{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:S2IS34JQIZBIHDDOF4OFODPYZF","short_pith_number":"pith:S2IS34JQ","canonical_record":{"source":{"id":"0910.4615","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-10-24T03:36:03Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"e963aadf030fdbbd557105cfc7b95c86980206084bd3d6a3e563516640c416b8","abstract_canon_sha256":"2d2a40b732f49046d6dd94da17a8801811a3f40438d8e0499dc1fe9cd59ec3d8"},"schema_version":"1.0"},"canonical_sha256":"96912df1304642838c6e2f1c570df8c963ecd3fe2b87c836a18994f3b9647366","source":{"kind":"arxiv","id":"0910.4615","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.4615","created_at":"2026-05-18T02:11:11Z"},{"alias_kind":"arxiv_version","alias_value":"0910.4615v1","created_at":"2026-05-18T02:11:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.4615","created_at":"2026-05-18T02:11:11Z"},{"alias_kind":"pith_short_12","alias_value":"S2IS34JQIZBI","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"S2IS34JQIZBIHDDO","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"S2IS34JQ","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:S2IS34JQIZBIHDDOF4OFODPYZF","target":"record","payload":{"canonical_record":{"source":{"id":"0910.4615","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-10-24T03:36:03Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"e963aadf030fdbbd557105cfc7b95c86980206084bd3d6a3e563516640c416b8","abstract_canon_sha256":"2d2a40b732f49046d6dd94da17a8801811a3f40438d8e0499dc1fe9cd59ec3d8"},"schema_version":"1.0"},"canonical_sha256":"96912df1304642838c6e2f1c570df8c963ecd3fe2b87c836a18994f3b9647366","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:11:11.970769Z","signature_b64":"14tRnWzbGRDY7D5YWQKX3GjSThEJRSCi6xN9kUeuplX1M8jGdaoN07K97dLaK6roG0zVIvL/at/PGrHfd/bvBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"96912df1304642838c6e2f1c570df8c963ecd3fe2b87c836a18994f3b9647366","last_reissued_at":"2026-05-18T02:11:11.969933Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:11:11.969933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0910.4615","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":"2026-05-18T02:11:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zd81q427yRwsPouODocYxo/oOu/LGTZxv4sbSQzLeMkVsREygi/NAl+b4cxRckETadcvgZNZS+uRn6ICk2OnDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T19:58:45.368443Z"},"content_sha256":"1e3718595100bc8074e1e7a69fcebef5998d4be4df460d27db08488dd64a2ef3","schema_version":"1.0","event_id":"sha256:1e3718595100bc8074e1e7a69fcebef5998d4be4df460d27db08488dd64a2ef3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:S2IS34JQIZBIHDDOF4OFODPYZF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The McKean-Vlasov Equation in Finite Volume","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Lincoln Chayes, Vladislav Panferov","submitted_at":"2009-10-24T03:36:03Z","abstract_excerpt":"We study the McKean--Vlasov equation on the finite tori of length scale $L$ in $d$--dimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in \\cite{GP} and \\cite{KM}. Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, $\\theta^{\\sharp}$ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for $\\theta < \\theta^{\\sharp}$ and prove, abstractly, that a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.4615","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":"2026-05-18T02:11:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lpU9BRbdhGjzjtooiBNomPr22xgsIR4fnoQgNGGGrX049O8VSwLE70Lhjh7eMkHVRgSouPxnBUy2oBfyIZfFBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T19:58:45.368788Z"},"content_sha256":"4c6074c39e3591684834f0a94b509bc3f7b089a641dd0ed260cc3ec2b11c728f","schema_version":"1.0","event_id":"sha256:4c6074c39e3591684834f0a94b509bc3f7b089a641dd0ed260cc3ec2b11c728f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S2IS34JQIZBIHDDOF4OFODPYZF/bundle.json","state_url":"https://pith.science/pith/S2IS34JQIZBIHDDOF4OFODPYZF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S2IS34JQIZBIHDDOF4OFODPYZF/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":"2026-06-27T19:58:45Z","links":{"resolver":"https://pith.science/pith/S2IS34JQIZBIHDDOF4OFODPYZF","bundle":"https://pith.science/pith/S2IS34JQIZBIHDDOF4OFODPYZF/bundle.json","state":"https://pith.science/pith/S2IS34JQIZBIHDDOF4OFODPYZF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S2IS34JQIZBIHDDOF4OFODPYZF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:S2IS34JQIZBIHDDOF4OFODPYZF","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":"2d2a40b732f49046d6dd94da17a8801811a3f40438d8e0499dc1fe9cd59ec3d8","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-10-24T03:36:03Z","title_canon_sha256":"e963aadf030fdbbd557105cfc7b95c86980206084bd3d6a3e563516640c416b8"},"schema_version":"1.0","source":{"id":"0910.4615","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.4615","created_at":"2026-05-18T02:11:11Z"},{"alias_kind":"arxiv_version","alias_value":"0910.4615v1","created_at":"2026-05-18T02:11:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.4615","created_at":"2026-05-18T02:11:11Z"},{"alias_kind":"pith_short_12","alias_value":"S2IS34JQIZBI","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"S2IS34JQIZBIHDDO","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"S2IS34JQ","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:4c6074c39e3591684834f0a94b509bc3f7b089a641dd0ed260cc3ec2b11c728f","target":"graph","created_at":"2026-05-18T02:11:11Z","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":"We study the McKean--Vlasov equation on the finite tori of length scale $L$ in $d$--dimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in \\cite{GP} and \\cite{KM}. Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, $\\theta^{\\sharp}$ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for $\\theta < \\theta^{\\sharp}$ and prove, abstractly, that a","authors_text":"Lincoln Chayes, Vladislav Panferov","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-10-24T03:36:03Z","title":"The McKean-Vlasov Equation in Finite Volume"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.4615","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":"sha256:1e3718595100bc8074e1e7a69fcebef5998d4be4df460d27db08488dd64a2ef3","target":"record","created_at":"2026-05-18T02:11:11Z","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":"2d2a40b732f49046d6dd94da17a8801811a3f40438d8e0499dc1fe9cd59ec3d8","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-10-24T03:36:03Z","title_canon_sha256":"e963aadf030fdbbd557105cfc7b95c86980206084bd3d6a3e563516640c416b8"},"schema_version":"1.0","source":{"id":"0910.4615","kind":"arxiv","version":1}},"canonical_sha256":"96912df1304642838c6e2f1c570df8c963ecd3fe2b87c836a18994f3b9647366","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"96912df1304642838c6e2f1c570df8c963ecd3fe2b87c836a18994f3b9647366","first_computed_at":"2026-05-18T02:11:11.969933Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:11:11.969933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"14tRnWzbGRDY7D5YWQKX3GjSThEJRSCi6xN9kUeuplX1M8jGdaoN07K97dLaK6roG0zVIvL/at/PGrHfd/bvBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:11:11.970769Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.4615","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1e3718595100bc8074e1e7a69fcebef5998d4be4df460d27db08488dd64a2ef3","sha256:4c6074c39e3591684834f0a94b509bc3f7b089a641dd0ed260cc3ec2b11c728f"],"state_sha256":"a69e31d688d44d4f79716cd2f3ed41f37e53a68449d630d817cb2709e9c214bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xj92EAViLS3bUtmtueLtjTu6gKCXoHWTFYZO+jDFcnl6zIRPmpUY6vRVJH7d3TizlP6ZjTc9NDZLmWm3ikvnBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T19:58:45.370644Z","bundle_sha256":"2c2174b0ebccbf16867272198b0c83d32d5299751e52c0e7af020b61991efc71"}}