We extend the Variational Boundary Theorem (VBT)—proven in its general geometric form in the companion paper 1 — from the static, spherically symmetric (Schwarzschild) case to two physically important settings: stationary axially symmetric black holes (Kerr) and Friedmann–Robertson–Walker (FRW) cosmology. For Kerr black holes with mass M and angular momentum J = aM, |a| < M, we show that the entanglement map λ : (r+,∞)×S2 → Sp(2N,R)/U(N) reaches the Satake boundary ∂Sat precisely at the outer horizon r = r+, with entropy S = A+/(4ℓ2 P). The key new ingredient is the ZAMO (Zero Angular Momentum Observer) frame, which remains timelike throughout the ergosphere and defines a local temperature diverging as (r − r+)−1/2 — the same exponent as Schwarzschild. Numerical verification confirms this universality for all spin parameters and polar angles tested. For FRW cosmology, two Satake boundaries are identified: the cosmological (de Sitter) horizon,where the Gibbons–Hawking temperature undergoes Tolman blueshift, and the Big Bang singularity at t = 0, where the cosmological temperature T(t) → ∞. The hyperbolicity of the Vacuum Time Geometry field equations, stated as a proposition in 1, is here promoted to a full theorem using the wave map theory of Shatah–Struwe and Tao for targets of nonpositive curvature, combined with the quasilinear framework of Hughes–Kato–Marsden. All proofs are complete.
ignacio caldini (Thu,) studied this question.