Pólya's conjecture for Neumann eigenvalues of annuli

We prove Pólya’s inequality for every Neumann eigenvalue of every Euclidean circular 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. It is entirely analytic when the width is at most one quarter of the inner radius. For arbitrary radii, explicit high-frequency and small-hole bounds reduce the problem to a compact parameter region, which is verified by exact rational and outward interval certificates valid on whole parameter rectangles. Established transfer principles yield the Neumann inequality for annular products and congruent annular sectors.