""" Mobile Microgripper Scientific Shortcut Refinement Uses scipy and numpy for statistical and numerical analysis of mathematical shortcuts: - scipy.stats for 6.5σ confidence intervals - scipy.optimize for parameter optimization - numpy for precise numerical calculations - scipy.integrate for differential equation solving - scipy.linalg for matrix operations Applies 6.5σ bounds only to statistical quantities; numerical optimization and physical design claims still require separate measurement/provenance gates. """ import numpy as np import scipy.stats as stats import scipy.optimize as opt import scipy.integrate as integrate import scipy.linalg as la import math import json from typing import Dict, List, Tuple, Any, Callable # Physical Constants (CODATA 2022) KB = 1.380649e-23 # Boltzmann constant (J/K) H_BAR = 1.054571817e-34 # Reduced Planck constant (J·s) MU_0 = 4 * np.pi * 1e-7 # Vacuum permeability (T·m/A) LAMBDA_TORSION = 1e-9 # Interaction length (m) # Microgripper Parameters ARM_LENGTH_UM = 200.0 ARM_WIDTH_UM = 30.0 ARM_THICKNESS_UM = 15.0 HINGE_RADIUS_UM = 10.0 GAP_OPEN_UM = 100.0 GAP_CLOSED_UM = 20.0 MAGNETIC_COATING_UM = 5.0 SENSOR_THICKNESS_UM = 10.0 # Magnetic Parameters B_BASE = 1.2 # Tesla B_STEER = 0.3 # Tesla MU_PARTICLE = 8.6e-19 # A·m² (Fe₃O₄ nanoferrite) # Temperature Range T_MIN = 50.0 # K (critical temperature from buckyball-MOF spec) T_MAX = 300.0 # K (room temperature) class StatisticalValidation: """6.5σ statistical validation framework.""" @staticmethod def calculate_confidence_interval(data: np.ndarray, confidence: float = 0.99999998) -> Tuple[float, float]: """Calculate confidence interval using scipy.stats.""" mean = np.mean(data) std = np.std(data, ddof=1) n = len(data) # 6.5σ corresponds to ~99.99998% confidence z_score = stats.norm.ppf((1 + confidence) / 2) margin = z_score * (std / np.sqrt(n)) return (mean - margin, mean + margin) @staticmethod def hypothesis_test(sample_mean: float, population_mean: float, std_dev: float, n: int, alpha: float = 1e-6) -> Dict[str, Any]: """Two-tailed hypothesis test with 6.5σ significance.""" z_score = (sample_mean - population_mean) / (std_dev / np.sqrt(n)) p_value = 2 * (1 - stats.norm.cdf(abs(z_score))) # 6.5σ threshold: z ≈ 6.5, p ≈ 1e-10 significant = p_value < alpha return { "z_score": z_score, "p_value": p_value, "significant": significant, "sigma_level": abs(z_score), "meets_6_5_sigma": abs(z_score) >= 6.5 } class RefinedFAMMAnalysis: """Refined FAMM frustration analysis with scipy optimization.""" def __init__(self): self.phi_history = [] def frustration_function(self, temperature: float, field_strength: float) -> float: """Φ(T, B) = (τ_thermal + τ_steric) / τ_magnetic""" # Magnetic torque tau_magnetic = MU_PARTICLE * field_strength # Thermal torque tau_thermal = KB * temperature / LAMBDA_TORSION # Steric torque (from hinge geometry) k_steric = 1e-12 theta_offset = math.radians(14) theta_lattice = math.radians(60) tau_steric = k_steric * (1 - math.cos(theta_offset - theta_lattice)) phi = (tau_thermal + tau_steric) / tau_magnetic return phi def find_optimal_temperature(self, target_phi: float = 1.0, field_strength: float = B_BASE) -> float: """Find temperature that achieves target Φ using scipy.optimize.""" def objective(T): return abs(self.frustration_function(T, field_strength) - target_phi) # Optimize in range [T_MIN, T_MAX] result = opt.minimize_scalar(objective, bounds=(T_MIN, T_MAX), method='bounded') return result.x def find_optimal_field(self, target_phi: float = 1.0, temperature: float = T_MAX) -> float: """Find field strength that achieves target Φ using scipy.optimize.""" def objective(B): return abs(self.frustration_function(temperature, B) - target_phi) # Optimize in range [0.8, 2.0] Tesla (from buckyball-MOF spec) result = opt.minimize_scalar(objective, bounds=(0.8, 2.0), method='bounded') return result.x def monte_carlo_validation(self, n_samples: int = 10000) -> Dict[str, Any]: """Monte Carlo statistical interval for Φ; not physical validation.""" # Sample temperature and field with uncertainty T_samples = np.random.normal(T_MAX, 10.0, n_samples) # ±10K uncertainty B_samples = np.random.normal(B_BASE, 0.1, n_samples) # ±0.1T uncertainty phi_samples = np.array([self.frustration_function(T, B) for T, B in zip(T_samples, B_samples)]) # Calculate 6.5σ confidence interval ci_low, ci_high = StatisticalValidation.calculate_confidence_interval(phi_samples) # Test hypothesis: Φ >> 1 (thermal dominance) test_result = StatisticalValidation.hypothesis_test( np.mean(phi_samples), 1.0, np.std(phi_samples), n_samples ) return { "phi_mean": np.mean(phi_samples), "phi_std": np.std(phi_samples), "phi_ci_6_5_sigma": (ci_low, ci_high), "hypothesis_test": test_result, "samples": n_samples } class RefinedQuaternionOptimization: """Refined quaternion optimization using scipy.linalg.""" def __init__(self): self.quaternions = [] def quaternion_to_rotation_matrix(self, q: Tuple[float, float, float, float]) -> np.ndarray: """Convert quaternion to rotation matrix using scipy.linalg conventions.""" w, x, y, z = q R = np.array([ [1 - 2*(y**2 + z**2), 2*(x*y - w*z), 2*(x*z + w*y)], [2*(x*y + w*z), 1 - 2*(x**2 + z**2), 2*(y*z - w*x)], [2*(x*z - w*y), 2*(y*z + w*x), 1 - 2*(x**2 + y**2)] ]) return R def optimize_counter_rotation(self, theta_initial: float, phi_initial: float) -> Tuple[float, float]: """Optimize angles for zero net angular momentum using scipy.optimize.""" def net_momentum(angles): theta, phi = angles # Quaternion at layer N q_n = (math.cos(theta/2), math.sin(theta/2)*math.cos(phi), math.sin(theta/2)*math.sin(phi), math.sin(theta/2)*math.cos(theta)) # Quaternion at layer N-1 (counter-rotated) theta_minus_1 = -theta q_n_minus_1 = (math.cos(theta_minus_1/2), math.sin(theta_minus_1/2)*math.cos(phi), math.sin(theta_minus_1/2)*math.sin(phi), math.sin(theta_minus_1/2)*math.cos(theta_minus_1)) # Counter-rotation q_n_minus_1_conj = (q_n_minus_1[0], -q_n_minus_1[1], -q_n_minus_1[2], -q_n_minus_1[3]) counter_rot = ( q_n[0]*q_n_minus_1_conj[0] - q_n[1]*q_n_minus_1_conj[1] - q_n[2]*q_n_minus_1_conj[2] - q_n[3]*q_n_minus_1_conj[3], q_n[0]*q_n_minus_1_conj[1] + q_n[1]*q_n_minus_1_conj[0] + q_n[2]*q_n_minus_1_conj[3] - q_n[3]*q_n_minus_1_conj[2], q_n[0]*q_n_minus_1_conj[2] - q_n[1]*q_n_minus_1_conj[3] + q_n[2]*q_n_minus_1_conj[0] + q_n[3]*q_n_minus_1_conj[1], q_n[0]*q_n_minus_1_conj[3] + q_n[1]*q_n_minus_1_conj[2] - q_n[2]*q_n_minus_1_conj[1] + q_n[3]*q_n_minus_1_conj[0] ) # Net angular momentum angle = 2 * math.acos(max(-1, min(1, counter_rot[0]))) return angle**2 # Minimize squared angle # Optimize angles result = opt.minimize(net_momentum, x0=[theta_initial, phi_initial], bounds=[(0, np.pi), (0, 2*np.pi)], method='L-BFGS-B') return tuple(result.x) class RefinedPISTOptimization: """Refined PIST state space optimization with scipy integration.""" def __init__(self): self.state_space = [] def pist_dynamics(self, state: np.ndarray, t: float, phi_threshold: float) -> np.ndarray: """PIST state space dynamics as differential equation.""" k, t_coord, mass = state # State space pruning rate pruning_rate = phi_threshold * np.exp(-phi_threshold * t) return np.array([-pruning_rate * k, -pruning_rate * t_coord, -pruning_rate * mass]) def integrate_state_space(self, initial_state: Tuple[float, float, float], phi_threshold: float, t_max: float = 10.0) -> np.ndarray: """Integrate PIST dynamics using scipy.integrate.odeint.""" sol = integrate.odeint( self.pist_dynamics, initial_state, np.linspace(0, t_max, 100), args=(phi_threshold,) ) return sol[-1] # Final state def optimize_phi_threshold(self, target_reduction: float = 0.5) -> float: """Find optimal Φ threshold for target state space reduction.""" initial_state = (1000, 100, 6000) # Initial (k, t, mass) def objective(phi): final_state = self.integrate_state_space(initial_state, phi) reduction = 1 - (final_state[0] / initial_state[0]) return abs(reduction - target_reduction) result = opt.minimize_scalar(objective, bounds=(0.1, 0.9), method='bounded') return result.x class RefinedStringStarAnalysis: """Refined String-Star curvature analysis with scipy.linalg.""" def __init__(self): self.curvature_data = [] def calculate_curvature_tensor(self, radius: float) -> np.ndarray: """Calculate curvature tensor using differential geometry.""" # For spherical surface: R_ij = (1/R²) * g_ij g_ij = np.eye(3) # Metric tensor (simplified) R_ij = (1.0 / radius**2) * g_ij return R_ij def calculate_ricci_scalar(self, radius: float) -> float: """Calculate Ricci scalar (scalar curvature).""" # For sphere: R = 2/R² return 2.0 / (radius**2) def optimize_hinge_radius(self, target_curvature: float = 100.0) -> float: """Optimize hinge radius for target curvature.""" def objective(r): curvature = self.calculate_ricci_scalar(r) return abs(curvature - target_curvature) # Optimize in range [5, 20] μm result = opt.minimize_scalar(objective, bounds=(5e-6, 20e-6), method='bounded') return result.x def convert_numpy_types(obj): """Convert numpy types to Python native types for JSON serialization.""" if isinstance(obj, np.ndarray): return obj.tolist() elif isinstance(obj, (np.integer, np.int64, np.int32)): return int(obj) elif isinstance(obj, (np.floating, np.float64, np.float32)): return float(obj) elif isinstance(obj, np.bool_): return bool(obj) elif isinstance(obj, dict): return {k: convert_numpy_types(v) for k, v in obj.items()} elif isinstance(obj, list): return [convert_numpy_types(item) for item in obj] elif isinstance(obj, tuple): return tuple(convert_numpy_types(item) for item in obj) else: return obj def run_scientific_refinement() -> Dict[str, Any]: """Run refined scientific analysis using scipy packages.""" print("Running Scientific Refinement with scipy and numpy...") print("=" * 70) results = {} # 1. Refined FAMM Analysis print("\n1. Refined FAMM Frustration Analysis (scipy.optimize)") famm = RefinedFAMMAnalysis() # Find optimal parameters optimal_temp = famm.find_optimal_temperature(target_phi=1.0, field_strength=B_BASE) optimal_field = famm.find_optimal_field(target_phi=1.0, temperature=T_MAX) # Monte Carlo validation mc_results = famm.monte_carlo_validation(n_samples=10000) results["famm_refined"] = { "optimal_temperature_K": float(optimal_temp), "optimal_field_T": float(optimal_field), "monte_carlo_validation": convert_numpy_types(mc_results), "refined_claim": f"Assembly requires T < {optimal_temp:.1f}K or B > {optimal_field:.2f}T", "confidence": "Monte Carlo statistical interval; physical claim still requires measurement provenance" } print(f" Optimal Temperature: {optimal_temp:.2f} K") print(f" Optimal Field: {optimal_field:.3f} T") print(f" Φ Mean: {mc_results['phi_mean']:.2e}") print(f" Φ 6.5σ CI: [{mc_results['phi_ci_6_5_sigma'][0]:.2e}, {mc_results['phi_ci_6_5_sigma'][1]:.2e}]") print(f" Hypothesis Test Z-score: {mc_results['hypothesis_test']['z_score']:.2f}") print(f" Meets 6.5σ: {mc_results['hypothesis_test']['meets_6_5_sigma']}") # 2. Refined Quaternion Optimization print("\n2. Refined Quaternion Optimization (scipy.linalg + scipy.optimize)") quat = RefinedQuaternionOptimization() theta_initial = math.radians(45) phi_initial = math.radians(30) optimal_angles = quat.optimize_counter_rotation(theta_initial, phi_initial) results["quaternion_refined"] = { "optimal_theta_rad": float(optimal_angles[0]), "optimal_phi_rad": float(optimal_angles[1]), "optimal_theta_deg": float(math.degrees(optimal_angles[0])), "optimal_phi_deg": float(math.degrees(optimal_angles[1])), "refined_claim": "Counter-rotation achievable with optimized field angles", "confidence": "Numerical optimization convergence" } print(f" Optimal Theta: {math.degrees(optimal_angles[0]):.2f}°") print(f" Optimal Phi: {math.degrees(optimal_angles[1]):.2f}°") # 3. Refined PIST Optimization print("\n3. Refined PIST State Space Optimization (scipy.integrate)") pist = RefinedPISTOptimization() optimal_phi = pist.optimize_phi_threshold(target_reduction=0.5) results["pist_refined"] = { "optimal_phi_threshold": float(optimal_phi), "refined_claim": f"Φ threshold of {optimal_phi:.3f} achieves 50% state space reduction", "confidence": "Differential equation integration" } print(f" Optimal Φ Threshold: {optimal_phi:.3f}") # 4. Refined String-Star Analysis print("\n4. Refined String-Star Curvature Analysis (scipy.linalg)") ss = RefinedStringStarAnalysis() optimal_radius = ss.optimize_hinge_radius(target_curvature=100.0) results["string_star_refined"] = { "optimal_hinge_radius_m": float(optimal_radius), "optimal_hinge_radius_um": float(optimal_radius * 1e6), "refined_claim": f"Optimal hinge radius: {optimal_radius*1e6:.2f} μm for target curvature", "confidence": "Curvature tensor calculation" } print(f" Optimal Hinge Radius: {optimal_radius*1e6:.2f} μm") # Summary of refined claims print("\n" + "=" * 70) print("REFINED CLAIMS (domain-gated; statistical intervals are not physical validation):") refined_claims = [ f"FAMM: Assembly requires T < {optimal_temp:.1f}K or B > {optimal_field:.2f}T (Monte Carlo interval; needs physical measurement)", f"Quaternion: Counter-rotation achievable at θ={math.degrees(optimal_angles[0]):.1f}°, φ={math.degrees(optimal_angles[1]):.1f}° (numerical optimization)", f"PIST: Φ threshold {optimal_phi:.3f} achieves 50% reduction (ODE integration)", f"String-Star: Optimal hinge radius {optimal_radius*1e6:.2f} μm (curvature tensor)" ] for i, claim in enumerate(refined_claims, 1): print(f" {i}. {claim}") results["refined_claims"] = refined_claims results["summary"] = { "total_refined_claims": len(refined_claims), "validation_method": "scipy + numpy numerical analysis with statistical interval where applicable", "primary_refinement": "FAMM assembly conditions quantified with Monte Carlo", "secondary_refinement": "Quaternion angles optimized via L-BFGS-B", "tertiary_refinement": "PIST dynamics integrated via ODE solver" } return results if __name__ == "__main__": results = run_scientific_refinement() # Save results output_file = "/home/allaun/Documents/Research Stack/5-Applications/text-to-cad/models/mobile_microgripper_sci_refined.json" with open(output_file, 'w') as f: json.dump(convert_numpy_types(results), f, indent=2) print(f"\nResults saved to: {output_file}") print("\nScientific refinement complete!")