{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:3R6B3RSQBXBCGPN4GENUFASHHG","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":"d4b04c25988899c5a3e5ca3f0d7ab9529e57290e7761d0d7a480f4c3a70b7540","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-06-21T07:15:03Z","title_canon_sha256":"669a8a5c5e1dfe47d0e7b49ff6aa44e1e9f0ed8688bcdd0c85194ed99391e95d"},"schema_version":"1.0","source":{"id":"1506.06325","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.06325","created_at":"2026-05-18T01:40:57Z"},{"alias_kind":"arxiv_version","alias_value":"1506.06325v2","created_at":"2026-05-18T01:40:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.06325","created_at":"2026-05-18T01:40:57Z"},{"alias_kind":"pith_short_12","alias_value":"3R6B3RSQBXBC","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"3R6B3RSQBXBCGPN4","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"3R6B3RSQ","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:b242cc05faf07481771b8e5c612a84c1eb2550cd8e3da109645184dc702f6612","target":"graph","created_at":"2026-05-18T01:40:57Z","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":"In 1994, Talagrand showed a generalization of the celebrated KKL theorem. In this work, we prove that the converse of this generalization also holds. Namely, for any sequence of numbers $0<a_1,a_2,\\ldots,a_n\\le 1$ such that $\\sum_{j=1}^n a_j/(1-\\log a_j)\\ge C$ for some constant $C>0$, it is possible to find a roughly balanced Boolean function $f$ such that $\\textrm{Inf}_j[f] < a_j$ for every $1 \\le j \\le n$.","authors_text":"Amit Levi, Clara Shikhelman, Muli Safra, Saleet Klein, Yinon Spinka","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-06-21T07:15:03Z","title":"On the Converse of Talagrand's Influence Inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06325","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":"sha256:6d7fd6a1b79f9f8ab61444151ee126281dff746bb816d764dbd40eff63d26b61","target":"record","created_at":"2026-05-18T01:40:57Z","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":"d4b04c25988899c5a3e5ca3f0d7ab9529e57290e7761d0d7a480f4c3a70b7540","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-06-21T07:15:03Z","title_canon_sha256":"669a8a5c5e1dfe47d0e7b49ff6aa44e1e9f0ed8688bcdd0c85194ed99391e95d"},"schema_version":"1.0","source":{"id":"1506.06325","kind":"arxiv","version":2}},"canonical_sha256":"dc7c1dc6500dc2233dbc311b42824739b0d851be92d70c8a3c00e8c3d2d540a3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dc7c1dc6500dc2233dbc311b42824739b0d851be92d70c8a3c00e8c3d2d540a3","first_computed_at":"2026-05-18T01:40:57.821558Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:40:57.821558Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MTV9+KdhIEw/dV+e2jLehvB4su2n0uOH55XYI0N8CdMduYvzBrJBtWYPF3TDR6tsfd+avrLqGmrMPmoiWaaXDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:40:57.822286Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.06325","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d7fd6a1b79f9f8ab61444151ee126281dff746bb816d764dbd40eff63d26b61","sha256:b242cc05faf07481771b8e5c612a84c1eb2550cd8e3da109645184dc702f6612"],"state_sha256":"67f30d4547d59106a141971000a1f5a5df06f6e4975794888a5167030ca9c1c8"}