Boundary Maps from Kantowski–Sachs Collapse to Cosmological Expansion: Canonical Classification, Reduced Invariants, and Covariant Obstructions

Abstract

This record archives version v1.0 of the manuscript and its reproducibility package. We ask when a relation between a collapsing Kantowski–Sachs boundary state and an expanding homogeneous state can be more than a kinematic identification. For Einstein gravity with a massless scalar, we classify all real affine point maps that preserve the reduced Hamiltonian constraint. The symplectic maps form an E(2) wall stabilizer with an exact shear–scalar rotation. A fixed-axis contribution to the volume expansion is twice and opposite to the largest rotatable contribution, so every symplectic image of every admissible terminal stratum still collapses. Expansion in this class therefore requires anti-symplectic time reversal. At spacelike walls, the admitted group instead splits into collapse-preserving and uniformly expanding components, with neither continuity nor quantum implementability supplying a selector. A diagonal class-B negative control retains a timelike physical wall even where the reduced geometry contains a null candidate direction. Conversely, at the established critical exponential slope, Bianchi I realizes a genuinely null physical wall: exact positive-potential solutions asymptotically attain the finite parabolic endpoint, but its two screen components have identical ranges and remain unselected. The rotation preserves an exact quadratic Casimir of the full reduced system, but a scoped off-shell test finds no lift within the local derivative-free diffeomorphism-covariant point-transformation class; the ordinary scalar-shift current survives separately. Two dynamical case studies expose the remaining gap from an expanding image to a physical continuation. A spacelike pure-tension layer can create initial two-direction expansion, but sustained expansion requires independently supplied child stress. A published limiting-curvature model gives a reproducible nonsingular reduced continuation, but its proposed scalar-invariant description cannot represent the model’s secondary constraint on unrestricted sectional data. Together these results establish a sequence of distinct obstructions between canonical transformability and covariantly defined cosmological evolution. The archive contains the REVTeX manuscript source, compiled manuscript and PRD cover-letter PDFs, symbolic verification scripts, machine-readable outputs, and pinned pipeline dependencies.