{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:KDZBO3LEOU36IRQ5BA2LAAHBAW","short_pith_number":"pith:KDZBO3LE","canonical_record":{"source":{"id":"1507.08448","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-30T10:52:14Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"d18df0984dfa076aa1b3a8c77beaf0eb6e0e6cd4c746cc0b113579098a81b478","abstract_canon_sha256":"89e7a4de9b96aed38e07c1dbaf19f37da2ce704839fa427ba210b90e52cc02f4"},"schema_version":"1.0"},"canonical_sha256":"50f2176d647537e4461d0834b000e10592b536c3ae4aef6474480ae559ffd214","source":{"kind":"arxiv","id":"1507.08448","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08448","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08448v1","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08448","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"pith_short_12","alias_value":"KDZBO3LEOU36","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KDZBO3LEOU36IRQ5","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KDZBO3LE","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:KDZBO3LEOU36IRQ5BA2LAAHBAW","target":"record","payload":{"canonical_record":{"source":{"id":"1507.08448","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-30T10:52:14Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"d18df0984dfa076aa1b3a8c77beaf0eb6e0e6cd4c746cc0b113579098a81b478","abstract_canon_sha256":"89e7a4de9b96aed38e07c1dbaf19f37da2ce704839fa427ba210b90e52cc02f4"},"schema_version":"1.0"},"canonical_sha256":"50f2176d647537e4461d0834b000e10592b536c3ae4aef6474480ae559ffd214","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:06.422521Z","signature_b64":"zZCzxIWNQ4JE4rP1wBYK9G/yKJJ1hNRQpQ5If1wfG/TCEdDSx2xSoizIYp1EQ74k+hzBBu2F89LquIKOsugABw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50f2176d647537e4461d0834b000e10592b536c3ae4aef6474480ae559ffd214","last_reissued_at":"2026-05-18T01:36:06.421930Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:06.421930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.08448","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":"2026-05-18T01:36:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Yagtz+baSF4OABvX3ZBWFne0yw1u1MFxx81blmJk/81Uu14b9j+GVLkxIfqNywQUpq8PEnTQEk/8WPmipcvAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T16:30:24.462199Z"},"content_sha256":"4b98e1e1803693e2bfb34a780b9a51434c329a313c746e5b60697392d008b122","schema_version":"1.0","event_id":"sha256:4b98e1e1803693e2bfb34a780b9a51434c329a313c746e5b60697392d008b122"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:KDZBO3LEOU36IRQ5BA2LAAHBAW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Antoine Genitrini, C\\'ecile Mailler","submitted_at":"2015-07-30T10:52:14Z","abstract_excerpt":"This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable? asymptotically when n tends to infinity?\n  The model of random Boolean expressions developed in the present paper is the model of Boolean Catalan trees, already extensively studied in the literature for a constant sequence (k_n)_{n\\geq 1}. The fundamental breakthrough of this paper is to generalise the previous results to any (reasonable) sequence of integers ("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08448","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":"2026-05-18T01:36:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ChBRxF8Yi6cRWrMs4bLUScmIYYjED79J4WBtxo5ai9smFewLDUHr4Ra+0VUq2W+xmZv3RYbmdCBPhY5tSx9mBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T16:30:24.462777Z"},"content_sha256":"1f1d8f11b25130f8b49fab8fcc09e8ad2ffef7681586934f04fddfff15d64e42","schema_version":"1.0","event_id":"sha256:1f1d8f11b25130f8b49fab8fcc09e8ad2ffef7681586934f04fddfff15d64e42"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KDZBO3LEOU36IRQ5BA2LAAHBAW/bundle.json","state_url":"https://pith.science/pith/KDZBO3LEOU36IRQ5BA2LAAHBAW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KDZBO3LEOU36IRQ5BA2LAAHBAW/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":"2026-05-30T16:30:24Z","links":{"resolver":"https://pith.science/pith/KDZBO3LEOU36IRQ5BA2LAAHBAW","bundle":"https://pith.science/pith/KDZBO3LEOU36IRQ5BA2LAAHBAW/bundle.json","state":"https://pith.science/pith/KDZBO3LEOU36IRQ5BA2LAAHBAW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KDZBO3LEOU36IRQ5BA2LAAHBAW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:KDZBO3LEOU36IRQ5BA2LAAHBAW","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":"89e7a4de9b96aed38e07c1dbaf19f37da2ce704839fa427ba210b90e52cc02f4","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-30T10:52:14Z","title_canon_sha256":"d18df0984dfa076aa1b3a8c77beaf0eb6e0e6cd4c746cc0b113579098a81b478"},"schema_version":"1.0","source":{"id":"1507.08448","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08448","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08448v1","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08448","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"pith_short_12","alias_value":"KDZBO3LEOU36","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KDZBO3LEOU36IRQ5","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KDZBO3LE","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:1f1d8f11b25130f8b49fab8fcc09e8ad2ffef7681586934f04fddfff15d64e42","target":"graph","created_at":"2026-05-18T01:36:06Z","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":"This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable? asymptotically when n tends to infinity?\n  The model of random Boolean expressions developed in the present paper is the model of Boolean Catalan trees, already extensively studied in the literature for a constant sequence (k_n)_{n\\geq 1}. The fundamental breakthrough of this paper is to generalise the previous results to any (reasonable) sequence of integers (","authors_text":"Antoine Genitrini, C\\'ecile Mailler","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-30T10:52:14Z","title":"Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08448","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":"sha256:4b98e1e1803693e2bfb34a780b9a51434c329a313c746e5b60697392d008b122","target":"record","created_at":"2026-05-18T01:36:06Z","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":"89e7a4de9b96aed38e07c1dbaf19f37da2ce704839fa427ba210b90e52cc02f4","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-30T10:52:14Z","title_canon_sha256":"d18df0984dfa076aa1b3a8c77beaf0eb6e0e6cd4c746cc0b113579098a81b478"},"schema_version":"1.0","source":{"id":"1507.08448","kind":"arxiv","version":1}},"canonical_sha256":"50f2176d647537e4461d0834b000e10592b536c3ae4aef6474480ae559ffd214","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50f2176d647537e4461d0834b000e10592b536c3ae4aef6474480ae559ffd214","first_computed_at":"2026-05-18T01:36:06.421930Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:06.421930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zZCzxIWNQ4JE4rP1wBYK9G/yKJJ1hNRQpQ5If1wfG/TCEdDSx2xSoizIYp1EQ74k+hzBBu2F89LquIKOsugABw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:06.422521Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.08448","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4b98e1e1803693e2bfb34a780b9a51434c329a313c746e5b60697392d008b122","sha256:1f1d8f11b25130f8b49fab8fcc09e8ad2ffef7681586934f04fddfff15d64e42"],"state_sha256":"32b5f113e7bfa683f232d83e2187fc5393a8ff5bc75e600dc445b9392872f404"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xqQa9505/2atbn7ZJteRqQTDspiyLNMCtgze20ELc3I8bvqf8VPUkbUqiXUNrHmFimk/AYokc2CuMTaatVmYBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T16:30:24.465698Z","bundle_sha256":"5adb96a34586a3b01cee08813a189482315d625b7504d023ddd1d19223742732"}}