Local Smoothing Conjecture

The local smoothing conjecture is a conjecture in harmonic analysis about solutions of the wave equation. It predicts that if the $L^p$ size of a wave is averaged over a short interval of time, rather than measured at one fixed instant, the solution is smoother than the sharp fixed-time estimates allow, by almost $1/p$ of a derivative when $p$ is at or above a critical exponent. Christopher Sogge formulated it in 1991.^[1][2] It sits at the top of a chain of implications among central problems of the field: local smoothing implies the Bochner-Riesz conjecture, which implies the Fourier restriction conjecture, which implies the Kakeya conjecture.^[2][4] Larry Guth, Hong Wang and Ruixiang Zhang proved the conjecture in two spatial dimensions in a paper posted in September 2019 and published in the Annals of Mathematics in 2020.^[3] On 26 October 2026, OpenAI released a manuscript produced by an internal model that claims a proof in every spatial dimension, together with a Lean formalization. As of 30 October 2026 that claim is an unrefereed preprint.^[15][16]

Statement

Consider the wave equation $\partial_t^2 u = \Delta u$ on $\mathbb{R}^n \times \mathbb{R}$ with $n \ge 2$. Its solutions are built from the half-wave propagators $e^{\pm it\sqrt{-\Delta}}$, which act on initial data $f$ by multiplying the Fourier transform $\hat f(\xi)$ by $e^{\pm it|\xi|}$.^[2] Smoothness is measured with the $L^p$ Sobolev (Bessel potential) spaces $L^p_s(\mathbb{R}^n)$, which count $s$ derivatives in $L^p$.

Fixed-time estimates. For a single fixed time $t \neq 0$, the propagator is not bounded on $L^p$ when $p \neq 2$. Peral and Miyachi showed in 1980 that it maps $L^p_s$ to $L^p_{s - \bar s_p}$, a loss of

$\bar s_p = (n-1)\left|\tfrac{1}{2} - \tfrac{1}{p}\right|$

derivatives, and that this loss cannot be reduced. Seeger, Sogge and Elias Stein extended the bound to general Fourier integral operators in 1991, which covers wave equations on compact Riemannian manifolds.^[2][5][6]

Local smoothing. Sogge observed that averaging over time recovers part of this loss.^[2][7] In the form given by Beltran, Hickman and Sogge, the conjecture states that for $n \ge 2$

$\left( \int_1^2 \left\| e^{it\sqrt{-\Delta}} f \right\|_{L^p_{-\bar s_p + \sigma}(\mathbb{R}^n)}^p \, dt \right)^{1/p} \lesssim \|f\|_{L^p(\mathbb{R}^n)}$

for every $\sigma < 1/p$ when $\frac{2n}{n-1} \le p < \infty$, and for every $\sigma < \bar s_p$ when $2 < p \le \frac{2n}{n-1}$.^[2] Equivalently, the space-time norm on $\mathbb{R}^n \times [1,2]$ is controlled by $\max{0, \frac{n-1}{2} - \frac{n}{p}} + \varepsilon$ derivatives of the data for every $\varepsilon > 0$.^[16]

Several features follow from the definitions:^[2]

• At $p = 2$ there is no gain, since Plancherel's theorem makes the $L^2$ norm constant in time.
• A gain of $1/p$ derivatives would be best possible; an explicit example shows that no larger gain can hold.
• The hardest case is the critical exponent $p_c = \frac{2n}{n-1}$. The estimates for all other $p$ follow from that case by interpolation with the $L^2$ identity and an $L^\infty$ bound. At $p_c$ the fixed-time loss equals $1/p_c$, so the conjecture says the propagator is bounded on $L^{p_c}$ locally in time up to an arbitrarily small loss of $\varepsilon$ derivatives.

Intuition

A wave can briefly concentrate its energy, for example when an incoming spherical wave reaches its focus, and at that instant its $L^p$ norm can be much larger than that of the initial data. The conjecture says that such extreme focusing is short-lived, so that the size of the wave at a typical time is much smaller than at the worst time.^[15] Quanta Magazine described the link with Kakeya sets: a general solution can be broken into pieces that travel in different directions, each piece resembling one of the needles in a Kakeya set, and ruling out heavy overlap of the needles rules out the pieces conspiring to create a blowup. Ciprian Demeter compared the averaging to investing in the financial market over 40 years, in which crashes occur here and there but the long-run average is good.^[4]

Implications

Sogge observed that the local smoothing conjecture formally implies the Bochner-Riesz conjecture on $L^p$ convergence of Fourier summation. The Bochner-Riesz conjecture in turn implies the Fourier restriction conjecture for spheres and paraboloids, and the restriction conjecture implies the Kakeya conjecture. Beltran, Hickman and Sogge summarize the chain as:^[2]

$\text{Local smoothing} \Rightarrow \text{Bochner-Riesz} \Rightarrow \text{Restriction} \Rightarrow \text{Kakeya}$

The implications run in one direction only. Terence Tao wrote in November 2024 that the partial implications known in the reverse direction are inefficient, so that a full resolution of the Kakeya conjecture would not immediately resolve the restriction conjecture.^[13] Quanta noted in 2023 that the two-dimensional cases of the Kakeya, restriction and Bochner-Riesz problems had been settled decades before the two-dimensional local smoothing conjecture, which made local smoothing the most difficult of the group.^[4]

Non-sharp local smoothing estimates also have uses of their own. Mockenhaupt, Seeger and Sogge used them in 1992 to give a proof of Jean Bourgain's circular maximal theorem, and such estimates are applied to maximal and variational problems in harmonic analysis.^[2][8]

Partial results

Progress on the Euclidean conjecture took two forms: a positive but non-sharp gain at the critical exponent, or the sharp gain of almost $1/p$ for a restricted range of $p$.^[2]

Intermediate improvements at $p = 4$ in the plane were also obtained by Bourgain, by Tao and Vargas, and by J. Lee, and Garrigós and Seeger, and later Garrigós, Seeger and Schlag, improved the range of $p$ in Wolff's decoupling inequalities before Bourgain and Demeter's theorem.^[2]

According to Quanta, Wolff introduced decoupling in 1999 while working on this conjecture, and the technique later produced results in number theory and geometry as well as harmonic analysis. Sogge, of Johns Hopkins University, told the magazine: "Using decoupling results, you now have world records in very famous, important problems."^[4]

Guth, Wang and Zhang took a different route from decoupling. They proved a sharp reverse square function estimate for the light cone in $\mathbb{R}^3$, which implies the conjecture for $n = 2$. The paper was received by the Annals on 9 October 2019 and published online on 9 September 2020.^[3]

Variants

The same question can be posed for wave equations on compact Riemannian manifolds and for general Fourier integral operators. There the expected range of $p$ is different. In 1997 Minicozzi and Sogge built compact manifolds, including arbitrarily small smooth perturbations of the Euclidean metric, on which the gain of almost $1/p$ fails for $p$ below an explicit exponent, which exceeds the Euclidean critical exponent when $n \ge 3$. The obstruction comes from Kakeya and Nikodym compression of geodesics.^[2] Beltran, Hickman and Sogge extended Bourgain-Demeter decoupling to variable coefficients and showed that the resulting local smoothing estimates for Fourier integral operators are sharp in odd dimensions.^[2] Gan, He, Li and Wu proved the manifold version of the conjecture in every odd dimension $n \ge 3$.^[7]

Local smoothing first appeared in $L^2$ form for the Schrödinger equation in the late 1980s, in independent work of Sjölin and Vega; there, solutions localized in space and averaged in time gain half a derivative. Sogge identified the analogous $L^p$ phenomenon for the wave equation.^[2][7]

OpenAI's claimed proof (2026)

Earlier three-dimensional manuscript

The initial commit of OpenAI's public mathematics repository, openai/math, made on 6 October 2026 (see OpenAI Mathematics Collection), already contained a manuscript titled "Critical local smoothing for the three-dimensional wave equation", dated 24 September 2026 and credited to OpenAI. It claims the critical $L^3$ estimate in three spatial dimensions with every positive Sobolev loss, and states that this resolves the Euclidean conjecture for $n = 3$; it says the lossless case $\varepsilon = 0$ is not addressed.^[18]

October 2026 manuscript

On 26 October 2026, OpenAI announced results in mathematics, computer science and physics in a post titled "Working with academic communities to build scientific understanding". The mathematics result was a claimed proof of the local smoothing conjecture in every dimension. OpenAI wrote that the conjecture implies the Kakeya conjecture and Stein's restriction conjecture, that its models therefore also resolve those conjectures in all dimensions, and that "the new proof is based on a novel auxiliary partial differential equation."^[15] The same release included a claimed refutation of the Exponential Time Hypothesis.^[15][19]

The manuscript, "Local smoothing for the Euclidean wave equation", is dated 25 October 2026 and lists OpenAI as its author.^[16] Its abstract and main theorem claim:^[16]

• Sogge's local smoothing conjecture for the Euclidean wave equation in every spatial dimension $n \ge 2$, for $2 \le p < \infty$ and every $\varepsilon > 0$ (with $\varepsilon = 0$ allowed at $p = 2$);
• the Bochner-Riesz conjecture, maximal estimates for Bochner-Riesz means, and sharp bounds for Stein's square function;
• Fourier restriction estimates for the sphere, the compact paraboloid and the truncated cone throughout the interior of the conjectured range, and local smoothing for the Schrödinger equation;
• the Kakeya and Nikodym maximal conjectures, and in particular that every Euclidean Kakeya set has full Hausdorff dimension.

The proof is organized around a single "critical annular estimate" at $p = p_c$: for frequencies of size about $N$, the space-time norm is bounded by a constant times $[\log(2+N)]^{2/p}|f|{L^p}$, and interpolation and summation over frequency annuli then give the main theorem.^[16] To prove it, the manuscript translates the wave in an imaginary direction along light rays, which filters a sum of plane waves by direction, and builds from the continued wave a positive semidefinite tensor whose columns have zero divergence in the physical variables and two auxiliary time variables. On each fiber of the construction it solves a Dirichlet problem for a quaternionic Monge-Ampère-type equation, $\det H{\mathrm{fib}}(\lambda|\Phi|^2 + u) = \lambda\mu$, and defines an energy modeled on the classical Aubin-Mabuchi energy. A scalar differential inequality for this energy then carries the estimate between circles in the auxiliary variables at a cost of logarithmic factors. The manuscript cites Kaiser's work on complex spacetime and Denis Serre's compensated integrability as related constructions.^[16]

Timeline and release

OpenAI said it obtained the AI output on 18 October 2026 and formalized it in Lean the following day. It then consulted an expert about forming a collaboration and, on that expert's recommendation, released the AI output directly, as it had done for its 6 October release. In contrast, the computer science result was handed to a working group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams, and the physics result to a group led by David Gross.^[15] OpenAI's post states that it does "not seek academic credit for the discoveries produced by our internal models."^[15]

The repository's documentation describes the formalization as covering, for every $n \ge 2$ and $2 \le p < \infty$, the $L^p(\mathbb{R}^n \times [1,2])$ bound with Sobolev order $\max(0, (n-1)/2 - n/p) + \varepsilon$ for Schwartz initial data, together with Bochner-Riesz, spherical extension, Stein square function and Kakeya maximal bounds and the full Hausdorff dimension of Kakeya sets.^[17] These are stated as six theorem targets for the Comparator checking tool, with the permitted axioms limited to propext, Quot.sound and Classical.choice.^[17]

Reception and status

The manuscript is a preprint released on GitHub, not a refereed journal publication. OpenAI's repository describes its manuscripts as "best understood as a form of raw research data, rather than polished academic papers", and notes that results there are at different stages of verification.^[20] The repository's change log shows that OpenAI withdrew three earlier manuscripts on 7 October 2026 after a sign error was found, although those withdrawals concerned algebraic geometry papers, not this one.^[21]

Alex Iosevich, a harmonic analyst at the University of Rochester whose doctoral adviser was Sogge, told Al Jazeera that OpenAI appeared to have "cracked" the conjecture in a "historic development". He said the news "was a shock at first" and that he expected theoretical mathematics to "change drastically" because of AI.^[19] Trade coverage, such as Interesting Engineering, reported the claim as OpenAI's and noted that outside experts would review the 26 October results.^[22]

If the claim is confirmed, it would also settle the higher-dimensional Kakeya problem. Hong Wang and Joshua Zahl had proved the Kakeya set conjecture in three dimensions in 2025.^[15][23]

Continue reading

• KKakeya ConjectureThe Kakeya conjecture (also called the Kakeya set conjecture or Besicovitch set conjecture) is a problem in geometric measure theory and harmonic analysis.Mentioned 4 times above
• OOpenAI Mathematics CollectionThe OpenAI Mathematics Collection is a public collection of mathematical manuscripts and proof artifacts produced by an internal OpenAI model. OpenAI announced the release on October 6, 2026.Listed in See also
• EExponential Time HypothesisThe exponential time hypothesis (ETH) is a conjecture in computational complexity theory about how fast the 3-SAT problem can be solved.Listed in See also
• AAutomated Theorem ProvingAutomated theorem proving (ATP) is the use of computer programs to find or check mathematical proofs, expressing both the statement and the proof in a formal_logic so that correctness can be decided…Listed in See also
• MMathlibMathlib (formally mathlib4, the active Lean 4 incarnation) is the community-maintained, open-source unified library of formalized mathematics for the lean theorem prover, and it is the largest single library…Listed in See also
• OpenAIOpenAI is an American artificial intelligence research and deployment organization.Mentioned 15 times above

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Topics:AI for ScienceMathematicsOpenAI

References

1. ^1 ^2Christopher D. Sogge. Propagation of singularities and maximal functions in the plane. *Inventiones Mathematicae* 104 (1991), 349-376.
2. ^1 ^2 ^3 ^4 ^5 ^6 ^7 ^8 ^9 ^10 ^11 ^12 ^13 ^14 ^15 ^16 ^17 ^18 ^19 ^20 ^21 ^22David Beltran, Jonathan Hickman and Christopher D. Sogge. Sharp local smoothing estimates for Fourier integral operators. arXiv:1812.11616 (v2, 5 September 2019); published in *Geometric Aspects of Harmonic Analysis*, Springer INdAM Series 45 (2021).
3. ^1 ^2 ^3Larry Guth, Hong Wang and Ruixiang Zhang. A sharp square function estimate for the cone in R^3. *Annals of Mathematics* 192 (2020), no. 2, 551-581. Preprint arXiv:1909.10693, 24 September 2019.
4. ^1 ^2 ^3 ^4Jordana Cepelewicz. A Tower of Conjectures That Rests Upon a Needle. *Quanta Magazine*, 12 September 2023.
5. ^Juan C. Peral. L^p estimates for the wave equation. *Journal of Functional Analysis* 36 (1980), no. 1, 114-145.
6. ^Andreas Seeger, Christopher D. Sogge and Elias M. Stein. Regularity properties of Fourier integral operators. *Annals of Mathematics* 134 (1991), no. 2, 231-251.
7. ^1 ^2 ^3 ^4Shengwen Gan, Danqing He, Xiaochun Li and Shukun Wu. On local smoothing estimates for wave equations. arXiv:2502.05973 (v1 9 February 2025; v2 4 January 2026).
8. ^1 ^2Gerd Mockenhaupt, Andreas Seeger and Christopher D. Sogge. Wave front sets, local smoothing and Bourgain's circular maximal theorem. *Annals of Mathematics* 136 (1992), no. 1, 207-218.
9. ^Gerd Mockenhaupt, Andreas Seeger and Christopher D. Sogge. Local smoothing of Fourier integral operators and Carleson-Sjölin estimates. *Journal of the American Mathematical Society* 6 (1993), no. 1, 65-130.
10. ^Thomas Wolff. Local smoothing type estimates on L^p for large p. *Geometric and Functional Analysis* 10 (2000), 1237-1288.
11. ^Yaryong Heo, Fëdor Nazarov and Andreas Seeger. Radial Fourier multipliers in high dimensions. *Acta Mathematica* 206 (2011), no. 1, 55-92.
12. ^Jean Bourgain and Ciprian Demeter. The proof of the l^2 decoupling conjecture. arXiv:1403.5335 (2014); *Annals of Mathematics* 182 (2015), no. 1, 351-389.
13. ^Terence Tao. Post on Mathstodon. 17 November 2024.
14. ^Chuanwei Gao, Bochen Liu, Changxing Miao and Yakun Xi. Improved local smoothing estimate for the wave equation in higher dimensions. arXiv:2108.06870 (2021); *Journal of Functional Analysis* 284 (2023).
15. ^1 ^2 ^3 ^4 ^5 ^6 ^7OpenAI. Working with academic communities to build scientific understanding. 26 October 2026.
16. ^1 ^2 ^3 ^4 ^5 ^6OpenAI. Local smoothing for the Euclidean wave equation. OpenAI Math Release preprint, 25 October 2026.
17. ^1 ^2OpenAI. Lean formalization scope: Local smoothing for the Euclidean wave equation and Comparator configuration. openai/math repository, accessed 30 October 2026.
18. ^OpenAI. Critical local smoothing for the three-dimensional wave equation. OpenAI Math Release preprint dated 24 September 2026, committed 6 October 2026.
19. ^1 ^2John Power. AI makes discoveries in mathematics, computer science, physics, OpenAI says. Al Jazeera, 27 October 2026.
20. ^OpenAI. openai/math README. Accessed 30 October 2026.
21. ^OpenAI. openai/math release history. Accessed 30 October 2026.
22. ^Aamir Khollam. OpenAI's triple threat: New proofs challenge math, computing and quantum physics. *Interesting Engineering*, 26 October 2026.
23. ^Hong Wang and Joshua Zahl. Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions. arXiv:2502.17655, 24 February 2025.

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