{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:JXTPSUDCJLQZAAL4ZWF53RE4UH","short_pith_number":"pith:JXTPSUDC","canonical_record":{"source":{"id":"1203.1256","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-03-06T17:10:19Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"70dac5821ca6cf27b82c1318b46c18a7cee8ba0b127033818fd8c00f49f17d83","abstract_canon_sha256":"d9e5a4ff9e9298e3400cd5909c2e8325a939a48802eeae6fbf8c9aa336fd1f81"},"schema_version":"1.0"},"canonical_sha256":"4de6f950624ae190017ccd8bddc49ca1d084716c1af8ca7af36b796f4619afa1","source":{"kind":"arxiv","id":"1203.1256","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.1256","created_at":"2026-05-18T04:00:39Z"},{"alias_kind":"arxiv_version","alias_value":"1203.1256v1","created_at":"2026-05-18T04:00:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1256","created_at":"2026-05-18T04:00:39Z"},{"alias_kind":"pith_short_12","alias_value":"JXTPSUDCJLQZ","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JXTPSUDCJLQZAAL4","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JXTPSUDC","created_at":"2026-05-18T12:27:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:JXTPSUDCJLQZAAL4ZWF53RE4UH","target":"record","payload":{"canonical_record":{"source":{"id":"1203.1256","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-03-06T17:10:19Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"70dac5821ca6cf27b82c1318b46c18a7cee8ba0b127033818fd8c00f49f17d83","abstract_canon_sha256":"d9e5a4ff9e9298e3400cd5909c2e8325a939a48802eeae6fbf8c9aa336fd1f81"},"schema_version":"1.0"},"canonical_sha256":"4de6f950624ae190017ccd8bddc49ca1d084716c1af8ca7af36b796f4619afa1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:39.957034Z","signature_b64":"CbHjcesMX00HmbrdQxN9fbr2RheWkOzU7XdfS5uAVuYq3IbCXzQPhLLTAphj4LFXSTGtJuw9WWR6QoL3aIWJCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4de6f950624ae190017ccd8bddc49ca1d084716c1af8ca7af36b796f4619afa1","last_reissued_at":"2026-05-18T04:00:39.956386Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:39.956386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.1256","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":"2026-05-18T04:00:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DZsZmMoDbDZMsxias5bdZNCrWCYsvQ/MT1HCwZeAMBqxc2zq1ZMKV8vSBk+/MXqv9lYu3Ne0sEQXS52fZPNWAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T00:10:27.195923Z"},"content_sha256":"17d7611e658a498907a08514867096c7dd4373afc49da35ca654019fcb6a12bc","schema_version":"1.0","event_id":"sha256:17d7611e658a498907a08514867096c7dd4373afc49da35ca654019fcb6a12bc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:JXTPSUDCJLQZAAL4ZWF53RE4UH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Laplacian on planar graphs and graphs on surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Richard Kenyon","submitted_at":"2012-03-06T17:10:19Z","abstract_excerpt":"These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main goals are to discuss: (1) for \"circular\" planar networks, the characterization due to Colin de Verdi\\`ere of Dirichlet-to-Neumann operator; (2) The connections with the random spanning tree model; and (3) the characteristic polynomial of the Laplacian on an annulus and torus."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1256","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":"2026-05-18T04:00:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dl7+xISbA3UR27gekhoacAXyXWE/naNWAaTJy2QqhIkArfe8kaYopbJKjyRY5aslgbN/EP/r4BRyQqdUKtQyBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T00:10:27.196522Z"},"content_sha256":"1424f067e64f700d191bf2673a8807bbc20eb6efe0bccc56a27aa8d9eb6a3219","schema_version":"1.0","event_id":"sha256:1424f067e64f700d191bf2673a8807bbc20eb6efe0bccc56a27aa8d9eb6a3219"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JXTPSUDCJLQZAAL4ZWF53RE4UH/bundle.json","state_url":"https://pith.science/pith/JXTPSUDCJLQZAAL4ZWF53RE4UH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JXTPSUDCJLQZAAL4ZWF53RE4UH/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":"2026-06-12T00:10:27Z","links":{"resolver":"https://pith.science/pith/JXTPSUDCJLQZAAL4ZWF53RE4UH","bundle":"https://pith.science/pith/JXTPSUDCJLQZAAL4ZWF53RE4UH/bundle.json","state":"https://pith.science/pith/JXTPSUDCJLQZAAL4ZWF53RE4UH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JXTPSUDCJLQZAAL4ZWF53RE4UH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JXTPSUDCJLQZAAL4ZWF53RE4UH","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":"d9e5a4ff9e9298e3400cd5909c2e8325a939a48802eeae6fbf8c9aa336fd1f81","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-03-06T17:10:19Z","title_canon_sha256":"70dac5821ca6cf27b82c1318b46c18a7cee8ba0b127033818fd8c00f49f17d83"},"schema_version":"1.0","source":{"id":"1203.1256","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.1256","created_at":"2026-05-18T04:00:39Z"},{"alias_kind":"arxiv_version","alias_value":"1203.1256v1","created_at":"2026-05-18T04:00:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1256","created_at":"2026-05-18T04:00:39Z"},{"alias_kind":"pith_short_12","alias_value":"JXTPSUDCJLQZ","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JXTPSUDCJLQZAAL4","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JXTPSUDC","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:1424f067e64f700d191bf2673a8807bbc20eb6efe0bccc56a27aa8d9eb6a3219","target":"graph","created_at":"2026-05-18T04:00:39Z","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":"These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main goals are to discuss: (1) for \"circular\" planar networks, the characterization due to Colin de Verdi\\`ere of Dirichlet-to-Neumann operator; (2) The connections with the random spanning tree model; and (3) the characteristic polynomial of the Laplacian on an annulus and torus.","authors_text":"Richard Kenyon","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-03-06T17:10:19Z","title":"The Laplacian on planar graphs and graphs on surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1256","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":"sha256:17d7611e658a498907a08514867096c7dd4373afc49da35ca654019fcb6a12bc","target":"record","created_at":"2026-05-18T04:00:39Z","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":"d9e5a4ff9e9298e3400cd5909c2e8325a939a48802eeae6fbf8c9aa336fd1f81","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-03-06T17:10:19Z","title_canon_sha256":"70dac5821ca6cf27b82c1318b46c18a7cee8ba0b127033818fd8c00f49f17d83"},"schema_version":"1.0","source":{"id":"1203.1256","kind":"arxiv","version":1}},"canonical_sha256":"4de6f950624ae190017ccd8bddc49ca1d084716c1af8ca7af36b796f4619afa1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4de6f950624ae190017ccd8bddc49ca1d084716c1af8ca7af36b796f4619afa1","first_computed_at":"2026-05-18T04:00:39.956386Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:39.956386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CbHjcesMX00HmbrdQxN9fbr2RheWkOzU7XdfS5uAVuYq3IbCXzQPhLLTAphj4LFXSTGtJuw9WWR6QoL3aIWJCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:39.957034Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.1256","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:17d7611e658a498907a08514867096c7dd4373afc49da35ca654019fcb6a12bc","sha256:1424f067e64f700d191bf2673a8807bbc20eb6efe0bccc56a27aa8d9eb6a3219"],"state_sha256":"9bc29e62eaeb67f85da5d81ef3db1c9afa43980216dfde0f40cd809504600acb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lMDtwHA5ndcfaQf+M00pyvVR622+thKAL4L3X8J9X1zkkzaLF0hlJ6BYjHqm67SRgXpl/YWgnHBBENme9fcuCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T00:10:27.200837Z","bundle_sha256":"b97a3925f49c77ddec94cb452ce2e42fc87c29ee549d3fd4d151a4dcb05ad8d1"}}