{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:N5XXTQNTCYS4QWAWTTANZV7ZFW","short_pith_number":"pith:N5XXTQNT","schema_version":"1.0","canonical_sha256":"6f6f79c1b31625c858169cc0dcd7f92db112608ab4ac00ab088f81a4dd7518d0","source":{"kind":"arxiv","id":"2606.01627","version":1},"attestation_state":"computed","paper":{"title":"The geometry of the giant component of random geometric graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Christian Gorski, Karoline Dubin, Marcus Michelen","submitted_at":"2026-06-01T03:24:53Z","abstract_excerpt":"Consider a random geometric graph $G_M(n;r)$ whose vertex set consists of $n$ points chosen independently and uniformly from a Riemannian manifold $M$, with edges joining pairs of vertices whose distance in the metric $d_M$ is at most $r$. Let $\\Delta$ denote the expected average degree of the graph. As is the case for Erd\\H{o}s-R\\'enyi graphs, there is a critical value $\\Delta_c$, depending only on the dimension of $M$, such that if $\\Delta > \\Delta_c$ then $G_M(n;r)$ has a giant component. We show that whenever $\\Delta > \\Delta_c$, the giant component of $G_M(n;r)$, equipped with the graph d"},"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":"2606.01627","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-06-01T03:24:53Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"21788441968cbbb7f1566f3f93d74ab603f2951e6172ca8c550fbeb9cec1765d","abstract_canon_sha256":"0a09850ed2df74ab32aba9db312cfd96fa9386e0d886da35305b7d6d87d6c6b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:38.466443Z","signature_b64":"t93E0mxXWxiens6OAeE9eGBPnqs0K1NNq+/0+dozPsnHJW11rYNqdBvAWVfVfE6LTDdM2r6KBT0NqeSzlRcsDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f6f79c1b31625c858169cc0dcd7f92db112608ab4ac00ab088f81a4dd7518d0","last_reissued_at":"2026-06-02T02:04:38.466000Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:38.466000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The geometry of the giant component of random geometric graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Christian Gorski, Karoline Dubin, Marcus Michelen","submitted_at":"2026-06-01T03:24:53Z","abstract_excerpt":"Consider a random geometric graph $G_M(n;r)$ whose vertex set consists of $n$ points chosen independently and uniformly from a Riemannian manifold $M$, with edges joining pairs of vertices whose distance in the metric $d_M$ is at most $r$. Let $\\Delta$ denote the expected average degree of the graph. As is the case for Erd\\H{o}s-R\\'enyi graphs, there is a critical value $\\Delta_c$, depending only on the dimension of $M$, such that if $\\Delta > \\Delta_c$ then $G_M(n;r)$ has a giant component. We show that whenever $\\Delta > \\Delta_c$, the giant component of $G_M(n;r)$, equipped with the graph d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.01627","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/2606.01627/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":"2606.01627","created_at":"2026-06-02T02:04:38.466056+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.01627v1","created_at":"2026-06-02T02:04:38.466056+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.01627","created_at":"2026-06-02T02:04:38.466056+00:00"},{"alias_kind":"pith_short_12","alias_value":"N5XXTQNTCYS4","created_at":"2026-06-02T02:04:38.466056+00:00"},{"alias_kind":"pith_short_16","alias_value":"N5XXTQNTCYS4QWAW","created_at":"2026-06-02T02:04:38.466056+00:00"},{"alias_kind":"pith_short_8","alias_value":"N5XXTQNT","created_at":"2026-06-02T02:04:38.466056+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/N5XXTQNTCYS4QWAWTTANZV7ZFW","json":"https://pith.science/pith/N5XXTQNTCYS4QWAWTTANZV7ZFW.json","graph_json":"https://pith.science/api/pith-number/N5XXTQNTCYS4QWAWTTANZV7ZFW/graph.json","events_json":"https://pith.science/api/pith-number/N5XXTQNTCYS4QWAWTTANZV7ZFW/events.json","paper":"https://pith.science/paper/N5XXTQNT"},"agent_actions":{"view_html":"https://pith.science/pith/N5XXTQNTCYS4QWAWTTANZV7ZFW","download_json":"https://pith.science/pith/N5XXTQNTCYS4QWAWTTANZV7ZFW.json","view_paper":"https://pith.science/paper/N5XXTQNT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.01627&json=true","fetch_graph":"https://pith.science/api/pith-number/N5XXTQNTCYS4QWAWTTANZV7ZFW/graph.json","fetch_events":"https://pith.science/api/pith-number/N5XXTQNTCYS4QWAWTTANZV7ZFW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N5XXTQNTCYS4QWAWTTANZV7ZFW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N5XXTQNTCYS4QWAWTTANZV7ZFW/action/storage_attestation","attest_author":"https://pith.science/pith/N5XXTQNTCYS4QWAWTTANZV7ZFW/action/author_attestation","sign_citation":"https://pith.science/pith/N5XXTQNTCYS4QWAWTTANZV7ZFW/action/citation_signature","submit_replication":"https://pith.science/pith/N5XXTQNTCYS4QWAWTTANZV7ZFW/action/replication_record"}},"created_at":"2026-06-02T02:04:38.466056+00:00","updated_at":"2026-06-02T02:04:38.466056+00:00"}