Not even a piece fits in 3D?
An AI preprint claims a smooth surface metric that cannot be built in three-space even near one point
Covers result 334 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:18. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
A Riemannian metric is a rule for measuring lengths on a patch of surface. Gauss showed that the lengths alone fix the curvature, and our computer confirms this on six surfaces. So ask: near a chosen point, can every smooth rule be built as a small smooth surface in ordinary three-space, with every length exactly right? For analytic rules, Janet and Cartan said yes in the 1920s. For smooth rules, the answer is yes when the curvature is positive, negative, or changes sign cleanly.
The general smooth case stayed open. The paper cites Yau, who wrote in 2000 that it remains an open problem, and Ghomi’s survey, which lists it as Problem 1.9. In 1971 Pogorelov found a metric with no twice-differentiable local realization, but the metric was only finitely smooth, as was a later example by Nadirashvili and Yuan. Smooth counterexamples were claimed in 2002 preprints by Nadirashvili, one with Yuan, but never published in a journal, and in 2008 those two authors wrote that the basic question still remains.
Result 334 is one OpenAI paper, 19 pages, dated September 24 and made public October 6, 2026. It claims a smooth metric on a square such that no neighborhood of the center, however small, has a smooth isometric immersion into three-space. The metric can even match the flat plane to every order at the center: every derivative agrees, yet no small piece can be built.
What it doesn’t claim
Not claimed: a counterexample whose curvature has one sign, since the construction uses both and leaves those cases open; anything about less smooth placements; failure away from the center; or an explicit formula, because the key perturbations come from a category argument, not a written-down construction.
Where it stands
- Posted as a preprint
- OpenAI, 19 pages, dated September 24, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- The main theorem; not the flat-to-every-order refinement (docs)
Last checked October 19, 2026. The paper is unrefereed and unchanged since release. Lean formalizes the main theorem in about 450 files and 57,000 lines, not run by us; the flat-to-every-order refinement is not in it. Three outside projects re-ran the check and it passed, one with a second, independent kernel. We found no public comment from specialists. The paper cites the 2002 preprint by Nadirashvili and Yuan and leaves the discrepancy unresolved. It does not mention a second 2002 preprint by Nadirashvili, which the joint preprint describes as giving a smooth metric of this kind, and which we could not find. OSVerify has not reviewed it.
How the proof goes
- Heights must solve an equation. Any placement gives height functions: the distance along a fixed direction. Each must solve the Darboux equation, and our computer checks it on 36 heights of real surfaces. The equation is elliptic where the curvature is positive and hyperbolic where it is negative. The paper’s model patch has a negative oval in a positive sea; in our version it covers about a tenth of the square. It shows that after a generic tiny wiggle of the metric, no height of a certain shape solves the equation on the whole patch. The key is a thin, fast ripple whose effect grows in proportion to its width, while anything a surface could do to compensate is bounded by the width squared. Our toy ripple shows the linear part.
- A trap that every placement falls into. The paper then places negatively curved disks that shrink toward the center, each ringed by tiny obstructed patches in many directions, with sizes shrinking and strengths fading so fast that the metric stays flat to every order at the center. Any smooth placement near the center contains a whole disk. A saddle-shaped disk whose curvature fades to zero at its rim cannot have its bending vanish all along that rim, so at some rim point it bends in exactly one direction. Near that point, the height measured along the normal at that point has the forbidden shape on a whole patch. That is impossible, so no placement exists.
The numbers in the video
- Ours: curvature computed from lengths alone matches the embedded formula on 6 surfaces, and the height (Darboux) equation holds on 36 heights of real surfaces to about 10⁻⁶.
- Ours: in our version of the paper’s model patch, the hyperbolic (negatively curved) region covers 9.9% of the square, and every curvature zero is a clean sign change.
- Ours: a toy ripple’s forced effect, divided by its width, is the constant 0.01477, matching the prediction: linear in the width.
- Lean: 446 files and 56,586 lines in the import closure, on top of Mathlib (counted by us; not run by us).
Read the source
- A Smooth Metric with No Local Isometric Immersion into Three-Space (PDF), 19 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 334.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
A Riemannian metric is a rule for measuring lengths on a patch of surface. Gauss showed that the lengths alone fix the curvature, and our computer confirms this on six surfaces. So ask: near a chosen point, can every smooth rule be built as a small smooth surface in ordinary three-space, with every length exactly right? For analytic rules, Janet and Cartan said yes in the 1920s. For smooth rules, the answer is yes when the curvature is positive, negative, or changes sign cleanly. The general smooth case stayed open. The paper cites Yau, who wrote in 2000 that it remains an open problem, and Ghomi’s survey, which lists it as Problem 1.9. In 1971 Pogorelov found a metric with no twice-differentiable local realization, but the metric was only finitely smooth, as was a later example by Nadirashvili and Yuan. Smooth counterexamples were claimed in 2002 preprints by Nadirashvili, one with Yuan, but never published in a journal, and in 2008 those two authors wrote that the basic question still remains.
1:10The claim
Result 334 is one OpenAI paper, 19 pages, dated September 24 and made public October 6, 2026. It claims a smooth metric on a square such that no neighborhood of the center, however small, has a smooth isometric immersion into three-space. The metric can even match the flat plane to every order at the center: every derivative agrees, yet no small piece can be built. Not claimed: a counterexample whose curvature has one sign, since the construction uses both and leaves those cases open; anything about less smooth placements; failure away from the center; or an explicit formula, because the key perturbations come from a category argument, not a written-down construction.
1:58How the proof goes
Any placement gives height functions: the distance along a fixed direction. Each must solve the Darboux equation, and our computer checks it on 36 heights of real surfaces. The equation is elliptic where the curvature is positive and hyperbolic where it is negative. The paper’s model patch has a negative oval in a positive sea; in our version it covers about a tenth of the square. It shows that after a generic tiny wiggle of the metric, no height of a certain shape solves the equation on the whole patch. The key is a thin, fast ripple whose effect grows in proportion to its width, while anything a surface could do to compensate is bounded by the width squared. Our toy ripple shows the linear part. The paper then places negatively curved disks that shrink toward the center, each ringed by tiny obstructed patches in many directions, with sizes shrinking and strengths fading so fast that the metric stays flat to every order at the center. Any smooth placement near the center contains a whole disk. A saddle-shaped disk whose curvature fades to zero at its rim cannot have its bending vanish all along that rim, so at some rim point it bends in exactly one direction. Near that point, the height measured along the normal at that point has the forbidden shape on a whole patch. That is impossible, so no placement exists.
3:21Where it stands
The paper is unrefereed and unchanged since release. Lean formalizes the main theorem in about 450 files and 57,000 lines, not run by us; the flat-to-every-order refinement is not in it. Three outside projects re-ran the check and it passed, one with a second, independent kernel. We found no public comment from specialists. The paper cites the 2002 preprint by Nadirashvili and Yuan and leaves the discrepancy unresolved. It does not mention a second 2002 preprint by Nadirashvili, which the joint preprint describes as giving a smooth metric of this kind, and which we could not find. If it holds, some smooth surfaces cannot be built in three-space even in the smallest piece, while result 333 claims that four dimensions are always enough for closed surfaces.
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.