{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:E6SMJJX5AYW5WX334AKHWVWB4X","short_pith_number":"pith:E6SMJJX5","schema_version":"1.0","canonical_sha256":"27a4c4a6fd062ddb5f7be0147b56c1e5c449d0c71b1b93270e6e805f6a6ee631","source":{"kind":"arxiv","id":"1801.08223","version":2},"attestation_state":"computed","paper":{"title":"Reciprocal lower bound on modulus of curve families in metric surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Kai Rajala, Matthew Romney","submitted_at":"2018-01-24T22:30:26Z","abstract_excerpt":"We prove that any metric space $X$ homeomorphic to $\\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \\subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $\\zeta_1, \\zeta_2, \\zeta_3, \\zeta_4$ and let $\\Gamma(\\zeta_i, \\zeta_j; Q)$ denote the family of curves in $Q$ connecting $\\zeta_i$ and $\\zeta_j$; then $\\text{mod} \\Gamma(\\zeta_1, \\zeta_3; Q) \\text{mod} \\Gamma(\\zeta_2, \\zeta_4; Q) \\geq 1/\\kappa$ for $\\kappa = 2000^2\\cdot (4/\\pi)^2$. Th"},"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":"1801.08223","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-01-24T22:30:26Z","cross_cats_sorted":[],"title_canon_sha256":"720d300f37113fcc1a239d550895335125d0d60f54409449a0870295c0d27969","abstract_canon_sha256":"34907555fd02c842144fccd3579146273ea6e9cc21a1dfab2b777256e609b6b0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:30.049904Z","signature_b64":"w980TUBTeGzYPjlxl8btKdZgMtgkZ9MmiW2PSCLYL+JVsZGKxfUZKUfVIQTjFbRT4k5iln9IaynZ5sqZUTF1Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27a4c4a6fd062ddb5f7be0147b56c1e5c449d0c71b1b93270e6e805f6a6ee631","last_reissued_at":"2026-05-17T23:56:30.049545Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:30.049545Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reciprocal lower bound on modulus of curve families in metric surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Kai Rajala, Matthew Romney","submitted_at":"2018-01-24T22:30:26Z","abstract_excerpt":"We prove that any metric space $X$ homeomorphic to $\\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \\subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $\\zeta_1, \\zeta_2, \\zeta_3, \\zeta_4$ and let $\\Gamma(\\zeta_i, \\zeta_j; Q)$ denote the family of curves in $Q$ connecting $\\zeta_i$ and $\\zeta_j$; then $\\text{mod} \\Gamma(\\zeta_1, \\zeta_3; Q) \\text{mod} \\Gamma(\\zeta_2, \\zeta_4; Q) \\geq 1/\\kappa$ for $\\kappa = 2000^2\\cdot (4/\\pi)^2$. Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.08223","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1801.08223","created_at":"2026-05-17T23:56:30.049599+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.08223v2","created_at":"2026-05-17T23:56:30.049599+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.08223","created_at":"2026-05-17T23:56:30.049599+00:00"},{"alias_kind":"pith_short_12","alias_value":"E6SMJJX5AYW5","created_at":"2026-05-18T12:32:19.392346+00:00"},{"alias_kind":"pith_short_16","alias_value":"E6SMJJX5AYW5WX33","created_at":"2026-05-18T12:32:19.392346+00:00"},{"alias_kind":"pith_short_8","alias_value":"E6SMJJX5","created_at":"2026-05-18T12:32:19.392346+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/E6SMJJX5AYW5WX334AKHWVWB4X","json":"https://pith.science/pith/E6SMJJX5AYW5WX334AKHWVWB4X.json","graph_json":"https://pith.science/api/pith-number/E6SMJJX5AYW5WX334AKHWVWB4X/graph.json","events_json":"https://pith.science/api/pith-number/E6SMJJX5AYW5WX334AKHWVWB4X/events.json","paper":"https://pith.science/paper/E6SMJJX5"},"agent_actions":{"view_html":"https://pith.science/pith/E6SMJJX5AYW5WX334AKHWVWB4X","download_json":"https://pith.science/pith/E6SMJJX5AYW5WX334AKHWVWB4X.json","view_paper":"https://pith.science/paper/E6SMJJX5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.08223&json=true","fetch_graph":"https://pith.science/api/pith-number/E6SMJJX5AYW5WX334AKHWVWB4X/graph.json","fetch_events":"https://pith.science/api/pith-number/E6SMJJX5AYW5WX334AKHWVWB4X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E6SMJJX5AYW5WX334AKHWVWB4X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E6SMJJX5AYW5WX334AKHWVWB4X/action/storage_attestation","attest_author":"https://pith.science/pith/E6SMJJX5AYW5WX334AKHWVWB4X/action/author_attestation","sign_citation":"https://pith.science/pith/E6SMJJX5AYW5WX334AKHWVWB4X/action/citation_signature","submit_replication":"https://pith.science/pith/E6SMJJX5AYW5WX334AKHWVWB4X/action/replication_record"}},"created_at":"2026-05-17T23:56:30.049599+00:00","updated_at":"2026-05-17T23:56:30.049599+00:00"}