A limit on how long waves can stay focused
- What it is
- A manuscript claims that waves cannot stay extremely concentrated for long, with consequences for how line segments can fill space.
- Who did it
- OpenAI; the manuscript and formalization were produced by an unnamed internal model.
- What it could mean
- If the argument holds, it would settle major questions about wave concentration and Kakeya geometry across Euclidean dimensions. Practical improvements in imaging or simulation have not been demonstrated.
See the check plan
Evidence & validation
From announcement to evidence
Discovery recorded. State of Proof has not yet examined this claim.
Read the original work
Local smoothing for the Euclidean wave equation ↗See the proposed checks
Compare the Euclidean half-wave, Sobolev and Kakeya definitions with the six Comparator targets; trace one selected endpoint through its source proof before any isolated pinned replay. Preserve finite p, positive ε loss, and strict restriction ranges. Full Hausdorff dimension does not assert positive volume.
No proof docket yet
A docket is the public record of checks and open questions. This paper does not have one yet; the check plan above describes work still to do.
Explore existing proof dockets →
Waves, concentration and directions
The crest and fan introduce wave concentration and the many directions in Kakeya geometry. These invented forms are not a solution of the wave equation, a time trace, or a Kakeya construction.
Read the source · Introduction · Euclidean local smoothing and Kakeya consequences ↗- Source kind
- Mathematical research record
- Evidence reported
- Manuscript; independent examination not started
- Source date
- ; added 2026-10-26
- What it claims
- Claims the Euclidean half-wave estimate for n≥2 and finite 2≤p<∞, with Sobolev order max(0,(n−1)/2−n/p)+ε for every ε>0; ε=0 is allowed at p=2. Consequences include Kakeya maximal bounds with every positive ε loss, full Hausdorff dimension of every Euclidean Kakeya set, and spherical restriction in the strict conjectured range.
- Why this could matter
- A limit on how long waves can stay focused A wave can become very intense at one instant. The claim limits that intensity when averaged over time and implies that a set containing a line segment in every direction must have full Hausdorff dimension.
- If it holds up
- It would resolve major Euclidean wave and Kakeya questions within the stated exponent ranges, giving researchers a common foundation for related Fourier estimates.
- If it does not
- The failed estimate or formal correspondence would identify where the claimed chain from wave concentration to geometric consequences needs repair.
- Impact horizon
- Foundational · Harmonic analysis · Wave equations · Kakeya geometry
- Version
- 2026-10-25; October 25, 2026 manuscript; public release at Git commit 7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a.
- Why we tracked it
- source-reviewed intake only; independent examination has not started.
- Highest-risk dependency
- Analyst judgment: statement correspondence and every dependency connecting the auxiliary partial differential equation to the wave estimate need examination. The six advertised Comparator targets do not certify every additional manuscript consequence. The boundary includes positive ε loss except at p=2, Euclidean geometry, and strict restriction ranges; no State of Proof proof validation or replay has occurred.
- Available artifacts
- Pinned openai/math 7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a; paper.pdf (Git blob 0472605895a485dd0441ada1257485d325f16f55), lean/docs/079.md, lean/OAI/Analysis/Fourier/LocalSmoothing/Comparator.lean and lean/ComparatorChallenges/LocalSmoothing.json. Lean v4.34.1; Mathlib d13f23b723b8a846827a245b89c10fc7d3f11612. No build or checker run in this intake.
- Current boundary
- Intake record only; examination not started. State of Proof has only recorded this intake. The paper's full argument remains unexamined; the next bounded check is: Compare the Euclidean half-wave, Sobolev and Kakeya definitions with the six Comparator targets; trace one selected endpoint through its source proof before any isolated pinned replay. Preserve finite p, positive ε loss, and strict restriction ranges. Full Hausdorff dimension does not assert positive volume.