OpenAI Codex helps prove Spherical Hadwiger Conjecture
A blog post titled "Is mathematics about to enter the conservatory?" reports that a preprint titled "The Spherical Hadwiger Theorem" landed in the author's inbox the same week Claude finished formalizing a proof of Fermat's Last Theorem in Lean. The preprint, by Wang & Wu of Hunan University, proves the Spherical Hadwiger Conjecture, a piece of integral-geometric machinery that has been open since about 1974. The paper's final section carries a disclaimer: "During the preparation of this manuscript, OpenAI Codex was used to assist with developing proof details, identifying gaps and points requiring clarification, organizing and typesetting the manuscript, and editing the English. The authors reviewed and verified all AI-assisted mathematical content and suggested changes, made all final mathematical and editorial decisions, and take full responsibility for the manuscript."
The blog's author reads that disclaimer as leaving open the possibility that Codex did a substantial share of work that, until recently, required a research-level mathematician: developing proof details, finding and fixing gaps, and apparently writing the paper itself. The author calls the result polished and readable to a research mathematician in the field, but has not fully verified the proof personally; working through it with Claude Fable, the author says it "passes the sniff test" without being fully digested yet. The author states this is no knock on Wang & Wu, calls it a great paper worth digesting, and notes the authors followed the principles for AI use laid out in the Leiden Declaration.
The result lands close to home for the author, who tried to prove the same theorem in grad school and made a half-hearted attempt again with AI assistance earlier this year. The author connects this to an OpenAI report released around the GPT-4 launch on the labor impact of LLMs, which modeled the exposure of different professions and estimated that 100% of a mathematician's job was exposed to LLM disruption across three distinct labor models, a higher figure than for writers, translators, artists, and graphic designers. The author suggests the delay in AI reaching this level in mathematics likely owes less to the field being uniquely hard and more to research mathematics carrying less economic value and less training data than those other creative domains.
The author then draws a comparison to classical music, which society no longer supports the way it supports popular music: it has been institutionalized into conservatories, with a small segment of society funding the training and professional careers of a few top performers. The author asks whether mathematics could follow the same path, noting that pure mathematics is arguably already in a conservatory of sorts, the academy, where jobs outside the university remain scarce and results are understood by only a select few. The author's conclusion is that it is worth continuing to support the cultural work of research mathematics, but that the incentive structures behind it, like those in other creative fields, are ill-suited to what AI is bringing: the tools make it harder to tell whether a complex argument is even correct, let alone who deserves funding, tenure, and recognition for it.
Key facts
- A preprint by Wang & Wu of Hunan University proves the Spherical Hadwiger Conjecture, open since about 1974.
- The paper's own disclaimer credits OpenAI Codex with assisting on proof details, gap-finding, typesetting, and English editing, while stating the authors verified all AI-assisted mathematical content and take full responsibility for it.
- The blog's author has not fully verified the proof; working through it with Claude Fable, the author says it passes the sniff test but is not yet fully digested.
- The post cites an OpenAI GPT-4-era labor-impact whitepaper estimating that 100% of a mathematician's job was exposed to LLM disruption across three labor models, higher than for writers, translators, artists, and graphic designers.
- The author proposes research mathematics may follow classical music into an academy-only "conservatory" model as AI absorbs more of the actual proof work.
Why it matters
The disclaimer in the Wang & Wu preprint is a rare documented case of an AI tool credited with developing proof details and identifying gaps in a genuine, decades-old open conjecture, not just formatting or checking finished work. Paired with an older OpenAI estimate that put mathematicians at 100% labor exposure to LLMs, the story reads as evidence that research-level mathematical work, long assumed to be insulated by its difficulty, is starting to move the way writing and translation already have.
Who it affects
Research mathematicians and the academic departments that employ, fund, and grant tenure to them are the direct subject; Wang & Wu and the broader community working on integral-geometric problems are the immediate case. The piece also implicates the AI labs building the tools, OpenAI's Codex and Anthropic's Claude both appear in the post, and anyone setting policy on how AI assistance in research should be disclosed and credited.
How to use it
The preprint's own disclaimer is offered as a model of disclosure: it names the tool, states exactly what it was used for, and has the authors explicitly claim final responsibility for the mathematical content. The author notes Wang & Wu followed the AI-use principles set out in the Leiden Declaration, framing that as the kind of accountability other AI-assisted papers should match.
How solid is it
The blog author is explicit about the limits of the verification here: the proof has not been fully checked, only worked through with Claude Fable to the point of passing a "sniff test." The preprint's disclaimer itself states the human authors reviewed and verified all AI-assisted mathematical content and made the final mathematical and editorial decisions, but the source gives no indication of peer review or journal acceptance, and does not specify which Codex model version was used.
Risks and caveats
The author's central worry is not that the proof is wrong but that AI assistance makes it structurally harder to tell whether a complex argument is correct at all, and harder still to decide who deserves funding, tenure, or credit for it. The broader "conservatory" argument is the author's own framing, not an established consensus, and rests on an economic analogy to classical music rather than on new data about mathematics specifically.
“During the preparation of this manuscript, OpenAI Codex was used to assist with developing proof details, identifying gaps and points requiring clarification, organizing and typesetting the manuscript, and editing the English. The authors reviewed and verified all AI-assisted mathematical content and suggested changes, made all final mathematical and editorial decisions, and take full responsibility for the manuscript.”
— AI-use disclaimer in the Wang & Wu preprint