“We need ABP”: six PDE specialists publish GPT-6 Astra’s proofs of the kinetic Krylov–Safonov theorem and of Pucci’s 1966 conjecture, with full chats (Loher, Mooney, Mouhot, Niebel, Silvestre, Zhu)
Awaiting review
Importance: major (4 of 5)The takeaway
Six analysts, among them Clément Mouhot and Luis Silvestre, say the missing estimate behind a long-open regularity theorem for kinetic equations was found by GPT-6 Astra Ultra on 6 Sept 2026 with no human contribution. They wrote a human exposition and published the prompts, chats and AI manuscripts, and state that they claim no credit.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 12 October 2026
Your AI and this story
- GPT-6 Astra162 days after its cutoff
- Claude Opus 5.5101 days after its cutoff
- Gemini 3.8 Flash192 days after its cutoff
- Grok 4.7131 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 101 days before it.
Key facts
- Paper: ‘An Aleksandrov estimate for kinetic equations with rough coefficients’, arXiv 2610.13921 [math.AP], 9 Oct 2026; Amélie Loher, Connor Mooney, Clément Mouhot, Lukas Niebel, Luis Silvestre, Yuzhe Zhu
- Abstract: ‘We give an exposition of a proof, discovered by Astra, of an Aleksandrov estimate for kinetic equations with rough coefficients.’
- Paper: ‘The proof of the kinetic Krylov–Safonov theorem was found by OpenAI’s GPT-6 Astra Ultra on September 6, 2026.’
- Theorem 1.1: for the Kolmogorov operator Ku = ∂_t u + v·∇x u − tr(A D²_v u) with λI ≤ A ≤ ΛI and merely measurable A, Ku = f ≥ 0 in Q_1 and u = 0 on the incoming boundary imply sup u ≤ C‖f‖{L^p(Q_1)} for some p(n, λ, Λ) < ∞
- Why it was missing (paper): the expansion-of-positivity and covering steps of Krylov–Safonov were known to extend to kinetic equations; ‘the main obstacle to extending the argument has been the lack of an ABP inequality, or a suitable substitute, for equations of Kolmogorov type or equations in Carnot groups’
- Method: by duality the estimate becomes an L log L (entropy) bound for the adjoint equation, obtained ‘by regularising along an auxiliary flow and exploiting entropy dissipation’ (site); the paper explains it in the elliptic, parabolic and kinetic settings
- Site ‘Source material’ page: ‘Here we collect AI-generated results. We do not claim credit for these results. Anyone could have obtained them by prompting the AI about it.’
- Source 01, kinetic Krylov–Safonov theorem: 6 Sept 2026, OpenAI GPT-6 Astra Ultra, ‘Approx. 150% of a Pro 20x plan’, ‘Human contribution: None’; prompt, chat and manuscript published
Show 4 more
- Pucci’s conjecture (1966): the ABP estimate sup|u| ≤ C‖L_A u‖_{L^p(B_1)} for trace-one matrices with ellipticity δ holds if and only if p > max{n/2, n[1 − (n−1)δ]}. Site: ‘This conjecture was solved by Astra’. Sources 03–04: dimensions d ≠ 3 (22 Sept, GPT-6 Astra Ultra) and d = 3 (1 Oct, GPT-6 Astra); Source 05: parabolic Pucci conjecture (5 Oct)
- Other sources: adjoint L log L estimate on the Heisenberg group and step-2 Carnot groups (10 Sept, Astra Ultra); generalisation to Carnot groups (7 Oct, GPT-6 Pro)
- Remark 1.3 of the paper: ‘A similar proof of the Aleksandrov estimate was made public in [8] on September 14, 2026. The precise involvement of AI in their proof was not made public.’ ([8] = Z. Hao and X. Zhang, arXiv 2609.15534)
- Open: the optimal exponent in the kinetic case (‘kinetic Pucci conjecture’); the authors say they are preparing lecture-note proofs of the full kinetic Krylov–Safonov theorem and an exposition of the Pucci proof
What happened
The Krylov–Safonov theorem (1979–80) says that solutions of elliptic and parabolic equations in nondivergence form are Hölder continuous and satisfy a Harnack inequality even when the coefficients are only bounded and measurable. Its proof starts from the Aleksandrov–Bakelman–Pucci (ABP) maximum estimate. For kinetic equations, where diffusion acts only in the velocity variable and transport spreads it to space, no ABP estimate was known, and the nondivergence theory was stuck there.
The six authors write that the proof of the kinetic theorem “was found by OpenAI’s GPT-6 Astra Ultra on September 6, 2026”. The new idea is to work with the adjoint equation and prove an entropy-type (L log L) bound by regularising along an auxiliary flow. Their paper explains this argument in the elliptic and parabolic cases first, then for the Kolmogorov equation. The same approach, pushed further by the model, is reported on their site to settle Pucci’s conjecture from 1966 on the best exponent p in the elliptic ABP estimate, and its parabolic analogue.
The project site is called “We need ABP”. It publishes each AI manuscript with its prompt, research instructions, the full chat, the model, the date and the share of a ChatGPT Pro plan used, and invites others to contribute expositions. A paper by Hao and Zhang (arXiv 2609.15534, 14 Sept) made a similar proof of the Aleksandrov estimate public; the authors note that the role of AI in that proof was not made public.
Why it matters
It is a case where recognised specialists (Mouhot is at the University of Cambridge, Silvestre at the University of Chicago, Mooney at UC Irvine) attribute a result in their own field wholly to a model and refuse credit for it. It also shows a new division of labour: the model produces the proof, and the experts’ paper is an exposition that makes the idea usable by others. The statements about Pucci’s conjecture rest on AI-written manuscripts that the group has not yet published a human account of, so they should be read as claims under review.
What is disputed or not yet verified
| Verification | Unrefereed preprint: human exposition of the Aleksandrov estimate by six specialists; the Pucci-conjecture manuscripts are AI-generated and their human exposition is still in preparation |
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No independent expert reaction, press coverage or Hacker News discussion was found on 12 Oct 2026. The AI manuscripts and chats were not read for this entry; dates, models and the “Human contribution: None” labels are quoted from the site’s source list.
Sources
5 sources from 2 sites. Numbers match the chips in the text.
5 sources: 5 primary
Primary
- arXiv 2610.13921: An Aleksandrov estimate for kinetic equations with rough coefficientsarxiv.org, paper
- We need ABP: an open mathematical collaboration (project site)weneedabp.github.io, official
- We need ABP: source material (AI manuscripts, prompts, chats, models and dates)weneedabp.github.io, official
- We need ABP: elliptic page (statement of Pucci’s conjecture)weneedabp.github.io, official
- Hao & Zhang, Harnack estimates for nondivergence kinetic equations (arXiv 2609.15534)arxiv.org, paper
Changes
- Filed from the arXiv PDF and weneedabp.github.io (found by a manual arXiv scan)