{"paper":{"title":"Escalations and criteria over real quadratic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Giuliano Romeo, Jakub Kr\\'asensk\\'y","submitted_at":"2026-08-12T23:16:54Z","abstract_excerpt":"The famous 15-Theorem and 290-Theorem fully characterise universal quadratic forms over $\\mathbb{Q}$. Similar theorems exist for every totally real number field, but only over $\\mathbb{Q}(\\sqrt5)$ the criterion set is explicitly known. We study criterion sets both theoretically and computationally: We develop the method of escalation over number fields, thus providing a simple proof of finiteness of the criteria and, more importantly, a practical tool for computing them. We illustrate this by explicit computations for $\\mathbb{Q}(\\sqrt2)$ and $\\mathbb{Q}(\\sqrt3)$, obtaining a conjecture about "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.12648","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/2608.12648/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"}