A Mathematical Theory of AGI

We present a comprehensive synopsis of the mathematical theory (universal algebra, category and sheaf theory) of Artificial General Intelligence (AGI). The theory is built upon three complementary and ultimately equivalent pillars: (1) an axiomatic framework formalizing infinite metacognition via approximate homomorphisms and category-theoretic limits; (2) a cognitive free‑energy principle defined over generalized cognitive hierarchies; and (3) the Natural Balance Principle, a single‑axiom optimization functional that requires no prior structural assumptions. All three frameworks independently converge to the same unique architecture: an AGI system must possess a minimal algebraic carrier isomorphic to the four‑element Boolean algebra, necessarily embeds a unique self‑referential prediction module (MetaPred) that accompanies every learning process, and exhibits a strict isomorphism among all machine learning mechanisms. We provide algorithmic pseudocode for MetaPred and its mandatory self‑measurement cycle, demonstrating concrete implementability. Furthermore, we detail a profound structural alignment between the derived cognitive architecture and well‑established findings in neuroscience (predictive coding, global workspace, prefrontal metacognitive networks) and cognitive psychology (dual‑process theory, exploration–exploitation). Finally, the theory's variational formulation reveals a deep structural isomorphism with fundamental physics, including a cognitive Einstein equation and a least‑action principle. The convergence of the three theoretical pillars, coupled with cross‑disciplinary consistency, provides strong evidence that the initial cognitive intuitions capture the mathematical essence of general intelligence.