Big Bang as Operational Time Emergence: A Page-Wootters and Fisher-Bures Approach

The Variational Boundary Theorem (VBT) establishes the BekensteinHawking 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 nite 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 complementaryextensions. 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 the deepest Satake stratum S0. The area formula is replaced by a regularized stratum entropy eSFB Sk(Λ), a nite, mode-dependent FisherBures information quantity, independent of any area integration. Option 5 develops the operational layer: the PageWootters cosmological clock-rate eη is well-dened only on the open operational domain Top = {eη > 0}, whose boundary ∂Top consists of two structurally distinct pieces the past-IR boundary at S0 where eη → ∞ (the operational Big Bang) and the future-UV boundary at the Fock face {νk = 1}k 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 dierent ends of the cosmological trajectory and by dierent geometric mechanisms. The classical WCHis realized at S0 via singleton collapse (M2); rigorously, the VTG cosmological pullback metric with isotropic approach to S0 admits an isotropic singularity in the ScottEricksson 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 Cabcd|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-xed origin; the Fock face is identied geometrically as the single K-xed interior point at nite FisherBures 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 ColemanDe Luccia bounce reconnecting the Fock face to S0 is rigorously excluded (Theorem 9.1): the FisherBures distance dFB(o, S0) = ∞ is incompatible with nite 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 identied as two manifestations of a single structural xed-point condition at the Fock face. For Big Bang nucleosynthesis we adopt the Pitrou et al. [2018] PRIMAT predictions as the leading-adiabatic VTG output and obtain agreement with YP at 0.5σ, with D/H at 1.5σ, with 3He/H within upper bounds, but with a 45σ residual 7Li/H discrepancy. A four-channel sensitivity-matrix analysis isolates a unique viable VTG resolution: a non-adiabatic, selective eη-driven suppression of the 3He(α, γ)7Be rate by approximately 72% in the Gamow window. The numerical Tsallis-q inverse-resolution lands at q∗ = 0.91953, inside the phenomenological BertulaniMathai window without parameter adjustment; the residual selectivity question is recorded as a successor sub-problem with three structural candidate routes. Ination in VTG is identied with a quasi-stationary plateau of the two-loop β(T) at Tqs = 1/(4c1) ≈ 1.929. The bosonic-only e-fold count is the closed-form all-orders bound N(P-ii,∞) e (28) = 0.4029, two orders of magnitude below the phenomenological 60 e-folds; this redirects the 60 e-fold question to the coupled bosonic+fermionic sector through the Howe dual pair (Sp(56,R), O(p, q)) with canonical (p, q) = (9, 3), giving Nfull,coupled e (28, 9, 3) = 60.69 ≈ 61.2 ± 0.8.Two future research directions registered at the most-speculative end of the frame- work are: (B5) operational baryogenesis via POVM asymmetry on the FoldyWouthuysen- separated subspaces H(±) S realizing the three Sakharov conditions simultaneously from the single geometric structure of X; and (B1-CdL-β) the last tick as maximal coherence at theFock face, identifying the operational future boundary not with a thermodynamic crossing but with the unique pure U(N)-invariant state of the metaplectic representation, with the continuation to a new S′0 realized relationally through the bilateral support of the Page Wootters constraint kernel. The originally-proposed absolute-negative-temperature framing is retired in the v1.14 consolidation (NAT exceeds the VTG framework and has a live consistency dispute in the literature); the reformulated mechanism is consistent with Theorem 9.1 and is structurally distinct from Penrose's conformal CCC. Keywords: Variational Boundary Theorem; Big Bang; FisherBures metric; Satake compactication; symplectic symmetric space; PageWootters time; Weyl curvature hypothesis; cosmological lithium problem; ColemanDe Luccia; baryogenesis.