{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ZKRCLMYKYXPK6NG6G6GIYJHAVJ","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":"9b16baf5d282554665dcc260843c8795c172513b7a6ad1199acfb5e54a299a6c","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-06-01T17:56:17Z","title_canon_sha256":"2a7a4601e63fd3c854dd31baa636bec6d498f4ae82e8c0f6fc4977c877053e07"},"schema_version":"1.0","source":{"id":"2606.02567","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02567","created_at":"2026-06-02T03:05:10Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02567v1","created_at":"2026-06-02T03:05:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02567","created_at":"2026-06-02T03:05:10Z"},{"alias_kind":"pith_short_12","alias_value":"ZKRCLMYKYXPK","created_at":"2026-06-02T03:05:10Z"},{"alias_kind":"pith_short_16","alias_value":"ZKRCLMYKYXPK6NG6","created_at":"2026-06-02T03:05:10Z"},{"alias_kind":"pith_short_8","alias_value":"ZKRCLMYK","created_at":"2026-06-02T03:05:10Z"}],"graph_snapshots":[{"event_id":"sha256:5b78de61a37f706014f4b5716ca8d317d6b6510f546c520c3a81dcb4f6a87b6f","target":"graph","created_at":"2026-06-02T03:05:10Z","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/2606.02567/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that for any set of $n$ unit vectors $v_1,\\ldots,v_n$ in a real Hilbert space and positive numbers $p_1,\\ldots,p_n$ satisfying $\\sum_j p_j = 1$, there exists a unit vector $u$ such that\n  \\[\n  \\sum_{j=1}^n \\frac{p_j^2}{\\langle v_j, u\\rangle^2}\\leq 1.\n  \\]\n  This inequality is a weighted version of the strong polarization inequality. As immediate corollaries, it yields a polarization inequality for products of powers of linear functionals and a strengthening of Bang's classical plank theorem for Hilbert spaces. The proof follows the approach introduced by Mart\\'inez and Ortega-Moreno in","authors_text":"Dami\\'an Pinasco, Daniel Galicer, Oscar Ortega-Moreno","cross_cats":["cs.IT","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-06-01T17:56:17Z","title":"Strong Polarization and Entropy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02567","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":"sha256:22d42789ceb52095b90e3da13eb29db14445a086151e78e194ae5cacaa2851b3","target":"record","created_at":"2026-06-02T03:05:10Z","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":"9b16baf5d282554665dcc260843c8795c172513b7a6ad1199acfb5e54a299a6c","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-06-01T17:56:17Z","title_canon_sha256":"2a7a4601e63fd3c854dd31baa636bec6d498f4ae82e8c0f6fc4977c877053e07"},"schema_version":"1.0","source":{"id":"2606.02567","kind":"arxiv","version":1}},"canonical_sha256":"caa225b30ac5deaf34de378c8c24e0aa4ea944c4d9d21aa25272a152da542f72","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"caa225b30ac5deaf34de378c8c24e0aa4ea944c4d9d21aa25272a152da542f72","first_computed_at":"2026-06-02T03:05:10.326086Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T03:05:10.326086Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"izp0CayA/+DYFvqTKi4OUCOES0Aktp+LcZJeaziRFqqT0ewocWO+qcUI+dXEoBIUp4Hr7E33+0YYWTXVTp5LAg==","signature_status":"signed_v1","signed_at":"2026-06-02T03:05:10.326445Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.02567","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:22d42789ceb52095b90e3da13eb29db14445a086151e78e194ae5cacaa2851b3","sha256:5b78de61a37f706014f4b5716ca8d317d6b6510f546c520c3a81dcb4f6a87b6f"],"state_sha256":"f392bf7d9086e61db11d59780a1528a815039bc85fd68f8c04a58ed9781e1c2f"}