Angluin's condition

Formal theory shows infinite trivial output is provably necessary for AI systems generating valuable mathematics

A new arXiv paper models AI-assisted formal mathematics generation as a nested language-generation-in-the-limit problem, using a proof checker as a membership oracle and an adversarial enumeration of the mathematical literature as the signal for 'valuable' content. The authors prove a sharp dichotomy: generators emitting only finitely many trivial (correct but worthless) statements achieve at most α/2 coverage of unseen valuable mathematics, while allowing an infinite (but asymptotically vanishing) stream of trivia raises the optimum to 1−α/2. The central result is that a perfect verifier cannot substitute for mathematical taste, and the flood of certified-but-trivial output from AI proof systems is a provable mathematical necessity, not an engineering failure. The work formalizes the gap between formal verifiability and mathematical value, which is increasingly the binding constraint as AI-proof-assistant systems scale.