2026/07/28 by Yutian Li · 1 citation
Mathematics · #math.AP #math.CA #math.SP #msc:11P21 #msc:33C10 #msc:35P15
86 pages, including 39 pages of appendix
arxiv created 2026/07/29 · arxiv updated 2026/07/30
We prove Pólya's conjectured lower bound for the Neumann counting function of every Euclidean ball. If d≥2, R>0, and E≥0, then NBRd<(E) ≥ (ωd)/((2π)d)|BRd|Ed/2 = ((R√ E)d)/(2dΓ(\frac d2+1)2). For d≥3, the physical radial eigenfrequencies satisfy a Dini equation rather than Jν'(k)=0. A strict Robin comparison transfers a phase estimate for the zeros of Jν' to the physical Neumann spectrum. Variational bounds handle low frequencies, while estimates based on finitely many radial levels and on beta moments cover the intermediate range uniformly in the dimension. The remaining high-frequency estimate is explicit. All compact-range verifications are given as exact rational inequalities in the proof appendix. No numerical approximation or executable certificate is used as a premise of the proof.