Exactness as a Foundational Test: Finite-Union Normal Form in Persistence-Collapse Reconstruction

This paper develops a restricted metatheoretic reconstruction of a persistence--collapse vocabulary in ordinary metatheory. Its aim is classificatory: to test whether a vocabulary organized around persistence, collapse, coherence, threshold, fusion, and recurrence can be made into a bounded theorem-bearing fragment. The framework is explicitly structure-relative. Persistence is defined by recurrence, unsuccessful fusion is represented by a distinguished collapse value, and admissible clusterhood requires both local threshold admissibility and genuinely global successful fusion. The main result is classificatory: under exact support assumptions, the successful additive part of the fragment is isomorphic to finite disjoint union on nonempty finite supports. Under intrinsically exact atomic-support assumptions, the same normal form is recovered from the fragment's own persistent atoms. Thus exact persistence does not generate a new arithmetic; it rigidly collapses to finite-union normal form. The semantic component is correspondingly limited: it is a persistence-sensitive valuation fragment rather than a full logic. Boundary theorems show that recurrence, global admissibility, overlap-control, and the distinction between threshold collapse and persistence-requires-admissibility are indispensable for this reconstruction. The contribution is therefore methodological and classificatory: an initially speculative organizing vocabulary is converted into a precise restricted framework, the exact regime is classified, and the formal limits of that regime are made explicit.