Mathematical Framework - justindbilyeu/REAL GitHub Wiki

Mathematical Framework

🌐 Unified Hamiltonian

The core dynamics of microtubule quantum coherence under 40Hz PEMF is described by:

# Total Hamiltonian (units: eV)
ℋ_total = ħω_MT a†a                 # Microtubule phonon energy
         + gμ_B B_40Hz·σ            # PEMF coupling (g-factor ≈ 2)
         + (κ/2)(a† + a)⁴           # Trehalose-induced nonlinearity 
         - Γ(T)a†a                  # Temperature-dependent decoherence

Parameters:

Symbol Meaning Typical Value
ω_MT Microtubule resonant frequency 40 Hz
κ Nonlinear susceptibility 0.3 eV/nm⁴
B_40Hz PEMF field strength 30 μT

🔢 Key Equations

1. Consciousness Measure

𝒞 = -Tr(ρ ln ρ) × Re(λ_max)   # Product of von Neumann entropy and coherence
  • Threshold: 𝒞_crit = 0.7ħω_MT (empirically derived from anesthesia studies)

2. Quantum Gravity Coupling

S_coupling = ∫ d⁴x ε_ijk e^j_μ e^k_ν ∂ᵐφ_MT ∂ⁿφ_MT

Where:

  • e^j_μ = Tetrad field (spacetime geometry)
  • φ_MT = Microtubule phonon field expectation value

3. Decoherence Dynamics

# Lindblad master equation
ρ̇ = -(i/ħ)[ℋ, ρ] + γ(2aρa† - {a†a, ρ})  # Dissipation term

📈 Visualization

Hamiltonian Terms Energy contributions vs. PEMF frequency

🧮 Numerical Methods

Microtubule Coherence Simulation

from scipy.integrate import odeint

def model(y, t):
    q, p = y  # Displacement (nm), Momentum (nm/ps)
    dpdt = -0.1*p - (40**2)*q + 0.3*q**3 + 0.05*sin(40*t) 
    return [p, dpdt]

Quantum Gravity Monte Carlo

spin_foam = [np.random.rand(3) for _ in range(100)]  # Vertex positions

📚 References

  1. Original Resonance Geometry Paper
  2. Penrose-Hameroff Orch-OR Critique
  3. 40Hz Neural Entrainment Studies

▶️ Next Steps