CAT'S THEORY: Minimal Bases and Minded Worlds: The Ontological Closure Theorem: Triadic Structure as a Necessary Condition for Existence as Such

The preceding papers in this series established that the Triadic Meta-Law P × I × Pr ≠ 0 is a necessary structural invariant for minded worlds, a necessary condition for any sufficient emergentist base ontology, and a transcendental precondition on intelligibility as such. The present paper closes the remaining gap by proving a strictly stronger claim: triadic structure is a necessary condition for existence as such, independently of any epistemic or intelligibility framing. The Radically Unintelligible Reality thesis (RUR) is formalized and proved self-defeating. The Distinguishability Lemma proves that for any candidate X to count as something rather than nothing, X must instantiate P × I × Pr ≠ 0, secured fundamentally through the bifurcated Intent factor. The Ontological Closure Theorem follows: the distinction between intelligibility and being collapses under analysis. A Self-Refutation Lemma completes the closure. The series is closed.