Closure Formation as Admissible Instability: Seed-Field Competence, Spectral Admissibility, and the Rarity Cascade

Closure-class formation involves two structurally distinct questions: the conditions under which closure of a specified class is admissible, and the contingent events through which any specific closure realizes. This paper develops the conditions side. Five structural obstruction modes — sparse-feedback, fragment-local, cost-imbalanced, energetically-starved, dimensionally-trapped — characterize seed-field incompetence: the conditions under which a substrate region is excluded from supporting closure-class formation across its accessible perturbation regimes. The admissibility theorem characterizes which closure classes a competent region admits as a function of a parameter family indexing the substrate's coupling structure, boundary exposure, energetic and temporal accessibility, and dimensional access. The theorem is stated spectrally — admissibility iff the linearized closure-formation operator possesses an unstable mode satisfying structural and dynamical conditions — and equivalently in variational form via the closure-potential functional. The four-stage architecture for closure-class formation (competence, admissibility, embryo formation, basin completion) entails a cascade structure in which closure-formation rates compose multiplicatively across stages, so that progressively more demanding closure classes are progressively rarer by structural composition rather than stipulation. The cascade is observable through four ensemble signatures — critical slowing, hysteresis, post-collapse renucleation, basin-deepening — that specialize the early-warning-signal apparatus independently developed in tipping-point ecology and regime-shift detection to closure-class transitions across substrate types. Consciousness-class closure is treated as a further-conditioned special case in which the cascade structure entails the framework's predicted rarity without further stipulation.