{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:TRUJL2G27TZSQXTWV3RUMOIYUI","short_pith_number":"pith:TRUJL2G2","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"},"canonical_sha256":"9c6895e8dafcf3285e76aee3463918a239418e87a2a24f65c46d2ae846999ac2","source":{"kind":"arxiv","id":"1509.06092","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.06092","created_at":"2026-05-18T01:32:35Z"},{"alias_kind":"arxiv_version","alias_value":"1509.06092v1","created_at":"2026-05-18T01:32:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06092","created_at":"2026-05-18T01:32:35Z"},{"alias_kind":"pith_short_12","alias_value":"TRUJL2G27TZS","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TRUJL2G27TZSQXTW","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TRUJL2G2","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:TRUJL2G27TZSQXTWV3RUMOIYUI","target":"record","payload":{"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"},"canonical_sha256":"9c6895e8dafcf3285e76aee3463918a239418e87a2a24f65c46d2ae846999ac2","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"},"source_kind":"arxiv","source_id":"1509.06092","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:32:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0JAxNRr3PPZrkpeZhRx3lM0Aq2ruRmEJ8MAmYT7C7/Jm4Iddg/rBq7BtMukRqa++lTYfYp3bMGYimRl4jdjBBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T06:40:44.877223Z"},"content_sha256":"a268f4ea08dad66921c892efaa3acbbb1378caaa25582a6245289cda5a131d62","schema_version":"1.0","event_id":"sha256:a268f4ea08dad66921c892efaa3acbbb1378caaa25582a6245289cda5a131d62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:TRUJL2G27TZSQXTWV3RUMOIYUI","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:32:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/EyIuMz/MxMI9sL6AQEetVzBSiVNbKzAmW4scPSkwwYRnGivNf3QeKDFjLGooTdef3ngYkIfEL6gr0b08enkCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T06:40:44.877929Z"},"content_sha256":"b45f4783d7d944e040087addfeb9cc55b24ce76b0e61c5a70980fc179693c111","schema_version":"1.0","event_id":"sha256:b45f4783d7d944e040087addfeb9cc55b24ce76b0e61c5a70980fc179693c111"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/bundle.json","state_url":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T06:40:44Z","links":{"resolver":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI","bundle":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/bundle.json","state":"https://pith.science/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TRUJL2G27TZSQXTWV3RUMOIYUI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:TRUJL2G27TZSQXTWV3RUMOIYUI","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4de413fc0ee879a3f58ff5f4b782adf2467bd66245ed133db3baac683cd21995","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-09-21T01:48:58Z","title_canon_sha256":"c8f222e331cbf2bd8061bbb91bcf1b629a7975318c695dbe0a6d4244df589591"},"schema_version":"1.0","source":{"id":"1509.06092","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.06092","created_at":"2026-05-18T01:32:35Z"},{"alias_kind":"arxiv_version","alias_value":"1509.06092v1","created_at":"2026-05-18T01:32:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06092","created_at":"2026-05-18T01:32:35Z"},{"alias_kind":"pith_short_12","alias_value":"TRUJL2G27TZS","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TRUJL2G27TZSQXTW","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TRUJL2G2","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:b45f4783d7d944e040087addfeb9cc55b24ce76b0e61c5a70980fc179693c111","target":"graph","created_at":"2026-05-18T01:32:35Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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.","authors_text":"Oliver Knill","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-09-21T01:48:58Z","title":"Universality for Barycentric subdivision"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06092","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a268f4ea08dad66921c892efaa3acbbb1378caaa25582a6245289cda5a131d62","target":"record","created_at":"2026-05-18T01:32:35Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4de413fc0ee879a3f58ff5f4b782adf2467bd66245ed133db3baac683cd21995","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-09-21T01:48:58Z","title_canon_sha256":"c8f222e331cbf2bd8061bbb91bcf1b629a7975318c695dbe0a6d4244df589591"},"schema_version":"1.0","source":{"id":"1509.06092","kind":"arxiv","version":1}},"canonical_sha256":"9c6895e8dafcf3285e76aee3463918a239418e87a2a24f65c46d2ae846999ac2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9c6895e8dafcf3285e76aee3463918a239418e87a2a24f65c46d2ae846999ac2","first_computed_at":"2026-05-18T01:32:35.861138Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:35.861138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k/A4NBsB9JUKDDWDc9SFMNpA+0cmUGJ3fgZGhO+ElEfDB9QKFJjlErdyq25yx2g5qjXT06WYjzRH+bVTX0qmCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:35.861618Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.06092","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a268f4ea08dad66921c892efaa3acbbb1378caaa25582a6245289cda5a131d62","sha256:b45f4783d7d944e040087addfeb9cc55b24ce76b0e61c5a70980fc179693c111"],"state_sha256":"a64d05c689c1e21dd20654df22330b35229defa95e319927e42bf941c7f9d7f8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fX/Z+76+3JISJbzIbBHuYp1EhlySOdU2B4KiOszG0Iz/0CMa1i3mku/QIN6Fq09L5iMDF33QSPT+Aoo+nIP+Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T06:40:44.881261Z","bundle_sha256":"0a8623e6eebc77337252e252fe404fa38df53de2886a6e9e83397791c4351d2d"}}