{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:GFPYQXUDF4OHLM6OSTWG3EHTMQ","short_pith_number":"pith:GFPYQXUD","schema_version":"1.0","canonical_sha256":"315f885e832f1c75b3ce94ec6d90f3640c011117aacac050cb3683e62e2ad80a","source":{"kind":"arxiv","id":"2606.18061","version":1},"attestation_state":"computed","paper":{"title":"Principal minors of effective-resistance matrices and local resistance radii","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guangfu Wang","submitted_at":"2026-06-16T15:37:09Z","abstract_excerpt":"Let $G$ be a finite connected weighted graph and let $R$ be its effective-resistance matrix. For every nonempty vertex set $S$, we factor the cofactor sum and determinant of the principal resistance submatrix $R[S]$ into an enumerative term and a boundary potential-theoretic term. If $\\tau(G)$ is the weighted spanning tree enumerator and $\\kappa_G(S)$ is the weighted enumerator of $S$-rooted spanning forests, then \\[\n  \\cof R[S]=(-2)^{|S|-1}\\kappa_G(S)/\\tau(G). \\] After Kron reduction to $S$, with reduced Laplacian $K=L^S$, $Q=K^+$, and $q=\\diag(Q)$, the remaining normalized factor is \\[\n  \\de"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.18061","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-16T15:37:09Z","cross_cats_sorted":[],"title_canon_sha256":"9a351e61bcce2e7799d993de5e5be839a6edaa69d632abd50194ce28fe4ccbc2","abstract_canon_sha256":"97625ab397ff32f94149f75c21e04a463b654b32d103a9082e2be089e8426663"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:10:47.506529Z","signature_b64":"Ond4bUXz7RA0WmaRgCpjA1+nI8TcydCyasPeOwxu0EcGMSOVKsuBwUTOBoHOdgfk3VIjSps7zBdADbX+qytaDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"315f885e832f1c75b3ce94ec6d90f3640c011117aacac050cb3683e62e2ad80a","last_reissued_at":"2026-06-19T16:10:47.506161Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:10:47.506161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Principal minors of effective-resistance matrices and local resistance radii","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guangfu Wang","submitted_at":"2026-06-16T15:37:09Z","abstract_excerpt":"Let $G$ be a finite connected weighted graph and let $R$ be its effective-resistance matrix. For every nonempty vertex set $S$, we factor the cofactor sum and determinant of the principal resistance submatrix $R[S]$ into an enumerative term and a boundary potential-theoretic term. If $\\tau(G)$ is the weighted spanning tree enumerator and $\\kappa_G(S)$ is the weighted enumerator of $S$-rooted spanning forests, then \\[\n  \\cof R[S]=(-2)^{|S|-1}\\kappa_G(S)/\\tau(G). \\] After Kron reduction to $S$, with reduced Laplacian $K=L^S$, $Q=K^+$, and $q=\\diag(Q)$, the remaining normalized factor is \\[\n  \\de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.18061","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.18061/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.18061","created_at":"2026-06-19T16:10:47.506224+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.18061v1","created_at":"2026-06-19T16:10:47.506224+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.18061","created_at":"2026-06-19T16:10:47.506224+00:00"},{"alias_kind":"pith_short_12","alias_value":"GFPYQXUDF4OH","created_at":"2026-06-19T16:10:47.506224+00:00"},{"alias_kind":"pith_short_16","alias_value":"GFPYQXUDF4OHLM6O","created_at":"2026-06-19T16:10:47.506224+00:00"},{"alias_kind":"pith_short_8","alias_value":"GFPYQXUD","created_at":"2026-06-19T16:10:47.506224+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GFPYQXUDF4OHLM6OSTWG3EHTMQ","json":"https://pith.science/pith/GFPYQXUDF4OHLM6OSTWG3EHTMQ.json","graph_json":"https://pith.science/api/pith-number/GFPYQXUDF4OHLM6OSTWG3EHTMQ/graph.json","events_json":"https://pith.science/api/pith-number/GFPYQXUDF4OHLM6OSTWG3EHTMQ/events.json","paper":"https://pith.science/paper/GFPYQXUD"},"agent_actions":{"view_html":"https://pith.science/pith/GFPYQXUDF4OHLM6OSTWG3EHTMQ","download_json":"https://pith.science/pith/GFPYQXUDF4OHLM6OSTWG3EHTMQ.json","view_paper":"https://pith.science/paper/GFPYQXUD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.18061&json=true","fetch_graph":"https://pith.science/api/pith-number/GFPYQXUDF4OHLM6OSTWG3EHTMQ/graph.json","fetch_events":"https://pith.science/api/pith-number/GFPYQXUDF4OHLM6OSTWG3EHTMQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GFPYQXUDF4OHLM6OSTWG3EHTMQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GFPYQXUDF4OHLM6OSTWG3EHTMQ/action/storage_attestation","attest_author":"https://pith.science/pith/GFPYQXUDF4OHLM6OSTWG3EHTMQ/action/author_attestation","sign_citation":"https://pith.science/pith/GFPYQXUDF4OHLM6OSTWG3EHTMQ/action/citation_signature","submit_replication":"https://pith.science/pith/GFPYQXUDF4OHLM6OSTWG3EHTMQ/action/replication_record"}},"created_at":"2026-06-19T16:10:47.506224+00:00","updated_at":"2026-06-19T16:10:47.506224+00:00"}