More harmonic than flat space?
Two AI preprints claim counterexamples to integer-degree harmonic dimension comparison
Covers result 361 in OpenAI’s catalog: all 2 papers. 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
A harmonic function has zero Laplacian: in flat space, its value at each point is the average over any sphere around it. In ordinary three-space, the harmonic polynomials of degree at most k form a space of dimension (k + 1)²: our computer counts 1, 4, 9, 16 by exact linear algebra. Yau asked whether, on a complete manifold with nonnegative Ricci curvature, the harmonic functions growing at most like the k-th power of distance can ever outnumber those of Euclidean space.
Li and Tam settled linear growth in 1989, and Colding and Minicozzi proved these spaces are always finite-dimensional. Donnelly beat the Euclidean count in 2001, but at a growth rate strictly between one and two, in dimension five and up. For integer degrees the question stayed open: two preprints from this summer, by Lin, Wang and Xu and by Cai and Lai, each proved the bound in a special case; Cai and Lai described the integer case as still open.
Result 361 is two independent OpenAI papers, dated September 25 and 26 and made public October 6, 2026. The 35-page paper claims a complete metric with nonnegative Ricci curvature in 16 dimensions with more harmonic functions of growth at most 50,000 than Euclidean space, by about one part in 8,000. The 39-page paper claims, in three dimensions and for every large enough k, metrics with nonnegative Ricci curvature and at least c times (k + 1)² such functions, for any c below 9/4 of the volume ratio.
What it doesn’t claim
Not claimed: one metric for every k; each metric is built for its own k, and the three-dimensional bound needs k beyond an unspecified threshold. Nonnegative Ricci curvature is all they assume; this summer’s special cases also needed nonnegative sectional curvature or conformal flatness, so they are not contradicted. We found no competing claim.
Where it stands
- Posted as a preprint
- OpenAI, 35 + 39 pages, dated September 25 and 26, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- The 16-dimensional paper’s main theorem only; the 3D paper has no Lean proof (scope notes)
Last checked October 19, 2026. Both papers are unrefereed and unchanged since release. Only the 16-dimensional paper’s main theorem is formalized in Lean, about 37,000 lines, not run by us. The three-dimensional paper, the catalog’s headline, has no Lean proof; an uncatalogued three-dimensional Lean folder states a version but does not prove it. We found no specialist comment and no reported error. OSVerify has not reviewed it.
How the proof goes
- A shape that keeps changing. The trick in both papers is that the shape of the space at large distances keeps changing. On a fixed cone with a shrunken cross-section, harmonic functions grow faster, so fewer fit under degree k: our computer counts 81 against Euclidean space’s 121 at k = 10. But if the cross-section keeps changing, a harmonic function only needs its average growth rate to stay below k, and the papers make many modes take turns being slow.
- Two constructions. The three-dimensional paper steers eigenvalue crossings on nearly round spheres so that a whole band of modes cycles; each mode’s average rate is about two thirds of the band edge, divided by the cone’s opening, which is where 9/4 of the volume ratio comes from. Our computer reproduces that limit with round-sphere bands. The 16-dimensional paper uses Berger spheres and exact counting; our computer recomputes its selected dimension, a 59-digit number, exactly from the paper’s description.
The numbers in the video
- Ours: harmonic polynomials of degree at most k in ℝ³ number (k + 1)² — 1, 4, 9, 16, … — by exact linear algebra.
- Ours: on a fixed cone with volume ratio 0.81, only 81 modes grow slower than r¹⁰, against 121 in flat space.
- Ours: round-sphere bands at volume ratio 0.81 and k = 20,000 give 1.8223 against the limit 9v/4 = 1.8225.
- Lean: 36 files and 36,732 lines in the import closure of the 16-dimensional paper's comparator (counted by us; not run by us).
Read the source
- A counterexample to integer-degree harmonic dimension comparison (PDF), 35 pages.
- A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison (PDF), 39 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 361.
The 2 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
A harmonic function has zero Laplacian: in flat space, its value at each point is the average over any sphere around it. In ordinary three-space, the harmonic polynomials of degree at most k form a space of dimension (k + 1)²: our computer counts 1, 4, 9, 16 by exact linear algebra. Yau asked whether, on a complete manifold with nonnegative Ricci curvature, the harmonic functions growing at most like the k-th power of distance can ever outnumber those of Euclidean space. Li and Tam settled linear growth in 1989, and Colding and Minicozzi proved these spaces are always finite-dimensional. Donnelly beat the Euclidean count in 2001, but at a growth rate strictly between one and two, in dimension five and up. For integer degrees the question stayed open: two preprints from this summer, by Lin, Wang and Xu and by Cai and Lai, each proved the bound in a special case; Cai and Lai described the integer case as still open.
1:05The claim
Result 361 is two independent OpenAI papers, dated September 25 and 26 and made public October 6, 2026. The 35-page paper claims a complete metric with nonnegative Ricci curvature in 16 dimensions with more harmonic functions of growth at most 50,000 than Euclidean space, by about one part in 8,000. The 39-page paper claims, in three dimensions and for every large enough k, metrics with nonnegative Ricci curvature and at least c times (k + 1)² such functions, for any c below 9/4 of the volume ratio. Not claimed: one metric for every k; each metric is built for its own k, and the three-dimensional bound needs k beyond an unspecified threshold. Nonnegative Ricci curvature is all they assume; this summer’s special cases also needed nonnegative sectional curvature or conformal flatness, so they are not contradicted. We found no competing claim.
2:10How the proof goes
The trick in both papers is that the shape of the space at large distances keeps changing. On a fixed cone with a shrunken cross-section, harmonic functions grow faster, so fewer fit under degree k: our computer counts 81 against Euclidean space’s 121 at k = 10. But if the cross-section keeps changing, a harmonic function only needs its average growth rate to stay below k, and the papers make many modes take turns being slow. The three-dimensional paper steers eigenvalue crossings on nearly round spheres so that a whole band of modes cycles; each mode’s average rate is about two thirds of the band edge, divided by the cone’s opening, which is where 9/4 of the volume ratio comes from. Our computer reproduces that limit with round-sphere bands. The 16-dimensional paper uses Berger spheres and exact counting; our computer recomputes its selected dimension, a 59-digit number, exactly from the paper’s description.
3:13Where it stands
Both papers are unrefereed and unchanged since release. Only the 16-dimensional paper’s main theorem is formalized in Lean, about 37,000 lines, not run by us. The three-dimensional paper, the catalog’s headline, has no Lean proof; an uncatalogued three-dimensional Lean folder states a version but does not prove it. We found no specialist comment and no reported error. If they hold, flat space does not always have the most harmonic functions of a given integer growth among complete spaces with nonnegative Ricci curvature.
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.