Modulated Gravity (KMS-PMG): Rigorous Mathematical Proof and Verification Code

Modulated Gravity (KMS-PMG): Rigorous Mathematical Proof and Verification Code import numpy as np import matplotlib.pyplot as plt from scipy.integrate import solve_ivp # === CONSTANTS OF THE NON-SINGULAR KMS-PMG SYSTEM === M = 1.0 # Mass parameter L = 1.0 # Photon angular momentum alpha_scale = 0.2 # LOCK scale parameter theta_0 = 2.0944 # Boundary phase at horizon (2*pi/3 rad) r_ECA = 0.05 * M # Transition boundary scale Lambda_ECA = 3.0 / (r_ECA**2) # Effective cosmological constant density def A_smooth(r): """ Analytically smoothed g_00 metric coefficient. Transitions from Schwarzschild (r >> r_ECA) to de Sitter (r -> 0) dynamically. """ # Classical Schwarzschild term regularized at the center schwarzschild_part = (2.0 * M * r**2) / (r**3 + 2.0 * M * r_ECA**2) # de Sitter core contribution active only near the center de_sitter_part = (Lambda_ECA / 3.0) * r**2 * np.exp(-(r / r_ECA)**2) return 1.0 - schwarzschild_part - de_sitter_part def V_R_smooth(r): """ Computes the continuous real effective potential without a hard cutoff. """ A = A_smooth(r) # Smoothly evolving phase term mapping the internal vacuum modulation phase = theta_0 * np.exp(-(r - 2.0 * M) / alpha_scale) if r > 0 else theta_0 # Effective potential based on smooth metric and phase potential = A * (np.cos(phase) / (4.0 * (np.cos(phase / 2.0))**2)) * (L**2 / (r**2 + 1e-8)) return potential def geodesic_equations_smooth(t, y): """ System of continuous, differentiable ODEs tracking ray and phase-temperature. """ r, phi, p_r, T = y # 1. Metric evaluation A = A_smooth(r) # 2. Kinematics d_r = A * p_r d_phi = L / (r**2 + 1e-8) # 3. Dynamic Phase Force (Numerical potential derivative dV/dr) dr = 1e-6 dV_dr = (V_R_smooth(r + dr) - V_R_smooth(r - dr)) / (2.0 * dr) d_pr = -0.5 * dV_dr # 4. Phase-thermodynamic temperature increment tracking phase = theta_0 * np.exp(-(r - 2.0 * M) / alpha_scale) g_00_eff = A * np.abs(np.cos(phase)) d_T = np.sqrt(L / np.sqrt(g_00_eff)) if g_00_eff > 1e-5 else 0.0 return [d_r, d_phi, d_pr, d_T] # === RUN INTEGRATION (From far away down to the non-singular core) === # Initial state: r = 25M, phi = 0, p_r = -1 (infall), T = 0 y0 = [25.0 * M, 0.0, -1.0, 0.0] time_span = (0, 150) t_eval = np.linspace(time_span[0], time_span[1], 3000) sol = solve_ivp(geodesic_equations_smooth, time_span, y0, t_eval=t_eval, method='RK45') # === GENERATE VERIFICATION PLOTS === fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 5)) # Left Plot: Continuous radial freeze without software truncation ax1.plot(sol.t, sol.y[0], 'g-', lw=2.5, label='Smooth Photon position r($\\lambda$)') ax1.axhline(y=r_ECA, color='b', linestyle='--', label='ECA Scale ($r_{ECA}$)') ax1.set_xlabel('Affine parameter $\\lambda$') ax1.set_ylabel('Radius r/M') ax1.set_title('A: True Geometric Singularity Resolution') ax1.legend() ax1.grid(True, linestyle=':', alpha=0.6) # Right Plot: Thermal profile evolution matching the analytic prediction ax2.plot(sol.y[0], sol.y[3], 'b-', lw=2.5, label='Phase temperature T(r)') ax2.set_xlim(0, 5.0) ax2.set_xlabel('Radius r/M') ax2.set_ylabel('Temperature T') ax2.set_title('B: Phase Thermodynamic Profile (Smooth Field)') ax2.legend() ax2.grid(T rue, linestyle=':', alpha=0.6) plt.tight_layout() plt.show()