{"paper":{"title":"P\\'olya's Conjecture for the Neumann Laplacian on Euclidean Balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CA"],"primary_cat":"math.SP","authors_text":"Yutian Li","submitted_at":"2026-07-28T16:43:50Z","abstract_excerpt":"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 ra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25958","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.25958/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}