{"schema":"postcutoff/event@1","as_of":"2026-10-27T13:45:00+01:00","url":"https://postcutoff.com/e/2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim/","md":"https://postcutoff.com/e/2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims local smoothing and Kakeya in all dimensions","deck":null,"takeaway":"A 59-page OpenAI manuscript dated 25 Oct claims Sogge's local smoothing conjecture for the wave equation in every dimension n ≥ 2. The stated corollaries include the Bochner–Riesz conjecture, Fourier restriction throughout the interior of the conjectured range, the Kakeya and Nikodym maximal conjectures, and full Hausdorff dimension for every Euclidean Kakeya set. That last one is the Kakeya set conjecture, proved in ℝ³ only in 2025 (Wang–Zahl). OpenAI lists Lean comparator statements for the main estimate and the corollaries. No expert has assessed it yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: Working with academic communities to build scientific understanding (26 Oct 2026)","url":"https://openai.com/index/building-scientific-understanding-with-academic-communities/","type":"official","group":"primary","domain":"openai.com"},{"n":2,"title":"OpenAI (paper, PDF): Local smoothing for the Euclidean wave equation (25 Oct 2026)","url":"https://github.com/openai/math/blob/main/preprints/Local-smoothing-for-the-Euclidean-wave-equation-October-25-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"Lean scope note for family 079","url":"https://github.com/openai/math/blob/main/lean/docs/079.md","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"openai/math history.md (26 Oct 2026 changelog)","url":"https://github.com/openai/math/blob/main/history.md","type":"official","group":"primary","domain":"github.com"},{"n":5,"title":"Wang & Zahl (2025): the Kakeya set conjecture in three dimensions (arXiv 2502.17655)","url":"https://arxiv.org/abs/2502.17655","type":"paper","group":"primary","domain":"arxiv.org"},{"n":6,"title":"PANews: OpenAI announces three new AI science discoveries","url":"https://panews.io/articles/01a17a35-769f-778a-a1fc-704ce7e3b9e2","type":"press","group":"press","domain":"panews.io"},{"n":7,"title":"James Leng on X: 'The proof is absolutely shocking' (27 Oct)","url":"https://x.com/JamesLeng15/status/2114875830228906072","type":"discussion","group":"reactions","domain":"x.com"},{"n":8,"title":"oai-math-revised: AGENT_NOTES for the local smoothing paper","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Local-smoothing-for-the-Euclidean-wave-equation-October-25-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":5,"filed":"2026-10-26","updated":"2026-10-27","orgs":["OpenAI"],"title":"OpenAI claims Sogge's local smoothing conjecture in every dimension, and with it the Kakeya conjecture (full Hausdorff dimension in ℝⁿ), Bochner–Riesz and most of Stein's restriction programme (Lean-formalized per OpenAI)","summary":"On 26 Oct 2026 OpenAI added \"Local smoothing for the Euclidean wave equation\" (59 pages, credited to \"OpenAI\", dated 25 Oct) to family 079 of github.com/openai/math, which until then claimed only the three-dimensional case. The paper claims Sogge's local smoothing conjecture in every spatial dimension n ≥ 2. It derives the Bochner–Riesz conjecture, sharp Stein square-function bounds, Fourier restriction for the sphere, paraboloid and cone in the interior of the conjectured range, and the Kakeya and Nikodym maximal conjectures: \"every Euclidean Kakeya set has full Hausdorff dimension\". OpenAI says these statements are formalized in Lean. The blog post announcing it names it as the maths result among three. It is unrefereed and unreviewed.","key_facts":["Repository changelog (history.md, 26 Oct, verbatim): 'Local smoothing for the Euclidean wave equation. The proof uses an auxiliary partial differential equation to establish Sogge’s local smoothing for waves in every spatial dimension at least two. It also yields the Kakeya maximal and dimension conjectures and Stein’s spherical restriction estimates throughout the interior of the conjectured range. We are releasing the model output directly, as we did in our October 6 release.'","Theorem 1.1 (paper): for n ≥ 2, 2 ≤ p < ∞, ε > 0, ‖e^{-iT|D|}f‖_{Lᵖ(ℝⁿ×[1,2])} ≤ C‖J^{s₊(p)+ε}f‖_{Lᵖ}, with s₊(p) = max(0, (n−1)/2 − n/p); a focusing wave shows the threshold is sharp (Prop. 8.1)","Abstract (verbatim): 'We deduce the Bochner–Riesz conjecture, maximal estimates for these means, and sharp bounds for Stein’s square function. We obtain Fourier restriction estimates for the sphere, the compact paraboloid, and the truncated cone throughout the interior of the conjectured range, as well as local smoothing for the Schrödinger equation. We also prove the Kakeya and Nikodym maximal conjectures. In particular, every Euclidean Kakeya set has full Hausdorff dimension.'","Lean scope note (lean/docs/079.md): covers the half-wave local smoothing bound for every n ≥ 2 and 2 ≤ p < ∞; uniform Bochner–Riesz bounds, spherical extension and Stein square-function bounds in their strict ranges; an Lⁿ Kakeya maximal bound with δ^(−ε) loss; and 'Every subset of ℝⁿ containing a unit segment in each direction has Hausdorff dimension n'. Comparator file: ComparatorChallenges/LocalSmoothing.lean. We have not run the check","Before this, family 079 held only 'Critical local smoothing for the three-dimensional wave equation' (165 pp., 24 Sept), with no Lean link as of 19 Oct; family 074 claimed the 3D Kakeya maximal and 4D Kakeya dimension conjectures (manuscript only)","Human state of the art: the Kakeya set conjecture is proved in ℝ² (Davies 1971) and ℝ³ (Hong Wang and Joshua Zahl, 2025); it is open in ℝⁿ for n ≥ 4. Local smoothing was known in 2+1 dimensions (Guth–Wang–Zhang, 2020) and in partial ranges elsewhere. Local smoothing is known to imply Bochner–Riesz, which implies restriction, which implies Kakeya","Other results in the same OpenAI announcement: 'The Exponential Time Hypothesis is False' (separate entry) and, an existence claim for 4D quantum field theories including QCD and pure Yang–Mills, reviewed by a group led by David Gross with the AI artifacts withheld (entry 2026-10-26-openai-yang-mills-qcd-existence-claim)","Unlike the ETH paper, the changelog names no review group for this paper. No expert reaction found on the evening of 26 Oct","Dates (per the oai-math-revised hub's summary of OpenAI's blog post, which returns 403 to us): the AI output was obtained on 18 Oct and formalized in Lean the following day","First expert reaction found: James Leng on X, 27 Oct: 'The proof is absolutely shocking. It is only ~25 pages and primarily uses elliptic PDEs. It notably does not use much of the mathematics developed by the harmonic analysis lineage.' No named review group, follow-up paper or public check by Kakeya/restriction specialists by 27 Oct; Quanta's math-and-AI dispatch (latest item 22 Oct) predates the release"],"key_numbers":[],"tags":["math","harmonic-analysis","local-smoothing","kakeya","bochner-riesz","fourier-restriction","openai","openai-math-release","lean","unverified","claim"],"science":{"field":"mathematics","subfield":"harmonic analysis (local smoothing, Bochner–Riesz, Fourier restriction, Kakeya)","problem":"Sogge's local smoothing conjecture (1991) in all dimensions; consequences: Bochner–Riesz, Stein's restriction conjecture (interior of range), Kakeya maximal and set conjectures in ℝⁿ","result":"Claimed proof of local smoothing for the Euclidean wave equation in every dimension n ≥ 2 with ε loss, with Bochner–Riesz, restriction in the interior of the range, Nikodym/Kakeya maximal bounds and full Hausdorff dimension of Kakeya sets as corollaries","open_since":"1991","ai_system":["OpenAI internal model (openai/math release)"],"human_role":"Credited to 'OpenAI'; released as 'model output directly', with no named human authors and no designated reviewers","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed manuscript; OpenAI lists Lean comparator statements for the main estimate and the corollaries (not checked by us); no expert assessment yet"},"status":"pending","shock":"Kakeya in dimensions ≥ 4 and Bochner–Riesz in dimensions ≥ 3 were among the central open problems of harmonic analysis. The 3D Kakeya set conjecture (Wang–Zahl 2025, cited for Hong Wang's 2026 Fields Medal) would become a special case."},"body_md":"## What happened\n\nOpenAI's 26 Oct repository update replaced its September three-dimensional local smoothing claim with a much larger one: Sogge's\nconjecture for the wave equation in every dimension. OpenAI says the proof \"uses an auxiliary partial differential equation\". The\npaper is 59 pages, against 165 for the three-dimensional version.\n\nThe implications run down a well-known chain. Local smoothing implies the Bochner–Riesz conjecture, Bochner–Riesz implies Stein's\nrestriction conjecture, and restriction implies the Kakeya conjecture. So the paper also claims Kakeya-type results in every\ndimension, including that every set in ℝⁿ containing a unit segment in every direction has Hausdorff dimension n. Humans had\nproved that only in the plane (1971) and, in 2025, in three dimensions.\n\nOpenAI says each of these statements has a Lean comparator target. The three-dimensional families had no formal proof when we\nchecked on 19 Oct. The changelog says this is \"model output directly\", released as on 6 Oct, and names no outside reviewers\n(the ETH paper released the same day has a working group).\n\n## Why it matters\n\nIf correct, this settles a large part of Euclidean harmonic analysis in one paper. Formal verification of statements this deep\nin analysis would also be a first. The open questions are whether the Lean definitions (Sobolev norms, Hausdorff dimension,\nKakeya maximal operator) match the standard ones, and whether the paper's argument holds up when experts read it. Expect\nreactions from the harmonic-analysis community, including Guth, Wang, Zahl and Tao, in the coming days.","disputed":[{"label":"Verification","text":"Unrefereed manuscript; OpenAI lists Lean comparator statements for the main estimate and the corollaries (not checked by us); no expert assessment yet","sources":[]}],"related":[{"id":"2026-10-26-openai-yang-mills-qcd-existence-claim","url":"https://postcutoff.com/e/2026-10-26-openai-yang-mills-qcd-existence-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims 4D Yang–Mills and QCD existence","deck":null,"takeaway":"The physics third of OpenAI's 26 Oct announcement claims that certain interacting 4D quantum field theories exist mathematically, among them pure Yang–Mills and quantum chromodynamics. That is the existence half of the Yang–Mills Millennium problem, not the mass gap. A group led by David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, is reviewing it. At the group's request the AI-produced material is not public while they write their own account. Nothing can be checked yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":1,"filed":"2026-10-27","updated":"2026-10-27","orgs":["OpenAI"]},{"id":"2026-10-26-openai-eth-false-subexponential-sat-claim","url":"https://postcutoff.com/e/2026-10-26-openai-eth-false-subexponential-sat-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims ETH is false","deck":null,"takeaway":"Anthropic independently posted a different Lean-checked refutation the next day (separate entry). OpenAI's model output claims a deterministic algorithm that counts the satisfying assignments of any k-CNF in time 2^(n/(log n)^(1/2+o(1))). If correct, it refutes ETH, SETH, Gap-ETH and NSETH, the assumptions behind hundreds of fine-grained hardness results. OpenAI says the main theorem is formalized in Lean. A working group led by ETH's proposers, Russell Impagliazzo and Ramamohan Paturi, with Ryan Williams, will examine the proof. No expert has assessed it in public yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":19,"official":7,"filed":"2026-10-26","updated":"2026-10-27","orgs":["OpenAI"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":77,"official":13,"filed":"2026-10-07","updated":"2026-10-25","orgs":["OpenAI"]},{"id":"2026-10-06-openai-release-3d-restriction-bochner-riesz-claims","url":"https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims 3D restriction and Bochner–Riesz conjectures","deck":null,"takeaway":"Beside quasi-RH and Unique Games, OpenAI's 6 Oct catalogue claims a whole cluster of central harmonic-analysis conjectures in three dimensions: Stein's restriction conjecture for the sphere, the diagonal extension conjecture, Bochner–Riesz, Sogge's local smoothing and the planar endpoint of Carleson's Schrödinger problem. None has a Lean formalization and no expert review has been published; the first arXiv paper to cite one appeared on 15 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":7,"filed":"2026-10-19","updated":"2026-10-26","orgs":["OpenAI"]},{"id":"2026-07-23-fields-medals-2026","url":"https://postcutoff.com/e/2026-07-23-fields-medals-2026/","date":"2026-07-23","date_precision":"day","short_title":"2026 Fields Medals go to Hong Wang, Yu Deng, John Pardon and Jacob Tsimerman at ICM Philadelphia","deck":null,"takeaway":"At the opening of the International Congress of Mathematicians in Philadelphia on July 23, 2026, the IMU awarded the Fields Medals to Hong Wang, Yu Deng, John Pardon and Jacob Tsimerman.","category":"milestone","category_label":"Milestones","importance":2,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":4,"official":1,"filed":"2026-10-02","updated":"2026-10-04","orgs":["International Mathematical Union"]}],"people":[],"posts":[{"id":"2026-10-26-gdb-three-scientific-discoveries","title":"Greg Brockman announces OpenAI's three discoveries (Kakeya/local smoothing, ETH false, 4D QFT)","url":"https://postcutoff.com/p/2026-10-26-gdb-three-scientific-discoveries/"},{"id":"2026-10-26-danintheory-three-discoveries","title":"Dan Roberts (OpenAI) announces the three discoveries in maths, CS and physics","url":"https://postcutoff.com/p/2026-10-26-danintheory-three-discoveries/"}],"videos":[],"models":[],"changes":[{"date":"2026-10-27","type":"edit","text":"First expert reaction (James Leng), discovery/formalization dates via the community hub"},{"date":"2026-10-27","type":"edit","text":"Linked the Yang–Mills/QCD entry"},{"date":"2026-10-26","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-26","run":null,"sources_read":null,"updated":"2026-10-27","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":179,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":118,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":209,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":148,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-local-smoothing-kakeya"}