{"paper":{"title":"Integral Planes and Unit-Norm Polytopes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CO","authors_text":"Daniele Corradetti","submitted_at":"2026-05-18T15:21:13Z","abstract_excerpt":"We introduce and study integral planes associated with crystallographic and non-crystallographic integral systems in real composition algebras. For an integral order $\\Order$ in such an algebra we define the plane $\\Order^{2}$ with quadratic form $Q(x,y)=\\NN(x)+\\NN(y)$, the axis shell, the balanced shell, and the corresponding unit-normalised spherical polytopes. For ten crystallographic orders we recover, in one uniform construction, the orthogonal-direct-sum root systems $2A_{1}$, $A_{2}\\oplus A_{2}$, $4A_{1}$, $D_{4}\\oplus D_{4}$, $16A_{1}$, and $E_{8}\\oplus E_{8}$ (with classical-polytope "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.18538","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/2605.18538/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"}