What was already known
Erdős asked the question for every irrational α. Earlier work proved it for at least one irrational α. A human-proved consecutive-prime string theorem supplies the deep input for the new universal deduction.
AI involvement
The paper credits the deduction entirely to an internal OpenAI model. Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke and Gregory Valiant digested and edited it; the prime-string theorem is earlier human work.
The paper attributes its Section 4 proof entirely to the internal model and describes the human authors' role as digesting and editing the write-up.
Importance
The deduction gives a universal negative answer to Erdős’s well-distribution question.
Limitations
The paper does not explicitly report a completed check of its Section 4 argument. A later Lean proof of the same theorem has not been mapped to this exact deduction and its cited prime theorem.
Technical verification
The paper does not explicitly report a completed check of its Section 4 deduction. Formal Conjectures points to Monticone's Lean formalization of the same theorem, but the reviewed record does not map that code's proof steps and assumptions to the OpenAI argument. A separate proof of the theorem does not itself verify the assessed AI proof, so no checking method is assigned to this revision.
Human understanding
The five human authors say they digested and clarified the proof; Section 4 explains its complete mechanism, from rational approximation to narrow blocks of consecutive prime fractional parts. No central gap in human explanation is reported.
Correction and publication history
The deduction gives a universal negative answer to Erdős’s well-distribution question.
Read this published revision