{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:CU5NECIQHGAEFAAY467INAEONV","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":"62a8502f4bd3625cae82f4b7a4d9cfcd5ec19ad898d1a71649e9e8bea7a61343","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2023-07-28T08:03:58Z","title_canon_sha256":"43ab7a613a8b339096fdabe114bd5598791154bab45b489d5fc8f24f0d9db9fb"},"schema_version":"1.0","source":{"id":"2307.15380","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2307.15380","created_at":"2026-07-05T07:27:07Z"},{"alias_kind":"arxiv_version","alias_value":"2307.15380v3","created_at":"2026-07-05T07:27:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.15380","created_at":"2026-07-05T07:27:07Z"},{"alias_kind":"pith_short_12","alias_value":"CU5NECIQHGAE","created_at":"2026-07-05T07:27:07Z"},{"alias_kind":"pith_short_16","alias_value":"CU5NECIQHGAEFAAY","created_at":"2026-07-05T07:27:07Z"},{"alias_kind":"pith_short_8","alias_value":"CU5NECIQ","created_at":"2026-07-05T07:27:07Z"}],"graph_snapshots":[{"event_id":"sha256:b06613b97971b2056af782d3dfa6ec33273760ebce87da47929b137b91c0524a","target":"graph","created_at":"2026-07-05T07:27:07Z","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/2307.15380/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A joint of a set of lines $\\mathcal{L}$ in $\\mathbb{F}^d$ is a point that is contained in $d$ lines with linearly independent directions. The joints problem asks for the maximum number of joints that are formed by $L$ lines. Guth and Katz showed that the number of joints is at most $O(L^{3/2})$ in $\\mathbb{R}^3$ using polynomial method. This upper bound is met by the construction given by taking the joints and the lines to be all the $d$-wise intersections and all the $(d-1)$-wise intersections of $M$ hyperplanes in general position. Furthermore, this construction is conjectured to be optimal.","authors_text":"Hung-Hsun Hans Yu, Ting-Wei Chao","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2023-07-28T08:03:58Z","title":"Tight Bound and Structural Theorem for Joints"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.15380","kind":"arxiv","version":3},"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":"sha256:64b8800d872e0ba868fb75c406f597f7bd9fe239ffa75c28ed6ee7e31598655d","target":"record","created_at":"2026-07-05T07:27:07Z","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":"62a8502f4bd3625cae82f4b7a4d9cfcd5ec19ad898d1a71649e9e8bea7a61343","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2023-07-28T08:03:58Z","title_canon_sha256":"43ab7a613a8b339096fdabe114bd5598791154bab45b489d5fc8f24f0d9db9fb"},"schema_version":"1.0","source":{"id":"2307.15380","kind":"arxiv","version":3}},"canonical_sha256":"153ad209103980428018e7be86808e6d44158cca777d8ac833b984e46206217e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"153ad209103980428018e7be86808e6d44158cca777d8ac833b984e46206217e","first_computed_at":"2026-07-05T07:27:07.301469Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:27:07.301469Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"86bbWwri3cLr24AtG3lIVVZITKZC9zqYP2uFFxoTHyft8uzWyV5rcGgqdhG7x3ixj3aXBUZyyxHwce884bVMCg==","signature_status":"signed_v1","signed_at":"2026-07-05T07:27:07.302009Z","signed_message":"canonical_sha256_bytes"},"source_id":"2307.15380","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:64b8800d872e0ba868fb75c406f597f7bd9fe239ffa75c28ed6ee7e31598655d","sha256:b06613b97971b2056af782d3dfa6ec33273760ebce87da47929b137b91c0524a"],"state_sha256":"5a5e731b99509153d61a50ee98db105da2f3fed07325d02830d097fc3cba2d0d"}