{"paper":{"title":"On the P\\'olya Frequency Order of the de Bruijn--Newman Kernel: Certified Failure at Order Five","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Wojciech Michalowski","submitted_at":"2026-02-23T19:59:05Z","abstract_excerpt":"We prove that the classical de Bruijn--Newman kernel $K(u)=\\Phi(|u|)$ is not a P\\'olya frequency function of order $5$ (PF$_5$). At $(u_0,h)=(0.01,0.05)$ we exhibit an explicit $5\\times5$ Toeplitz minor whose determinant is rigorously enclosed in $[-1.8472496\\times10^{-9},-1.8472225\\times10^{-9}]$. The certificate uses 80-digit outward-rounded interval arithmetic and a proved truncation bound for the theta series. Eight further configurations are certified by both an explicit Leibniz expansion and an independent interval determinant computation. At the central configuration the determinants $D"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.20313","kind":"arxiv","version":2},"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/2602.20313/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"}