{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:2IAG6XMLSUNIGJ7ZGZC54MG4XD","short_pith_number":"pith:2IAG6XML","canonical_record":{"source":{"id":"2605.30331","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2026-05-28T17:57:58Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"b20ae1c747bd206e952635ba1cbf8b43d318158ae0f53a8236bed316997b9b04","abstract_canon_sha256":"41b0265fbd0191918094db9b658e02c9845ff49d5294cb008afe2f40387d8e72"},"schema_version":"1.0"},"canonical_sha256":"d2006f5d8b951a8327f93645de30dcb8e54f580e9b985e8b7a1047c294df0af6","source":{"kind":"arxiv","id":"2605.30331","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.30331","created_at":"2026-05-29T02:06:16Z"},{"alias_kind":"arxiv_version","alias_value":"2605.30331v1","created_at":"2026-05-29T02:06:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.30331","created_at":"2026-05-29T02:06:16Z"},{"alias_kind":"pith_short_12","alias_value":"2IAG6XMLSUNI","created_at":"2026-05-29T02:06:16Z"},{"alias_kind":"pith_short_16","alias_value":"2IAG6XMLSUNIGJ7Z","created_at":"2026-05-29T02:06:16Z"},{"alias_kind":"pith_short_8","alias_value":"2IAG6XML","created_at":"2026-05-29T02:06:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:2IAG6XMLSUNIGJ7ZGZC54MG4XD","target":"record","payload":{"canonical_record":{"source":{"id":"2605.30331","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2026-05-28T17:57:58Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"b20ae1c747bd206e952635ba1cbf8b43d318158ae0f53a8236bed316997b9b04","abstract_canon_sha256":"41b0265fbd0191918094db9b658e02c9845ff49d5294cb008afe2f40387d8e72"},"schema_version":"1.0"},"canonical_sha256":"d2006f5d8b951a8327f93645de30dcb8e54f580e9b985e8b7a1047c294df0af6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T02:06:16.767286Z","signature_b64":"gM89ISu0/vqJckA6oHt+ZG22KLrOVBUKgH8fuiImNIfik00DbTkUOpoomVT5zw39X0zaXLhdMoMxP1MQc1gyAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2006f5d8b951a8327f93645de30dcb8e54f580e9b985e8b7a1047c294df0af6","last_reissued_at":"2026-05-29T02:06:16.766890Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T02:06:16.766890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.30331","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":"2026-05-29T02:06:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ENxmZRlCrFkljJOnIVahY/aEp3F6XFn+55OSaoUJYEPrbOaNx5jeMqkc4z8e0Lf6X5lL4PTFIMdB5xEe5oq+CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T03:44:22.598404Z"},"content_sha256":"b498ad5353f645a1a8181fc6fa5646f94325267b764a6e195d1cff4af967ce43","schema_version":"1.0","event_id":"sha256:b498ad5353f645a1a8181fc6fa5646f94325267b764a6e195d1cff4af967ce43"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:2IAG6XMLSUNIGJ7ZGZC54MG4XD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Majorization precursors to supermodularity and subadditivity on the majorization lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Alexander St\\'evins, Michael G. Jabbour, Nicolas J. Cerf, Serge Deside","submitted_at":"2026-05-28T17:57:58Z","abstract_excerpt":"We establish two structural majorization relations, which we call precursors, underlying the properties of supermodularity and subadditivity on the lattice induced by majorization. These are precursors in that they immediately imply that all sums of concave functions, which we dub sum-concave functions, are supermodular and subadditive on the majorization lattice. Using these majorization relations, we then show the supermodularity and subadditivity (in the lattice-theoretic sense) of Tsallis entropies (for all $\\alpha$) and R\\'enyi entropies (for all $\\alpha > 1$), also recovering these prope"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.30331","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.30331/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-05-29T02:06:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f7nZdHAMMwHYnLLCcts7hvwEL8pZ8tMKyp+CmHNkgFxhCC206+ljB8DsHBm2LT8vlH6yALtvrZEg4XklVK5kDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T03:44:22.598773Z"},"content_sha256":"435944736e6ac8efc24650e5cba8c3120a1d42beff97946589b872fe0962b683","schema_version":"1.0","event_id":"sha256:435944736e6ac8efc24650e5cba8c3120a1d42beff97946589b872fe0962b683"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2IAG6XMLSUNIGJ7ZGZC54MG4XD/bundle.json","state_url":"https://pith.science/pith/2IAG6XMLSUNIGJ7ZGZC54MG4XD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2IAG6XMLSUNIGJ7ZGZC54MG4XD/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":"2026-06-02T03:44:22Z","links":{"resolver":"https://pith.science/pith/2IAG6XMLSUNIGJ7ZGZC54MG4XD","bundle":"https://pith.science/pith/2IAG6XMLSUNIGJ7ZGZC54MG4XD/bundle.json","state":"https://pith.science/pith/2IAG6XMLSUNIGJ7ZGZC54MG4XD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2IAG6XMLSUNIGJ7ZGZC54MG4XD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:2IAG6XMLSUNIGJ7ZGZC54MG4XD","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":"41b0265fbd0191918094db9b658e02c9845ff49d5294cb008afe2f40387d8e72","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2026-05-28T17:57:58Z","title_canon_sha256":"b20ae1c747bd206e952635ba1cbf8b43d318158ae0f53a8236bed316997b9b04"},"schema_version":"1.0","source":{"id":"2605.30331","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.30331","created_at":"2026-05-29T02:06:16Z"},{"alias_kind":"arxiv_version","alias_value":"2605.30331v1","created_at":"2026-05-29T02:06:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.30331","created_at":"2026-05-29T02:06:16Z"},{"alias_kind":"pith_short_12","alias_value":"2IAG6XMLSUNI","created_at":"2026-05-29T02:06:16Z"},{"alias_kind":"pith_short_16","alias_value":"2IAG6XMLSUNIGJ7Z","created_at":"2026-05-29T02:06:16Z"},{"alias_kind":"pith_short_8","alias_value":"2IAG6XML","created_at":"2026-05-29T02:06:16Z"}],"graph_snapshots":[{"event_id":"sha256:435944736e6ac8efc24650e5cba8c3120a1d42beff97946589b872fe0962b683","target":"graph","created_at":"2026-05-29T02:06:16Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.30331/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish two structural majorization relations, which we call precursors, underlying the properties of supermodularity and subadditivity on the lattice induced by majorization. These are precursors in that they immediately imply that all sums of concave functions, which we dub sum-concave functions, are supermodular and subadditive on the majorization lattice. Using these majorization relations, we then show the supermodularity and subadditivity (in the lattice-theoretic sense) of Tsallis entropies (for all $\\alpha$) and R\\'enyi entropies (for all $\\alpha > 1$), also recovering these prope","authors_text":"Alexander St\\'evins, Michael G. Jabbour, Nicolas J. Cerf, Serge Deside","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2026-05-28T17:57:58Z","title":"Majorization precursors to supermodularity and subadditivity on the majorization lattice"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.30331","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:b498ad5353f645a1a8181fc6fa5646f94325267b764a6e195d1cff4af967ce43","target":"record","created_at":"2026-05-29T02:06:16Z","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":"41b0265fbd0191918094db9b658e02c9845ff49d5294cb008afe2f40387d8e72","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2026-05-28T17:57:58Z","title_canon_sha256":"b20ae1c747bd206e952635ba1cbf8b43d318158ae0f53a8236bed316997b9b04"},"schema_version":"1.0","source":{"id":"2605.30331","kind":"arxiv","version":1}},"canonical_sha256":"d2006f5d8b951a8327f93645de30dcb8e54f580e9b985e8b7a1047c294df0af6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d2006f5d8b951a8327f93645de30dcb8e54f580e9b985e8b7a1047c294df0af6","first_computed_at":"2026-05-29T02:06:16.766890Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-29T02:06:16.766890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gM89ISu0/vqJckA6oHt+ZG22KLrOVBUKgH8fuiImNIfik00DbTkUOpoomVT5zw39X0zaXLhdMoMxP1MQc1gyAg==","signature_status":"signed_v1","signed_at":"2026-05-29T02:06:16.767286Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.30331","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b498ad5353f645a1a8181fc6fa5646f94325267b764a6e195d1cff4af967ce43","sha256:435944736e6ac8efc24650e5cba8c3120a1d42beff97946589b872fe0962b683"],"state_sha256":"4d5dbe90901a64fb7d4d16a9309afc8be2eb3e20980b80a7a4bd432970b82aca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LYw3m2GgbgwcE/oePQyDgbS8nBvwnngmDgEIQTSflIs4pOwWc2nF7PGD5/aVsgTle2L29OG3kUfqj4ryul3cBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T03:44:22.600684Z","bundle_sha256":"335bd458175e412e851ba0b07f2a3cca82b599716fc160ca5a52f41244aaf87f"}}