{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:65HNJ3QOZB56SOZMP3RGPNRYS6","short_pith_number":"pith:65HNJ3QO","canonical_record":{"source":{"id":"1101.3083","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-01-16T17:55:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f046d87cfbddd651055af97c9ea388b55c0889df8c9460ad967a4b7910579dc0","abstract_canon_sha256":"9ca9dd11972f2c1d6fbdea4d1c5494060e075a9974933f7c681540a3f9a73e00"},"schema_version":"1.0"},"canonical_sha256":"f74ed4ee0ec87be93b2c7ee267b638979f86219e72acdd44c50b94d1d975a33a","source":{"kind":"arxiv","id":"1101.3083","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.3083","created_at":"2026-05-18T03:32:45Z"},{"alias_kind":"arxiv_version","alias_value":"1101.3083v1","created_at":"2026-05-18T03:32:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3083","created_at":"2026-05-18T03:32:45Z"},{"alias_kind":"pith_short_12","alias_value":"65HNJ3QOZB56","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"65HNJ3QOZB56SOZM","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"65HNJ3QO","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:65HNJ3QOZB56SOZMP3RGPNRYS6","target":"record","payload":{"canonical_record":{"source":{"id":"1101.3083","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-01-16T17:55:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f046d87cfbddd651055af97c9ea388b55c0889df8c9460ad967a4b7910579dc0","abstract_canon_sha256":"9ca9dd11972f2c1d6fbdea4d1c5494060e075a9974933f7c681540a3f9a73e00"},"schema_version":"1.0"},"canonical_sha256":"f74ed4ee0ec87be93b2c7ee267b638979f86219e72acdd44c50b94d1d975a33a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:45.712830Z","signature_b64":"ggyiOMSFcNGe7rTLnGBfWeZRv2H1l4+79Ebdnfie5LPSdavkO/KAd7LhiJFz5NNnehbcwg4cNU2Z9SeEeKUGAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f74ed4ee0ec87be93b2c7ee267b638979f86219e72acdd44c50b94d1d975a33a","last_reissued_at":"2026-05-18T03:32:45.711937Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:45.711937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1101.3083","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:32:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9AuhEcVn+SXA8Q0Gc4qcVCAFv4WRmmuheIUSNYUjwr+ENR3iJ2EA5nLKOo2wiWU/cd6GHtdZK9Z9nCO4PBI/BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T21:53:22.230013Z"},"content_sha256":"1686b400d8504c8e9236627ff532ea6c175ed37d0e2388858d27a480e22598d1","schema_version":"1.0","event_id":"sha256:1686b400d8504c8e9236627ff532ea6c175ed37d0e2388858d27a480e22598d1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:65HNJ3QOZB56SOZMP3RGPNRYS6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharpness in the k-nearest neighbours random geometric graph model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Mark Walters, Victor Falgas-Ravry","submitted_at":"2011-01-16T17:55:49Z","abstract_excerpt":"Let $S_{n,k}$ denote the random geometric graph obtained by placing points in a square box of area $n$ according to a Poisson process of intensity 1 and joining each point to its $k$ nearest neighbours. Balister, Bollob\\'as, Sarkar and Walters conjectured that for every $0< \\epsilon <1$ and all $n$ sufficiently large there exists $C=C(\\epsilon)$ such that whenever the probability $S_{n,k}$ is connected is at least $\\epsilon $ then the probability $S_{n,k+C}$ is connected is at least $1-\\epsilon $. In this paper we prove this conjecture.\n  As a corollary we prove that there is a constant $C'$ s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3083","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:32:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1M9m9sXRuqg52t23xzfa4/lXs6xK3sCYQJgMT+HOt2qdvkW6AVpSt8vCQ5m1ze85p1wltbqZsJQ0lFtzIKIfDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T21:53:22.231485Z"},"content_sha256":"6536619f919c6778dbdf8649a7b8a8b596e4f1edc7f9b84ce8f94c1a8923aee5","schema_version":"1.0","event_id":"sha256:6536619f919c6778dbdf8649a7b8a8b596e4f1edc7f9b84ce8f94c1a8923aee5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/65HNJ3QOZB56SOZMP3RGPNRYS6/bundle.json","state_url":"https://pith.science/pith/65HNJ3QOZB56SOZMP3RGPNRYS6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/65HNJ3QOZB56SOZMP3RGPNRYS6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T21:53:22Z","links":{"resolver":"https://pith.science/pith/65HNJ3QOZB56SOZMP3RGPNRYS6","bundle":"https://pith.science/pith/65HNJ3QOZB56SOZMP3RGPNRYS6/bundle.json","state":"https://pith.science/pith/65HNJ3QOZB56SOZMP3RGPNRYS6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/65HNJ3QOZB56SOZMP3RGPNRYS6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:65HNJ3QOZB56SOZMP3RGPNRYS6","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":"9ca9dd11972f2c1d6fbdea4d1c5494060e075a9974933f7c681540a3f9a73e00","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-01-16T17:55:49Z","title_canon_sha256":"f046d87cfbddd651055af97c9ea388b55c0889df8c9460ad967a4b7910579dc0"},"schema_version":"1.0","source":{"id":"1101.3083","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.3083","created_at":"2026-05-18T03:32:45Z"},{"alias_kind":"arxiv_version","alias_value":"1101.3083v1","created_at":"2026-05-18T03:32:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3083","created_at":"2026-05-18T03:32:45Z"},{"alias_kind":"pith_short_12","alias_value":"65HNJ3QOZB56","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"65HNJ3QOZB56SOZM","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"65HNJ3QO","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:6536619f919c6778dbdf8649a7b8a8b596e4f1edc7f9b84ce8f94c1a8923aee5","target":"graph","created_at":"2026-05-18T03:32:45Z","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 $S_{n,k}$ denote the random geometric graph obtained by placing points in a square box of area $n$ according to a Poisson process of intensity 1 and joining each point to its $k$ nearest neighbours. Balister, Bollob\\'as, Sarkar and Walters conjectured that for every $0< \\epsilon <1$ and all $n$ sufficiently large there exists $C=C(\\epsilon)$ such that whenever the probability $S_{n,k}$ is connected is at least $\\epsilon $ then the probability $S_{n,k+C}$ is connected is at least $1-\\epsilon $. In this paper we prove this conjecture.\n  As a corollary we prove that there is a constant $C'$ s","authors_text":"Mark Walters, Victor Falgas-Ravry","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-01-16T17:55:49Z","title":"Sharpness in the k-nearest neighbours random geometric graph model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3083","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":"sha256:1686b400d8504c8e9236627ff532ea6c175ed37d0e2388858d27a480e22598d1","target":"record","created_at":"2026-05-18T03:32:45Z","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":"9ca9dd11972f2c1d6fbdea4d1c5494060e075a9974933f7c681540a3f9a73e00","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-01-16T17:55:49Z","title_canon_sha256":"f046d87cfbddd651055af97c9ea388b55c0889df8c9460ad967a4b7910579dc0"},"schema_version":"1.0","source":{"id":"1101.3083","kind":"arxiv","version":1}},"canonical_sha256":"f74ed4ee0ec87be93b2c7ee267b638979f86219e72acdd44c50b94d1d975a33a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f74ed4ee0ec87be93b2c7ee267b638979f86219e72acdd44c50b94d1d975a33a","first_computed_at":"2026-05-18T03:32:45.711937Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:45.711937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ggyiOMSFcNGe7rTLnGBfWeZRv2H1l4+79Ebdnfie5LPSdavkO/KAd7LhiJFz5NNnehbcwg4cNU2Z9SeEeKUGAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:45.712830Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.3083","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1686b400d8504c8e9236627ff532ea6c175ed37d0e2388858d27a480e22598d1","sha256:6536619f919c6778dbdf8649a7b8a8b596e4f1edc7f9b84ce8f94c1a8923aee5"],"state_sha256":"c443a7cd8e2f8cfd7d854e29453e58781509f5061cc6e316564463a24ecaaeea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FXiguB6Ya/aKul92afR1o9gnrmgSd4K+XGYMd/1QC3ndkQeuLVZctgwiiDfw39vayGbV7+ITs5DwOecMih6dCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T21:53:22.233453Z","bundle_sha256":"83772fa0286772b46b46e4a65fe87fd24e8ea2715689fed19ffcafefaee6cead"}}