{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:XSWFYTJIJEXM34RT7T6PIWNQAO","short_pith_number":"pith:XSWFYTJI","canonical_record":{"source":{"id":"0901.0055","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2008-12-31T04:30:15Z","cross_cats_sorted":["math.CO","math.IT","math.NT","math.PR"],"title_canon_sha256":"fe95bebb3a223657221fbcc1b09a8be7bcb3ce84782c59daca0963dde19226d1","abstract_canon_sha256":"db992577af5cd54b86e1ad781db25255bf41ea89254afae42382635acf3e3062"},"schema_version":"1.0"},"canonical_sha256":"bcac5c4d28492ecdf233fcfcf459b003989932104d1dac5ed796a3e06f18cc7d","source":{"kind":"arxiv","id":"0901.0055","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.0055","created_at":"2026-05-18T03:54:26Z"},{"alias_kind":"arxiv_version","alias_value":"0901.0055v3","created_at":"2026-05-18T03:54:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.0055","created_at":"2026-05-18T03:54:26Z"},{"alias_kind":"pith_short_12","alias_value":"XSWFYTJIJEXM","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"XSWFYTJIJEXM34RT","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"XSWFYTJI","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:XSWFYTJIJEXM34RT7T6PIWNQAO","target":"record","payload":{"canonical_record":{"source":{"id":"0901.0055","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2008-12-31T04:30:15Z","cross_cats_sorted":["math.CO","math.IT","math.NT","math.PR"],"title_canon_sha256":"fe95bebb3a223657221fbcc1b09a8be7bcb3ce84782c59daca0963dde19226d1","abstract_canon_sha256":"db992577af5cd54b86e1ad781db25255bf41ea89254afae42382635acf3e3062"},"schema_version":"1.0"},"canonical_sha256":"bcac5c4d28492ecdf233fcfcf459b003989932104d1dac5ed796a3e06f18cc7d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:26.483668Z","signature_b64":"IOCUrKN+EeatSCn3E2VoPwXIAiy1WyGhUaIfdmrohqI1DOy4rjz67Db9o61U0ZGUPKfxLvFbfYxzHR1bu4V9BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bcac5c4d28492ecdf233fcfcf459b003989932104d1dac5ed796a3e06f18cc7d","last_reissued_at":"2026-05-18T03:54:26.482984Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:26.482984Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0901.0055","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:54:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9B2m3aJ3PF340NRbZt9/KVyTdS04Qd5HWUBzpR6Y/jTBhTD1MzkRc7l35sH+2K70O4IytFg9d9kBBKm7pGxuAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T18:27:41.206063Z"},"content_sha256":"3d06d2ad18ab7fdab5ff42951972781fe480e58b3c64bea4142437ab1a8fe471","schema_version":"1.0","event_id":"sha256:3d06d2ad18ab7fdab5ff42951972781fe480e58b3c64bea4142437ab1a8fe471"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:XSWFYTJIJEXM34RT7T6PIWNQAO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Entropy and set cardinality inequalities for partition-determined functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.IT","math.NT","math.PR"],"primary_cat":"cs.IT","authors_text":"Adam Marcus, Mokshay Madiman, Prasad Tetali","submitted_at":"2008-12-31T04:30:15Z","abstract_excerpt":"A new notion of partition-determined functions is introduced, and several basic inequalities are developed for the entropy of such functions of independent random variables, as well as for cardinalities of compound sets obtained using these functions. Here a compound set means a set obtained by varying each argument of a function of several variables over a set associated with that argument, where all the sets are subsets of an appropriate algebraic structure so that the function is well defined. On the one hand, the entropy inequalities developed for partition-determined functions imply entro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.0055","kind":"arxiv","version":3},"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":"2026-05-18T03:54:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dXV9Gl4tn8uwDB5kpKFlSBaGto5OfEzFCQA50rFfvhi0KzZAz+3Obsil9UP94Zv3PPXZdLAxtbv5NZWODletDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T18:27:41.206946Z"},"content_sha256":"fca54c1089b34fe1472ce6ab8e73b625ec6648bf488dfb012f610c2b747e49fe","schema_version":"1.0","event_id":"sha256:fca54c1089b34fe1472ce6ab8e73b625ec6648bf488dfb012f610c2b747e49fe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XSWFYTJIJEXM34RT7T6PIWNQAO/bundle.json","state_url":"https://pith.science/pith/XSWFYTJIJEXM34RT7T6PIWNQAO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XSWFYTJIJEXM34RT7T6PIWNQAO/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":"2026-06-09T18:27:41Z","links":{"resolver":"https://pith.science/pith/XSWFYTJIJEXM34RT7T6PIWNQAO","bundle":"https://pith.science/pith/XSWFYTJIJEXM34RT7T6PIWNQAO/bundle.json","state":"https://pith.science/pith/XSWFYTJIJEXM34RT7T6PIWNQAO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XSWFYTJIJEXM34RT7T6PIWNQAO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:XSWFYTJIJEXM34RT7T6PIWNQAO","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":"db992577af5cd54b86e1ad781db25255bf41ea89254afae42382635acf3e3062","cross_cats_sorted":["math.CO","math.IT","math.NT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2008-12-31T04:30:15Z","title_canon_sha256":"fe95bebb3a223657221fbcc1b09a8be7bcb3ce84782c59daca0963dde19226d1"},"schema_version":"1.0","source":{"id":"0901.0055","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.0055","created_at":"2026-05-18T03:54:26Z"},{"alias_kind":"arxiv_version","alias_value":"0901.0055v3","created_at":"2026-05-18T03:54:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.0055","created_at":"2026-05-18T03:54:26Z"},{"alias_kind":"pith_short_12","alias_value":"XSWFYTJIJEXM","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"XSWFYTJIJEXM34RT","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"XSWFYTJI","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:fca54c1089b34fe1472ce6ab8e73b625ec6648bf488dfb012f610c2b747e49fe","target":"graph","created_at":"2026-05-18T03:54:26Z","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":"A new notion of partition-determined functions is introduced, and several basic inequalities are developed for the entropy of such functions of independent random variables, as well as for cardinalities of compound sets obtained using these functions. Here a compound set means a set obtained by varying each argument of a function of several variables over a set associated with that argument, where all the sets are subsets of an appropriate algebraic structure so that the function is well defined. On the one hand, the entropy inequalities developed for partition-determined functions imply entro","authors_text":"Adam Marcus, Mokshay Madiman, Prasad Tetali","cross_cats":["math.CO","math.IT","math.NT","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2008-12-31T04:30:15Z","title":"Entropy and set cardinality inequalities for partition-determined functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.0055","kind":"arxiv","version":3},"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":"sha256:3d06d2ad18ab7fdab5ff42951972781fe480e58b3c64bea4142437ab1a8fe471","target":"record","created_at":"2026-05-18T03:54:26Z","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":"db992577af5cd54b86e1ad781db25255bf41ea89254afae42382635acf3e3062","cross_cats_sorted":["math.CO","math.IT","math.NT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2008-12-31T04:30:15Z","title_canon_sha256":"fe95bebb3a223657221fbcc1b09a8be7bcb3ce84782c59daca0963dde19226d1"},"schema_version":"1.0","source":{"id":"0901.0055","kind":"arxiv","version":3}},"canonical_sha256":"bcac5c4d28492ecdf233fcfcf459b003989932104d1dac5ed796a3e06f18cc7d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bcac5c4d28492ecdf233fcfcf459b003989932104d1dac5ed796a3e06f18cc7d","first_computed_at":"2026-05-18T03:54:26.482984Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:54:26.482984Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IOCUrKN+EeatSCn3E2VoPwXIAiy1WyGhUaIfdmrohqI1DOy4rjz67Db9o61U0ZGUPKfxLvFbfYxzHR1bu4V9BA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:54:26.483668Z","signed_message":"canonical_sha256_bytes"},"source_id":"0901.0055","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3d06d2ad18ab7fdab5ff42951972781fe480e58b3c64bea4142437ab1a8fe471","sha256:fca54c1089b34fe1472ce6ab8e73b625ec6648bf488dfb012f610c2b747e49fe"],"state_sha256":"7d5d9675c30586d55eae0a553ff1ecac070c66dc182b09d27ef9f16254b9956a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"liASJqs32rIHOsneWA+FFodhg9tKM3dLClqRs239idqe4jM0fvkd5uD5WC7TwDmcCyt07D50O4Yz4YchV4GbAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T18:27:41.210943Z","bundle_sha256":"0f3b831d5fbbb14f002c0dd2c38ef8dd2cbb93355fbf089bb33fd0c7f7291d0f"}}