The Complete Vacuum Metric from Foam Dynamics: Schwarzschild and Kerr

The complete Schwarzschild metric gₜt = −c² (1−2GM/rc²), gᵣr = (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 (gₜt × gᵣr = −c² for null geodesics). The angular metric follows from Oₕ transitivity on the foam vacuum. Newton's constant G = c³lP²/ħ is derived from the foam cell size lP. These results are recovered in the continuum limit assuming covariance holds; the emergence of Lorentz invariance from discrete Oₕ symmetry remains an open problem.