The Recursive Classification of Discovery: From Finite Sampling to Epistemic Boundary

This paper introduces a universal classification of discovery grounded in recursion. All discoveries are shown to fall into two categories: finite sampling discoveries, which generate bounded truths within existing frameworks, and epistemic boundary breakthroughs, which connect finite samples to infinite recursion and restructure the ontology of knowledge. The framework operationalizes this classification through three tests—Extension ($T_1$), Unification ($T_2$), and Repair ($T_3$)—and formalizes necessary and sufficient conditions for boundary crossing. Case studies in mathematics, physics, biology, artificial intelligence, and philosophy demonstrate how discoveries can be systematically classified, while appendices provide formal lemmas and theorems (including the “Glass of Water from the River Lemma” and Theorem 𝒴-Opt.2). By offering predictive claims, a minimal toy model, and an evaluation plan, the paper transforms the epistemic boundary from a conceptual metaphor into a testable scientific construct.