This paper proves a finite-dimensional structural theorem for boundary accessibility compression. A boundary channel determines which distinctions of a total carrier remain distinguishable at a boundary and which become invisible under boundary access. The resulting compression invariant measures the difference between the total distinction structure and the distinction structure accessible at the boundary. The main theorem provides exact dimension formulas linking accessible distinctions, boundary-invisible distinctions, and the corresponding accessibility quotient. Compression vanishes exactly when the boundary law separates all states of the total carrier. The paper distinguishes three notions that are often conflated: full-state separation, restricted-sector accessibility, and physical recovery. A restricted-state theorem shows how separation on a chosen class differs from compression on the full carrier, thereby keeping accessibility, reconstruction, and recovery conceptually distinct. The theorem is applied to partial trace, channel composition, projective measurement, and finite history selection. In each case, the compression invariant can be computed exactly. The results show that compression is monotone under channel composition, quantify the distinguishable structure retained under nonselective measurement, and characterize the loss of independent distinctions induced by finite history quotients before any additional boundary compression is applied. As a holographic application, the paper formulates holographic reduction as exact boundary representation on an accessibility quotient. A boundary description carries precisely the distinctions separated by its accessible observable structure, while the remaining distinctions lie in boundary-invisible fibers. The theorem therefore provides a finite-dimensional criterion for determining when a boundary law is complete, when it is compressed, and what additional structure is required if physical reconstruction is claimed. A companion sequel extends the one-step accessibility-compression theorem developed here to recursive channel chains and establishes the corresponding criterion for source recognition through composed accessibility quotients and survivor invariants. License note: Distributed under CC BY-NC-ND 4.0.
Salimah Meghani (Sat,) studied this question.