Geometric Vacuum Selection from Constrained Spacetime Foliations

This work proposes an infrared vacuum-selection mechanism within General Relativity based on a preferred spacetime foliation defined by a scalar time-ordering field Θ. A non-local, leafwise constraint fixes the asymptotic normalized average of the squared extrinsic curvature K² = K⏛⏜K^μν over causal domains to a constant scale K0². The constraint acts as an infrared boundary condition on the foliation: for generic asymptotically flat slicings, Minkowski space yields ⟨K²⟩ → 0 in the infinite-volume limit and is therefore not selected as an infrared vacuum for K0² > 0. De Sitter spacetime is instead admitted, with the curvature scale determined geometrically by H² = K0²/3. Because the constraint is controlled by horizon-scale averaging, local gravitational physics remains effectively unchanged, recovering Schwarzschild and post-Newtonian limits. The manuscript also develops the finite-domain/causal formulation needed to discuss realistic cosmological eras and is accompanied by a Lean formalization artifact for the present formal scope. A full BRST/Faddeev-Popov treatment establishing gauge independence of physical residues, a fundamental UV cutoff or UV completion, and a complete cosmological-perturbation phenomenology remain outside the scope of this work. This version substantially expands the original proof-of-principle preprint and supersedes the earlier partial artifact snapshot.