Abstract

This paper proposes a necessary condition for genuine understanding—mathematical or otherwise—in both biological and artificial cognitive systems. We argue that understanding requires two complementary capacities: recursive meta-cognitive hierarchization, formalized as the Variable-Level Condition (VLC), which demands that a system be able to treat the levels of its own cognitive hierarchy as variables subject to dynamic reconstitution without a predetermined ceiling; and Dynamic Computational Self-Monitoring (DCSM), a real-time mechanism for tracking cognitive level, estimating marginal epistemic yield of further level reconstitution, and applying dynamic termination criteria that are themselves subject to meta-cognitive revision. Together, these constitute the DCSM Criterion. We show that VLC generates computational explosion of exponential order in any finite system, and that the Paradox of Unbounded Understanding is resolved through principled finitude rather than actual infinite recursion. Applying the DCSM Criterion to current AI architectures, we argue that they occupy an asymptotic trajectory toward understanding: they partially satisfy conditions (a) and (b) but fail condition (c)—operative level reconstitution—in a way that may represent a principled rather than merely technical limitation. The paper situates this finding within a broader research program at the intersection of epistemology, cognitive science, and the theory of computation.