--- id: "2026-10-16-smith-toda-v4-angeltveit-improves-openai" url: "https://postcutoff.com/e/2026-10-16-smith-toda-v4-angeltveit-improves-openai/" as_of: "2026-10-19T09:28:00+02:00" date: "2026-10-16" date_precision: day category: science importance: 4 confidence: high status: [Event confirmed, Awaiting review] verification: Preprint only sources: 3 editor: human editor human_review: null version: "2026-10-19" --- As of: 2026-10-19 09:28 CEST. Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/ Canonical page: https://postcutoff.com/e/2026-10-16-smith-toda-v4-angeltveit-improves-openai/ # Smith–Toda V(4) for p ≥ 11, after OpenAI's p = 1009 Full title: Smith–Toda complex V(4) exists for all primes p ≥ 11: Angeltveit extends and simplifies OpenAI's AI-written p = 1009 construction, with ChatGPT's help Vigleik Angeltveit (ANU) posted arXiv 2610.23215 on 16 Oct 2026. It proves that the Smith–Toda complex V(4), a finite p-local spectrum with BP-homology BP∗/(p, v1, v2, v3, v4), exists for every prime p ≥ 11. This improves "the recently announced result by OpenAI that V(4) exists at the prime 1009" (openai/math family 308). He replaces three sections of OpenAI's argument with a reference to Toda, a simpler May spectral sequence computation and a Toda-bracket trick. He found no mathematical errors in the OpenAI papers, but called them "a bit suspicious", and speculates that their ultrafilters were copied in by the AI model from an irrelevant reference. ChatGPT was "used extensively as an interactive research assistant" for his own paper. ## 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 ([2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov](https://postcutoff.com/e/2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov/index.md)), 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 ## Your AI and this story - GPT-6 Astra (training cutoff April 2026): 169 days after its cutoff - Claude Opus 5.5 (training cutoff June 2026): 108 days after its cutoff - Gemini 4 Argon (training cutoff March 2026): 199 days after its cutoff - Grok 4.7 (training cutoff May 2026): 138 days after its cutoff ## Sources 1. [arXiv 2610.23215: The Smith-Toda complex V(4) exists for p ≥ 11 (Angeltveit)](https://arxiv.org/abs/2610.23215) (arxiv.org, paper) 2. [OpenAI math release: A finite Smith–Toda complex V(4) at the prime 1009 (PDF)](https://github.com/openai/math/blob/main/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf) (github.com, paper) 3. [OpenAI math release: Finite Smith–Toda Complexes at Varying Primes (PDF)](https://github.com/openai/math/blob/main/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf) (github.com, paper) ## Changes - 2026-10-19: Removed an unsourced background sentence on V(4)'s prior status - 2026-10-19 (filed): Created ## Related - 2026-10-19: [AI-assisted results in the 19 Oct 2026 arXiv listing](https://postcutoff.com/e/2026-10-19-ai-assisted-results-october-19-listing/index.md) - 2026-10-16: [Brennan's conjecture: experts confirm and shorten OpenAI's proof](https://postcutoff.com/e/2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov/index.md) - 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)