Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 361-02. A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison
Manuscript map · lines 9335–9344
OpenAI@2026-10-07#rem:oai-p-361-02
Authors: OpenAI. Family 361.
Title: A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison.
Abstract: For every $4/9{}lt v{}lt 1$ and $1{}lt c{}lt 9v/4$, all sufficiently large integers k admit a complete smooth metric on ℝ3 with nonnegative Ricci curvature, asymptotic volume ratio v, and at least $c(k+1){}2$ 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 $1{}lt c{}lt 9/4$, these metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. The metric may depend on k.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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