The Alignment Trap: Complexity Barriers
This work suggests that achieving safe, highly capable AI may be inherently impossible, posing a foundational challenge for the entire AI field.
The paper argues that AI alignment is fundamentally impossible due to logical contradictions, presenting five mathematical proofs showing that safe policies have measure zero, verification is computationally hard, training data is unobtainable, safety rules exceed network capacity, and optimization is anti-aligned.
This paper argues that AI alignment is not merely difficult, but is founded on a fundamental logical contradiction. We first establish The Enumeration Paradox: we use machine learning precisely because we cannot enumerate all necessary safety rules, yet making ML safe requires examples that can only be generated from the very enumeration we admit is impossible. This paradox is then confirmed by a set of five independent mathematical proofs, or "pillars of impossibility." Our main results show that: (1) Geometric Impossibility: The set of safe policies has measure zero, a necessary consequence of projecting infinite-dimensional world-context requirements onto finite-dimensional models. (2) Computational Impossibility: Verifying a policy's safety is coNP-complete, even for non-zero error tolerances. (3) Statistical Impossibility: The training data required for safety (abundant examples of rare disasters) is a logical contradiction and thus unobtainable. (4) Information-Theoretic Impossibility: Safety rules contain more incompressible, arbitrary information than any feasible network can store. (5) Dynamic Impossibility: The optimization process for increasing AI capability is actively hostile to safety, as the gradients for the two objectives are generally anti-aligned. Together, these results demonstrate that the pursuit of safe, highly capable AI is not a matter of overcoming technical hurdles, but of confronting fundamental, interlocking barriers. The paper concludes by presenting a strategic trilemma that these impossibilities force upon the field. A formal verification of the core theorems in Lean4 is currently in progress.