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 establishes that point‑like concentration of deformation is structurally inadmissible in any continuous finite‑capacity substrate whose admissibility boundary is non‑attainable. Building on H1—the Boundary‑Response Necessity Theorem, which proves that admissibility‑preserving response must diverge as local capacity approaches zero—the analysis demonstrates that any attempted concentration of deformation onto regions of vanishing measure necessarily forces divergent structural response. Since rupture is excluded by continuity, boundary attainment is excluded by global admissibility closure, and point‑like concentration induces boundary‑response divergence, admissible near‑saturation continuation within this route class must proceed through redistribution over regions of nonzero measure. The paper converts an implicit admissibility consequence into an explicit structural theorem and identifies finite‑capacity cores as the distributed, high‑but‑finite deformation support structures that replace singular or point‑supported concentration within the inherited admissibility architecture. The result is purely structural: no dynamics, constitutive laws, characteristic length scales, stress models, or model‑specific physical assumptions are introduced. H2 therefore functions as a hardening result internal to the substrate framework rather than as a replacement for general relativity, quantum field theory, or existing compact‑object models. More broadly, the paper strengthens the admissibility branch of the substrate program by converting singularity avoidance from a purely negative exclusion statement into a positive structural continuation result. Once rupture, boundary attainment, and finite‑response point concentration are excluded, admissible high‑deformation continuation must remain distributed over nonzero‑measure support. Version 1.3 replaces Version 1.2 and incorporates finalized upstream substrate‑framework alignment, updated DOI synchronization across the inherited reference chain, refined admissibility‑layer clarification, hostile‑referee scope hardening, and tightened continuation‑language constraints. No theorem, proof, or mathematical changes were introduced.