OpenAI's local-smoothing proof claim: scope, Lean evidence and independent checks

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Evidence status: Provisional proof claim with public formalization · Peer reviewed: No. How the evidence labels work.

The problem: how much can time averaging smooth a wave?

A wave can concentrate strongly at one instant. Local smoothing asks how controlling it over a time interval changes the regularity needed in the initial data. This connects estimates for wave equations to harmonic analysis and geometric questions about thin tubes pointing in many directions.

Earlier work settled important cases. Guth, Hong Wang and Ruixiang Zhang's published 2+1-dimensional result concerns two spatial dimensions plus time. OpenAI's new claim concerns every Euclidean spatial dimension at least two. Those are different scopes.

What the new manuscript actually claims

The 25 October manuscript claims Sogge's local-smoothing conjecture for the Euclidean half-wave on Rⁿ, n ≥ 2. For Schwartz initial data, times [1,2] and 2 ≤ p < ∞, its stated Sobolev order is max(0,(n−1)/2−n/p)+ε, with ε > 0; p = 2 permits zero loss.

The paper also claims Kakeya and Nikodym maximal estimates, full Hausdorff dimension for Euclidean Kakeya sets, and multiplier and restriction consequences in specified ranges. These are claims in a single linked argument family. They are not separate independently confirmed discoveries.

Why the result would matter

If correct and faithful to the intended analytic statements, the result would settle a central wave-concentration conjecture and several related problems in all dimensions. The manuscript's proposed mechanism uses an auxiliary partial differential equation and a conserved current.

The conclusion needs its qualifiers. The setting is Euclidean; epsilon losses and strict or interior exponent ranges remain in the stated consequences. The release does not justify saying that every endpoint, arbitrary manifold or variable-coefficient version has been solved.

What AI contributed, according to OpenAI

OpenAI's announcement attributes the natural-language proof to internal models, reporting output on 18 October and formalization on 19 October, before public release on 26 October. The displayed manuscript credit is OpenAI.

This is a substantive AI role reported by the institution. The release does not name a model version or allocate every formalization step between people and models. It would be misleading to fill in those details or describe Kingy as having tested the proof system.

Public formalization and independent assessment are separate checks

The released Lean scope 079 documents a half-wave theorem and associated geometric and analytic consequences. A Comparator theorem target and its configuration make the intended formal target inspectable.

Public code is evidence that can be examined. It is not our successful rebuild receipt or a completed semantic audit. Kingy did not rebuild Lean or run Comparator. No completed independent specialist correctness verdict was located in the bounded source check on 26 October, so the tracker retains provisional status and does not equate evidence tier 1 with correctness certification.

Revision history and what could change the status

The manuscript is dated 25 October 2026; the public announcement and this record's first source review are dated 26 October. This landing explanation adds context without rewriting those dates. Later independent examination, a reproducible formal check, a correction or a journal decision would be distinct updates requiring sources.

The historical Wang–Zahl three-dimensional Kakeya theorem has a different authorship, scope and evidence history. Its specialist support should not be transferred to this new all-dimensional claim. See the tracker methodology for the distinction between a manuscript, formalization, independent check and peer review.

Landing explanation reviewed 2026-10-26. The record's own last-checked date and append-only history are listed separately below.

Stable ID

math-local-smoothing-all-dimensions-openai-2026

Revision

math-local-smoothing-all-dimensions-openai-2026.v1

Field

Mathematics · Harmonic analysis, partial differential equations and geometric measure theory

Evidence

Tier 1 · Peer reviewed: No

Record state

Provisional · Provisional proof claim with public formalization

Last checked

2026-10-26

Record details

Problem or result

Sogge's local smoothing conjecture for the Euclidean half-wave, with related harmonic-analysis consequences

Authors

OpenAI (displayed manuscript credit)

Institutions

OpenAI

Result date

Public release October 26, 2026; manuscript dated October 25. OpenAI reports natural-language output October 18 and formalization October 19.

Theorem scope

For Schwartz data, n ≥ 2 and 2 ≤ p < ∞, the spacetime Lᵖ norm on Rⁿ × [1,2] is bounded by the spatial Sobolev norm of order max(0, (n−1)/2 − n/p) + ε, with ε > 0; ε = 0 is also permitted at p = 2.

Sources

1. Official announcement: OpenAI announcement and external examination arrangements
2. Primary preprint: Local smoothing for the Euclidean wave equation October 25 2026 — pinned manuscript
3. Formalization scope: Documented Lean scope 079 — pinned revision
4. Formal target: LocalSmoothing Comparator theorem statement — pinned revision
5. Formal configuration: LocalSmoothing Comparator configuration — pinned revision

Correction and revision history

1. 2026-10-26 — Added after primary-source, scope, status, AI-role and limitation review.

Machine-readable: JSON v1 · CSV v1 · Schema v1

The dataset's publication fields describe each stored revision. Live landing publication status reports current indexability separately. This reviewed landing page is indexable.