{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:ZFLF6S4S7KGD3PMU66O5TGRFWU","short_pith_number":"pith:ZFLF6S4S","schema_version":"1.0","canonical_sha256":"c9565f4b92fa8c3dbd94f79dd99a25b50c7fa89a40dc6f48ca00dff2c314eec1","source":{"kind":"arxiv","id":"1207.3605","version":2},"attestation_state":"computed","paper":{"title":"There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.CO","authors_text":"Boris Springborn, G\\\"unter Rote, Ivan Izmestiev, John M. Sullivan, Robert B. Kusner","submitted_at":"2012-07-16T09:02:29Z","abstract_excerpt":"There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the oth"},"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":"1207.3605","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-16T09:02:29Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"0e1b0807a915d8fbe0cad8eebd3e49c706fb6ab4073fd44c4c7fd584118f7837","abstract_canon_sha256":"a951088b9b14024ecc1ecfe50fef137cb45333227cbe180978c498a5743bac82"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:21.898415Z","signature_b64":"h9joMKUY4ofn3kAJ+KF+OvkrTs05AV5g8CcxBjTFnxkFJgQCcKzpoNA050WweXbBEsy4bYLrH5NVAHHkDWzWCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9565f4b92fa8c3dbd94f79dd99a25b50c7fa89a40dc6f48ca00dff2c314eec1","last_reissued_at":"2026-05-18T03:13:21.897789Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:21.897789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.CO","authors_text":"Boris Springborn, G\\\"unter Rote, Ivan Izmestiev, John M. Sullivan, Robert B. Kusner","submitted_at":"2012-07-16T09:02:29Z","abstract_excerpt":"There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the oth"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3605","kind":"arxiv","version":2},"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":"1207.3605","created_at":"2026-05-18T03:13:21.897870+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.3605v2","created_at":"2026-05-18T03:13:21.897870+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3605","created_at":"2026-05-18T03:13:21.897870+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZFLF6S4S7KGD","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZFLF6S4S7KGD3PMU","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZFLF6S4S","created_at":"2026-05-18T12:27:30.460161+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/ZFLF6S4S7KGD3PMU66O5TGRFWU","json":"https://pith.science/pith/ZFLF6S4S7KGD3PMU66O5TGRFWU.json","graph_json":"https://pith.science/api/pith-number/ZFLF6S4S7KGD3PMU66O5TGRFWU/graph.json","events_json":"https://pith.science/api/pith-number/ZFLF6S4S7KGD3PMU66O5TGRFWU/events.json","paper":"https://pith.science/paper/ZFLF6S4S"},"agent_actions":{"view_html":"https://pith.science/pith/ZFLF6S4S7KGD3PMU66O5TGRFWU","download_json":"https://pith.science/pith/ZFLF6S4S7KGD3PMU66O5TGRFWU.json","view_paper":"https://pith.science/paper/ZFLF6S4S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.3605&json=true","fetch_graph":"https://pith.science/api/pith-number/ZFLF6S4S7KGD3PMU66O5TGRFWU/graph.json","fetch_events":"https://pith.science/api/pith-number/ZFLF6S4S7KGD3PMU66O5TGRFWU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZFLF6S4S7KGD3PMU66O5TGRFWU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZFLF6S4S7KGD3PMU66O5TGRFWU/action/storage_attestation","attest_author":"https://pith.science/pith/ZFLF6S4S7KGD3PMU66O5TGRFWU/action/author_attestation","sign_citation":"https://pith.science/pith/ZFLF6S4S7KGD3PMU66O5TGRFWU/action/citation_signature","submit_replication":"https://pith.science/pith/ZFLF6S4S7KGD3PMU66O5TGRFWU/action/replication_record"}},"created_at":"2026-05-18T03:13:21.897870+00:00","updated_at":"2026-05-18T03:13:21.897870+00:00"}