{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:GUKN5M5KWUCQKSHHLJAHFRBEV7","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":"06d3c928f594c24499eed500421e02a75e48e273fc4dbe826d687643a0026114","cross_cats_sorted":["math.CO","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-09-19T15:01:43Z","title_canon_sha256":"68937c6800c5649f71e254faf73053925d9ffdc516a40bfaaf390bd68f53609c"},"schema_version":"1.0","source":{"id":"1309.6651","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.6651","created_at":"2026-05-18T03:12:10Z"},{"alias_kind":"arxiv_version","alias_value":"1309.6651v1","created_at":"2026-05-18T03:12:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6651","created_at":"2026-05-18T03:12:10Z"},{"alias_kind":"pith_short_12","alias_value":"GUKN5M5KWUCQ","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"GUKN5M5KWUCQKSHH","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"GUKN5M5K","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:594ee3dda4cdf422cd644ffb9c2f0ea665c2cf32d4551f9b10658b30c212f5eb","target":"graph","created_at":"2026-05-18T03:12:10Z","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 study a random system of $cn$ linear equations over $n$ variables in GF(2), where each equation contains exactly $r$ variables; this is equivalent to $r$-XORSAT. \\cite{ikkm,amxor} determined the clustering threshold, $c^*_r$: if $c=c^*_r+\\e$ for any constant $\\e>0$, then \\aas the solutions partition into well-connected, well-separated {\\em clusters} (with probability tending to 1 as $n\\rightarrow\\infty$). This is part of a general clustering phenomenon which is hypothesized to arise in most of the commonly studied models of random constraint satisfaction problems, via sophisticated but most","authors_text":"Michael Molloy, Pu Gao","cross_cats":["math.CO","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-09-19T15:01:43Z","title":"Inside the clustering threshold for random linear equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6651","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":"sha256:5bbd46f94eed08157dacec8251a4b55b0351cc718387925c45b686d78602e722","target":"record","created_at":"2026-05-18T03:12:10Z","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":"06d3c928f594c24499eed500421e02a75e48e273fc4dbe826d687643a0026114","cross_cats_sorted":["math.CO","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-09-19T15:01:43Z","title_canon_sha256":"68937c6800c5649f71e254faf73053925d9ffdc516a40bfaaf390bd68f53609c"},"schema_version":"1.0","source":{"id":"1309.6651","kind":"arxiv","version":1}},"canonical_sha256":"3514deb3aab5050548e75a4072c424afde5ac4bfcfefd247cd61c9d57a97d61c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3514deb3aab5050548e75a4072c424afde5ac4bfcfefd247cd61c9d57a97d61c","first_computed_at":"2026-05-18T03:12:10.994383Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:10.994383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pSXHtGs42anxvN/ARED6oJTN9cdK3rMGChN+smprM9MTjWKcF5593teyKbfl65YkAFG8l503wLQxn0ED13CsCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:10.995267Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.6651","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5bbd46f94eed08157dacec8251a4b55b0351cc718387925c45b686d78602e722","sha256:594ee3dda4cdf422cd644ffb9c2f0ea665c2cf32d4551f9b10658b30c212f5eb"],"state_sha256":"ebcdce175affb05d60c7877dfe2e40c91f7c79412e09c1f76429911f9de714dd"}