P\'olya's Conjecture for the Neumann Eigenvalues on Euclidean Balls
We prove P\'olya's conjectured lower bound for the Neumann eigenvalue counting function of Euclidean balls. If $B_R^d\subset\mathbb R^d$ is the ball of radius $R$, then, for every $d\ge2$, $R>0$, and $E\ge0$, $$ N_{B_R^d}^{<}(E) \ge \frac{\omega_d}{(2\pi)^d}|B_R^d|E^{d/2} = \frac{(R\sqrt E)^d}{2^d\Gamma(\frac d2+1)^2}, $$ where $\omega_d$ is the volume of the unit $d$-ball and $N_{B_R^d}^{<}(E)$ counts Neumann eigenvalues strictly below $E$. Combined with the Dirichlet theorem for balls, this settles both P\'olya inequalities for Euclidean balls in every dimension $d\ge2$. In the disk case, the proof replaces a computer-assisted finite-frequency step by explicit Rayleigh--Ritz estimates. In dimensions $d\ge3$, the radial Neumann condition is a Dini condition rather than a derivative-zero Bessel condition. A strict comparison with an auxiliary Robin problem transfers a derivative-zero Bessel phase estimate to the physical Neumann spectrum. The problem then becomes a comparison between a multiplicity-weighted phase staircase and an integral equal to the Weyl term. Variational trial spaces control low frequencies; finitely many radial levels and beta-integral estimates cover the intermediate range; and a uniform phase estimate treats high frequencies. All finite computations for $2\le d\le6$ are printed in the paper. For $d\ge7$, one compact two-parameter estimate is verified in exact rational arithmetic by the ancillary program.