Pólya's Conjecture for the Neumann Laplacian on Euclidean Balls

We prove P\'olya's conjectured lower bound for the Neumann counting function of every Euclidean ball. If $d\ge2$, $R>0$, and $E\ge0$, then \begin{equation*} N_{B_R^d}^{<}(E) \ge \frac{\omega_d}{(2\pi)^d}|B_R^d|E^{d/2} = \frac{(R\sqrt E)^d}{2^d\Gamma(\frac d2+1)^2}. \end{equation*} The radial boundary condition in dimensions $d\ge3$ is a Dini condition, not a derivative-zero condition. A strict Robin comparison first reduces it to a Bessel phase estimate. 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. The finite rational calculations form part of the proof appendix.