{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:RXWSXMLJFS5E4LERWSAIP3OICM","short_pith_number":"pith:RXWSXMLJ","canonical_record":{"source":{"id":"1308.1084","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-05T19:54:54Z","cross_cats_sorted":["cs.CC","cs.DM","math.CO"],"title_canon_sha256":"d2496ee5b0f50454b816e01fc2728ce824349c2bcfcaa6f9c9a5035c7ce18295","abstract_canon_sha256":"a69436a37b500fe7f6dff3a9e8b5c12c0af3864f8ab3ea24893932ad6787679c"},"schema_version":"1.0"},"canonical_sha256":"8ded2bb1692cba4e2c91b48087edc813292449a4e9f016f804563647e37078c2","source":{"kind":"arxiv","id":"1308.1084","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.1084","created_at":"2026-05-18T02:43:36Z"},{"alias_kind":"arxiv_version","alias_value":"1308.1084v2","created_at":"2026-05-18T02:43:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.1084","created_at":"2026-05-18T02:43:36Z"},{"alias_kind":"pith_short_12","alias_value":"RXWSXMLJFS5E","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"RXWSXMLJFS5E4LER","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"RXWSXMLJ","created_at":"2026-05-18T12:27:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:RXWSXMLJFS5E4LERWSAIP3OICM","target":"record","payload":{"canonical_record":{"source":{"id":"1308.1084","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-05T19:54:54Z","cross_cats_sorted":["cs.CC","cs.DM","math.CO"],"title_canon_sha256":"d2496ee5b0f50454b816e01fc2728ce824349c2bcfcaa6f9c9a5035c7ce18295","abstract_canon_sha256":"a69436a37b500fe7f6dff3a9e8b5c12c0af3864f8ab3ea24893932ad6787679c"},"schema_version":"1.0"},"canonical_sha256":"8ded2bb1692cba4e2c91b48087edc813292449a4e9f016f804563647e37078c2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:36.892344Z","signature_b64":"beXsqPvEsqDPfKmH03ztFyqCqmXWbmLWKvavKAYiXag9iLy6ToJ3IvdutrxEdCIx0jBZWJJG205y/zyt9AhABg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ded2bb1692cba4e2c91b48087edc813292449a4e9f016f804563647e37078c2","last_reissued_at":"2026-05-18T02:43:36.891669Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:36.891669Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.1084","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:43:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S1F5+EXP2fkYGzTjnVbu55SrAz/QcLIWPZhsclF+ENv2E9ArBvI9O+gk4zOrvYAdTUloibQm44ciAWKIZz+AAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T23:06:17.633564Z"},"content_sha256":"de76a1df4a5600a6bd8d431d5fcc87bbfb8a17e1765bbaef9dfcb00293ef2ca6","schema_version":"1.0","event_id":"sha256:de76a1df4a5600a6bd8d431d5fcc87bbfb8a17e1765bbaef9dfcb00293ef2ca6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:RXWSXMLJFS5E4LERWSAIP3OICM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Sharp Thresholds in Random Geometric Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DM","math.CO"],"primary_cat":"math.PR","authors_text":"Milan Bradonji\\'c, Will Perkins","submitted_at":"2013-08-05T19:54:54Z","abstract_excerpt":"We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.1084","kind":"arxiv","version":2},"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":"2026-05-18T02:43:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R2pZ4Aip9peAac3o0yM2viNAdpY8TIxQXsw5woXH+tsHfQJuFDzjOkdQMQFeJomAiVHLi7mD1Lq7XOPkgbpjDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T23:06:17.633910Z"},"content_sha256":"03afc23ff08dc95a91c26fb52f02ba0b5323304817492d23882676230a3c316d","schema_version":"1.0","event_id":"sha256:03afc23ff08dc95a91c26fb52f02ba0b5323304817492d23882676230a3c316d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RXWSXMLJFS5E4LERWSAIP3OICM/bundle.json","state_url":"https://pith.science/pith/RXWSXMLJFS5E4LERWSAIP3OICM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RXWSXMLJFS5E4LERWSAIP3OICM/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":"2026-06-08T23:06:17Z","links":{"resolver":"https://pith.science/pith/RXWSXMLJFS5E4LERWSAIP3OICM","bundle":"https://pith.science/pith/RXWSXMLJFS5E4LERWSAIP3OICM/bundle.json","state":"https://pith.science/pith/RXWSXMLJFS5E4LERWSAIP3OICM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RXWSXMLJFS5E4LERWSAIP3OICM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:RXWSXMLJFS5E4LERWSAIP3OICM","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":"a69436a37b500fe7f6dff3a9e8b5c12c0af3864f8ab3ea24893932ad6787679c","cross_cats_sorted":["cs.CC","cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-05T19:54:54Z","title_canon_sha256":"d2496ee5b0f50454b816e01fc2728ce824349c2bcfcaa6f9c9a5035c7ce18295"},"schema_version":"1.0","source":{"id":"1308.1084","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.1084","created_at":"2026-05-18T02:43:36Z"},{"alias_kind":"arxiv_version","alias_value":"1308.1084v2","created_at":"2026-05-18T02:43:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.1084","created_at":"2026-05-18T02:43:36Z"},{"alias_kind":"pith_short_12","alias_value":"RXWSXMLJFS5E","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"RXWSXMLJFS5E4LER","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"RXWSXMLJ","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:03afc23ff08dc95a91c26fb52f02ba0b5323304817492d23882676230a3c316d","target":"graph","created_at":"2026-05-18T02:43:36Z","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":"We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability.","authors_text":"Milan Bradonji\\'c, Will Perkins","cross_cats":["cs.CC","cs.DM","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-05T19:54:54Z","title":"On Sharp Thresholds in Random Geometric Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.1084","kind":"arxiv","version":2},"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:de76a1df4a5600a6bd8d431d5fcc87bbfb8a17e1765bbaef9dfcb00293ef2ca6","target":"record","created_at":"2026-05-18T02:43:36Z","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":"a69436a37b500fe7f6dff3a9e8b5c12c0af3864f8ab3ea24893932ad6787679c","cross_cats_sorted":["cs.CC","cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-08-05T19:54:54Z","title_canon_sha256":"d2496ee5b0f50454b816e01fc2728ce824349c2bcfcaa6f9c9a5035c7ce18295"},"schema_version":"1.0","source":{"id":"1308.1084","kind":"arxiv","version":2}},"canonical_sha256":"8ded2bb1692cba4e2c91b48087edc813292449a4e9f016f804563647e37078c2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ded2bb1692cba4e2c91b48087edc813292449a4e9f016f804563647e37078c2","first_computed_at":"2026-05-18T02:43:36.891669Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:43:36.891669Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"beXsqPvEsqDPfKmH03ztFyqCqmXWbmLWKvavKAYiXag9iLy6ToJ3IvdutrxEdCIx0jBZWJJG205y/zyt9AhABg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:43:36.892344Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.1084","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de76a1df4a5600a6bd8d431d5fcc87bbfb8a17e1765bbaef9dfcb00293ef2ca6","sha256:03afc23ff08dc95a91c26fb52f02ba0b5323304817492d23882676230a3c316d"],"state_sha256":"0eabb841c5a72ee22cfca8fc1424e0e7db8999621de35bfcc0c468aa1d6009e4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kpCYVZxFVRRYCgpZa1K7+ZAHDvaLAmZSGF2kmjSzFVxkfjDzk7hL16j/NGsbkfYoU2wIwIdtLA3ZLNZp/03CBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T23:06:17.635818Z","bundle_sha256":"2ec6c2e8d99c0533bfcad33725c503a87d6f456fc154313f2f172dfed623798b"}}