{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:BSOQODD74VLTL3YNI2K32ELKT6","short_pith_number":"pith:BSOQODD7","canonical_record":{"source":{"id":"1508.02027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-08-09T14:20:48Z","cross_cats_sorted":["math.CO","math.SP"],"title_canon_sha256":"6ad196b2522d13a174f3e41046b261f72254775f7040f1affa20c1d1265430f3","abstract_canon_sha256":"78b2e24ec7e8a47902aff36336eb6d7f43a81c367ca07cb6110dbedc544b5121"},"schema_version":"1.0"},"canonical_sha256":"0c9d070c7fe55735ef0d4695bd116a9f942e85e2d2e4905d5e70c51069a55ad3","source":{"kind":"arxiv","id":"1508.02027","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02027","created_at":"2026-05-18T01:35:34Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02027v1","created_at":"2026-05-18T01:35:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02027","created_at":"2026-05-18T01:35:34Z"},{"alias_kind":"pith_short_12","alias_value":"BSOQODD74VLT","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BSOQODD74VLTL3YN","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BSOQODD7","created_at":"2026-05-18T12:29:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:BSOQODD74VLTL3YNI2K32ELKT6","target":"record","payload":{"canonical_record":{"source":{"id":"1508.02027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-08-09T14:20:48Z","cross_cats_sorted":["math.CO","math.SP"],"title_canon_sha256":"6ad196b2522d13a174f3e41046b261f72254775f7040f1affa20c1d1265430f3","abstract_canon_sha256":"78b2e24ec7e8a47902aff36336eb6d7f43a81c367ca07cb6110dbedc544b5121"},"schema_version":"1.0"},"canonical_sha256":"0c9d070c7fe55735ef0d4695bd116a9f942e85e2d2e4905d5e70c51069a55ad3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:34.448577Z","signature_b64":"jGBBph+s4Xc04Dr2F7Ns2MWdM+rNdi2+fEc+zBjsX2T9+LNPcXRShL1YjqhiSL6rXRmiO+ifcaI/mihq4z8KBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c9d070c7fe55735ef0d4695bd116a9f942e85e2d2e4905d5e70c51069a55ad3","last_reissued_at":"2026-05-18T01:35:34.448025Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:34.448025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.02027","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":"2026-05-18T01:35:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1YGzReDbf2F1z8VO9jmlXP8Sf8lPHpD7hh14oAhu9UoHpbcK0KZo9ngvlHr4y64LRc0CkK5MKMufZsEW7uTlDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T22:14:35.923225Z"},"content_sha256":"a1c0fde8261333e80082db2c54d7c0e4ab6000f614c294a296263901f379cad4","schema_version":"1.0","event_id":"sha256:a1c0fde8261333e80082db2c54d7c0e4ab6000f614c294a296263901f379cad4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:BSOQODD74VLTL3YNI2K32ELKT6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The graph spectrum of barycentric refinements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.SP"],"primary_cat":"cs.DM","authors_text":"Oliver Knill","submitted_at":"2015-08-09T14:20:48Z","abstract_excerpt":"Given a finite simple graph G, let G' be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If L(0)=0<L(1) <= L(2) ... <= L(n) are the eigenvalues of the Laplacian of G, define the spectral function F(x) as the function F(x) = L([n x]) on the interval [0,1], where [r] is the floor function giving the largest integer smaller or equal than r. The graph G' is known to be homotopic to G with Euler characteristic chi(G')=chi(G) and dim(G') >= dim(G). Let G(m) be the sequ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02027","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":"2026-05-18T01:35:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LPrAu3ArHfUd9hDwitD1OhqYD4XPR0J5EPdqyOQZtK2qkLPEivYvvc5V6tCjC2xrCPgylSs5JmNGRFluLE1PDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T22:14:35.923752Z"},"content_sha256":"d385c04752c857048a8fb68724a883cc6653eea592b88a0ed545d86d2fa13166","schema_version":"1.0","event_id":"sha256:d385c04752c857048a8fb68724a883cc6653eea592b88a0ed545d86d2fa13166"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BSOQODD74VLTL3YNI2K32ELKT6/bundle.json","state_url":"https://pith.science/pith/BSOQODD74VLTL3YNI2K32ELKT6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BSOQODD74VLTL3YNI2K32ELKT6/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":"2026-05-27T22:14:35Z","links":{"resolver":"https://pith.science/pith/BSOQODD74VLTL3YNI2K32ELKT6","bundle":"https://pith.science/pith/BSOQODD74VLTL3YNI2K32ELKT6/bundle.json","state":"https://pith.science/pith/BSOQODD74VLTL3YNI2K32ELKT6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BSOQODD74VLTL3YNI2K32ELKT6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:BSOQODD74VLTL3YNI2K32ELKT6","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":"78b2e24ec7e8a47902aff36336eb6d7f43a81c367ca07cb6110dbedc544b5121","cross_cats_sorted":["math.CO","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-08-09T14:20:48Z","title_canon_sha256":"6ad196b2522d13a174f3e41046b261f72254775f7040f1affa20c1d1265430f3"},"schema_version":"1.0","source":{"id":"1508.02027","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02027","created_at":"2026-05-18T01:35:34Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02027v1","created_at":"2026-05-18T01:35:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02027","created_at":"2026-05-18T01:35:34Z"},{"alias_kind":"pith_short_12","alias_value":"BSOQODD74VLT","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BSOQODD74VLTL3YN","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BSOQODD7","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:d385c04752c857048a8fb68724a883cc6653eea592b88a0ed545d86d2fa13166","target":"graph","created_at":"2026-05-18T01:35:34Z","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":"Given a finite simple graph G, let G' be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If L(0)=0<L(1) <= L(2) ... <= L(n) are the eigenvalues of the Laplacian of G, define the spectral function F(x) as the function F(x) = L([n x]) on the interval [0,1], where [r] is the floor function giving the largest integer smaller or equal than r. The graph G' is known to be homotopic to G with Euler characteristic chi(G')=chi(G) and dim(G') >= dim(G). Let G(m) be the sequ","authors_text":"Oliver Knill","cross_cats":["math.CO","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-08-09T14:20:48Z","title":"The graph spectrum of barycentric refinements"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02027","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:a1c0fde8261333e80082db2c54d7c0e4ab6000f614c294a296263901f379cad4","target":"record","created_at":"2026-05-18T01:35:34Z","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":"78b2e24ec7e8a47902aff36336eb6d7f43a81c367ca07cb6110dbedc544b5121","cross_cats_sorted":["math.CO","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-08-09T14:20:48Z","title_canon_sha256":"6ad196b2522d13a174f3e41046b261f72254775f7040f1affa20c1d1265430f3"},"schema_version":"1.0","source":{"id":"1508.02027","kind":"arxiv","version":1}},"canonical_sha256":"0c9d070c7fe55735ef0d4695bd116a9f942e85e2d2e4905d5e70c51069a55ad3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0c9d070c7fe55735ef0d4695bd116a9f942e85e2d2e4905d5e70c51069a55ad3","first_computed_at":"2026-05-18T01:35:34.448025Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:34.448025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jGBBph+s4Xc04Dr2F7Ns2MWdM+rNdi2+fEc+zBjsX2T9+LNPcXRShL1YjqhiSL6rXRmiO+ifcaI/mihq4z8KBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:34.448577Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.02027","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a1c0fde8261333e80082db2c54d7c0e4ab6000f614c294a296263901f379cad4","sha256:d385c04752c857048a8fb68724a883cc6653eea592b88a0ed545d86d2fa13166"],"state_sha256":"f046035eb58ebe0739244db3ec0aaf039f4c9e6fba71f2b236cff8909c90033e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5YMarSRKMVARTrY/hSM8UJVzzcc6Yjs62jL9Tfqs4C7ZNAGGcEnU4Fb55HoiSVWZLh2nBU9TXZ/pqHBv3eesDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T22:14:35.926706Z","bundle_sha256":"c54c2aa1d8ef91c50c300f559ad4e7839c3078130eddece22afa50c3cc5f209a"}}