{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:A5ODDMCEHRUVBJGZAKED6HZAMK","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":"cf8131549c80a6acf0884e07d35e1c3e17e4f9dc5856575b4ef327c855797602","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-10-02T09:05:47Z","title_canon_sha256":"f262c76653e42be7ae166bfa9fcfbb8540febc13e5e902accb3de6a23dc83eb3"},"schema_version":"1.0","source":{"id":"1110.0160","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.0160","created_at":"2026-05-18T03:40:25Z"},{"alias_kind":"arxiv_version","alias_value":"1110.0160v2","created_at":"2026-05-18T03:40:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.0160","created_at":"2026-05-18T03:40:25Z"},{"alias_kind":"pith_short_12","alias_value":"A5ODDMCEHRUV","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"A5ODDMCEHRUVBJGZ","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"A5ODDMCE","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:e41304a06c9bbf55cbe7bd0427e612aa6c78407ce59dc2704a282dabaf1868b9","target":"graph","created_at":"2026-05-18T03:40:25Z","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":"A sorting network is a shortest path from 12..n to n..21 in the Cayley graph of the symmetric group S(n) generated by nearest-neighbor swaps. A pattern is a sequence of swaps that forms an initial segment of some sorting network. We prove that in a uniformly random n-element sorting network, any fixed pattern occurs in at least cn^2 disjoint space-time locations, with probability tending to 1 exponentially fast as n tends to infinity. Here c is a positive constant which depends on the choice of pattern. As a consequence, the probability that the uniformly random sorting network is geometricall","authors_text":"Alexander E. Holroyd, Omer Angel, Vadim Gorin","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-10-02T09:05:47Z","title":"A pattern theorem for random sorting networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.0160","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":"sha256:1e4c4e54ef33bdc94346a43d1a799ab513f81492e6e6aefe8e2f9ab166694acb","target":"record","created_at":"2026-05-18T03:40:25Z","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":"cf8131549c80a6acf0884e07d35e1c3e17e4f9dc5856575b4ef327c855797602","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-10-02T09:05:47Z","title_canon_sha256":"f262c76653e42be7ae166bfa9fcfbb8540febc13e5e902accb3de6a23dc83eb3"},"schema_version":"1.0","source":{"id":"1110.0160","kind":"arxiv","version":2}},"canonical_sha256":"075c31b0443c6950a4d902883f1f2062b1b37c7664772b90b22121a5c566d39d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"075c31b0443c6950a4d902883f1f2062b1b37c7664772b90b22121a5c566d39d","first_computed_at":"2026-05-18T03:40:25.791384Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:40:25.791384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3JXGjycDHmS5j/f7oSmu1Iw0iMWPXkAGDdOsVoQhSeC6C/o0DPBITN7cH4W0A5rHq4tmuslkVsO+cHCGTo2ZAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:40:25.792121Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.0160","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1e4c4e54ef33bdc94346a43d1a799ab513f81492e6e6aefe8e2f9ab166694acb","sha256:e41304a06c9bbf55cbe7bd0427e612aa6c78407ce59dc2704a282dabaf1868b9"],"state_sha256":"ea2574827db25a8237d4bee663bfa8fcc8517e3479a5bdcc78ce8424bb75bc69"}