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It will be shown that $\\beta_{p, p+j}(K[\\mathcal{O}(P)]) = \\beta_{p, p+j}(K[\\mathcal{C}(P)])$ for all $j \\geq 0$, where $p$ is the projective dimension of $K[\\mathcal{O}(P)]$ (and that of $K[\\mathcal{C}(P)]$). In particular, $K[\\mathcal{O}(P)]$ is level if and only if $K[\\mathcal{C}(P)]$ is level."},"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":"1809.08462","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-22T18:03:40Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"6b0c95701f6714ee985938d3901c7b989910bf709ebd7c5e5c321dfcf63cdcd1","abstract_canon_sha256":"84f28fadfcb39b7e9d7916ba6762237277cf16f72682da96bf7eba2d1750e107"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:31.268474Z","signature_b64":"jvMCuDHTxTUrDQpLcESN9dAKQxlgRqSxATMte/SXnfau/adEAo/z31mu5oLuNFuH4Rxv/seoT+LT3CIV/rhHBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7bc2270602af317c0aa15d8e15578c68f851b22734dec2ad0678bb554c9d3654","last_reissued_at":"2026-05-17T23:57:31.267838Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:31.267838Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Levelness of toric rings arising from order and chain polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Akihiro Higashitani, Takayuki Hibi","submitted_at":"2018-09-22T18:03:40Z","abstract_excerpt":"Let $K[\\mathcal{O}(P)]$ denote the toric ring of the order polytope $\\mathcal{O}(P)$ of a finite partially ordered set $P$ and $K[\\mathcal{C}(P)]$ that of the chain polytope $\\mathcal{C}(P)$. It will be shown that $\\beta_{p, p+j}(K[\\mathcal{O}(P)]) = \\beta_{p, p+j}(K[\\mathcal{C}(P)])$ for all $j \\geq 0$, where $p$ is the projective dimension of $K[\\mathcal{O}(P)]$ (and that of $K[\\mathcal{C}(P)]$). In particular, $K[\\mathcal{O}(P)]$ is level if and only if $K[\\mathcal{C}(P)]$ is level."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.08462","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1809.08462","created_at":"2026-05-17T23:57:31.267920+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.08462v3","created_at":"2026-05-17T23:57:31.267920+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.08462","created_at":"2026-05-17T23:57:31.267920+00:00"},{"alias_kind":"pith_short_12","alias_value":"PPBCOBQCV4YX","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_16","alias_value":"PPBCOBQCV4YXYCVB","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_8","alias_value":"PPBCOBQC","created_at":"2026-05-18T12:32:46.962924+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PPBCOBQCV4YXYCVBLWHBKV4MND","json":"https://pith.science/pith/PPBCOBQCV4YXYCVBLWHBKV4MND.json","graph_json":"https://pith.science/api/pith-number/PPBCOBQCV4YXYCVBLWHBKV4MND/graph.json","events_json":"https://pith.science/api/pith-number/PPBCOBQCV4YXYCVBLWHBKV4MND/events.json","paper":"https://pith.science/paper/PPBCOBQC"},"agent_actions":{"view_html":"https://pith.science/pith/PPBCOBQCV4YXYCVBLWHBKV4MND","download_json":"https://pith.science/pith/PPBCOBQCV4YXYCVBLWHBKV4MND.json","view_paper":"https://pith.science/paper/PPBCOBQC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.08462&json=true","fetch_graph":"https://pith.science/api/pith-number/PPBCOBQCV4YXYCVBLWHBKV4MND/graph.json","fetch_events":"https://pith.science/api/pith-number/PPBCOBQCV4YXYCVBLWHBKV4MND/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PPBCOBQCV4YXYCVBLWHBKV4MND/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PPBCOBQCV4YXYCVBLWHBKV4MND/action/storage_attestation","attest_author":"https://pith.science/pith/PPBCOBQCV4YXYCVBLWHBKV4MND/action/author_attestation","sign_citation":"https://pith.science/pith/PPBCOBQCV4YXYCVBLWHBKV4MND/action/citation_signature","submit_replication":"https://pith.science/pith/PPBCOBQCV4YXYCVBLWHBKV4MND/action/replication_record"}},"created_at":"2026-05-17T23:57:31.267920+00:00","updated_at":"2026-05-17T23:57:31.267920+00:00"}