{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:YFR32NW6PJKCD7F75FKQR5VLCJ","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":"a05b2db8681157790f948df2f5a2fb755939bb9172187477269b966591869894","cross_cats_sorted":["math.CO","math.IT","math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-07-31T05:43:56Z","title_canon_sha256":"bf81d01be2e0baf1490c057e969cdbbd31633fde2c42b4954ee9225ea737e2cb"},"schema_version":"1.0","source":{"id":"2607.29042","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.29042","created_at":"2026-08-03T01:18:10Z"},{"alias_kind":"arxiv_version","alias_value":"2607.29042v1","created_at":"2026-08-03T01:18:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.29042","created_at":"2026-08-03T01:18:10Z"},{"alias_kind":"pith_short_12","alias_value":"YFR32NW6PJKC","created_at":"2026-08-03T01:18:10Z"},{"alias_kind":"pith_short_16","alias_value":"YFR32NW6PJKCD7F7","created_at":"2026-08-03T01:18:10Z"},{"alias_kind":"pith_short_8","alias_value":"YFR32NW6","created_at":"2026-08-03T01:18:10Z"}],"graph_snapshots":[{"event_id":"sha256:5dd10dd725188bd393acf6e88b015a495099789886713b45061553d2e5216908","target":"graph","created_at":"2026-08-03T01:18:10Z","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/2607.29042/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $X,X'$ be independent and identically distributed discrete real-valued random variables of finite Shannon entropy, and write $H(X)$ for the Shannon entropy of $X$. We prove that \\[\n  \\max\\{H(X+X'),\\,H(XX')\\} \\ge \\frac87 H(X)-O(\\log H(X)). \\] This is the entropic analog of the celebrated sum-product phenomenon, and answers a question of Goh, which simply asked for a coefficient strictly larger than 1. An example by the author, Gavalakis, and Kontoyiannis showed the coefficient cannot exceed $\\frac43$. Previous work by Gavalakis, Goh, and Kontoyiannis was able to prove a result of a weaker f","authors_text":"Rupert Li","cross_cats":["math.CO","math.IT","math.PR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-07-31T05:43:56Z","title":"The Entropic Sum-Product Phenomenon"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.29042","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":"sha256:8d86956b9b6c45ade08397a62854a85396a6a621c25da94b65ea8083fcf454c5","target":"record","created_at":"2026-08-03T01:18:10Z","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":"a05b2db8681157790f948df2f5a2fb755939bb9172187477269b966591869894","cross_cats_sorted":["math.CO","math.IT","math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.IT","submitted_at":"2026-07-31T05:43:56Z","title_canon_sha256":"bf81d01be2e0baf1490c057e969cdbbd31633fde2c42b4954ee9225ea737e2cb"},"schema_version":"1.0","source":{"id":"2607.29042","kind":"arxiv","version":1}},"canonical_sha256":"c163bd36de7a5421fcbfe95508f6ab125516a6a772e420642650d6c9f2d3e492","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c163bd36de7a5421fcbfe95508f6ab125516a6a772e420642650d6c9f2d3e492","first_computed_at":"2026-08-03T01:18:10.455508Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-03T01:18:10.455508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"e/hpbL7OvB7ZvPHal7sUSLoOv0S9dFO3vpXRE9Cqik2Hv5rqATbUaF93NIU7l4cLAQ17f9WWnJJRb58Kg+LyAA==","signature_status":"signed_v1","signed_at":"2026-08-03T01:18:10.457037Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.29042","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8d86956b9b6c45ade08397a62854a85396a6a621c25da94b65ea8083fcf454c5","sha256:5dd10dd725188bd393acf6e88b015a495099789886713b45061553d2e5216908"],"state_sha256":"aa1b46519cb35e7deeb9f6b15d82efb58bdd80d3d4a1d19397c261a76b12ed21"}