P\'olya's conjecture for Neumann eigenvalues of annuli

Abstract

We prove strict P\'olya inequalities for every positive Neumann eigenvalue of every Euclidean annulus. The proof preserves the exact Weyl area term through a radial action identity and combines variational bounds with uniform Bessel-phase and angular-sum estimates. Explicit polynomial positivity arguments and angular-layer comparisons cover the remaining finite spectral ranges. We also obtain a positive counting surplus uniform in the radius ratio, an explicit subtraction of order $\sqrt{k}/(R^2-r^2)$ for all $k\ge1604$, and a positive square-root correction over the entire spectrum with a universal, non-explicit coefficient. The square-root order is sharp, and the first positive pair and the low-energy count admit optimal constants expressed by the first positive zero of $J_1'$. Neumann inequalities for annular products and congruent annular sectors follow by established transfer principles.