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We prove that the normalized length of the power-weighted shortest path between $x, y$ through $\\mathcal X_n$ converges almost surely to a constant multiple of the Riemannian distance between $x,y$ under the metric tensor $g_p = f^{2(1-p)/d} g_1$, where $p > 1$ is the power parameter."},"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":"1202.0045","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-31T22:43:00Z","cross_cats_sorted":[],"title_canon_sha256":"436eef6fe08f1a4f9098b989efac1856921279da840505c3f37f96675a22c77a","abstract_canon_sha256":"7078f58c1cee765d4107d9d07c60695956dc783c936c3d73c469be499e7b3e18"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:17.378725Z","signature_b64":"tX1zbLYitbDZSXZ5VWLEe4rcF4oKt6XxXPt3WaBwTAGEZZaAgI0/6D0MjM6x9+sKSKqG76pcRxX1gCcidrR7Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d6d3c5b77cbb5a6ec2c754c5355d2468e4800cde28c8de78a23eb77a2774d95","last_reissued_at":"2026-05-18T01:00:17.378081Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:17.378081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shortest Path through Random Points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alfred O. 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We prove that the normalized length of the power-weighted shortest path between $x, y$ through $\\mathcal X_n$ converges almost surely to a constant multiple of the Riemannian distance between $x,y$ under the metric tensor $g_p = f^{2(1-p)/d} g_1$, where $p > 1$ is the power parameter."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0045","kind":"arxiv","version":5},"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":"1202.0045","created_at":"2026-05-18T01:00:17.378194+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.0045v5","created_at":"2026-05-18T01:00:17.378194+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.0045","created_at":"2026-05-18T01:00:17.378194+00:00"},{"alias_kind":"pith_short_12","alias_value":"TVWTYW3XZO22","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"TVWTYW3XZO22N3BM","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"TVWTYW3X","created_at":"2026-05-18T12:27:23.164592+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/TVWTYW3XZO22N3BMOVGFGVOSI2","json":"https://pith.science/pith/TVWTYW3XZO22N3BMOVGFGVOSI2.json","graph_json":"https://pith.science/api/pith-number/TVWTYW3XZO22N3BMOVGFGVOSI2/graph.json","events_json":"https://pith.science/api/pith-number/TVWTYW3XZO22N3BMOVGFGVOSI2/events.json","paper":"https://pith.science/paper/TVWTYW3X"},"agent_actions":{"view_html":"https://pith.science/pith/TVWTYW3XZO22N3BMOVGFGVOSI2","download_json":"https://pith.science/pith/TVWTYW3XZO22N3BMOVGFGVOSI2.json","view_paper":"https://pith.science/paper/TVWTYW3X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.0045&json=true","fetch_graph":"https://pith.science/api/pith-number/TVWTYW3XZO22N3BMOVGFGVOSI2/graph.json","fetch_events":"https://pith.science/api/pith-number/TVWTYW3XZO22N3BMOVGFGVOSI2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TVWTYW3XZO22N3BMOVGFGVOSI2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TVWTYW3XZO22N3BMOVGFGVOSI2/action/storage_attestation","attest_author":"https://pith.science/pith/TVWTYW3XZO22N3BMOVGFGVOSI2/action/author_attestation","sign_citation":"https://pith.science/pith/TVWTYW3XZO22N3BMOVGFGVOSI2/action/citation_signature","submit_replication":"https://pith.science/pith/TVWTYW3XZO22N3BMOVGFGVOSI2/action/replication_record"}},"created_at":"2026-05-18T01:00:17.378194+00:00","updated_at":"2026-05-18T01:00:17.378194+00:00"}