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Also, above the satisfiability threshold, where a random formula is with high probability (whp) unsatisfiable, the unsatisfiability is whp due to a large \"minimal unsatisfiable subformula\" (MUF).\n  By contrast, we show that for the (rare) unsatisfiable formulae below the pure literal threshold, the unsatisfiability is whp due to a unique MUF with smallest possible \"excess\", failing this whp due to a unique MU"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1008.1260","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-08-06T19:45:23Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"17cc108b1d396797f816e68f4a0e11c4dcb93be8be73652ea52d61604616ce17","abstract_canon_sha256":"ffd04678e37c1d68b55d34826fb43fdd3aee2984c1ce67a5393f4d11ba3d0fd7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:33.857542Z","signature_b64":"CkqN3tt+WUFz8MknM2vWGuhDyt9yIQe94PPdsufOi7uWxeD1Y1xPwzsal7KleDX+UypBrSjMtdT/B/fgNxqtBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1532135126bf95c7a49280ca6e09365b5414b64fa5e63c9a7ab7dbf5aff16c9","last_reissued_at":"2026-05-18T04:42:33.855863Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:33.855863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Structure of random r-SAT below the pure literal threshold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Alexander D. 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