactification_radius']:>14.4e} " f"{effects['kk_mass_GeV']:>14.4e}") # ================================================================ # QUESTION 5: Thermodynamic Cost # ================================================================ print("\n" + "=" * 70) print(" Q5: Thermodynamic Navigation Cost") print("=" * 70) cost = nav_result.navigation_cost if cost is not None: print(f"\n Free energy change (Jarzynski): {cost.free_energy_change:.6e}") print(f" Mean work: {np.mean(cost.work_values):.6e}") print(f" Dissipated work: {cost.dissipated_work:.6e}") print(f" Entropy production: {cost.entropy_production:.6e}") print(f" Landauer cost: {cost.landauer_cost:.6e}") print("\n Physics consistency:") print(f" Entropy production >= 0: " f"{'PASS' if cost.entropy_production >= -1e-15 else 'FAIL'}") if cost.crooks_verification is not None: print(f"\n Crooks fluctuation theorem:") for k, v in cost.crooks_verification.items(): print(f" {k}: {v:.6e}") # ================================================================ # Observable coordinate dynamics # ================================================================ print("\n" + "=" * 70) print(" Observable Coordinate Dynamics (Tr(P_d @ rho) * R_d)") print("=" * 70) # Show coordinates at 5 time snapshots snapshots = [0, n_steps // 4, n_steps // 2, 3 * n_steps // 4, n_steps] print() header = f" {'Dim':>4s}" for s in snapshots: header += f" {'t=' + str(s):>14s}" print(header) print(" " + "-" * (4 + 16 * len(snapshots))) for d in range(11): row = f" {d:>4d}" for s in snapshots: c = nav_result.states[s].coordinates[d] row += f" {c:>14.6e}" print(row) # Count dimensions with actual coordinate evolution initial_c = nav_result.states[0].coordinates final_c = nav_result.states[-1].coordinates active_dims = sum( 1 for d in range(11) if abs(final_c[d] - initial_c[d]) > 1e-15 ) # ================================================================ # Holographic & Chaos # ================================================================ print("\n" + "=" * 70) print(" Holographic Metrics") print("=" * 70) holo = nav_result.holographic_metrics if holo: print(f"\n BH entropy: {holo.get('bh_entropy', 0):.6e}") print(f" Scrambling time: {holo.get('scrambling_time', 0):.6e}") print(f" RT entanglement: {holo.get('rt_entanglement_entropy', 0):.6e}") print("\n" + "=" * 70) print(" Chaos Analysis") print("=" * 70) chaos = nav_result.chaos_metrics if chaos: print(f"\n Max Lyapunov: {chaos.get('lyapunov_max', 0):.6e}") print(f" Is chaotic: {chaos.get('is_chaotic', False)}") print(f" MSS bound satisfied: {chaos.get('mss_bound_satisfied', True)}") # ================================================================ # Summary # ================================================================ print("\n" + "=" * 70) print(" Navigation Summary") print("=" * 70) print(f"\n Hilbert space: {dim_q} levels (vs 4 in toy system)") print(f" FCE fidelity: {nav_result.total_fidelity:.6f}") print(f" Active dimensions: {nav_result.transition_state.active_dimensions}") print(f" Dims with coord evolution: {active_dims}/11") print(f" Genuine spectral curvature: {genuine_count}/11 dimensions") print(f" Hausdorff dimension: " f"{nav_result.evolution_result.hausdorff_dimension:.4f}") print(f" Trajectory health: " f"{nav_result.evolution_result.trajectory_health}") # KK coordinate comparison: initial vs final print(f"\n KK momentum dynamics (compact dims):") print(f" {'Dim':>4s} {'Initial

':>14s} {'Final

':>14s} " f"{'Delta':>14s} {'Active?':>8s}") print(" " + "-" * 60) for d in range(7): p_init = float(np.real(np.trace( P_ops[d] @ nav_result.states[0].quantum_state ))) p_final = float(np.real(np.trace( P_ops[d] @ nav_result.states[-1].quantum_state ))) delta = p_final - p_init active = "YES" if abs(delta) > 1e-10 else "no" print(f" {d+4:>4d} {p_init:>14.6e} {p_final:>14.6e} " f"{delta:>14.6e} {active:>8s}") print("\n" + "=" * 70) print(" KK Tower Navigation Complete") print(" All 5 core questions answered with physical KK structure.") print("=" * 70) if __name__ == '__main__': main()