{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:TRUJL2G27TZSQXTWV3RUMOIYUI","short_pith_number":"pith:TRUJL2G2","schema_version":"1.0","canonical_sha256":"9c6895e8dafcf3285e76aee3463918a239418e87a2a24f65c46d2ae846999ac2","source":{"kind":"arxiv","id":"1509.06092","version":1},"attestation_state":"computed","paper":{"title":"Universality for Barycentric subdivision","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.SP","authors_text":"Oliver Knill","submitted_at":"2015-09-21T01:48:58Z","abstract_excerpt":"The spectrum of the Laplacian of successive Barycentric subdivisions of a graph converges exponentially fast to a limit which only depends on the clique number of the initial graph and not on the graph itself. The proof uses an explicit linear operator mapping the clique vector of a graph to the clique vector of the Barycentric refinement. The eigenvectors of its transpose produce integral geometric invariants for which Euler characteristic is one example."},"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":"1509.06092","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-09-21T01:48:58Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"c8f222e331cbf2bd8061bbb91bcf1b629a7975318c695dbe0a6d4244df589591","abstract_canon_sha256":"4de413fc0ee879a3f58ff5f4b782adf2467bd66245ed133db3baac683cd21995"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:35.861618Z","signature_b64":"k/A4NBsB9JUKDDWDc9SFMNpA+0cmUGJ3fgZGhO+ElEfDB9QKFJjlErdyq25yx2g5qjXT06WYjzRH+bVTX0qmCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c6895e8dafcf3285e76aee3463918a239418e87a2a24f65c46d2ae846999ac2","last_reissued_at":"2026-05-18T01:32:35.861138Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:35.861138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universality for Barycentric subdivision","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.SP","authors_text":"Oliver Knill","submitted_at":"2015-09-21T01:48:58Z","abstract_excerpt":"The spectrum of the Laplacian of successive Barycentric subdivisions of a graph converges exponentially fast to a limit which only depends on the clique number of the initial graph and not on the graph itself. The proof uses an explicit linear operator mapping the clique vector of a graph to the clique vector of the Barycentric refinement. The eigenvectors of its transpose produce integral geometric invariants for which Euler characteristic is one example."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06092","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":"1509.06092","created_at":"2026-05-18T01:32:35.861207+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.06092v1","created_at":"2026-05-18T01:32:35.861207+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06092","created_at":"2026-05-18T01:32:35.861207+00:00"},{"alias_kind":"pith_short_12","alias_value":"TRUJL2G27TZS","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TRUJL2G27TZSQXTW","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TRUJL2G2","created_at":"2026-05-18T12:29:42.218222+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1907.09092","citing_title":"The counting matrix of a simplicial complex","ref_index":5,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI","json":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI.json","graph_json":"https://pith.science/api/pith-number/TRUJL2G27TZSQXTWV3RUMOIYUI/graph.json","events_json":"https://pith.science/api/pith-number/TRUJL2G27TZSQXTWV3RUMOIYUI/events.json","paper":"https://pith.science/paper/TRUJL2G2"},"agent_actions":{"view_html":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI","download_json":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI.json","view_paper":"https://pith.science/paper/TRUJL2G2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.06092&json=true","fetch_graph":"https://pith.science/api/pith-number/TRUJL2G27TZSQXTWV3RUMOIYUI/graph.json","fetch_events":"https://pith.science/api/pith-number/TRUJL2G27TZSQXTWV3RUMOIYUI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/action/storage_attestation","attest_author":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/action/author_attestation","sign_citation":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/action/citation_signature","submit_replication":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/action/replication_record"}},"created_at":"2026-05-18T01:32:35.861207+00:00","updated_at":"2026-05-18T01:32:35.861207+00:00"}