{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:UIAGEXPUEVFUHFQ5NJ2IWMR5MH","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":"32dfd517faff8bb167c63eca7e868d79fee15208e68b84313430c38793af0a38","cross_cats_sorted":["math.MG","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T17:58:24Z","title_canon_sha256":"b9a364169fe667749443a43d2027a7ce8851b1e5e9433add31a944c72cbb227d"},"schema_version":"1.0","source":{"id":"2605.21480","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.21480","created_at":"2026-05-21T02:05:39Z"},{"alias_kind":"arxiv_version","alias_value":"2605.21480v1","created_at":"2026-05-21T02:05:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21480","created_at":"2026-05-21T02:05:39Z"},{"alias_kind":"pith_short_12","alias_value":"UIAGEXPUEVFU","created_at":"2026-05-21T02:05:39Z"},{"alias_kind":"pith_short_16","alias_value":"UIAGEXPUEVFUHFQ5","created_at":"2026-05-21T02:05:39Z"},{"alias_kind":"pith_short_8","alias_value":"UIAGEXPU","created_at":"2026-05-21T02:05:39Z"}],"graph_snapshots":[{"event_id":"sha256:ba216a9b27b29f179785cf93b86a37c8ee7dd6000d484afa3f52d73de6394dd3","target":"graph","created_at":"2026-05-21T02:05:39Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.21480/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A metric probability space $M$ admits thresholds if the random geometric graph on $M$ has a threshold for every monotone graph property. We connect the existence of thresholds to the uniform expansion of $M$ and prove that all standard tori, spheres, and cubes admit thresholds.","authors_text":"Bhargav Narayanan","cross_cats":["math.MG","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T17:58:24Z","title":"Thresholds for geometric graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21480","kind":"arxiv","version":1},"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":"sha256:8f01c931d44d9be06ec78c102acce72891d7663a24edacf1a55f99503a6653ce","target":"record","created_at":"2026-05-21T02:05:39Z","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":"32dfd517faff8bb167c63eca7e868d79fee15208e68b84313430c38793af0a38","cross_cats_sorted":["math.MG","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-20T17:58:24Z","title_canon_sha256":"b9a364169fe667749443a43d2027a7ce8851b1e5e9433add31a944c72cbb227d"},"schema_version":"1.0","source":{"id":"2605.21480","kind":"arxiv","version":1}},"canonical_sha256":"a200625df4254b43961d6a748b323d61f5920dc1ff92f8959d3c4d7d00d62ff2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a200625df4254b43961d6a748b323d61f5920dc1ff92f8959d3c4d7d00d62ff2","first_computed_at":"2026-05-21T02:05:39.226640Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T02:05:39.226640Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0HtHie/2Uti7Ab1uVGENKdBq22FEJhrwHE3hrkDCrWnR3u/ahZqzU+7kFvkqZC+/oqKsQR2raN9/NOhSPCCFBg==","signature_status":"signed_v1","signed_at":"2026-05-21T02:05:39.227345Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.21480","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8f01c931d44d9be06ec78c102acce72891d7663a24edacf1a55f99503a6653ce","sha256:ba216a9b27b29f179785cf93b86a37c8ee7dd6000d484afa3f52d73de6394dd3"],"state_sha256":"5b04ac556eeb57f3c1046debaaa0d22114dda06f2b34f908133d41150ffe5580"}