{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:TVWTYW3XZO22N3BMOVGFGVOSI2","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":"7078f58c1cee765d4107d9d07c60695956dc783c936c3d73c469be499e7b3e18","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-31T22:43:00Z","title_canon_sha256":"436eef6fe08f1a4f9098b989efac1856921279da840505c3f37f96675a22c77a"},"schema_version":"1.0","source":{"id":"1202.0045","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.0045","created_at":"2026-05-18T01:00:17Z"},{"alias_kind":"arxiv_version","alias_value":"1202.0045v5","created_at":"2026-05-18T01:00:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.0045","created_at":"2026-05-18T01:00:17Z"},{"alias_kind":"pith_short_12","alias_value":"TVWTYW3XZO22","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"TVWTYW3XZO22N3BM","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"TVWTYW3X","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:be47965469c3a724f9e059006b6d20fa9fa50e47c04c9cb43102ff81ec29f8b4","target":"graph","created_at":"2026-05-18T01:00:17Z","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 $(M,g_1)$ be a complete $d$-dimensional Riemannian manifold for $d > 1$. Let $\\mathcal X_n$ be a set of $n$ sample points in $M$ drawn randomly from a smooth Lebesgue density $f$ supported in $M$. Let $x,y$ be two points in $M$. 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.","authors_text":"Alfred O. Hero III, Steven B. Damelin, Sung Jin Hwang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-31T22:43:00Z","title":"Shortest Path through Random Points"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0045","kind":"arxiv","version":5},"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":"sha256:725faedfda1cd36f49fff18101a4c960fd732b55530d92ed20d7691dd7b861ae","target":"record","created_at":"2026-05-18T01:00:17Z","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":"7078f58c1cee765d4107d9d07c60695956dc783c936c3d73c469be499e7b3e18","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-31T22:43:00Z","title_canon_sha256":"436eef6fe08f1a4f9098b989efac1856921279da840505c3f37f96675a22c77a"},"schema_version":"1.0","source":{"id":"1202.0045","kind":"arxiv","version":5}},"canonical_sha256":"9d6d3c5b77cbb5a6ec2c754c5355d2468e4800cde28c8de78a23eb77a2774d95","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d6d3c5b77cbb5a6ec2c754c5355d2468e4800cde28c8de78a23eb77a2774d95","first_computed_at":"2026-05-18T01:00:17.378081Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:00:17.378081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tX1zbLYitbDZSXZ5VWLEe4rcF4oKt6XxXPt3WaBwTAGEZZaAgI0/6D0MjM6x9+sKSKqG76pcRxX1gCcidrR7Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:00:17.378725Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.0045","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:725faedfda1cd36f49fff18101a4c960fd732b55530d92ed20d7691dd7b861ae","sha256:be47965469c3a724f9e059006b6d20fa9fa50e47c04c9cb43102ff81ec29f8b4"],"state_sha256":"d52ce745fa65f3e619c2638ada36bd14eb6cc8d24098763208233808a9fb9df6"}