Result 361, Differential geometry

Failure of integer-degree harmonic dimension comparison

Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ3 has at least (k+2)2(k+2)^2 independent harmonic functions of growth at most k, exceeding the Euclidean count (k+1)2(k+1)^2. The metric may depend on k.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Harmonic functions model equilibrium, such as steady temperature profiles. These unreviewed manuscripts claim that even under nonnegative Ricci curvature, curved spaces can support more independent, slowly growing equilibria than Euclidean space.

What changes?

For every volume ratio v strictly between 4/9 and 1 and factor c strictly between 1 and 9v/4, the three-dimensional manuscript reports complete smooth metrics on three-dimensional space with nonnegative Ricci curvature. For all sufficiently large integers k, they support at least c times (k+1) squared independent harmonic functions growing no faster than distance to the power k at infinity. Euclidean space has only (k+1) squared. The metric may depend on k; v measures large-ball volume relative to Euclidean space.

What does that help mathematicians do?

For each fixed factor c strictly between 1 and 9/4, the abstract also claims examples arbitrarily close to Euclidean geometry in global bi-Lipschitz distance: all distances differ by multiplicative factors arbitrarily close to one. Thus near-Euclidean distance geometry and nonnegative Ricci curvature do not force the exact Euclidean dimension bound. Researchers cannot use that bound to limit the number of independent polynomial-growth equilibria under these assumptions alone.

Are there practical applications?

The immediate value is foundational, in understanding how geometry constrains solutions of the Laplace equation, which defines harmonic functions. These examples test proposed comparison principles: any replacement for the rejected bound must change its assumptions or conclusion. The supplied abstracts report no practical computational or physical benefit.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

A counterexample to integer-degree harmonic dimension comparison

September 25, 2026 35 pages Main result formalized in Lean

For some even n ≥ 8 and integer k ≥ 2, we construct a complete smooth metric on ℝn with nonnegative Ricci curvature whose space of real harmonic functions of pointwise polynomial growth at most k has dimension larger than the Euclidean harmonic-polynomial dimension. The metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. This answers the integer-degree form of Yau's dimension comparison question in the negative.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026,
  author = {{OpenAI}},
  title = {{A counterexample to integer-degree harmonic dimension comparison}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf}{OAI:A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026}},
  year = {2026}
}

A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison

September 26, 2026 39 pages

For every 4/9<v<14/9\lt v\lt 1 and 1<c<9v/41\lt c\lt 9v/4, all sufficiently large integers k admit a complete smooth metric on ℝ3 with nonnegative Ricci curvature, asymptotic volume ratio v, and at least c(k+1)2c(k+1)^2 linearly independent real harmonic functions of pointwise growth at most k. This answers Yau's integer-degree dimension comparison question negatively in dimension three, with a fixed-factor excess over the Euclidean count. For each 1<c<9/41\lt c\lt 9/4, these metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. The metric may depend on k.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026,
  author = {{OpenAI}},
  title = {{A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf}{OAI:A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/361.md.

Failure of integer-degree harmonic dimension comparison

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result disproves the proposed Euclidean upper comparison for dimensions of harmonic functions with integer polynomial growth. It constructs one complete smooth metric with nonnegative Ricci curvature on an even-dimensional Euclidean space, Euclidean near the origin and with asymptotic volume ratio strictly between zero and one, admitting more independent harmonic functions of the prescribed growth degree than the Euclidean count. The construction uses dimension 1616 and degree 5000050000. Separate tangent-cone and cone-spectrum conclusions are not included.

Comparator links

Result Comparator statement
Counterexample to harmonic dimension comparison HarmonicGrowth.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.