{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:UJ7WIVVQ2H66ERZA45HXAHBOQ4","short_pith_number":"pith:UJ7WIVVQ","schema_version":"1.0","canonical_sha256":"a27f6456b0d1fde24720e74f701c2e87132aff0f854e5f6ff561256e7af3c77a","source":{"kind":"arxiv","id":"1612.01360","version":1},"attestation_state":"computed","paper":{"title":"Completely regular codes in the infinite hexagonal grid","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Anastasia Yu. Vasil'eva (Sobolev Institute of Mathematics, Denis S. Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia), Sergey V. Avgustinovich (Sobolev Institute of Mathematics","submitted_at":"2016-11-30T21:00:02Z","abstract_excerpt":"A set $C$ of vertices of a simple graph is called a completely regular code if for each $i=0$, $1$, $2$, \\ldots and $j = i-1$, $i$, $i+1$, all vertices at distance $i$ from $C$ have the same number $s_{ij}$ of neighbors at distance $j$ from $C$. We characterize the completely regular codes in the infinite hexagonal grid graph."},"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":"1612.01360","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-11-30T21:00:02Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7ce53136f3b4a9d8d89308781f9ecad5418cc4c2fd0b65b6caa218c601fecf23","abstract_canon_sha256":"7cde5e4bba0f6b8fdd9ac3271276eb5803c6b9016621abf91fff226d8b6adbac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:52.858394Z","signature_b64":"AEDQmYtmya6XHMoSx6um3uSPvb/1uvMuLyVksuI8eknceRzz1NjiT9t2/3nMrSk6CmLf2uiOJfrXabia37SADQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a27f6456b0d1fde24720e74f701c2e87132aff0f854e5f6ff561256e7af3c77a","last_reissued_at":"2026-05-18T00:55:52.857886Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:52.857886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Completely regular codes in the infinite hexagonal grid","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Anastasia Yu. Vasil'eva (Sobolev Institute of Mathematics, Denis S. Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia), Sergey V. Avgustinovich (Sobolev Institute of Mathematics","submitted_at":"2016-11-30T21:00:02Z","abstract_excerpt":"A set $C$ of vertices of a simple graph is called a completely regular code if for each $i=0$, $1$, $2$, \\ldots and $j = i-1$, $i$, $i+1$, all vertices at distance $i$ from $C$ have the same number $s_{ij}$ of neighbors at distance $j$ from $C$. We characterize the completely regular codes in the infinite hexagonal grid graph."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01360","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":""},"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":"1612.01360","created_at":"2026-05-18T00:55:52.857967+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.01360v1","created_at":"2026-05-18T00:55:52.857967+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.01360","created_at":"2026-05-18T00:55:52.857967+00:00"},{"alias_kind":"pith_short_12","alias_value":"UJ7WIVVQ2H66","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"UJ7WIVVQ2H66ERZA","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"UJ7WIVVQ","created_at":"2026-05-18T12:30:46.583412+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/UJ7WIVVQ2H66ERZA45HXAHBOQ4","json":"https://pith.science/pith/UJ7WIVVQ2H66ERZA45HXAHBOQ4.json","graph_json":"https://pith.science/api/pith-number/UJ7WIVVQ2H66ERZA45HXAHBOQ4/graph.json","events_json":"https://pith.science/api/pith-number/UJ7WIVVQ2H66ERZA45HXAHBOQ4/events.json","paper":"https://pith.science/paper/UJ7WIVVQ"},"agent_actions":{"view_html":"https://pith.science/pith/UJ7WIVVQ2H66ERZA45HXAHBOQ4","download_json":"https://pith.science/pith/UJ7WIVVQ2H66ERZA45HXAHBOQ4.json","view_paper":"https://pith.science/paper/UJ7WIVVQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.01360&json=true","fetch_graph":"https://pith.science/api/pith-number/UJ7WIVVQ2H66ERZA45HXAHBOQ4/graph.json","fetch_events":"https://pith.science/api/pith-number/UJ7WIVVQ2H66ERZA45HXAHBOQ4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UJ7WIVVQ2H66ERZA45HXAHBOQ4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UJ7WIVVQ2H66ERZA45HXAHBOQ4/action/storage_attestation","attest_author":"https://pith.science/pith/UJ7WIVVQ2H66ERZA45HXAHBOQ4/action/author_attestation","sign_citation":"https://pith.science/pith/UJ7WIVVQ2H66ERZA45HXAHBOQ4/action/citation_signature","submit_replication":"https://pith.science/pith/UJ7WIVVQ2H66ERZA45HXAHBOQ4/action/replication_record"}},"created_at":"2026-05-18T00:55:52.857967+00:00","updated_at":"2026-05-18T00:55:52.857967+00:00"}