Foundational Admissibility: Closure as a Selection Principle on Cosmological Possibility Space Paper 79 of the NEMS Suite

This paper develops the Foundational Admissibility arc of the NEMS program. Earlier NEMS results showed what closure implies: admissible initiality, structural irreversibility, internal grounding of realized history, no foundational external runner, and a wide family of logical, physical, and epistemic consequences. The present paper isolates the first selection step relevant to cosmological possibility space by proving that closure compatibility and foundational viability are equivalent. The formal setting is the unified closure architecture developed for the grand unification theorem. On that base, we define (\FndViable(U)) as the conjunction of three conditions: closure-admissible initiality, structural irreversibility, and closure-realized history. We define (\ClosureCompat(U)) as nonemptiness of the cosmological closure schema. The main theorem is the theorem of foundational admissibility: [ \FndViable(U)⟺ \ClosureCompat(U), ] Proved by foundational_admissibility (\ClosureCompat⇒ \FndViable) and foundationally_viable_implies_closure_compatible (\FndViable⇒ \ClosureCompat); plus the failure-form contrapositive \neg \FndViable(U)⇒ \neg \ClosureCompat(U). This gives the first mathematically explicit sieve step on cosmological possibility space: closure compatibility and foundational viability coincide at the level of the unified closure architecture, so viability failure excludes closure compatibility and closure-compatible frameworks are exactly the foundationally viable ones. The paper then defines the first stages of a post-admissibility cascade: survivor compatibility, probabilistic admissibility, and physics-architecture admissibility. The conceptual significance is that closure is no longer merely a theorem generator inside a favored model class. It becomes the first selection principle in a larger classification program. Trust boundary. The equivalence (\FndViable(U)⟺ \ClosureCompat(U)) and named lemmas are machine-checked in nems-lean . Later-stage cascade predicates in this paper are defined formally but the summit selection program continues beyond this equivalence; see .