{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:YJVWZLYW65IPQHB7MKCYWWA65S","short_pith_number":"pith:YJVWZLYW","canonical_record":{"source":{"id":"1805.01380","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-03T15:39:18Z","cross_cats_sorted":[],"title_canon_sha256":"d3560d0ff1d48cf18d2ebcdb1968cb05b5e0ddadb880425769f0be1b8ca58f28","abstract_canon_sha256":"9d2c291305408c1cf5c5b5ad7d7b2474f7c557504943b2f029c1f93139e09911"},"schema_version":"1.0"},"canonical_sha256":"c26b6caf16f750f81c3f62858b581eeca4910a584335677c8d46d2c6cb3a159d","source":{"kind":"arxiv","id":"1805.01380","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.01380","created_at":"2026-05-18T00:16:51Z"},{"alias_kind":"arxiv_version","alias_value":"1805.01380v1","created_at":"2026-05-18T00:16:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.01380","created_at":"2026-05-18T00:16:51Z"},{"alias_kind":"pith_short_12","alias_value":"YJVWZLYW65IP","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"YJVWZLYW65IPQHB7","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"YJVWZLYW","created_at":"2026-05-18T12:33:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:YJVWZLYW65IPQHB7MKCYWWA65S","target":"record","payload":{"canonical_record":{"source":{"id":"1805.01380","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-03T15:39:18Z","cross_cats_sorted":[],"title_canon_sha256":"d3560d0ff1d48cf18d2ebcdb1968cb05b5e0ddadb880425769f0be1b8ca58f28","abstract_canon_sha256":"9d2c291305408c1cf5c5b5ad7d7b2474f7c557504943b2f029c1f93139e09911"},"schema_version":"1.0"},"canonical_sha256":"c26b6caf16f750f81c3f62858b581eeca4910a584335677c8d46d2c6cb3a159d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:51.570085Z","signature_b64":"PzanglQ9UYsWmevBHsWpi8smYllaRdKJLsyI8hZXajxnMz12IUydbcJ2vBfskPtriBBTN8uW6iBrJtbJ9LkYDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c26b6caf16f750f81c3f62858b581eeca4910a584335677c8d46d2c6cb3a159d","last_reissued_at":"2026-05-18T00:16:51.569560Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:51.569560Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.01380","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":"2026-05-18T00:16:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KlDgVcOCwng1P9Bsh1fSMtWGqTgL6rColXh3ktXUNY9areiABzsZiLyWDaMuSp7dPuktlCMZpJXkMuLeJ8/kDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T13:29:15.787664Z"},"content_sha256":"2aece4c9a87f0c40d250b7a2d79bae3bba3d4c3d151df6fe293fa0aeb9206328","schema_version":"1.0","event_id":"sha256:2aece4c9a87f0c40d250b7a2d79bae3bba3d4c3d151df6fe293fa0aeb9206328"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:YJVWZLYW65IPQHB7MKCYWWA65S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Resistors in dual networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Martina Furrer, Norbert Hungerb\\\"uhler, Simon Jantschgi","submitted_at":"2018-05-03T15:39:18Z","abstract_excerpt":"Let $G$ be a finite plane multigraph and $G'$ its dual. Each edge $e$ of $G$ is interpreted as a resistor of resistance $R_e$, and the dual edge $e'$ is assigned the dual resistance $R_{e'}:=1/R_e$. Then the equivalent resistance $r_e$ over $e$ and the equivalent resistance $r_{e'}$ over $e'$ satisfy $r_e/R_e+r_{e'}/R_{e'}=1$. We provide a graph theoretic proof of this relation by expressing the resistances in terms of sums of weights of spanning trees in $G$ and $G'$ respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01380","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":"2026-05-18T00:16:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4Hu6JbqZWq1l5/D47UQolm99caIpRHQUYIaCIw8qZaiCXua4GL+jSIXB/GFmar1ugzPEaSnwqE79fwP31vIVAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T13:29:15.788008Z"},"content_sha256":"865d9e8c53b2de4dabe0c326f3f748491ce2bb4b7a04c768a5aa37cf88fdf94d","schema_version":"1.0","event_id":"sha256:865d9e8c53b2de4dabe0c326f3f748491ce2bb4b7a04c768a5aa37cf88fdf94d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YJVWZLYW65IPQHB7MKCYWWA65S/bundle.json","state_url":"https://pith.science/pith/YJVWZLYW65IPQHB7MKCYWWA65S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YJVWZLYW65IPQHB7MKCYWWA65S/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":"2026-06-24T13:29:15Z","links":{"resolver":"https://pith.science/pith/YJVWZLYW65IPQHB7MKCYWWA65S","bundle":"https://pith.science/pith/YJVWZLYW65IPQHB7MKCYWWA65S/bundle.json","state":"https://pith.science/pith/YJVWZLYW65IPQHB7MKCYWWA65S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YJVWZLYW65IPQHB7MKCYWWA65S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:YJVWZLYW65IPQHB7MKCYWWA65S","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":"9d2c291305408c1cf5c5b5ad7d7b2474f7c557504943b2f029c1f93139e09911","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-03T15:39:18Z","title_canon_sha256":"d3560d0ff1d48cf18d2ebcdb1968cb05b5e0ddadb880425769f0be1b8ca58f28"},"schema_version":"1.0","source":{"id":"1805.01380","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.01380","created_at":"2026-05-18T00:16:51Z"},{"alias_kind":"arxiv_version","alias_value":"1805.01380v1","created_at":"2026-05-18T00:16:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.01380","created_at":"2026-05-18T00:16:51Z"},{"alias_kind":"pith_short_12","alias_value":"YJVWZLYW65IP","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"YJVWZLYW65IPQHB7","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"YJVWZLYW","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:865d9e8c53b2de4dabe0c326f3f748491ce2bb4b7a04c768a5aa37cf88fdf94d","target":"graph","created_at":"2026-05-18T00:16:51Z","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":"Let $G$ be a finite plane multigraph and $G'$ its dual. Each edge $e$ of $G$ is interpreted as a resistor of resistance $R_e$, and the dual edge $e'$ is assigned the dual resistance $R_{e'}:=1/R_e$. Then the equivalent resistance $r_e$ over $e$ and the equivalent resistance $r_{e'}$ over $e'$ satisfy $r_e/R_e+r_{e'}/R_{e'}=1$. We provide a graph theoretic proof of this relation by expressing the resistances in terms of sums of weights of spanning trees in $G$ and $G'$ respectively.","authors_text":"Martina Furrer, Norbert Hungerb\\\"uhler, Simon Jantschgi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-03T15:39:18Z","title":"Resistors in dual networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01380","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":"sha256:2aece4c9a87f0c40d250b7a2d79bae3bba3d4c3d151df6fe293fa0aeb9206328","target":"record","created_at":"2026-05-18T00:16:51Z","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":"9d2c291305408c1cf5c5b5ad7d7b2474f7c557504943b2f029c1f93139e09911","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-03T15:39:18Z","title_canon_sha256":"d3560d0ff1d48cf18d2ebcdb1968cb05b5e0ddadb880425769f0be1b8ca58f28"},"schema_version":"1.0","source":{"id":"1805.01380","kind":"arxiv","version":1}},"canonical_sha256":"c26b6caf16f750f81c3f62858b581eeca4910a584335677c8d46d2c6cb3a159d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c26b6caf16f750f81c3f62858b581eeca4910a584335677c8d46d2c6cb3a159d","first_computed_at":"2026-05-18T00:16:51.569560Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:51.569560Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PzanglQ9UYsWmevBHsWpi8smYllaRdKJLsyI8hZXajxnMz12IUydbcJ2vBfskPtriBBTN8uW6iBrJtbJ9LkYDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:51.570085Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.01380","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2aece4c9a87f0c40d250b7a2d79bae3bba3d4c3d151df6fe293fa0aeb9206328","sha256:865d9e8c53b2de4dabe0c326f3f748491ce2bb4b7a04c768a5aa37cf88fdf94d"],"state_sha256":"e1b6b9aca81e3c3c48b5fb17ba89a98ef098d882bf992307424dcee7136d261b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q4YSWo9s8w9wG2ne7ksw/e866uGgBC4jpcMwqCD/U7vmLwzxjStzNxxtHtS4kQlf4d054h0+RUatoeFc3zyeCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T13:29:15.789997Z","bundle_sha256":"f3c99e3c2e69d6ecc1bc2f6145f048963a3fde8b01da48e31dc725227433558d"}}