Carleson's pointwise convergence problem for the Schrödinger equation
From OpenAI Aftermath
Carleson's pointwise convergence problem for the Schrödinger equation asks for the least Sobolev regularity s such that the free Schrödinger evolution eitΔf converges to f almost everywhere as t tends to 0 for every f in Hs(ℝn). After Bourgain's counterexample for s below n/(2(n+1)) and the positive results of Du, Guth and Li and of Du and Zhang above it, the endpoint itself remained open. Two manuscripts in OpenAI's public mathematics collection (github.com/openai/math), both dated 24 September 2026, claim convergence at the endpoint s = n/(2(n+1)) in every dimension n ≥ 2. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introduction of the planar manuscript:
- Carleson introduced the problem and proved convergence in one dimension at s = 1/4, which Dahlberg and Kenig showed is necessary. Sjölin and Vega independently obtained convergence for s > 1/2 in higher dimensions. In the plane, the connection with Fourier restriction developed by Bourgain and by Moyua, Vargas and Vega, the bilinear approach of Tao and Vargas, Tao's bilinear restriction theorem and Lee's work lowered the threshold to s > 3/8; in dimensions n ≥ 3 Bourgain obtained s > 1/2 − 1/(4n).
- Bourgain's later counterexample forces s ≥ n/(2(n+1)) for n ≥ 2 but does not exclude equality. Du, Guth and Li proved planar convergence and a local L3 maximal estimate for s > 1/3, using polynomial partitioning, wave packets, refined Strichartz estimates and decoupling; Du, Guth, Li and Zhang treated higher dimensions for s > (n+1)/(2(n+2)), and Du and Zhang established convergence for s > n/(2(n+1)) when n ≥ 3.
The original results
- Claim (plane). Theorem 1.1 of Endpoint convergence for the planar Schrödinger equation states that for initial data in H1/3(ℝ2) the Schrödinger evolution, defined by first taking the limit of Gaussian regularizations, exists on one full-measure spatial set for every 0 < t < 1 and converges almost everywhere to the initial data as t tends to 0. The underlying maximal estimate has local L1 output; the manuscript states that the stronger endpoint local L3 maximal estimate is not asserted.
- Claim (higher dimensions). Endpoint pointwise convergence for the Schrödinger equation in higher dimensions states the same conclusion for initial data in Hn/(2(n+1))(ℝn) for every n ≥ 3. Together with Bourgain's counterexample, the two claims identify n/(2(n+1)) as the exact threshold.
- Method, as described: the central estimate is a local maximal estimate at the exact Sobolev exponent, proved first at unit frequency with wave packets on a measurable time graph. Multilinear restriction and the refined fractal restriction estimate of Du and Zhang reduce a saturated transverse configuration to weighted incidences between points and tubes; a robust projection theorem and polynomial partitioning then give a fixed power saving, which is propagated to sparse packet estimates. A time tree recovers the Sobolev square norm, and the pointwise representative is constructed directly rather than by intersecting time-dependent exceptional sets.
- Sources (openai/math repository, commit
7ef5c0c): Endpoint convergence for the planar Schrödinger equation; Endpoint pointwise convergence for the Schrödinger equation in higher dimensions; both dated 24 September 2026.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | The Sobolev endpoint for Schrödinger convergence, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the evolution and its representative, the exact endpoint claim, why summing frequencies is the endpoint problem, the geometric input to square summation and the construction of one exceptional set, with proof outlines on packet arrays, broad and narrow configurations, the linearizing graph, the time tree and the order of limits. | Blog article; it states that its source review covers statements, introductions and the proof mechanisms it describes and is not an independent verification. |
| 2026-10-14 | AI Proved What?, episode 080 Exactly enough?, remade version of 26 October 2026 | Reading (exposition) | Video explainer of the claims and their history. The episode states that the planar paper does not claim the stronger endpoint maximal estimate with local L3 output, and that, if the papers hold, convergence for every f in Hs holds exactly when s ≥ n/(2(n+1)). It reports that the higher-dimensional paper relies on lemmas and the time-tree mechanism of the planar one and proves its own version of a packet estimate for which Singh and Singh Parmar found a mismatch in published work on Falconer's problem. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as preprints: OpenAI, both dated September 24, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: not fully, with uncatalogued Lean code proving the planar theorem from the paper's short-time distribution bound but not the main estimate, and nothing for the higher-dimensional paper. |
| 2026-10-16 | J. Jardine, Lattice24, referee reviews rank_047 (plane) and rank_069 (higher dimensions) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap in either manuscript. The planar review has no notes. For the higher-dimensional manuscript it adds one minor note, which it states is a limit on verification and not a demonstrated error: the lossless weak estimate depends on a new frame theorem of about 1800 lines and on a transverse saving imported from the planar companion, neither verified line by line, and the higher-dimensional extension of the companion's lemmas is asserted rather than rederived. | Part of an audit that its README describes as complete for the collection's manuscripts without a Lean formalization. The reviews are published under CC BY 4.0. |
Notes
- The background summarizes the planar manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
