This paper is the fourth installment in a five‑paper substrate‑framework program. It builds on three upstream works: The Displacement Framework (Partin, 2026a), Toward a Formal No‑Tear Theorem in a Continuous Substrate Framework (Partin, 2026b), and The Charged Fabric (Partin, 2026c). Together, these upstream layers establish the global accounting structure, the continuity‑based admissibility condition, and the microphysical deformation and gradient‑response framework that precede the bounded‑deformation result developed here. The present paper introduces a bounded‑deformation admissibility axis for a continuous substrate framework. Continuity alone excludes rupture but does not by itself exclude unbounded local compression. To address this gap, the paper imposes a finite upper bound on admissible local deformation, proves a deformation‑limit theorem excluding unbounded local deformation within the admissible class, and derives a corollary excluding zero‑volume configurations whose realization would require divergent deformation. The result is structural and non‑dynamical. Together with the upstream continuity‑based admissibility condition, it establishes a two‑axis admissibility structure: continuity without tear and finite deformation without inadmissible zero‑volume concentration. One downstream paper remains, addressing saturation response and admissibility‑boundary behavior as the final installment of the program.
William T Partin (Mon,) studied this question.