Smith–Toda V(4) for p ≥ 11, after OpenAI’s p = 1009
Event confirmedAwaiting review
Importance: major (4 of 5)The takeaway
A homotopy theorist checked two Smith–Toda papers from OpenAI’s maths release, found no errors but ‘strange choices’ (a suspiciously high prime, needless ultrafilters), and widened the result from one prime to all primes p ≥ 11.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 19 October 2026
Your AI and this story
- GPT-6 Astra169 days after its cutoff
- Claude Opus 5.5108 days after its cutoff
- Gemini 4 Argon199 days after its cutoff
- Grok 4.7138 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 108 days before it.
Key facts
- Paper: Vigleik Angeltveit, ‘The Smith-Toda complex V(4) exists for p ≥ 11’, arXiv 2610.23215 [math.AT], submitted 16 Oct 2026
- Theorem 1.1: for p ≥ 7, v4 is a permanent cycle in the Adams–Novikov spectral sequence for π∗V(3); for p ≥ 11 there is a finite p-local spectrum X with BP∗X ≅ BP∗/(p, v1, v2, v3, v4)
- OpenAI papers addressed: ‘A finite Smith–Toda complex V(4) at the prime 1009’ and ‘Finite Smith–Toda Complexes at Varying Primes’ (V(n) exists at some prime for every n), both dated 23 Sept 2026, family 308 of github.com/openai/math, which has no Lean scope note
- Critique: ‘We thought both of these papers looked a bit suspicious. We have not found any mathematical errors, but both of them make some strange choices… 1009 is suspiciously high.’ The second paper ‘uses ultrafilters throughout… We speculate that ultrafilters came from one of the references, and the AI model left them in’
- Simplifications: OpenAI’s Section 4 replaced by a citation of Toda; Section 6 by ‘a much simpler May spectral sequence calculation’; Section 7 by ‘a simple Toda bracket juggling trick’; the hardest step of the OpenAI paper gets ‘a much shorter argument (Theorem 3.8)’
- AI disclosure: ‘ChatGPT (OpenAI) was used extensively as an interactive research assistant… including for checking calculations, locating relevant literature, and exploring simplifications of the arguments in the OpenAI Smith-Toda papers. In particular, conversations with ChatGPT helped lead to the reformulation of the final Adams-Novikov obstruction in terms of Toda’s classical bracket-juggling argument. The author independently checked the mathematical arguments’
- No Hacker News thread or high-reach X post found on 19 Oct morning
What happened
Smith–Toda complexes V(n) are basic building blocks in chromatic homotopy theory: finite spectra whose Brown–Peterson homology is BP∗ modulo (p, v1, …, vn). Per Angeltveit’s background section (citing Toda), V(3) exists for p ≥ 7 and has the ring structure needed here only for p ≥ 11.
OpenAI’s 6 Oct release contained two papers on the question (family 308). Angeltveit worked through the first, kept its overall plan and two differentials that must vanish, and found shorter routes for the hard parts. That gave the construction for every prime from 11 up. Conversations with ChatGPT helped him find the Toda-bracket reformulation of the last obstruction.
Why it matters
With the Brennan paper posted the same day (Brennan’s conjecture: experts confirm and shorten OpenAI’s proof), this is a second case of a specialist checking an OpenAI release paper, finding it correct, and quickly improving it. Angeltveit’s remarks about the odd prime 1009 and leftover ultrafilters are a rare concrete description of what an AI-written paper looks like to an expert: valid, but shaped by its search process rather than by the natural statement.
What is disputed or not yet verified
| Verification | Unrefereed arXiv preprint by a specialist; OpenAI papers expert-read, no errors found |
|---|
Sources
3 sources from 2 sites. Numbers match the chips in the text.
3 sources: 3 primary
Primary
Changes
- Removed an unsourced background sentence on V(4)’s prior status
- Filed