Finite-Capacity Cores: Structural Necessity of Distributed Configurations

This paper establishes that point‑like concentration of deformation is structurally inadmissible within a continuous finite‑capacity substrate possessing a non‑attainable admissibility boundary. Building on the previously established Boundary‑Response Necessity Theorem, which proves that admissibility‑preserving response must diverge as local capacity approaches zero, the analysis shows that any attempted concentration of deformation onto regions of vanishing measure necessarily induces divergent structural response. Since rupture is excluded by continuity, boundary attainment is excluded by global admissibility closure, and point‑like concentration forces boundary‑response divergence, admissible near‑saturation continuation must proceed through redistribution over regions of nonzero measure. This result converts an implicit admissibility consequence into an explicit structural theorem and defines finite‑capacity cores as the distributed, high‑but‑finite deformation configurations that replace singular concentration within the framework. The paper introduces no dynamics, constitutive laws, characteristic length scales, or model‑specific physical assumptions; it is a purely structural result internal to the substrate framework.