{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2000:7L26HWV4ROTBDIDNS5ASMWJDW3","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":"731219709a0bc71e209020c7c4cf67c2d8927642ff661a42a3a25f1096bbc9f7","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2000-05-13T15:44:27Z","title_canon_sha256":"da1dc24058525fa5762c2b2d52ee15ef7976c6d9f8936645042d32cdfb0f105a"},"schema_version":"1.0","source":{"id":"math/0005136","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0005136","created_at":"2026-05-18T03:53:18Z"},{"alias_kind":"arxiv_version","alias_value":"math/0005136v2","created_at":"2026-05-18T03:53:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0005136","created_at":"2026-05-18T03:53:18Z"},{"alias_kind":"pith_short_12","alias_value":"7L26HWV4ROTB","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_16","alias_value":"7L26HWV4ROTBDIDN","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_8","alias_value":"7L26HWV4","created_at":"2026-05-18T12:25:49Z"}],"graph_snapshots":[{"event_id":"sha256:db5aad4c3a8d7e461d6a83be8a811a5fdaa07828ebcd895f71153643659be85c","target":"graph","created_at":"2026-05-18T03:53:18Z","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":"There has been much recent interest in the satisfiability of random Boolean formulas. A random k-SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2-SAT this happens when m/n --> 1, for 3-SAT the critical ratio is thought to be m/n ~ 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called \\nu=\\nu_k (the smaller the value of \\nu the sharp","authors_text":"David B. Wilson","cross_cats":[],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2000-05-13T15:44:27Z","title":"On the critical exponents of random k-SAT"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0005136","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:a28642183c0453c65529f33f8a35f3023b93d48c4a74f0f5c16d8cc16248059b","target":"record","created_at":"2026-05-18T03:53:18Z","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":"731219709a0bc71e209020c7c4cf67c2d8927642ff661a42a3a25f1096bbc9f7","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2000-05-13T15:44:27Z","title_canon_sha256":"da1dc24058525fa5762c2b2d52ee15ef7976c6d9f8936645042d32cdfb0f105a"},"schema_version":"1.0","source":{"id":"math/0005136","kind":"arxiv","version":2}},"canonical_sha256":"faf5e3dabc8ba611a06d9741265923b6d2d3bf5a0bd215a0f09cf112036db4ab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"faf5e3dabc8ba611a06d9741265923b6d2d3bf5a0bd215a0f09cf112036db4ab","first_computed_at":"2026-05-18T03:53:18.063048Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:53:18.063048Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"V7xo8i5BNjQ1wTExUFSiFsSLM+oeHi84p/g0xXUZi5vbv9M5xIipUKfERIjUcIUEbfPwbCrtiLtUXKVQAlmuCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:53:18.063942Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0005136","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a28642183c0453c65529f33f8a35f3023b93d48c4a74f0f5c16d8cc16248059b","sha256:db5aad4c3a8d7e461d6a83be8a811a5fdaa07828ebcd895f71153643659be85c"],"state_sha256":"85b1a2d7396e84fbb2c359aed06cd2aaf717e01c13ab946ae1c4c088bd592200"}