{"schema":"postcutoff/event@1","as_of":"2026-10-19T22:14:00+02:00","url":"https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/","md":"https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/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":"2026-10-06-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":[{"n":1,"title":"OpenAI math catalogue (CONTENTS.md, families 074–080)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI: Elliptic capacity propagation and Fourier restriction to the sphere (24 Sept 2026, PDF)","url":"https://github.com/openai/math/blob/main/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"OpenAI: Diagonal Fourier extension for positively curved surfaces in three dimensions (PDF)","url":"https://github.com/openai/math/blob/main/preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":4,"title":"OpenAI: Bochner–Riesz multipliers in three dimensions (PDF)","url":"https://github.com/openai/math/blob/main/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":5,"title":"OpenAI: Critical local smoothing for the three-dimensional wave equation (PDF)","url":"https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":6,"title":"OpenAI: Endpoint convergence for the planar Schrödinger equation (PDF)","url":"https://github.com/openai/math/blob/main/preprints/Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":7,"title":"arXiv 2610.22634: Probabilistic Fourier extension and the Schrödinger equation (Korobenko, Sawyer)","url":"https://arxiv.org/abs/2610.22634","type":"paper","group":"primary","domain":"arxiv.org"}],"official":7,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI"],"title":"OpenAI's maths release also claims the 3D Fourier restriction, Bochner–Riesz and local smoothing conjectures (families 077–080): the core of Stein's harmonic-analysis programme in three dimensions, manuscript only","summary":"OpenAI's 6 Oct 2026 release of AI-written mathematics (github.com/openai/math) contains four harmonic-analysis families that claim central open problems in three dimensions. Family 077 claims Stein's Fourier restriction conjecture for the sphere in its bounded-data form (the extension operator maps L∞(S²) to Lᵖ(ℝ³) for every p > 3, 103 pages) and the diagonal Lᵖ → Lᵖ form for all compact positively curved surfaces (25 pages). Family 078 claims the 3D Bochner–Riesz conjecture (114 pages), 079 Sogge's local smoothing conjecture for the 3D wave equation (165 pages) and 080 the sharp Sobolev endpoint of Carleson's Schrödinger convergence problem (83 pages for the planar H^(1/3) case). The papers are dated 24 Sept and credited to \"OpenAI\". The catalogue gives none of them a Lean link, and we found no expert assessment. The first citation on arXiv is Korobenko and Sawyer (15 Oct), who call the diagonal theorem \"claimed\" and build a probabilistic analogue for the Schrödinger equation.","key_facts":["Family 077 (catalogue): 'Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions … bounded from Lᵖ(Σ) to Lᵖ(ℝ³) for every p > 3.' Two papers dated 24 Sept 2026: 'Elliptic capacity propagation and Fourier restriction to the sphere' (103 pp.; 'the Fourier extension operator maps L∞(S²) boundedly into Lᵖ(ℝ³) for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp') and 'Diagonal Fourier extension for positively curved surfaces in three dimensions' (25 pp.)","Prior state of the art, as the sphere paper itself summarises it: Guth's polynomial partitioning gave the bounded-data estimate for p > 13/4, Hong Wang's broom method p > 42/13, and Wang–Wu the diagonal estimate for p > 22/7. The claim closes the remaining gap to p > 3","Dependencies: the diagonal paper 'uses two theorem inputs': the packet propagation theorem of the sphere paper and 'The Kakeya maximal theorem [12, Theorem 1.1]', i.e. OpenAI's family 074 manuscript 'The Kakeya maximal conjecture in three dimensions' (23 Sept). The restriction claim therefore stands or falls with the Kakeya maximal claim, which Levent Alpöge linked to a scooping dispute","Family 078: 'Resolves the three-dimensional Bochner–Riesz conjecture in its strict range' (paper 'Bochner–Riesz multipliers in three dimensions', 114 pp.; main estimate: Tδ bounded on L³(ℝ³) for every δ > 0, then interpolation and duality)","Family 079: 'Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions' (paper 'Critical local smoothing for the three-dimensional wave equation', 165 pp.; critical L³ estimate with every positive Sobolev loss)","Family 080: 'Proves almost-everywhere convergence e^{itΔ}f → f … for every f ∈ H^{n/(2(n+1))}(ℝⁿ) and every dimension n ≥ 2', the sharp Sobolev endpoint of Carleson's problem (planar paper 'Endpoint convergence for the planar Schrödinger equation', 83 pp.)","Verification: the catalogue entries for 074, 077, 078, 079 and 080 carry no Lean link (073 Falconer and 075 L log L Fourier convergence do), lean/docs/077.md returns 404, and neither 077 PDF mentions Lean. Korobenko–Sawyer's remark that OpenAI proved the diagonal case 'with L∃∀N certification' is not supported by the repository as checked on 19 Oct","First follow-up: Lyudmila Korobenko (Reed College) and Eric T. Sawyer (McMaster), 'Probabilistic Fourier extension and the Schrödinger equation', arXiv 2610.22634 (submitted 15 Oct, 8 pp.): 'Just recently, OpenAI has claimed to prove the diagonal inequality p = q in the case of d = 3 dimensions'; their averaged Sobolev regularity result for the Schrödinger equation is the probabilistic counterpart of 'the deterministic analogue in three dimensions … obtained recently by OpenAI'. The paper has no AI-use statement","Coverage gap: our main release entry listed Kakeya maximal and Falconer but not these four families; a viral 7 Oct summary post had listed 'Fourier restriction conjecture — PARTIAL'. No press article, Hacker News thread or expert blog post on the restriction or Bochner–Riesz claims was found on 19 Oct"],"key_numbers":[],"tags":["math","harmonic-analysis","fourier-restriction","bochner-riesz","local-smoothing","kakeya","schrodinger","openai","openai-math-release","unverified"],"science":{"field":"mathematics","subfield":"harmonic analysis (Fourier restriction, Bochner–Riesz, local smoothing, dispersive PDE)","problem":"Stein's Fourier restriction conjecture in ℝ³ (sphere, bounded data; diagonal form for positively curved surfaces); 3D Bochner–Riesz conjecture; Sogge's local smoothing conjecture in 3D; Sobolev endpoint of Carleson's Schrödinger convergence problem","result":"Claimed full resolution of each in three dimensions (planar endpoint for Carleson) in OpenAI manuscripts of 24 Sept 2026, released 6 Oct; not formally verified and not yet reviewed","open_since":"1967","ai_system":["OpenAI internal model (openai/math release)"],"human_role":"Credited to 'OpenAI' with no named human authors; OpenAI marks the release's review status as unchecked","verification":{"key":"preprint","label":"Preprint only","text":"Unverified manuscripts; no Lean formalization listed for families 077–080; one arXiv follow-up (Korobenko–Sawyer) cites the diagonal claim without checking it"},"status":"pending","shock":"Restriction, Bochner–Riesz and local smoothing in 3D were among the main targets of harmonic analysis since Stein; recent human progress moved the restriction exponent from 13/4 to 22/7, and the claim jumps to the conjectured endpoint p > 3."},"body_md":"## What happened\n\nOpenAI's 6 Oct 2026 release (see the [main entry](/e/2026-10-06-openai-math-release-722-manuscripts/)) is so large that some of its\nbiggest claims went unnoticed here for two weeks. Families 077–080 sit in the analysis section of the catalogue, next to the Falconer\n(073) and Kakeya (074) families, and claim most of the classical \"Stein programme\" in three dimensions:\n\n- **Fourier restriction (077).** Elias Stein's restriction problem (late 1960s) asks for which exponents the Fourier extension\n  operator of the sphere is bounded. In ℝ³ the conjecture is that L∞(S²) → Lᵖ(ℝ³) holds for every p > 3. The best human results, as\n  OpenAI's own paper lists them, were p > 13/4 (Guth, polynomial partitioning), p > 42/13 (Hong Wang, \"brooms\") and the diagonal\n  estimate for p > 22/7 (Wang and Wu). OpenAI's 103-page paper claims the full range, and a 25-page companion claims the diagonal\n  Lᵖ(Σ) → Lᵖ(ℝ³) bound for every compact positively curved surface and every p > 3.\n- **Bochner–Riesz (078).** A 114-page paper claims boundedness of the Bochner–Riesz means on L³(ℝ³) for every positive order,\n  which by interpolation gives the conjectured range.\n- **Local smoothing (079).** A 165-page paper claims Sogge's local smoothing conjecture for the wave equation in three space\n  dimensions.\n- **Carleson's Schrödinger problem (080).** The catalogue claims almost-everywhere convergence at the sharp Sobolev exponent\n  n/(2(n+1)) in every dimension; the planar paper (83 pages) treats H^(1/3)(ℝ²).\n\nThe diagonal restriction paper says it depends on the sphere paper and on OpenAI's 3D Kakeya maximal theorem (family 074). These\nfour families have no Lean formalization in the repository, unlike the release's showcase results.\n\nOn 15 Oct Lyudmila Korobenko and Eric Sawyer posted the first arXiv paper we found that cites one of these claims. They prove a\nprobabilistic (\"martingale transform\") version of Fourier extension estimates for the Schrödinger equation, and describe OpenAI's\ndiagonal theorem as the deterministic analogue that has been \"claimed\". They do not check it.\n\n## Why it matters\n\nIf they hold, these papers settle, in dimension three, problems that have organised harmonic analysis since the 1970s, and that\nBourgain, Wolff, Tao, Guth, Wang and others advanced step by step. They are long, unformalized and unreviewed, and they rest on\nthe equally unreviewed Kakeya maximal manuscript. They are a natural next target for the kind of expert check that confirmed\nOpenAI's Brennan proof. Readers should treat them as claims.","disputed":[{"label":"Verification","text":"Unverified manuscripts; no Lean formalization listed for families 077–080; one arXiv follow-up (Korobenko–Sawyer) cites the diagonal claim without checking it","sources":[]}],"related":[{"id":"2026-10-19-ai-assisted-results-october-19-listing","url":"https://postcutoff.com/e/2026-10-19-ai-assisted-results-october-19-listing/","date":"2026-10-19","date_precision":"day","short_title":"AI-assisted results in the 19 Oct 2026 arXiv listing","deck":"Experts rework OpenAI's release papers (Deligne–Drinfeld, Hénon–Lane–Emden, Navier–Stokes), Wright's harmonic-analysis problems disproved with Codex, an Aaronson–Ambainis lower bound, and DeepMind's Busy Beaver agent","takeaway":"Monday's arXiv announcement brought the first wave of human follow-up papers on OpenAI's October maths release, alongside a dozen routine AI disclosures. The strongest items have their own entries: Brennan, Smith–Toda, lonely runner and tournament Hamiltonian cycles.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":16,"official":16,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","Harmonic","various mathematicians"]},{"id":"2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov","url":"https://postcutoff.com/e/2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov/","date":"2026-10-16","date_precision":"day","short_title":"Brennan's conjecture: experts confirm and shorten OpenAI's proof","deck":null,"takeaway":"The first famous result from OpenAI's October maths release to be checked by leading experts in its field: four complex analysts, including Fields medallist Stanislav Smirnov, found no gap. They say the AI combined known techniques in a way 'missed by the human experts'.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":4,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","University of Toronto","Tel Aviv University","Åbo Akademi University","University of Geneva"]},{"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":84,"official":13,"filed":"2026-10-07","updated":"2026-10-19","orgs":["OpenAI"]},{"id":"2026-10-06-quasi-riemann-hypothesis-openai","url":"https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the quasi-Riemann hypothesis","deck":"No zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized)","takeaway":"If correct, this is the largest advance on the zeros of ζ since the 19th century, and it would remove the Siegel-zero obstruction that runs through modern analytic number theory.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":9,"filed":"2026-10-07","updated":"2026-10-19","orgs":["OpenAI"]}],"people":[{"id":"levent-alpoge","name":"Levent Alpöge","url":"https://postcutoff.com/person/levent-alpoge/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"filed","text":"Created (catalogue and the five OpenAI PDFs read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-19","run":null,"sources_read":"Catalogue and the five OpenAI PDFs read","updated":"2026-10-19","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-3d-restriction-bochner"}