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

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 ${\mathbb{R}}^3$ 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 $k$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.