What was already known
Burr and Erdős asked whether such a basis exists. The positive-upper-density and exact-density variants are separate questions; a later DeepMind Lean proof addresses the same bounded-gap question through a separately credited argument.
AI involvement
The paper credits its proof entirely to an internal OpenAI model. Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke and Gregory Valiant digested and edited the explanation; the later DeepMind proof remains separate work.
The paper expressly attributes its Section 3 proof entirely to an internal OpenAI model and limits the named human authors' role to digesting and editorial clarification.
Importance
Resolves the bounded-gap partition question for additive bases.
Limitations
No completed check of this specific powers-of-five proof is documented. The later Lean proof of the theorem has not been shown to replay or check this construction.
Technical verification
The paper provides a complete proof, but its AI comment documents human digestion and editing, not an explicit completed check of the exact powers-of-five argument. The later DeepMind Lean proof addresses the same bounded-gap problem through a separately credited proof; it is not evidence that OpenAI's construction was checked. No checked method for the assessed argument is established.
Human understanding
The five named human authors say they digested the proof and revised its explanation; Section 3 clearly explains how the block construction covers all large integers and how the J_k intervals force arbitrarily long monochromatic gaps. No central unexplained step is reported.
Correction and publication history
Resolves the bounded-gap partition question for additive bases.
Read this published revision