The Infinite Choice Barrier -Part II: Understanding the Boundary

This paper is Part II of a trilogy derived from the works on the ICB ( ’The Infinite Choice Barrier: A Structural Limit of Algorithmic Cognition’ , 'AGI is Impossible'(Schlereth, 2025)), expanding upon specific sub-domains of the original proof. Building upon the conceptual diagnosis presented in Part I (“The Canary in the Algorith- mic Coal Mine”), this paper provides the rigorous mathematical proof of the Infinite Choice Barrier. (ICB)1 We demonstrate that the limitations of algorithmic cognition described in the prelude are not merely engineering bottlenecks, but formal inevitabilities derived from a triangulation of three fundamental mathematical domains. Specifically, we prove: (1) via Computability Theory (Rice’s Theorem), that semantic frame adequacy is undecidable from within a system; (2) via Information Theory, that entropy diverges in heavy- tailed decision spaces (α≤1), rendering probabilistic inference structurally unstable; and (3) via Algorithmic Complexity (Chaitin’s Incompleteness), that frame-transcendent insights are algorithmically unrecognizable. Furthermore, weunifytheseresultsthroughacategoricalproofusingSheafTheory, demon- strating that “Gluing Failures” between consistent local semantic sections are mathematically necessary in irreducibly infinite decision spaces. All core theorems presented herein, including the Sheaf-Theoretic impossibility of global section construction, have been formalized and mechanically verified using the Coq Proof Assistant (code provided in Appendices).