Finite-Capacity Cores: Structural Necessity of Distributed Configurations

This paper is the second entry in the Hardening Series (H‑series), a sequence of structural results within the finite‑capacity substrate framework. H2 examines whether high deformation can remain an admissible finite‑response continuation by concentrating onto regions whose measure tends toward zero. Building on H1 — Boundary Response Necessity: Enforcement at a Non‑Attainable Finite‑Capacity Boundary, H2 shows that when deformation on such shrinking support approaches the finite admissible bound, it generates moving locations of vanishing remaining capacity. Under H1’s boundary‑response enforcement premise, the corresponding boundary‑response magnitude must diverge, excluding point‑like concentration as a finite‑response continuation route. Within the explicitly restricted H2 near‑saturation route class, three continuation routes are excluded: rupture by inherited continuity direct boundary attainment by global admissibility closure finite‑response point‑like concentration by the H2 concentration‑exclusion theorem The remaining admissibility‑preserving finite‑response alternative within this route class is distributed high‑deformation support that remains uniformly non‑vanishing in measure. H2 identifies finite‑capacity cores as designated high‑but‑finite, nonzero‑measure support regions within admissible configurations. The result is purely structural: it derives no dynamics, constitutive law, characteristic radius, unique core profile, stress model, redistribution mechanism, or empirical prediction. H2 remains a hardening result internal to the substrate framework rather than a replacement for general relativity, quantum field theory, or existing compact‑object models. The resulting support restriction provides the controlled handoff to H3, which separately addresses preservation of admissible‑class membership under admissible continuation. Version 2.0 replaces Version 1.3 and substantially hardens the paper’s formal structure. It incorporates H1’s explicit enforcement premise and moving‑witness formulation, separates shrinking support from near‑saturation capacity collapse, formalizes the restricted H2 route class and its distributed‑support consequence, clarifies the status of finite‑capacity cores, and establishes explicit upstream and downstream paper interfaces while preserving H2’s original structural role.