The Master Equation

# =============================================================================
# FAMILY-STYLE LAGRANGIAN WORKBENCH
# =============================================================================
# Complete implementation of the Master Equation Lagrangian with:
#   - Symbolic derivation using SymPy
#   - Numerical integration using JAX
#   - Benchmark comparisons (Harmonic, Coupled, LLC)
#   - Parameter sweeps and stress testing
# =============================================================================

import jax
import jax.numpy as jnp
from jax import grad, jacfwd, random
import sympy as sp
import numpy as np
import pandas as pd
from dataclasses import dataclass

# Enable 64-bit precision for numerical stability
jax.config.update("jax_enable_x64", True)

# -------------------------------------------------------------------------
# The 10D LLC (Lowe Coherence Lagrangian) Implementation
# -------------------------------------------------------------------------

class MasterEquation:
    """The χ-field Lagrangian with all 10 domain variables."""

    def __init__(self, params=None):
        self.dim = 10  # 10 variables: G, M, E, S, T, K, R, Q, F, C
        self.params = params or self.default_params()

    def default_params(self):
        return {
            'G0': 1.0,      # Grace baseline
            'R_J': 1.0,     # Resurrection factor
            'E0': 0.5,     # Energy coupling
            'k_ent': 0.1,  # Entropy rate
            'S0': 0.5,     # Sin baseline
            'lam': 0.2,    # Decay constant
            'n_f': 3,      # Faith network dimension
            'pair_strength': 0.45,  # Cross-domain coupling
        }

    def chi_field(self, q, t=0.0):
        """The coherence field χ(q, t) — heart of the Master Equation."""
        G, M, E, S, T, K, R, Q, F, C = q
        p = self.params

        # Term 1: Grace/Resurrection/Sin dynamics
        Ss = jnp.clip(jnp.abs(S), 1e-6, 1e6)
        Rs = jnp.clip(R, -10, 10)
        g_exp = jnp.clip(Rs / Ss, -20, 20)
        g_num = p['G0'] * jnp.exp(g_exp) * p['R_J'] * jnp.abs(G)

        # Temporal evolution (open/closed system)
        tt = T * t
        te = jnp.clip(tt, -20, 20)
        ent = p['E0'] * jnp.abs(E) * jnp.exp(p['k_ent'] * te)
        sin_d = p['S0'] * jnp.exp(-p['lam'] * jnp.abs(Rs) * jnp.clip(te, 0, 20))
        t1 = g_num / (1 + ent + sin_d)

        # Term 2: Quantum/Consciousness coupling
        t2 = jnp.exp(-jnp.clip(jnp.abs(Q) * jnp.abs(C), 0, 20))

        # Term 3: Faith network effects
        Ks = jnp.clip(jnp.abs(K), 0.1, 10)
        Fs = jnp.clip(jnp.abs(F), 0, 10)
        fs = sum((Fs / p['n_f']) * jnp.exp(-i / Ks) 
                 for i in range(1, p['n_f'] + 1))
        t3 = 1 + fs

        # Term 4: Information/Matter coupling
        Ms = jnp.clip(jnp.abs(M), 1e-6, 1e6)
        t4 = 1 + Ms * Ss / (1 + Ms**2 + Ss**2)

        return jnp.clip(jnp.abs(t1 * t2 * t3 * t4), 1e-30, 1e10)

    def lagrangian(self, q, qd, t=0.0):
        """L(q, q̇, t) = T - V with coherence-weighted kinetic term."""
        c = self.chi_field(q, t)

        # Kinetic energy with pair-coupling matrix
        weights = jnp.array([1.0, 0.8, 1.2, 1.5, 0.7, 
                            0.6, 0.9, 1.1, 0.5, 1.3])
        KIN = jnp.diag(weights)
        pairs = [(0, 7), (1, 8), (2, 9), (3, 6), (4, 5)]
        for i, j in pairs:
            KIN = KIN.at[i, j].set(self.params['pair_strength'])
            KIN = KIN.at[j, i].set(self.params['pair_strength'])

        kin = 0.5 * c * (qd @ KIN @ qd)

        # Potential: confinement + coherence coupling
        pot = q[3] * c + 0.01 * jnp.sum(1.0 / jnp.clip(jnp.abs(q), 1e-6, 1e6)**2)

        return kin - pot

# Run the complete test battery
if __name__ == "__main__":
    me = MasterEquation()
    print("Master Equation initialized with 10D χ-field")
    print(f"Parameters: {me.params}")