Finite-Capacity Cores: Structural Necessity of Distributed Configurations

This paper examines a structural question that arises in any framework that treats finite deformation capacity as a genuine boundary condition. If rupture and direct boundary attainment are excluded, can increasingly concentrated deformation remain admissible simply by being confined to progressively smaller regions? H2 shows that it cannot. Building on the boundary‑enforcement result established in H1, the analysis demonstrates that point‑like concentration—shrinking the region of high deformation while driving that deformation toward the finite‑capacity limit—still produces an approach to the admissibility boundary. Under the inherited enforcement condition, such concentration cannot remain finite‑response and is therefore excluded as a continuation route. The result shifts the structural question from whether point‑like concentration can be excluded to what forms of support remain admissible once that route is closed. Within the deliberately restricted route class considered here, admissible finite‑response continuation requires high deformation to remain distributed over support that does not vanish in measure rather than collapse toward point‑like support. This motivates the notion of a finite‑capacity core: a designated high‑but‑finite region of deformation whose support remains genuinely extended rather than singular or vanishing in measure. The argument is intentionally structural rather than dynamical. It does not introduce a force law, constitutive model, characteristic scale, unique core geometry, redistribution mechanism, or empirical prediction. Its philosophical significance lies in clarifying how finitude, continuity, admissibility, and spatial concentration constrain one another. Once finite capacity is treated as a genuine structural boundary, avoiding boundary violation restricts not only how much deformation is admissible but also the forms of support through which near‑boundary configurations can persist. This paper—H2, Finite‑Capacity Cores: Structural Necessity of Distributed Configurations—is the second paper in the Hardening Series supporting the parent framework The Displacement Framework for Horizon‑Bounded Cosmological Accounting in a Timeless 4D Manifold. H2 is limited to the support‑restriction layer. Its immediate upstream result is H1, Boundary Response Necessity: Enforcement at a Non‑Attainable Finite‑Capacity Boundary, which establishes the enforcement condition used here. Its immediate downstream result is reserved for H3, which separately examines whether admissible‑class membership is preserved under admissible continuation.