OpenAI, the Partition Principle, and Mathematics

What it claims

The critique is about presentation, not correctness. The author, a set theorist who works on the Axiom of Choice, says OpenAI's preprint on the Partition Principle cannot be followed. They argue that an expert in the area should be able to skim a paper on this topic and grasp its strategy and construction, and that this one does not allow that. They decline to verify it, and they withhold the bottle of whisky their Problems page offers for a solution. The post never says the proof is wrong.

Evidence

The specific complaints, which the summary does not give:

• Odd lemmas. Lemma 7.4 and Lemma 8.1 are named as results the author would not expect in a paper like this. Other theorems are described as "stated in a very strange manner."
• Terminology. The author calls it "a bit off."
• References:
• One of the author's own papers on the Bristol model is cited for the basic introduction of symmetric extensions. The author says it is the wrong source for that.
• At least three unpublished, unrefereed, non-arXiv lecture notes are cited. One is the author's own, used for a proposition the author says appears in published papers by them and by others.
• Scale. The post says OpenAI released "some hundreds" of such "solutions." The press release calls them "progress," while media coverage calls them "solutions." The author compares this to a denial of service on reviewers' time.

Limits

These are the points that matter if you plan to cite or rely on the result:

• The math was not checked. The author says they took only "a brief look" at the preprint and does not plan to read it properly. The post therefore neither confirms nor refutes that the Partition Principle fails to imply the Axiom of Choice.
• The Lean code was not reviewed. OpenAI also released a Lean formalization. The author skipped it, saying it was "enormous" and that they know little about using Lean. For an engineer, the Lean artifact is the obvious independent check, and this post gives no signal on it.
• No expert endorsement. The author's remedy is for OpenAI to consult domain mathematicians, at least on writing quality. As of this post, no such input is mentioned.
• No preprint details. The page gives no link to the preprint and no identifier for it in the fetched text.