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In particular, if the toric ideal of a $(0,1)$-polytope is generated by quadratic binomials, then every clique of its 1-skeleton is the vertex set of a face. For $(0,1)$-polytopes satisfying condition (E), we characterize this clique-face property in terms of divisibility by quadratic monomials appear"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.20430","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-18T16:10:05Z","cross_cats_sorted":[],"title_canon_sha256":"954d872c742e15f0aeafcf709c8a4186982a41dba9aef3aff3f0241480f41c14","abstract_canon_sha256":"a854f972da6950d8ae74f23d1b5e785daf75b7c87a39a08fe7530fa8ed99ce77"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:13:12.249450Z","signature_b64":"AgbchYgSXPtK1UlxiTm6xAJRskavGnPzIWhb3sRwIRez7plCpkn3KNQA0yAe0y3djxQhvjqLlBQn62nLl6QxAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"224445223041b60c6235050afa75142daa53f8f90b49bcdaccf79c7bd2fd6a8b","last_reissued_at":"2026-06-19T16:13:12.249047Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:13:12.249047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Simplex faces and quadratic toric ideals of lattice polytopes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aki Mori, Hidefumi Ohsugi","submitted_at":"2026-06-18T16:10:05Z","abstract_excerpt":"We study interactions between simplex faces of lattice polytopes and quadratic generation of toric ideals. 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