Every surface fits in 4D?
An AI preprint claims every closed surface isometrically immerses in four-dimensional space
Covers result 333 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
3:30. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
A surface with a metric knows its own lengths and angles. Can you place it in flat space so every length is exactly right? A flat torus cannot sit smoothly in ordinary three-space: any closed surface there has a point of positive curvature, and our computer finds the doughnut’s outer rim curved at 0.64. In four dimensions a Clifford torus does it exactly; our computer matches its lengths to about 10 decimal places.
The conditions are three equations, one for each independent entry of the metric. In three-space there are three unknown coordinate functions; in four-space, four. Nash’s method needs much more room: in 1956 his bound for surfaces was 17 dimensions, and Gromov later brought closed surfaces down to 5. Small patches fit in four, by Poznyak in 1973. In 1999 Gromov wrote that it was unknown whether every surface with a smooth metric can be isometrically immersed in four-space, later attributing the question to Chern around 1950.
Result 333 is one OpenAI paper, 41 pages, dated September 23 and made public October 6, 2026. It claims that every closed smooth Riemannian surface admits a smooth isometric immersion into four-dimensional space: orientable or not, with any curvature. The surface may cross itself.
What it doesn’t claim
Not claimed: an embedding without self-crossings, which for the round projective plane is impossible in four-space; surfaces that are not closed; analytic or lower-regularity versions; or an explicit formula.
Where it stands
- Posted as a preprint
- OpenAI, 41 pages, dated September 23, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- The full theorem (docs)
Last checked October 19, 2026. The paper is unrefereed and unchanged since release. Lean formalizes the full theorem with Mathlib’s manifolds, including the classical inputs it uses, in about 1,880 files and 168,000 lines of its own, on top of Mathlib, not run by us. Two outside projects re-ran the check and it passed. We found no public comment from specialists and no reported error. A separate AI-assisted follow-up claims embeddings for orientable surfaces; it is unreviewed. OSVerify has not reviewed it.
How the proof goes
- No room for the usual tools. In four-space, the usual free-map tools are unavailable: they need five independent derivative directions, and four-space has room for only four. Our computer confirms the count. So the paper starts by drawing the surface very small on a round three-sphere, so all its lengths are strictly too short, with a normal direction pointing inward even when the surface is not orientable.
- Add the missing length, one direction at a time. The missing metric is cut into finitely many pieces, each a single direction squared, using three directions 120 degrees apart. Each piece is added by a fast spiral in a plane at right angles to the other direction and its bending, with errors cancelled to any chosen order and then removed by an exact smooth correction that only needs the bending to be nonzero. Our toy spiral, on a flat sheet, shows the basic effect: the metric error falls fourfold each time the scale halves, while the curvature doubles.
The numbers in the video
- Ours: a Clifford-type torus in 4-space matches its flat lengths to about 10⁻¹⁰; a torus of revolution in 3-space has curvature 0.64 at its outer rim.
- Ours: free maps need 5 independent derivative directions; 4-space allows at most 4.
- Ours: a toy spiral on a flat sheet: the metric error falls fourfold each time the scale halves, while the curvature doubles.
- Lean: 1,883 files and 167,865 lines of OpenAI’s own code in the import closure, on top of Mathlib (counted by us; not run by us).
Read the source
- Smooth isometric immersions of closed surfaces into Euclidean four-space (PDF), 41 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 333.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
A surface with a metric knows its own lengths and angles. Can you place it in flat space so every length is exactly right? A flat torus cannot sit smoothly in ordinary three-space: any closed surface there has a point of positive curvature, and our computer finds the doughnut’s outer rim curved at 0.64. In four dimensions a Clifford torus does it exactly; our computer matches its lengths to about 10 decimal places. The conditions are three equations, one for each independent entry of the metric. In three-space there are three unknown coordinate functions; in four-space, four. Nash’s method needs much more room: in 1956 his bound for surfaces was 17 dimensions, and Gromov later brought closed surfaces down to 5. Small patches fit in four, by Poznyak in 1973. In 1999 Gromov wrote that it was unknown whether every surface with a smooth metric can be isometrically immersed in four-space, later attributing the question to Chern around 1950.
1:05The claim
Result 333 is one OpenAI paper, 41 pages, dated September 23 and made public October 6, 2026. It claims that every closed smooth Riemannian surface admits a smooth isometric immersion into four-dimensional space: orientable or not, with any curvature. The surface may cross itself. Not claimed: an embedding without self-crossings, which for the round projective plane is impossible in four-space; surfaces that are not closed; analytic or lower-regularity versions; or an explicit formula.
1:43How the proof goes
In four-space, the usual free-map tools are unavailable: they need five independent derivative directions, and four-space has room for only four. Our computer confirms the count. So the paper starts by drawing the surface very small on a round three-sphere, so all its lengths are strictly too short, with a normal direction pointing inward even when the surface is not orientable. The missing metric is cut into finitely many pieces, each a single direction squared, using three directions 120 degrees apart. Each piece is added by a fast spiral in a plane at right angles to the other direction and its bending, with errors cancelled to any chosen order and then removed by an exact smooth correction that only needs the bending to be nonzero. Our toy spiral, on a flat sheet, shows the basic effect: the metric error falls fourfold each time the scale halves, while the curvature doubles.
2:39Where it stands
The paper is unrefereed and unchanged since release. Lean formalizes the full theorem with Mathlib’s manifolds, including the classical inputs it uses, in about 1,880 files and 168,000 lines of its own, on top of Mathlib, not run by us. Two outside projects re-ran the check and it passed. We found no public comment from specialists and no reported error. A separate AI-assisted follow-up claims embeddings for orientable surfaces; it is unreviewed. If it holds, every closed surface, whatever its curvature, can be realized with exactly its own lengths in four-dimensional space, crossing itself if needed, settling the four-dimensional immersion question 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.