OpenAI · 361-02 · 2026-09-26 · Differential geometry

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

preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026

Family 361. Counterexamples to Yau's harmonic dimension bound.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

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 $R^{3}$ 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 $k$ .

For every $4/9 < v < 1$ and $1 < c < 9 v /4$ , all sufficiently large integers k admit a complete smooth metric on ℝ3 with nonnegative Ricci curvature, asymptotic volume ratio v, and at least $c (k + 1)^{2}$ 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 $1 < c < 9/4$ , these metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. The metric may depend on k.

Preprint source and Lean scope

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A Lean scope note is stored for family 361, but it does not name this preprint. It names 361-01. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.