{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:SYG7U3SNNH52YTKSLJHOSEJPD7","short_pith_number":"pith:SYG7U3SN","canonical_record":{"source":{"id":"1504.08316","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-04-30T17:48:21Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"45e65e1ee96da94d44842e33a9bf3e7164b9ada24cf84dd4ca64a01da2f4bce8","abstract_canon_sha256":"fab0c7c91c24031b0a477de40a4b28eb01a70af300cfbfaa43aea910da6f0c95"},"schema_version":"1.0"},"canonical_sha256":"960dfa6e4d69fbac4d525a4ee9112f1ff90b3ee5b0bb2802003edfd05c08dab4","source":{"kind":"arxiv","id":"1504.08316","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.08316","created_at":"2026-05-18T02:17:22Z"},{"alias_kind":"arxiv_version","alias_value":"1504.08316v1","created_at":"2026-05-18T02:17:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.08316","created_at":"2026-05-18T02:17:22Z"},{"alias_kind":"pith_short_12","alias_value":"SYG7U3SNNH52","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"SYG7U3SNNH52YTKS","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"SYG7U3SN","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:SYG7U3SNNH52YTKSLJHOSEJPD7","target":"record","payload":{"canonical_record":{"source":{"id":"1504.08316","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-04-30T17:48:21Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"45e65e1ee96da94d44842e33a9bf3e7164b9ada24cf84dd4ca64a01da2f4bce8","abstract_canon_sha256":"fab0c7c91c24031b0a477de40a4b28eb01a70af300cfbfaa43aea910da6f0c95"},"schema_version":"1.0"},"canonical_sha256":"960dfa6e4d69fbac4d525a4ee9112f1ff90b3ee5b0bb2802003edfd05c08dab4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:22.653635Z","signature_b64":"YTIesw/rhYcCIEh3Q1mIl/MReIMuQw7nHmSFryYwW6aRl1r4vjwLk+auvmWvFxoMM85nbQCLUN6ggKVc/ZJCBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"960dfa6e4d69fbac4d525a4ee9112f1ff90b3ee5b0bb2802003edfd05c08dab4","last_reissued_at":"2026-05-18T02:17:22.652972Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:22.652972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.08316","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":"2026-05-18T02:17:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5xfezA3LNIy1hlMOGtjNpY9Ae1X4qcKK2YApdxrC6aU/VpXOnT1VC0aAdz5FeVMYvcx0RoB5lJrRGiiq2GkzAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T21:45:25.976039Z"},"content_sha256":"ba2470b4590f9c2ba8eccb1b945d461b5c9fccd91c44a5d12ef70a4a3926278a","schema_version":"1.0","event_id":"sha256:ba2470b4590f9c2ba8eccb1b945d461b5c9fccd91c44a5d12ef70a4a3926278a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:SYG7U3SNNH52YTKSLJHOSEJPD7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Concentration of the number of solutions of random planted CSPs and Goldreich's one-way candidates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.PR","authors_text":"Emmanuel Abbe, Katherine Edwards","submitted_at":"2015-04-30T17:48:21Z","abstract_excerpt":"This paper shows that the logarithm of the number of solutions of a random planted $k$-SAT formula concentrates around a deterministic $n$-independent threshold. Specifically, if $F^*_{k}(\\alpha,n)$ is a random $k$-SAT formula on $n$ variables, with clause density $\\alpha$ and with a uniformly drawn planted solution, there exists a function $\\phi_k(\\cdot)$ such that, besides for some $\\alpha$ in a set of Lesbegue measure zero, we have $ \\frac{1}{n}\\log Z(F^*_{k}(\\alpha,n)) \\to \\phi_k(\\alpha)$ in probability, where $Z(F)$ is the number of solutions of the formula $F$. This settles a problem lef"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.08316","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":"2026-05-18T02:17:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QdzykYEIn9JeEMm4vpoma9+nokXDVnbu/BuX/dfcMpafWn+0Vm3Jg5TXZu6omLbs4r8BFLwQ0gVyJGPbw3rVAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T21:45:25.976789Z"},"content_sha256":"129fccafe993833749a2c0abe29143d969c72172385d30d99306069e7eb9f793","schema_version":"1.0","event_id":"sha256:129fccafe993833749a2c0abe29143d969c72172385d30d99306069e7eb9f793"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SYG7U3SNNH52YTKSLJHOSEJPD7/bundle.json","state_url":"https://pith.science/pith/SYG7U3SNNH52YTKSLJHOSEJPD7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SYG7U3SNNH52YTKSLJHOSEJPD7/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":"2026-05-27T21:45:25Z","links":{"resolver":"https://pith.science/pith/SYG7U3SNNH52YTKSLJHOSEJPD7","bundle":"https://pith.science/pith/SYG7U3SNNH52YTKSLJHOSEJPD7/bundle.json","state":"https://pith.science/pith/SYG7U3SNNH52YTKSLJHOSEJPD7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SYG7U3SNNH52YTKSLJHOSEJPD7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:SYG7U3SNNH52YTKSLJHOSEJPD7","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":"fab0c7c91c24031b0a477de40a4b28eb01a70af300cfbfaa43aea910da6f0c95","cross_cats_sorted":["cs.CC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-04-30T17:48:21Z","title_canon_sha256":"45e65e1ee96da94d44842e33a9bf3e7164b9ada24cf84dd4ca64a01da2f4bce8"},"schema_version":"1.0","source":{"id":"1504.08316","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.08316","created_at":"2026-05-18T02:17:22Z"},{"alias_kind":"arxiv_version","alias_value":"1504.08316v1","created_at":"2026-05-18T02:17:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.08316","created_at":"2026-05-18T02:17:22Z"},{"alias_kind":"pith_short_12","alias_value":"SYG7U3SNNH52","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"SYG7U3SNNH52YTKS","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"SYG7U3SN","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:129fccafe993833749a2c0abe29143d969c72172385d30d99306069e7eb9f793","target":"graph","created_at":"2026-05-18T02:17:22Z","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 paper shows that the logarithm of the number of solutions of a random planted $k$-SAT formula concentrates around a deterministic $n$-independent threshold. Specifically, if $F^*_{k}(\\alpha,n)$ is a random $k$-SAT formula on $n$ variables, with clause density $\\alpha$ and with a uniformly drawn planted solution, there exists a function $\\phi_k(\\cdot)$ such that, besides for some $\\alpha$ in a set of Lesbegue measure zero, we have $ \\frac{1}{n}\\log Z(F^*_{k}(\\alpha,n)) \\to \\phi_k(\\alpha)$ in probability, where $Z(F)$ is the number of solutions of the formula $F$. This settles a problem lef","authors_text":"Emmanuel Abbe, Katherine Edwards","cross_cats":["cs.CC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-04-30T17:48:21Z","title":"Concentration of the number of solutions of random planted CSPs and Goldreich's one-way candidates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.08316","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:ba2470b4590f9c2ba8eccb1b945d461b5c9fccd91c44a5d12ef70a4a3926278a","target":"record","created_at":"2026-05-18T02:17:22Z","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":"fab0c7c91c24031b0a477de40a4b28eb01a70af300cfbfaa43aea910da6f0c95","cross_cats_sorted":["cs.CC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-04-30T17:48:21Z","title_canon_sha256":"45e65e1ee96da94d44842e33a9bf3e7164b9ada24cf84dd4ca64a01da2f4bce8"},"schema_version":"1.0","source":{"id":"1504.08316","kind":"arxiv","version":1}},"canonical_sha256":"960dfa6e4d69fbac4d525a4ee9112f1ff90b3ee5b0bb2802003edfd05c08dab4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"960dfa6e4d69fbac4d525a4ee9112f1ff90b3ee5b0bb2802003edfd05c08dab4","first_computed_at":"2026-05-18T02:17:22.652972Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:22.652972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YTIesw/rhYcCIEh3Q1mIl/MReIMuQw7nHmSFryYwW6aRl1r4vjwLk+auvmWvFxoMM85nbQCLUN6ggKVc/ZJCBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:22.653635Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.08316","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba2470b4590f9c2ba8eccb1b945d461b5c9fccd91c44a5d12ef70a4a3926278a","sha256:129fccafe993833749a2c0abe29143d969c72172385d30d99306069e7eb9f793"],"state_sha256":"13471dfebb28a21dc1659eb8e056ea294a0cec17b4bbabe01a44687b6487d6ae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2XPl24qj+jy5mvfADS4kJ76ASAlS7zzRoDud/hESilMVkM6XOi5zgkkXya5QqxbyVLtr2MEYwU8shN7ASoWYAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T21:45:25.980482Z","bundle_sha256":"3b4bcaaae0208b1b0e015075dade4ea7dbdffe1bf7a9f9d98e86a5491a6f2fed"}}