Boundary Response Necessity: Divergent Resistance Near the Admissibility Boundary

This manuscript is H1 in the four-paper Hardening Series supporting the parent Displacement Framework for Horizon-Bounded Cosmological Accounting in a Timeless 4D Manifold (Partin, 2026). The parent paper establishes the admissible class, finite local capacity, continuity restrictions, and global non-attainability of the finite-capacity boundary. It does not specify the enforcement mechanism by which continuous admissible evolution respects that boundary. H1 begins exactly where the parent paper stops: it formalizes the response-enforcement condition required to protect the inherited non-attainable boundary under continuous admissible approach. Earlier formulations treated non-attainability as though it were sufficient, together with continuity, to force divergence of a response functional. It is not. An open set may possess a non-attainable boundary while a bounded scalar functional remains well-defined along sequences approaching it. Version 2.0 repairs this gap by introducing an explicit enforcement premise: finite-response interiority, the requirement that every finite-response sublevel of the designated functional remain uniformly separated from vanishing local capacity. Under this premise, H1 proves a clean equivalence: finite-response interiority holds if and only if every admissible boundary-approaching sequence forces the response functional to diverge. The theorem is stated for both moving-point sequences and continuous paths, removing the fixed-point ambiguity present in earlier drafts and making the downstream H2 concentration arguments formally legitimate. The paper reinstates the structural-cost functional, geometric interpretation, and consequence material from earlier versions, but now each statement is explicitly conditioned on the enforcement premise and derived from Theorem 4.1 rather than asserted outright. No dynamics, constitutive laws, characteristic scales, or specific response formulas are introduced. The result is structural: it identifies the exact limiting behavior required of any response functional assigned to enforce the inherited non-attainable boundary. H1 therefore completes the enforcement gap left open by the parent Displacement Framework and supplies the first hardening result needed for the subsequent papers in the sequence: H2 (exclusion of finite-response point-like concentration and distributed finite-capacity support), H3 (preservation of admissible class membership under admissible continuation), and H4 (global non-rupturing redistribution).