Saturation Limits of the Displacement Field: Bounded Deformation, Admissibility Under Limits, and the Exclusion of Singular Collapse

This paper is the fourth installment in a five-paper substrate-framework program. It builds on three upstream papers: 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 works 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. It shows that 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 collapse configurations that 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 collapse. One downstream paper remains, addressing saturation response and near-boundary behavior as the final installment of the program.