Pólya's inequality for three-dimensional spherical shells and balls with small spherical holes

This record contains the full author preprint and computational supplementary material for “Pólya's inequality for three-dimensional spherical shells and balls with small spherical holes”. The paper proves the Dirichlet Pólya inequality for every concentric spherical shell in three dimensions, at every spectral frequency. It also establishes sufficient conditions for balls with finitely many small spherical holes at arbitrary admissible positions. The computational supplement contains 11 Python/C++ source files, five certificate files, an English program guide, reproduction instructions, verification records, and two supplementary mathematical notes in PDF and LaTeX formats. All nine mandatory verification steps were successfully replayed during preparation of the package. The computations verify finite arithmetic and continuous certificate coverage using the analytic inputs established or cited in the manuscript.