OpenAI · 361-01 · 2026-09-25 · Differential geometry
A counterexample to integer-degree harmonic dimension comparison
preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026
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 , a complete smooth metric on has at least independent harmonic functions of growth at most , exceeding the Euclidean count . The metric may depend on .
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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 361: Failure of integer-degree harmonic dimension comparison. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.