The Variational Boundary Theorem (VBT) establishes the Bekenstein-Hawking entropy law S = A/ (4ℓ2 P) on the Satake boundary ∂Sat of the target manifold X = Sp (2N, R) /U (N) for congurations admitting a Killing horizon of finite cross-section area (Schwarzschild, Kerr, de Sitter). The extension of the VBT to the FRW Big Bang proves that the entangle-ment map λ reaches ∂Sat exactly at cosmic time t = 0, but the candidate cross-section has vanishing 3-volume: the area formula does not apply by construction, not by calculation. We close this open problem by developing two parallel and ultimately complementary extensions. Option 1 reformulates the Big Bang as the IR endpoint of the renormalization-group ow of the σ-model on X. The variational domain has no before t = 0 not because such a region is censored, but because the parametric ow terminates structurally at thedeepest Satake stratum S0. The area formula is replaced by a regularized stratum entropy eSFB Sk (Λ), a finite, mode-dependent Fisher-Bures information quantity, independent of any area integration. Option 5 develops the operational layer: the Page-Wootters cosmological clock-rate eη is well-defined only on the open operational domain Top = eη > 0, whoseboundary ∂Top consists of two structurally distinct pieces if the past-IR boundary at S0 where eη → ∞ (the operational Big Bang) and the future-UV boundary at the Fock face νk = 1k where eη → 0 (the operational Big Freeze). A central structural result is that the classical Weyl Curvature Hypothesis (WCH) of Penrose and the quantum qWCH of Kiefer are both automatic, but at different ends of the cosmological trajectory and by difierent geometric mechanisms. The classical WCH is realized at S0 via singleton collapse (M2) ; rigorously, the VTG cosmological pullback metric with isotropic approach to S0 admits an isotropic singularity in the Scott-Ericksson sense with conformal factor equal to the FRW scale factor, the singularity surface Σ is umbilic in the unphysical spacetime via the Tod Fuchsian framework, and the full Weyl tensor C abcd|S0 = 0 follows (Theorem 7. 5). The quantum qWCH is realized at the asymptotic future Fock face via maximal U (N) -symmetry (M1) on the K-fixed origin; the Fock face is identified geometrically as the single K-fixed interior point at finite Fisher-Bures distance, operator-theoretically as the unique U (N) -invariant state in L2 (RN) via the metaplectic representation, and dynamically as the unitary distinguished boundary of the closed oscillator semigroup. A Coleman-De Luccia bounce reconnecting the Fock face to S0 is rigorously excluded (Theorem 9. 1): the FisherBures distance dFB (o, S0) = ∞ is incompatible with finite Euclidean action, providing a fourth converging argument for the selection of the cosmological direction. The empirical near-vanishing of the Standard Model Higgs quartic coupling λSM (MPl) ≈ 0 and of the cosmological constant Λ are identified as two manifestations of a single structural fixed-point condition at the Fock face.
ignacio caldini (Sat,) studied this question.