Continuous gradients in every dimension?
An AI preprint claims interior C¹,α estimates for infinity-harmonic functions in three and more dimensions
Covers result 377 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
3:52. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Fix heights along the edge of a region and fill in the inside as flatly as possible, so that the steepest slope is as small as it can be, not just overall but on every smaller piece, given its own edge values. Aronsson studied these absolutely minimizing extensions in the 1960s, and, as Jensen proved in 1993, they solve the infinity-Laplace equation. Our computer shows why the extra condition matters: for one set of edge heights, two extensions both reach the smallest possible overall slope, 1.3, yet disagree at the centre, +1 against −0.8. The absolutely minimizing one is unique, and a tug-of-war rule on a grid approximates it.
How smooth are these functions? Aronsson’s example, x^(4/3) − y^(4/3), solves the equation, and, as the paper notes and our computer confirms, its gradient changes like the distance to the power one third exactly, and no better. In the plane, gradients were shown continuous in 2005 and Hölder continuous in 2008. In three and more dimensions, Evans and Smart proved in 2011 that solutions are differentiable everywhere, and wrote that it remains an open problem whether they are continuously differentiable.
Result 377, the last in the release, is one OpenAI paper, 46 pages, dated October 4 and made public October 6, 2026. It claims that in every dimension three and up there is a positive exponent and a constant such that every bounded infinity-harmonic function on the unit ball has, on the half ball, a bounded gradient that is Hölder continuous with that exponent, all controlled by how much the function varies. So these functions are continuously differentiable in three and more dimensions, the question Evans and Smart left open.
What it doesn’t claim
Not claimed: the exponent’s value, which is not explicit and at most one third; the conjectured optimal one third itself; anything at the boundary; or the plane. A special case, for potentials in convex rings, was published in 2025, and the paper builds on a method of Evans and Savin.
Where it stands
- Posted as a preprint
- OpenAI, 46 pages, dated October 4, 2026; unchanged since release
- Peer reviewed
- Not yet; no specialist verdict
- Formalized in Lean
- None, catalogued or uncatalogued
Last checked October 19, 2026. The paper is unrefereed and unchanged since release. It has no Lean proof, catalogued or uncatalogued. We found no specialist verdict, and a paper posted three days after the release, with new planar estimates for these functions, does not cite it. OSVerify has not reviewed it.
How the proof goes
- Flatness, bad scales and a limit. First, add a dummy linear variable, and use a three-point midpoint estimate to show solutions look nearly affine at small scales, uniformly. Then measure flatness in thin cylinders. If the flatness did not improve at a power rate, the paper picks bad scales and applies Evans and Savin’s lopsided rescaling, producing in the limit an entire, non-affine solution of a simpler equation that has good fits at every centre and scale. Our computer checks the rescaling identity exactly.
- A new Liouville theorem. The new step is a Liouville theorem: such a limit must be affine. Its proof takes about twenty-nine of the 46 pages, with an extremal defect, a fair-choice walk with a drift inequality, contact localization and a rigidity argument. The contradiction gives a power rate, and the gradient exponent is that rate divided by two plus the rate, which can never exceed one third. Our computer confirms the bookkeeping: a rate of one would give exactly one third.
The numbers in the video
- Ours: in Aronsson’s |x|^{4/3} − |y|^{4/3} the two terms of the infinity Laplacian are exactly ±64/81; its gradient changes like (4/3)h^{1/3}, as the paper notes.
- Ours: two extensions of the same edge heights both have slope 1.3 overall but differ at the centre (+1.0 vs −0.8); the harmonic extension’s slope is 1.59.
- Ours: a tug-of-war grid iteration stays within 0.030 of Aronsson’s function; plain averaging is off by 0.160.
- Ours: the exponent α = γ/(2+γ) never exceeds 1/3, with equality exactly at γ = 1; the rescaling identity (3.15) holds exactly in 4 trials.
Read the source
- Uniform Interior C^{1,α} Estimates for Infinity-Harmonic Functions (PDF), 46 pages.
- The collection’s catalog, where this is result 377.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Fix heights along the edge of a region and fill in the inside as flatly as possible, so that the steepest slope is as small as it can be, not just overall but on every smaller piece, given its own edge values. Aronsson studied these absolutely minimizing extensions in the 1960s, and, as Jensen proved in 1993, they solve the infinity-Laplace equation. Our computer shows why the extra condition matters: for one set of edge heights, two extensions both reach the smallest possible overall slope, 1.3, yet disagree at the centre, +1 against −0.8. The absolutely minimizing one is unique, and a tug-of-war rule on a grid approximates it. How smooth are these functions? Aronsson’s example, x^(4/3) − y^(4/3), solves the equation, and, as the paper notes and our computer confirms, its gradient changes like the distance to the power one third exactly, and no better. In the plane, gradients were shown continuous in 2005 and Hölder continuous in 2008. In three and more dimensions, Evans and Smart proved in 2011 that solutions are differentiable everywhere, and wrote that it remains an open problem whether they are continuously differentiable.
1:22The claim
Result 377, the last in the release, is one OpenAI paper, 46 pages, dated October 4 and made public October 6, 2026. It claims that in every dimension three and up there is a positive exponent and a constant such that every bounded infinity-harmonic function on the unit ball has, on the half ball, a bounded gradient that is Hölder continuous with that exponent, all controlled by how much the function varies. So these functions are continuously differentiable in three and more dimensions, the question Evans and Smart left open. Not claimed: the exponent’s value, which is not explicit and at most one third; the conjectured optimal one third itself; anything at the boundary; or the plane. A special case, for potentials in convex rings, was published in 2025, and the paper builds on a method of Evans and Savin.
2:20How the proof goes
First, add a dummy linear variable, and use a three-point midpoint estimate to show solutions look nearly affine at small scales, uniformly. Then measure flatness in thin cylinders. If the flatness did not improve at a power rate, the paper picks bad scales and applies Evans and Savin’s lopsided rescaling, producing in the limit an entire, non-affine solution of a simpler equation that has good fits at every centre and scale. Our computer checks the rescaling identity exactly. The new step is a Liouville theorem: such a limit must be affine. Its proof takes about twenty-nine of the 46 pages, with an extremal defect, a fair-choice walk with a drift inequality, contact localization and a rigidity argument. The contradiction gives a power rate, and the gradient exponent is that rate divided by two plus the rate, which can never exceed one third. Our computer confirms the bookkeeping: a rate of one would give exactly one third.
3:19Where it stands
The paper is unrefereed and unchanged since release. It has no Lean proof, catalogued or uncatalogued. We found no specialist verdict, and a paper posted three days after the release, with new planar estimates for these functions, does not cite it. If it holds, infinity-harmonic functions in every dimension have Hölder continuous gradients: the first regularity beyond differentiability for general solutions in three and more dimensions.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.