Categorical Reconstruction on the Non-Archimedean Skeleton from Coordinate-Free Renormalized Tail Orbits: Theta Cosheaves, Wall-Kernel Monodromy, and Finite-Stratum Recovery

The previous paper in this series NonArchSkeleton2026 reconstructed, from coordinate-free renormalized tail orbits in a positive finite-type regime, the completed theta algebra \ (\), the positive potential, the compact mirror package, and the non-Archimedean skeleton \ W B \ together with its wall-corrected degeneration. What remained open was a sharper categorical question: can the orbit determine not only the algebra and the skeleton, but also a canonical category living on that skeleton and encoding wall transport at the level of perfect complexes? This paper gives a positive answer, in a deliberately explicit scope. We work with the regular polyhedral skeleton \ (B\) and the completed monoid-chart package already recovered in the previous paper, and impose two extra hypotheses only where they are genuinely needed: flat face inclusions for categorical descent, and a simple-wall unimodular condition for the one-skeleton monodromy theorem. Within that regime the orbit canonically determines a constructible cosheaf of small idempotent-complete pretriangulated dg categories \ C_^orb \ on \ (B\). On each maximal cell \ (\), the value is the perfect derived category of the completed chart algebra \ ( (_) \) ; on each codimension-one wall the gluing is implemented by an orbit-determined invertible wall-kernel bimodule. The first main theorem is an orbit-to-theta-cosheaf closure theorem: the entire cellwise categorical package is determined up to Morita equivalence by the orbit. The second theorem identifies the homotopy colimit of this cosheaf with the global perfect category \ ( () \), so the orbit determines the global theta category together with a skeletal presentation by local chart categories. The third theorem upgrades the one-skeleton to a weak categorical schober: wall-kernel monodromy along a transverse loop is exactly the categorical path-ordered product of the orbit wall data, and in the rank-one simple-wall case the induced action on truncated coefficient spaces is triangular with explicit binomial coefficients. The fourth theorem is quantitative: on any bounded face window and monomial cutoff, finitely many orbit probes recover the truncated incidence poset, local multiplication tables, wall kernels, and categorical monodromy representation with explicit asymptotic error bounds. The point is conceptual as well as technical. The orbit does not merely recover a mirror algebra and the polyhedral set on which that mirror collapses. It also reconstructs a canonical algebraic theta-category on the skeleton itself. We state this sharply: the paper does not claim a full comparison with wrapped Fukaya or microlocal sheaf categories. What it proves is an intrinsic, orbit-determined categorical skeleton theorem, strong enough to support such a comparison in a later step.