{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:ELJJLUQHLVPYTY7UBIIWA4OUKV","short_pith_number":"pith:ELJJLUQH","canonical_record":{"source":{"id":"1310.7982","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-29T23:25:09Z","cross_cats_sorted":[],"title_canon_sha256":"991178bf57005b0c6e52d5d9155957b880a3bdfb088af7c2c2a1f15e26fe93f3","abstract_canon_sha256":"a9ab8197a1bdf81c4c330dedaba936e81b70514d927bf3a3ab288cbcaf0f2c0e"},"schema_version":"1.0"},"canonical_sha256":"22d295d2075d5f89e3f40a116071d45558d964a3e3d7e08a0bff9032afae4949","source":{"kind":"arxiv","id":"1310.7982","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.7982","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"arxiv_version","alias_value":"1310.7982v2","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.7982","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"pith_short_12","alias_value":"ELJJLUQHLVPY","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"ELJJLUQHLVPYTY7U","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"ELJJLUQH","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:ELJJLUQHLVPYTY7UBIIWA4OUKV","target":"record","payload":{"canonical_record":{"source":{"id":"1310.7982","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-29T23:25:09Z","cross_cats_sorted":[],"title_canon_sha256":"991178bf57005b0c6e52d5d9155957b880a3bdfb088af7c2c2a1f15e26fe93f3","abstract_canon_sha256":"a9ab8197a1bdf81c4c330dedaba936e81b70514d927bf3a3ab288cbcaf0f2c0e"},"schema_version":"1.0"},"canonical_sha256":"22d295d2075d5f89e3f40a116071d45558d964a3e3d7e08a0bff9032afae4949","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:24.643234Z","signature_b64":"nbsf5eVLuL+UjOhqt1k0qyalt56o3KZkjs6TVFs7fprVzHxwODecA9PjUGu5vaP3M3yHhAlTiNo99BZS1RZdCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22d295d2075d5f89e3f40a116071d45558d964a3e3d7e08a0bff9032afae4949","last_reissued_at":"2026-05-18T02:48:24.642793Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:24.642793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.7982","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:48:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/6kZZz1dvzuMnf4vwJ99Mg39uS5F+NaIWjKMEEHmutmefU8E5QJPYFrPjAJJx1i9p1/d52gjy/IcFcx1aL/uAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T03:36:33.631406Z"},"content_sha256":"10983922d07fa5137dc7fdd9ace39b74be371b78da261cb571c82c24e10a9b76","schema_version":"1.0","event_id":"sha256:10983922d07fa5137dc7fdd9ace39b74be371b78da261cb571c82c24e10a9b76"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:ELJJLUQHLVPYTY7UBIIWA4OUKV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Log-Concavity of the Partition Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Igor Pak, Stephen DeSalvo","submitted_at":"2013-10-29T23:25:09Z","abstract_excerpt":"We prove that the partition function $p(n)$ is log-concave for all $n>25$. We then extend the results to resolve two related conjectures by Chen. The proofs are based on Lehmer's estimates on the remainders of the Hardy--Ramanujan and the Rademacher series for $p(n)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7982","kind":"arxiv","version":2},"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":"2026-05-18T02:48:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/stZgO8Fk/LOOIYv7CIZ8LYkH5MIlQc2Er7VU9EoUEIWANGJCyltrn+Whd1sMrMYI+rJa8gURbtci3AusgNPDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T03:36:33.631743Z"},"content_sha256":"36c492576178bee1ea416a81616c599997838f57be2973639bb08e67dce46289","schema_version":"1.0","event_id":"sha256:36c492576178bee1ea416a81616c599997838f57be2973639bb08e67dce46289"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ELJJLUQHLVPYTY7UBIIWA4OUKV/bundle.json","state_url":"https://pith.science/pith/ELJJLUQHLVPYTY7UBIIWA4OUKV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ELJJLUQHLVPYTY7UBIIWA4OUKV/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":"2026-05-31T03:36:33Z","links":{"resolver":"https://pith.science/pith/ELJJLUQHLVPYTY7UBIIWA4OUKV","bundle":"https://pith.science/pith/ELJJLUQHLVPYTY7UBIIWA4OUKV/bundle.json","state":"https://pith.science/pith/ELJJLUQHLVPYTY7UBIIWA4OUKV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ELJJLUQHLVPYTY7UBIIWA4OUKV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ELJJLUQHLVPYTY7UBIIWA4OUKV","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":"a9ab8197a1bdf81c4c330dedaba936e81b70514d927bf3a3ab288cbcaf0f2c0e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-29T23:25:09Z","title_canon_sha256":"991178bf57005b0c6e52d5d9155957b880a3bdfb088af7c2c2a1f15e26fe93f3"},"schema_version":"1.0","source":{"id":"1310.7982","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.7982","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"arxiv_version","alias_value":"1310.7982v2","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.7982","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"pith_short_12","alias_value":"ELJJLUQHLVPY","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"ELJJLUQHLVPYTY7U","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"ELJJLUQH","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:36c492576178bee1ea416a81616c599997838f57be2973639bb08e67dce46289","target":"graph","created_at":"2026-05-18T02:48:24Z","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 prove that the partition function $p(n)$ is log-concave for all $n>25$. We then extend the results to resolve two related conjectures by Chen. The proofs are based on Lehmer's estimates on the remainders of the Hardy--Ramanujan and the Rademacher series for $p(n)$.","authors_text":"Igor Pak, Stephen DeSalvo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-29T23:25:09Z","title":"Log-Concavity of the Partition Function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7982","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":"sha256:10983922d07fa5137dc7fdd9ace39b74be371b78da261cb571c82c24e10a9b76","target":"record","created_at":"2026-05-18T02:48:24Z","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":"a9ab8197a1bdf81c4c330dedaba936e81b70514d927bf3a3ab288cbcaf0f2c0e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-29T23:25:09Z","title_canon_sha256":"991178bf57005b0c6e52d5d9155957b880a3bdfb088af7c2c2a1f15e26fe93f3"},"schema_version":"1.0","source":{"id":"1310.7982","kind":"arxiv","version":2}},"canonical_sha256":"22d295d2075d5f89e3f40a116071d45558d964a3e3d7e08a0bff9032afae4949","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"22d295d2075d5f89e3f40a116071d45558d964a3e3d7e08a0bff9032afae4949","first_computed_at":"2026-05-18T02:48:24.642793Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:48:24.642793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nbsf5eVLuL+UjOhqt1k0qyalt56o3KZkjs6TVFs7fprVzHxwODecA9PjUGu5vaP3M3yHhAlTiNo99BZS1RZdCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:48:24.643234Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.7982","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:10983922d07fa5137dc7fdd9ace39b74be371b78da261cb571c82c24e10a9b76","sha256:36c492576178bee1ea416a81616c599997838f57be2973639bb08e67dce46289"],"state_sha256":"88e0bfac9353c4712ba51c98d29fe797dd6db745d2c6688124d013f8d44661dd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WGE2SHy1UTfpCCyaydU0KUMqtYU63hqaoVxYoR0h0GmiLP1BelnnYAIv9qBGe+IeckwM3zPLpRknX8nb03QrBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T03:36:33.633686Z","bundle_sha256":"c1140c351d8a5f461ba2ebbb3af357c39ca5166812d80a31e470743f5ac0eb7b"}}