{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:H6K4JSTRFD6KPGR4MUAB263EMG","short_pith_number":"pith:H6K4JSTR","canonical_record":{"source":{"id":"0909.0641","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2009-09-03T16:22:01Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"025ed2c7ba6257dcb24557f755e04f5967f4e75151f496dcc8374e870b9c03c4","abstract_canon_sha256":"53977d750dfc72a94cdaffa9ff7d9eed8ca0523f7b05ecefd62219c97cca839a"},"schema_version":"1.0"},"canonical_sha256":"3f95c4ca7128fca79a3c65001d7b6461bfd760eb3ede0abbac0946208fd0a9b3","source":{"kind":"arxiv","id":"0909.0641","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.0641","created_at":"2026-05-18T04:39:00Z"},{"alias_kind":"arxiv_version","alias_value":"0909.0641v2","created_at":"2026-05-18T04:39:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.0641","created_at":"2026-05-18T04:39:00Z"},{"alias_kind":"pith_short_12","alias_value":"H6K4JSTRFD6K","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"H6K4JSTRFD6KPGR4","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"H6K4JSTR","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:H6K4JSTRFD6KPGR4MUAB263EMG","target":"record","payload":{"canonical_record":{"source":{"id":"0909.0641","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2009-09-03T16:22:01Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"025ed2c7ba6257dcb24557f755e04f5967f4e75151f496dcc8374e870b9c03c4","abstract_canon_sha256":"53977d750dfc72a94cdaffa9ff7d9eed8ca0523f7b05ecefd62219c97cca839a"},"schema_version":"1.0"},"canonical_sha256":"3f95c4ca7128fca79a3c65001d7b6461bfd760eb3ede0abbac0946208fd0a9b3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:00.214909Z","signature_b64":"FB6uDphrB2C9VZ5u4xJGhoCUYeEzXAsh3N9Peo5/E4iymv8TglLeovI89VtEXinMfcV+1zU7roFzIcz+WY8WCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f95c4ca7128fca79a3c65001d7b6461bfd760eb3ede0abbac0946208fd0a9b3","last_reissued_at":"2026-05-18T04:39:00.214508Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:00.214508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0909.0641","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":"2026-05-18T04:39:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lnNByCnM/cQCnjqhXlngMI9aq0SMW+8YERl5bhf/vNRz16y4Yzp0lPrQ4mxxGmvce4XqKqGY+jCKkhMGweXVAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T21:09:21.048713Z"},"content_sha256":"d0361b4b0a66b5cc007ac17d662e80f04bb3558dd8c2be00c8b7ce5472395798","schema_version":"1.0","event_id":"sha256:d0361b4b0a66b5cc007ac17d662e80f04bb3558dd8c2be00c8b7ce5472395798"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:H6K4JSTRFD6KPGR4MUAB263EMG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Monotonicity, thinning and discrete versions of the Entropy Power Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Oliver Johnson, Yaming Yu","submitted_at":"2009-09-03T16:22:01Z","abstract_excerpt":"We consider the entropy of sums of independent discrete random variables, in analogy with Shannon's Entropy Power Inequality, where equality holds for normals. In our case, infinite divisibility suggests that equality should hold for Poisson variables. We show that some natural analogues of the Entropy Power Inequality do not in fact hold, but propose an alternative formulation which does always hold. The key to many proofs of Shannon's Entropy Power Inequality is the behaviour of entropy on scaling of continuous random variables. We believe that R\\'{e}nyi's operation of thinning discrete rand"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.0641","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":"2026-05-18T04:39:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2yZk3Q2DAfN4xCOs94rTC71RzCFT54y+IKt66L2VuwLafQGrib4LPnU/4amzEJK5WoZKwehRKW/2hwK3WwalAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T21:09:21.049468Z"},"content_sha256":"2ee64227b003f2418f4a2825df13f1cd50b8141fd8b1e000304228fbc3b794b8","schema_version":"1.0","event_id":"sha256:2ee64227b003f2418f4a2825df13f1cd50b8141fd8b1e000304228fbc3b794b8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H6K4JSTRFD6KPGR4MUAB263EMG/bundle.json","state_url":"https://pith.science/pith/H6K4JSTRFD6KPGR4MUAB263EMG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H6K4JSTRFD6KPGR4MUAB263EMG/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":"2026-06-05T21:09:21Z","links":{"resolver":"https://pith.science/pith/H6K4JSTRFD6KPGR4MUAB263EMG","bundle":"https://pith.science/pith/H6K4JSTRFD6KPGR4MUAB263EMG/bundle.json","state":"https://pith.science/pith/H6K4JSTRFD6KPGR4MUAB263EMG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H6K4JSTRFD6KPGR4MUAB263EMG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:H6K4JSTRFD6KPGR4MUAB263EMG","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":"53977d750dfc72a94cdaffa9ff7d9eed8ca0523f7b05ecefd62219c97cca839a","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2009-09-03T16:22:01Z","title_canon_sha256":"025ed2c7ba6257dcb24557f755e04f5967f4e75151f496dcc8374e870b9c03c4"},"schema_version":"1.0","source":{"id":"0909.0641","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.0641","created_at":"2026-05-18T04:39:00Z"},{"alias_kind":"arxiv_version","alias_value":"0909.0641v2","created_at":"2026-05-18T04:39:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.0641","created_at":"2026-05-18T04:39:00Z"},{"alias_kind":"pith_short_12","alias_value":"H6K4JSTRFD6K","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"H6K4JSTRFD6KPGR4","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"H6K4JSTR","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:2ee64227b003f2418f4a2825df13f1cd50b8141fd8b1e000304228fbc3b794b8","target":"graph","created_at":"2026-05-18T04:39:00Z","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 consider the entropy of sums of independent discrete random variables, in analogy with Shannon's Entropy Power Inequality, where equality holds for normals. In our case, infinite divisibility suggests that equality should hold for Poisson variables. We show that some natural analogues of the Entropy Power Inequality do not in fact hold, but propose an alternative formulation which does always hold. The key to many proofs of Shannon's Entropy Power Inequality is the behaviour of entropy on scaling of continuous random variables. We believe that R\\'{e}nyi's operation of thinning discrete rand","authors_text":"Oliver Johnson, Yaming Yu","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2009-09-03T16:22:01Z","title":"Monotonicity, thinning and discrete versions of the Entropy Power Inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.0641","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:d0361b4b0a66b5cc007ac17d662e80f04bb3558dd8c2be00c8b7ce5472395798","target":"record","created_at":"2026-05-18T04:39:00Z","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":"53977d750dfc72a94cdaffa9ff7d9eed8ca0523f7b05ecefd62219c97cca839a","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2009-09-03T16:22:01Z","title_canon_sha256":"025ed2c7ba6257dcb24557f755e04f5967f4e75151f496dcc8374e870b9c03c4"},"schema_version":"1.0","source":{"id":"0909.0641","kind":"arxiv","version":2}},"canonical_sha256":"3f95c4ca7128fca79a3c65001d7b6461bfd760eb3ede0abbac0946208fd0a9b3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3f95c4ca7128fca79a3c65001d7b6461bfd760eb3ede0abbac0946208fd0a9b3","first_computed_at":"2026-05-18T04:39:00.214508Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:00.214508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FB6uDphrB2C9VZ5u4xJGhoCUYeEzXAsh3N9Peo5/E4iymv8TglLeovI89VtEXinMfcV+1zU7roFzIcz+WY8WCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:00.214909Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.0641","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d0361b4b0a66b5cc007ac17d662e80f04bb3558dd8c2be00c8b7ce5472395798","sha256:2ee64227b003f2418f4a2825df13f1cd50b8141fd8b1e000304228fbc3b794b8"],"state_sha256":"bd5e680d850ec5a62bb2ca53bb43d48ad47f03112e2c096c691f9317f7d95347"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GTVTNZb6b9+UIbBJPWglijJ9zH3TVF7ikvoy/WykGQ23kaIMmXawhqEcn+EQvafWfhscdDxJ5O4QnWgfmjxDAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T21:09:21.053474Z","bundle_sha256":"9be42b06f44361a8bd9b829f0b493a95b7bb1140cb3ab06a05dd020f2b0b33b6"}}