AGI is Mathematically Impossible 2: When Entropy Returns - Infinite Choice Barrier, Shannon Revisited

!Author’s Note (Update - June 18, 2025) Subsequent to the completion of this paper, a widely discussed study by Shojaee et al. at Apple (The Illusion of Thinking, 2025) presented empirical evidence that several advanced reasoning models (e.g., Claude 3.7 Sonnet, DeepSeek-R1) exhibit a collapse in reasoning effort and accuracy as task complexity increases — despite sufficient inference budget. Notably, the observations they report closely mirror the predictions derived in Chapters 6 and 7 of this paper, particularly regarding entropy divergence in heavy-tailed decision spaces and the increased instability introduced by deeper model architectures. While developed independently and prior to these results, the framework presented here may be seen as a theoretical explanation for the empirical phenomena documented in their study. Abstract This paper presents a second, independent proof that Artificial General Intelligence (AGI) is mathematically impossible. While Part 1 established the Infinite Choice Barrier (ICB) through computability theory, this paper presents an information-theoretic proof that general algorithmic reasoning faces a fundamental structural limit — the Infinite Choice Barrier (ICB). In decision spaces governed by heavy-tailed distributions with tail exponent \alpha \leq 1, entropy does not converge, and additional information increases uncertainty rather than reducing it. We define this phenomenon as Information Opens, Entropy Rises (IOpenER) and show that such spaces are undecidable by algorithmic systems, regardless of compute scale or architectural depth. We also introduce the Layer Hypothesis, which predicts that deeper models produce more semantic divergence under structural instability. Taken together, these results offer a formal limit on the reasoning capacity of current AI systems. Notably, key predictions derived here have since been empirically confirmed in Shojaee et al. (Apple, 2025), which documents collapse behaviors in frontier reasoning models consistent with this framework. (Note: Sections 2.-4. are identical to those in the first paper "AGI is Impossible - Here is The Proof")