Boundary Accessibility Quotients and Apparent Information Loss: A Finite-Dimensional Structural Theorem for Exterior Information Recovery

This paper proves a finite-dimensional boundary-accessibility theorem for apparent information loss. A boundary channel does not merely produce a reduced state; it induces an accessibility quotient on the total state carrier. Distinct total states become operationally indistinguishable whenever they define the same boundary law, and recovery is possible only for distinctions separated by the resulting quotient. The main result gives an exact Heisenberg-range criterion for complete recovery. A boundary channel separates all total density states if and only if its Heisenberg adjoint has full operator range. Equivalently, pulled-back boundary effects span the full real space of total observables. A restricted version characterizes recovery on arbitrary state classes, code sectors, or history sectors. A dimension obstruction follows: no channel into a smaller boundary Hilbert space can be informationally complete for all total states. The theorem is applied to total unitary evolution followed by boundary restriction. Unitary dynamics preserve total-state distinctions, but do not make a noninjective boundary channel injective. In the canonical tensor-factor model, exterior partial trace is noninjective whenever the complementary factor is nontrivial. Consequently, mixed boundary states, increased boundary entropy, and reconstruction failure are compatible with total purity and total unitarity. As an application, the black-hole information problem is reformulated as a theorem of boundary accessibility. Apparent information loss is not identified with failure of total unitarity, but with failure of the boundary-accessible quotient to separate total states. A model claiming exterior recovery must either make the boundary channel injective on the relevant carrier or state class, or explicitly supply additional recovery data such as a code subspace, complementary access, records, histories, islands, a recovery map, a selected quotient section, or another carrier extension. More broadly, the theorem identifies apparent information loss as an accessibility-level quotient phenomenon. Total unitarity acts on the total carrier, while exterior recovery is governed by the distinctions separated by the boundary-accessible carrier. The boundary law recovers exactly those distinctions, and no others, unless additional recovery structure is supplied. License note: Distributed under CC BY-NC-ND 4.0.