{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ZGDYTR4WMUJ6HYRGFVXCHIBAXI","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":"a9e580d7663a26c8851f869b609e96d38691440c77af3e6f75ec9150d37cb275","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.OC","submitted_at":"2026-08-07T19:01:13Z","title_canon_sha256":"ae8f269c84253c2e7a2e7962419f6da67d99a2ecd0ead5f30f5011c6d4d5ef82"},"schema_version":"1.0","source":{"id":"2608.07716","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.07716","created_at":"2026-08-11T00:15:46Z"},{"alias_kind":"arxiv_version","alias_value":"2608.07716v1","created_at":"2026-08-11T00:15:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.07716","created_at":"2026-08-11T00:15:46Z"},{"alias_kind":"pith_short_12","alias_value":"ZGDYTR4WMUJ6","created_at":"2026-08-11T00:15:46Z"},{"alias_kind":"pith_short_16","alias_value":"ZGDYTR4WMUJ6HYRG","created_at":"2026-08-11T00:15:46Z"},{"alias_kind":"pith_short_8","alias_value":"ZGDYTR4W","created_at":"2026-08-11T00:15:46Z"}],"graph_snapshots":[{"event_id":"sha256:bb8c25a2c34daa7c61e38284aed93f49cc4cda7fd2a74d9930fdb2f373014234","target":"graph","created_at":"2026-08-11T00:15:46Z","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/2608.07716/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The cosine measure of a set of vectors in $\\mathbb{R}^n$ measures how well the set covers all directions in $\\mathbb{R}^n$. It identifies the direction furthest, in angle, from the set. It is used in the convergence theory of various optimization algorithms, but also highlights interesting geometric properties of sets. For example, the cosine measure of a set $S$ is greater than zero if and only if given any $\\mathcal{C}^1$ function $f$ at a point $\\mathbf{x}$ where the gradient is nonzero, $S$ must contain a descent direction of $f$ at $\\mathbf{x}$. In this paper, we examine the question of w","authors_text":"Chayne Planiden, Gabriel Jarry-Bolduc, Warren Hare","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.OC","submitted_at":"2026-08-07T19:01:13Z","title":"The cosine measure of a function at a point"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.07716","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":"sha256:6ae6210e9d3f4209da7e44eada56487d2515783927a7d5223fa8b92eb09afdb7","target":"record","created_at":"2026-08-11T00:15:46Z","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":"a9e580d7663a26c8851f869b609e96d38691440c77af3e6f75ec9150d37cb275","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.OC","submitted_at":"2026-08-07T19:01:13Z","title_canon_sha256":"ae8f269c84253c2e7a2e7962419f6da67d99a2ecd0ead5f30f5011c6d4d5ef82"},"schema_version":"1.0","source":{"id":"2608.07716","kind":"arxiv","version":1}},"canonical_sha256":"c98789c7966513e3e2262d6e23a020ba134a55cdcccf0fea32d78ddd776ba7ec","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c98789c7966513e3e2262d6e23a020ba134a55cdcccf0fea32d78ddd776ba7ec","first_computed_at":"2026-08-11T00:15:46.168454Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-11T00:15:46.168454Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WXajRD+3eIy2faVFA3cYa0BYP9WudV/pWDgjx84ezZVgZFCcsqkvcsjgU0rNgz3soqHSCZIePpZrQ792U3LVAg==","signature_status":"signed_v1","signed_at":"2026-08-11T00:15:46.170713Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.07716","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6ae6210e9d3f4209da7e44eada56487d2515783927a7d5223fa8b92eb09afdb7","sha256:bb8c25a2c34daa7c61e38284aed93f49cc4cda7fd2a74d9930fdb2f373014234"],"state_sha256":"1152c140bd45fbfe13460ee8d86c3bc3b80987e4f6c590d081d1cce4dbeec2ca"}