{"paper":{"title":"Non-crystallographic systems of integers over composition algebras","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A weak golden octonion order arises from Cayley-Dickson doubling of the icosian ring, carrying a 240-element H4⊕H4 shell and remaining self-dual under the polar norm.","cross_cats":[],"primary_cat":"math.RA","authors_text":"Daniele Corradetti","submitted_at":"2026-05-14T17:00:48Z","abstract_excerpt":"In this work we revisit classical systems of integers inside the real normed division algebras from the point of view of finite norm shells and root systems. Building on the icosian framework of Moody--Patera and on the integral root-system viewpoint of Chen--Moody--Patera and of Johnson, we isolate the precise axiomatic ingredients of the non-crystallographic analogue: an order over the golden ring \\(\\Zphi\\) together with a distinguished finite root shell whose Cartan coefficients lie in \\(\\Zphi\\). We show that the usual Gaussian, Eisenstein, Hamilton, Hurwitz and Coxeter--Dickson examples ar"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct a weak golden octonion order by Cayley--Dickson doubling of the icosian ring; the resulting free rank-8 Zφ-order has a 240-element finite shell of type H4⊕H4 and its multiplication is genuinely octonionic. We prove (i) that this weak double is self-dual with respect to the polar norm pairing, hence has no strict norm-integral overorder, and (ii) that the first trace-integral discriminant tower over it contains no octonion-stable nonzero isotropic gluing.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that the Cayley-Dickson doubling applied to the icosian ring preserves a genuinely octonionic multiplication while producing exactly the claimed 240-element H4⊕H4 shell and satisfying the self-duality and discriminant-tower properties without additional hidden constraints on the order.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new free rank-8 Z[φ]-order is built by doubling the icosian ring, equipped with a 240-element H4⊕H4 finite root shell, shown to be self-dual under the polar norm and free of octonion-stable nonzero isotropic gluings in its first trace-integral discriminant tower.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A weak golden octonion order arises from Cayley-Dickson doubling of the icosian ring, carrying a 240-element H4⊕H4 shell and remaining self-dual under the polar norm.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3fde6606ac4058e2d017c62aac1e8a9f68a29b64e08947a66677579eb8acac70"},"source":{"id":"2605.15075","kind":"arxiv","version":1},"verdict":{"id":"f39b370e-fe50-4de3-b6a1-c18b11b5ef94","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:32:11.598063Z","strongest_claim":"We construct a weak golden octonion order by Cayley--Dickson doubling of the icosian ring; the resulting free rank-8 Zφ-order has a 240-element finite shell of type H4⊕H4 and its multiplication is genuinely octonionic. We prove (i) that this weak double is self-dual with respect to the polar norm pairing, hence has no strict norm-integral overorder, and (ii) that the first trace-integral discriminant tower over it contains no octonion-stable nonzero isotropic gluing.","one_line_summary":"A new free rank-8 Z[φ]-order is built by doubling the icosian ring, equipped with a 240-element H4⊕H4 finite root shell, shown to be self-dual under the polar norm and free of octonion-stable nonzero isotropic gluings in its first trace-integral discriminant tower.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that the Cayley-Dickson doubling applied to the icosian ring preserves a genuinely octonionic multiplication while producing exactly the claimed 240-element H4⊕H4 shell and satisfying the self-duality and discriminant-tower properties without additional hidden constraints on the order.","pith_extraction_headline":"A weak golden octonion order arises from Cayley-Dickson doubling of the icosian ring, carrying a 240-element H4⊕H4 shell and remaining self-dual under the polar norm."},"references":{"count":24,"sample":[{"doi":"","year":1998,"title":"Baake, A guide to mathematical quasicrystals, in Quasicrystals, eds","work_id":"1029d3c1-5fd2-4539-8e96-fb5398599721","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"M. Baake and U. Grimm, Aperiodic Order, Cambridge University Press, 2013","work_id":"66740269-f37d-45b9-9e5d-e56334ba9f28","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"L. Chen, R. V. Moody and J. Patera, Non-crystallographic root systems, Fields Institute Monographs, Volume 10, 1998, 135--178","work_id":"93aef1b0-d64b-4753-a348-563adf732772","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1999,"title":"J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, 1999","work_id":"c2dfc9aa-3617-4942-bcfb-a78051fef556","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters/CRC Press, 2003","work_id":"468cd8f6-9d95-4da7-b3cb-955c772e1ba8","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":24,"snapshot_sha256":"099611fad034a19a54fa7a8e05d63cd2f905fe65e7b7446c856067b4aa80b425","internal_anchors":0},"formal_canon":{"evidence_count":3,"snapshot_sha256":"624b4592ce3b75923d9643e0b1d4f76695b64d6d9e87b0299f1e679afa6352ee"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}