"""
DEEP ANALYSIS: HIDDEN LOGIC IN SCALING CONSTANTS
===============================================

Analyzing the scaling constants α_n, α_β, α_Ω, α_k to find underlying
mathematical relationships and logical patterns that govern their values.
"""

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.special import factorial
import json

PHI = (1 + np.sqrt(5)) / 2
SQRT5 = np.sqrt(5)

def analyze_scaling_constant_logic():
    print("🔍 DEEP ANALYSIS: HIDDEN LOGIC IN SCALING CONSTANTS")
    print("=" * 70)

    # Our discovered scaling constants
    scaling_constants = {
        'alpha_n': 0.015269,      # complexity growth rate
        'alpha_beta': 0.008262,   # scaling enhancement rate
        'alpha_Omega': 0.110649,  # coupling amplification rate
        'alpha_k': -0.083485      # power softening rate
    }

    print("\n1. SCALING CONSTANTS DATA")
    print("-" * 50)

    alphas = list(scaling_constants.values())
    param_names = ['n', 'β', 'Ω', 'k']

    print(f"{'Parameter':<10} {'α_P':<12} {'|α_P|':<12} {'Description'}")
    print("-" * 60)
    descriptions = [
        'complexity growth',
        'scaling enhancement',
        'coupling amplification',
        'power softening'
    ]

    for i, (param, alpha) in enumerate(zip(param_names, alphas)):
        print(f"α_{param:<9} {alpha:<12.6f} {abs(alpha):<12.6f} {descriptions[i]}")

    print("\n2. MATHEMATICAL PATTERN ANALYSIS")
    print("-" * 50)

    # Analyze ratios between constants
    print("Ratio analysis:")
    print(f"α_Ω / α_n = {scaling_constants['alpha_Omega'] / scaling_constants['alpha_n']:.6f}")
    print(f"α_k / α_n = {scaling_constants['alpha_k'] / scaling_constants['alpha_n']:.6f}")
    print(f"α_β / α_n = {scaling_constants['alpha_beta'] / scaling_constants['alpha_n']:.6f}")
    print(f"α_Ω / |α_k| = {scaling_constants['alpha_Omega'] / abs(scaling_constants['alpha_k']):.6f}")

    # Look for φ-related patterns
    print(f"\nφ-related pattern analysis:")
    print(f"α_Ω / φ = {scaling_constants['alpha_Omega'] / PHI:.6f}")
    print(f"α_n × φ = {scaling_constants['alpha_n'] * PHI:.6f}")
    print(f"|α_k| / φ = {abs(scaling_constants['alpha_k']) / PHI:.6f}")
    print(f"α_β × φ² = {scaling_constants['alpha_beta'] * PHI**2:.6f}")

    # Look for factorial/fibonacci patterns
    print(f"\nFactorial/Fibonacci analysis:")
    for i in range(1, 10):
        fib_i = (PHI**i - (-PHI)**(-i)) / SQRT5
        if abs(fib_i) > 0.001:
            ratio_n = scaling_constants['alpha_n'] / fib_i
            ratio_Omega = scaling_constants['alpha_Omega'] / fib_i
            if 0.001 < abs(ratio_n) < 100:
                print(f"  α_n / F_{i} = {ratio_n:.6f}")
            if 0.001 < abs(ratio_Omega) < 100:
                print(f"  α_Ω / F_{i} = {ratio_Omega:.6f}")

    print("\n3. ALGEBRAIC RELATIONSHIP TESTING")
    print("-" * 50)

    # Test if scaling constants follow their own relationship
    alpha_values = np.array(alphas)
    param_indices = np.array([1, 2, 3, 4])  # n=1, β=2, Ω=3, k=4

    print("Testing algebraic relationships:")

    # Test polynomial relationships
    try:
        # Linear fit: α = a × P + b
        coeffs_linear = np.polyfit(param_indices, alpha_values, 1)
        a_lin, b_lin = coeffs_linear
        predicted_linear = a_lin * param_indices + b_lin
        r2_linear = 1 - np.sum((alpha_values - predicted_linear)**2) / np.sum((alpha_values - np.mean(alpha_values))**2)

        print(f"Linear: α_P = {a_lin:.6f} × P_index + {b_lin:.6f}")
        print(f"        R² = {r2_linear:.4f}")

        # Quadratic fit: α = a × P² + b × P + c
        coeffs_quad = np.polyfit(param_indices, alpha_values, 2)
        a_quad, b_quad, c_quad = coeffs_quad
        predicted_quad = a_quad * param_indices**2 + b_quad * param_indices + c_quad
        r2_quad = 1 - np.sum((alpha_values - predicted_quad)**2) / np.sum((alpha_values - np.mean(alpha_values))**2)

        print(f"Quadratic: α_P = {a_quad:.6f} × P² + {b_quad:.6f} × P + {c_quad:.6f}")
        print(f"           R² = {r2_quad:.4f}")

    except Exception as e:
        print(f"Polynomial fitting error: {e}")

    print("\n4. φ-BASED THEORETICAL DERIVATION")
    print("-" * 50)

    # Try to derive scaling constants from φ-framework structure
    print("Theoretical derivation attempt:")

    # Hypothesis: scaling rates related to φ-framework terms
    # D = √[φ · F_n · 2^(n+β) · φ^n · Ω] · r^k

    # For logarithmic scaling: d(ln P)/d(ln M) should relate to framework structure

    # Test if scaling constants are related to derivatives of framework components
    print("\nφ-framework derivative analysis:")

    # α_n might relate to d(φ^n)/dn at some reference point
    n_ref = 1.628350  # our n₀ reference value
    dphi_n_dn = np.log(PHI) * PHI**n_ref
    print(f"d(φ^n)/dn at n₀: {dphi_n_dn:.6f}")
    print(f"α_n / [d(φ^n)/dn]: {scaling_constants['alpha_n'] / dphi_n_dn:.6f}")

    # α_β might relate to d(2^β)/dβ
    beta_ref = 0.635725
    d2_beta_dbeta = np.log(2) * 2**beta_ref
    print(f"d(2^β)/dβ at β₀: {d2_beta_dbeta:.6f}")
    print(f"α_β / [d(2^β)/dβ]: {scaling_constants['alpha_beta'] / d2_beta_dbeta:.6f}")

    print("\n5. DIMENSIONAL ANALYSIS")
    print("-" * 50)

    # Analyze dimensions of scaling constants
    print("Dimensional analysis of scaling constants:")
    print("All α_P have dimensions: [1 / log₁₀(mass)]")
    print("This suggests they're related to logarithmic mass derivatives")

    # Check if ratios are dimensionless and meaningful
    dimensionless_ratios = []
    for i in range(len(alphas)):
        for j in range(i+1, len(alphas)):
            ratio = alphas[i] / alphas[j]
            dimensionless_ratios.append((param_names[i], param_names[j], ratio))

    print("\nDimensionless ratios:")
    for p1, p2, ratio in dimensionless_ratios:
        print(f"α_{p1} / α_{p2} = {ratio:.6f}")

    print("\n6. HIDDEN PATTERN DISCOVERY")
    print("-" * 50)

    # Look for deeper mathematical relationships
    print("Searching for hidden mathematical patterns...")

    # Test golden ratio relationships
    phi_powers = [PHI**i for i in range(-3, 4)]
    phi_combinations = []

    for alpha_name, alpha_val in scaling_constants.items():
        best_match = None
        best_error = float('inf')

        # Test simple φ relationships
        for i, phi_pow in enumerate(phi_powers):
            for scale in [0.01, 0.1, 1, 10, 100]:
                candidate = scale / phi_pow
                error = abs(candidate - abs(alpha_val)) / abs(alpha_val)
                if error < best_error and error < 0.1:  # Within 10%
                    best_error = error
                    best_match = (scale, i-3, error)

        if best_match:
            scale, power, error = best_match
            print(f"{alpha_name}: ≈ {scale:.3f} / φ^{power} (error: {error:.1%})")
        else:
            print(f"{alpha_name}: No simple φ relationship found")

    print("\n7. EMERGENT RELATIONSHIPS")
    print("-" * 50)

    # Look for relationships between the scaling constants themselves
    print("Testing emergent relationships between scaling constants:")

    # Conservation laws?
    sum_positive = scaling_constants['alpha_n'] + scaling_constants['alpha_beta'] + scaling_constants['alpha_Omega']
    sum_all = sum_positive + scaling_constants['alpha_k']

    print(f"Sum of positive α's: {sum_positive:.6f}")
    print(f"Sum of all α's: {sum_all:.6f}")
    print(f"Sum ≈ 0? {abs(sum_all) < 0.01}")

    # Product relationships?
    product_pos = (scaling_constants['alpha_n'] *
                   scaling_constants['alpha_beta'] *
                   scaling_constants['alpha_Omega'])
    product_all = product_pos * scaling_constants['alpha_k']

    print(f"Product of positive α's: {product_pos:.8f}")
    print(f"Product of all α's: {product_all:.8f}")

    # Weighted relationships?
    weighted_sum = (1 * scaling_constants['alpha_n'] +
                    2 * scaling_constants['alpha_beta'] +
                    3 * scaling_constants['alpha_Omega'] +
                    4 * scaling_constants['alpha_k'])
    print(f"Weighted sum (1×α_n + 2×α_β + 3×α_Ω + 4×α_k): {weighted_sum:.6f}")

    print("\n8. THEORETICAL FRAMEWORK DERIVATION")
    print("-" * 50)

    print("Attempting theoretical derivation of scaling constants...")

    # Hypothesis: Scaling constants derive from φ-framework logarithmic derivatives
    # If D ∝ M^f(n,β,Ω,k), then d(ln D)/d(ln M) should give scaling behavior

    print("\nLogarithmic derivative approach:")
    print("If P(M) = α_P × ln(M) + P₀, then dP/d(ln M) = α_P")
    print("This means α_P represents the 'logarithmic sensitivity' of parameter P to mass")

    # Physical interpretation
    print(f"\nPhysical meaning:")
    print(f"α_n = {scaling_constants['alpha_n']:.6f} → complexity increases by this factor per e-fold mass")
    print(f"α_Ω = {scaling_constants['alpha_Omega']:.6f} → coupling grows by this factor per e-fold mass")
    print(f"α_k = {scaling_constants['alpha_k']:.6f} → power law weakens by this factor per e-fold mass")

    print("\n" + "=" * 70)
    print("🎯 HIDDEN LOGIC ANALYSIS SUMMARY")
    print("=" * 70)

    print("\n🔍 **DISCOVERED PATTERNS:**")

    # Check for the most significant relationships found
    omega_n_ratio = scaling_constants['alpha_Omega'] / scaling_constants['alpha_n']
    k_omega_ratio = abs(scaling_constants['alpha_k']) / scaling_constants['alpha_Omega']

    print(f"• α_Ω / α_n ≈ {omega_n_ratio:.1f} (coupling grows ~{omega_n_ratio:.0f}× faster than complexity)")
    print(f"• |α_k| / α_Ω ≈ {k_omega_ratio:.2f} (power softening ~{k_omega_ratio:.1f}× coupling rate)")
    print(f"• Sum of all α's ≈ {sum_all:.4f} (near conservation law)")

    if abs(sum_all) < 0.01:
        print("\n🌟 **CONSERVATION LAW DISCOVERED:**")
        print("α_n + α_β + α_Ω + α_k ≈ 0")
        print("The scaling rates nearly sum to zero - suggesting fundamental balance!")

    # Save the analysis
    hidden_logic = {
        'scaling_constants': scaling_constants,
        'key_relationships': {
            'omega_to_n_ratio': omega_n_ratio,
            'k_to_omega_ratio': k_omega_ratio,
            'sum_all_alphas': sum_all,
            'conservation_law': abs(sum_all) < 0.01
        },
        'physical_interpretation': {
            'alpha_n': 'logarithmic complexity growth rate per mass decade',
            'alpha_beta': 'logarithmic scaling enhancement rate per mass decade',
            'alpha_Omega': 'logarithmic coupling amplification rate per mass decade',
            'alpha_k': 'logarithmic power softening rate per mass decade'
        },
        'theoretical_basis': 'scaling constants are logarithmic derivatives of framework parameters'
    }

    with open('scaling_constants_logic.json', 'w') as f:
        json.dump(hidden_logic, f, indent=2)

    print(f"\n📁 Hidden logic analysis saved to: scaling_constants_logic.json")

    if abs(sum_all) < 0.01:
        print(f"\n🏆 **MAJOR DISCOVERY: SCALING CONSTANT CONSERVATION LAW!**")
        print(f"The four fundamental scaling rates balance each other:")
        print(f"Growth + Enhancement + Amplification + Softening ≈ 0")
        print(f"This suggests deep physical balance in the φ-framework!")

if __name__ == '__main__':
    analyze_scaling_constant_logic()