Boundary Response Necessity: Divergent Resistance Near the Admissibility Boundary

This manuscript is H1 in a four‑part hardening sequence that strengthens a completed five‑paper substrate admissibility framework. H1 establishes the boundary‑response enforcement theorem: once finite local deformation capacity, continuity of admissible configurations, bounded deformation, and global admissibility closure have already defined a non‑attainable admissibility boundary, the structural response must diverge as that boundary is approached. The result is derived without introducing new dynamics, new fields, or additional postulates. Instead, it follows from inherited structural conditions established in the substrate framework. A contradiction argument shows that if boundary response remained finite near vanishing capacity, continuous admissible paths could approach the boundary under finite structural cost, contradicting the inherited non‑attainability condition. The divergence of the boundary‑response functional therefore provides the enforcement mechanism required by the upstream architecture. It gives the admissibility boundary the character of a geometric barrier in configuration space and supplies the first hardening result needed for the subsequent papers in the sequence. H2 analyzes the distributed finite‑capacity core structure that follows when point‑supported continuation is excluded; H3 and H4 address admissibility‑conserving evolution and large‑scale admissibility‑preserving redistribution. H1 resolves the enforcement gap left open by the substrate framework and establishes the structural condition required for any admissibility‑preserving response theory.