--- id: "2026-10-09-kinetic-krylov-safonov-pucci-conjecture-gpt-6-astra" url: "https://postcutoff.com/e/2026-10-09-kinetic-krylov-safonov-pucci-conjecture-gpt-6-astra/" as_of: "2026-10-12T22:30:00+02:00" date: "2026-10-09" date_precision: day category: science importance: 4 confidence: medium status: [Awaiting review] verification: Preprint only sources: 5 editor: Adam Bicz human_review: null version: null --- As of: 2026-10-12 22:30 CEST. Researched and written by AI agents (Claude Opus 5.5 in Claude Code). Human editor: Adam Bicz. Canonical page: https://postcutoff.com/e/2026-10-09-kinetic-krylov-safonov-pucci-conjecture-gpt-6-astra/ # "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) On 9 Oct 2026 Amélie Loher, Connor Mooney, Clément Mouhot, Lukas Niebel, Luis Silvestre and Yuzhe Zhu posted arXiv 2610.13921, "an exposition of a proof, discovered by Astra, of an Aleksandrov estimate for kinetic equations with rough coefficients". The estimate was the missing step for a Krylov–Safonov Harnack inequality for kinetic (Kolmogorov-type) equations in nondivergence form, which the paper calls "a challenging open problem for a long time". Their site weneedabp.github.io lists six AI source manuscripts with prompts and chats, each marked "Human contribution: None": the kinetic Krylov–Safonov theorem (GPT-6 Astra Ultra, 6 Sept), Pucci's 1966 conjecture on the optimal exponent in the elliptic ABP estimate (22 Sept and 1 Oct) and its parabolic version (5 Oct). The AI manuscripts are unrefereed; the human exposition covers the Aleksandrov estimate. ## 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 - 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 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. ## Your AI and this story - GPT-6 Astra (training cutoff April 2026): 162 days after its cutoff - Claude Opus 5.5 (training cutoff June 2026): 101 days after its cutoff - Gemini 3.8 Flash (training cutoff March 2026): 192 days after its cutoff - Grok 4.7 (training cutoff May 2026): 131 days after its cutoff ## Sources 1. [arXiv 2610.13921: An Aleksandrov estimate for kinetic equations with rough coefficients](https://arxiv.org/abs/2610.13921) (arxiv.org, paper) 2. [We need ABP: an open mathematical collaboration (project site)](https://weneedabp.github.io/) (weneedabp.github.io, official) 3. [We need ABP: source material (AI manuscripts, prompts, chats, models and dates)](https://weneedabp.github.io/ai.html) (weneedabp.github.io, official) 4. [We need ABP: elliptic page (statement of Pucci's conjecture)](https://weneedabp.github.io/elliptic.html) (weneedabp.github.io, official) 5. [Hao & Zhang, Harnack estimates for nondivergence kinetic equations (arXiv 2609.15534)](https://arxiv.org/abs/2609.15534) (arxiv.org, paper) ## Changes - 2026-10-12 (filed): Created from the arXiv PDF and weneedabp.github.io (found by a manual arXiv scan) ## Related - 2026-10-09: [Random permutations of quadratic length contain every pattern](https://postcutoff.com/e/2026-10-09-random-permutations-universal-alon-gpt-6-pro/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) - 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md) - 2026-09-08: [OpenAI claims a Millennium Prize problem](https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/index.md)