Cosmological Singularities and the Variational Boundary in Vacuum Time Geometry

We study the cosmological singularity as a structural feature of Vacuum Time Geometry (VTG), the framework developed in Papers I–II of this series. In VTG the dynamical variable is the entanglement map λ : M3,1 → Sp(2N,R)/U(N) between the vacuum-clock and matter; time emerges relationally through the Page–Wootters identification Δt = tP ΔS/˜η, with ΔS the entropy increment on the matter side per tick and ˜η the relational clock rate; the target carries the Fisher–Bures information metric gFB as the canonical Petz monotone-maximal quantifier of the relational tick; and the spacetime metric is the pullback gμν = gFBab (λ) ∂μλa ∂νλb — a derived structure, not a primary geometric object. Gravity is emergent as a gradient of relational-clock geometry, the Einstein field equations appear in the adiabatic limit through the Sakharov mechanism, and the cosmological singularity is identified with the Satake boundary ∂Sat at infinite Fisher–Bures distance, where the relational tick degenerates.The Variational Boundary Theorem (VBT) of Papers I–II states that a smooth critical point λ of the VTG action cannot be smoothly extended through ∂Sat; it has been applied there to Schwarzschild and Kerr horizons and to the Friedmann–Robertson–Walker Big Bang and de Sitter horizon. The present paper extends the VBT to all anisotropic spatially homogeneous Bianchi cosmologies (types I–IX) and to a structured class of inhomogeneous data. Section §A reduces the field equations on a Bianchi background to a coupled ODE system with anisotropic extrinsic curvature and proves, by Picard–Lindelöf theory together with a Bochner–Weitzenböck cascade adapted to the Bianchi context, that λ cannot reach ∂Sat before t = 0. The argument covers all nine Bianchi types — including Bianchi IX (Mixmaster) with infinitely oscillating Kasner epochs — and the intermediate Gowdy T3 case is closed almost everywhere through Ringström’s AVTD regime, with spikes a codimension-one set that does not affect theconclusion.