Strong-Field Klein-Gordon Equation in the CM Metric: Spatial Flatness, Effective Mass, and Eigenvalue Predictions

We derive the exact Klein-Gordon equation for a massive scalar field in the Clausius-Mossotti (CM) metric ds² = W^(1/3)c²dt² − W^(−1/3)(dr² + r²dΩ²), where W = (1−β²)/(1+2β²) and β² = 2GM/(rc²). Three structural properties distinguish this equation from the Schwarzschild case: (i) the spatial Laplacian is exactly flat — no metric function appears inside the radial derivative, a direct consequence of the CM identity g_tt × |g_rr| = 1; (ii) the effective particle mass μ_eff = μ W^(−1/6) increases near the gravitating source, opposite to GR where the effective mass decreases; (iii) the frequency term carries W^(−2/3), encoding gravitational time dilation into the wave's temporal oscillation alone. Numerical bound-state eigenvalues at gravitational coupling strengths α_g = 0.1, 0.3, and 0.5 show that CM bound states are systematically less tightly bound than GR, with the ground state shift reaching +18.5% at α_g = 0.5 and CM supporting three bound states compared to GR's four. The weaker binding is traced to the 4π+2π=6π geometric decomposition: geodesics use the ratio g_tt/|g_rr| (exponents add, giving full GR), while the d'Alembertian uses √(−g)×g^rr (exponents cancel, losing spatial confinement). Zero free parameters are introduced.