Browse › Differential geometry › 361 Failure of integer-degree harmonic dimension comparison
A counterexample to integer-degree harmonic dimension comparison
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Abstract
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.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 361: Failure of integer-degree harmonic dimension comparison (lean/docs/361.md, 2026). The release's scope notes say the Lean development constructs this paper's dimension-16, degree-50000 metric, including the Euclidean core and asymptotic volume ratio in (0,1); the tangent-cone and cone-spectrum conclusions are not formalized.
- OpenAI math release, OAI/Geometry/HarmonicGrowth/Main.lean (openai/math, 2026). This file holds the top-level Lean theorem
HarmonicCounterexample.main, whose docstring says it proves every clause of the manuscript's main theorem (n = 16, k = 50000). [AI-generated per source]
Possibly relevant literature (found by the agent, not checked)
- S. Lin, H. Wang, G. Xu, Eigenvalues on spheres (arXiv:2607.11544, July 2026). According to its abstract, this paper proves a sharp Euclidean dimension bound for polynomial-growth harmonic functions on certain complete three-manifolds, a positive integer-degree result under extra hypotheses alongside the Cai–Lai result this paper discusses.
- X.-T. Huang, On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth (arXiv:1705.05008, May 2017). This paper estimates h_d for large d under maximal volume growth and a unique tangent cone, which relates to this paper's point that its counterexample must have nonunique tangent cones; it is distinct from the Huang 2023 paper the manuscript cites.
Digests and expositions
- Latent Seminar, Harmonic dimension beyond the Euclidean count: changing links and averaged growth (Latent Seminar, 2026-10-07). This exposition compares both constructions, explains averaged growth and exact harmonic continuation, and reports a rerun of the first paper's finite arithmetic check while explicitly distinguishing it from verification of the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 361
Independent sites; their content is not reviewed here.
Cite
@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}
}From OpenAI's README for this paper:
Verification
Use Python 3.8 or later; the finite spectral-selection check uses only the standard library. From this paper directory, run with assertions enabled:
env -u PYTHONOPTIMIZE python3 verification/counting-check.py
Successful execution exits zero and prints exact count proves (3): True. This checks the finite selection at dimension 16 and degree 50000; the verification package retains its detailed scope and recorded output.
Lean formalization
OpenAI provides Lean files for this family: lean/docs/361.md. The Lean statements have not been checked by people against the paper's claims.
Family 361: 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 $(k+2)^2$ independent harmonic functions of growth at most k, exceeding the Euclidean count $(k+1)^2$. The metric may depend on k.
Other papers in this family:
- A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison · September 26, 2026