Axiom of Choice Has Geometric Correlate in Neural Proofs

A new study delves into the foundational differences between classical and constructive mathematics by examining how the Axiom of Choice manifests within neural proof embeddings. Using Lean 4's kernel-level tracking, researchers analyzed over 470,000 mathematical declarations from Mathlib, focusing on theorems' dependence on the Axiom of Choice. They trained a self-supervised proof encoder on constructive proofs and observed that classical proofs exhibit a distinct geometric signature in this proof space. This signature, measured by anomaly score, reconstruction loss, and density containment, declines predictably with the proof's distance from the axiom in the dependency graph, showing clear separation at shallow distances. The findings also have operational implications for AI theorem provers: Lean's `aesop` tactic solved constructive theorems 13 times faster than classical ones. A neural-guided hybrid prover reduced this gap to 5 times, and the geometric anomaly score proved predictive of `aesop` failure, even beyond proof length. This research establishes a tangible link between abstract mathematical axioms and the performance characteristics of neural theorem provers.