The Complete Vacuum Metric from Foam Dynamics: Schwarzschild and Kerr

The complete Schwarzschild metric g_tt = −c²(1−2GM/rc²), g_rr = (1−2GM/rc²)⁻¹ and the Kerr metric are recovered from Planck-scale foam dynamics in the infrared/continuum limit. The temporal component follows from the foam's relativistic Euler equation with equation of state P=ρc². The spatial component is derived by two independent routes: vacuum consistency (T_μν=0 ⟹ Birkhoff's theorem) and propagation isotropy (g_tt × g_rr = −c² for null geodesics). The angular metric follows from O_h transitivity on the foam vacuum. Newton's constant G = c³l_P²/ħ is derived from the foam cell size l_P. These results are recovered in the continuum limit assuming covariance holds; the emergence of Lorentz invariance from discrete O_h symmetry remains an open problem.