{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:CHS5Y4KC245KMP3FVFEU7YD4AZ","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":"ca2540a9617dfbfeb844de386f550e43edd95efb7bda4e3cd849c35154ccb697","cross_cats_sorted":["math.MG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-27T16:33:54Z","title_canon_sha256":"34d6fe2db1a1ea6ea1fdc43444fe8ac5fb039ab5dd5e4e35e9deeb2981faa36d"},"schema_version":"1.0","source":{"id":"2605.28709","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.28709","created_at":"2026-05-28T02:05:00Z"},{"alias_kind":"arxiv_version","alias_value":"2605.28709v1","created_at":"2026-05-28T02:05:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.28709","created_at":"2026-05-28T02:05:00Z"},{"alias_kind":"pith_short_12","alias_value":"CHS5Y4KC245K","created_at":"2026-05-28T02:05:00Z"},{"alias_kind":"pith_short_16","alias_value":"CHS5Y4KC245KMP3F","created_at":"2026-05-28T02:05:00Z"},{"alias_kind":"pith_short_8","alias_value":"CHS5Y4KC","created_at":"2026-05-28T02:05:00Z"}],"graph_snapshots":[{"event_id":"sha256:9f6a21dc9337be7aa74229e3019f049dc5ba7da7fb9c17a21f5b32240bf29340","target":"graph","created_at":"2026-05-28T02:05:00Z","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/2605.28709/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In 1974, Witsenhausen asked for the maximum possible density $\\alpha_n$ of a measurable subset $A$ of the unit sphere $\\mathbb{S}^{n-1}\\subset \\mathbb{R}^n$ such that $A$ contains no pair of orthogonal vectors. For $n=3$, the best known lower bound is $1 - 1/\\sqrt{2} = 0.29289\\dots$, obtained from the natural \"double cap\" construction of two opposite spherical caps, which is conjectured to be optimal for all $n$ by Gil Kalai. In this paper, we use a novel approach to establish an upper bound of $\\alpha_3\\le 0.2953$, improving the previous best known bound $0.2977$ due to Bekker et al. (2025). ","authors_text":"\\'Akos D\\'ucz, D\\'aniel Varga, Domonkos Czifra, M\\'at\\'e Matolcsi, P\\'al Zs\\'amboki","cross_cats":["math.MG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-27T16:33:54Z","title":"Improved bounds for the double cap conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28709","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:bada0af21a817312bbf592d9369e687d848846c59136a8bc00e7b0818bd5353b","target":"record","created_at":"2026-05-28T02:05:00Z","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":"ca2540a9617dfbfeb844de386f550e43edd95efb7bda4e3cd849c35154ccb697","cross_cats_sorted":["math.MG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-27T16:33:54Z","title_canon_sha256":"34d6fe2db1a1ea6ea1fdc43444fe8ac5fb039ab5dd5e4e35e9deeb2981faa36d"},"schema_version":"1.0","source":{"id":"2605.28709","kind":"arxiv","version":1}},"canonical_sha256":"11e5dc7142d73aa63f65a9494fe07c06418a83033d5df04d28ba76357d3ffa99","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"11e5dc7142d73aa63f65a9494fe07c06418a83033d5df04d28ba76357d3ffa99","first_computed_at":"2026-05-28T02:05:00.678420Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-28T02:05:00.678420Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LDWFiCVCU1Ly3uffE1qgDP3Pqs0BVu3JFDAjZb/DHVTpA7Kk1s+3H8uEBjjcNWIH311QqXUtbBVsF5gozHKvAw==","signature_status":"signed_v1","signed_at":"2026-05-28T02:05:00.678839Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.28709","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bada0af21a817312bbf592d9369e687d848846c59136a8bc00e7b0818bd5353b","sha256:9f6a21dc9337be7aa74229e3019f049dc5ba7da7fb9c17a21f5b32240bf29340"],"state_sha256":"a2a0aa7f86ce369e4474c2473effe7a69314ec07b8eb5fc099cf122b07cac6bc"}