{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:UULPBTSKZWVBGEABME37XSNR3V","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":"88a24e29fd367873e62ed59b083835900db3b48eaed286808ca8466b107ed979","cross_cats_sorted":["math.AG","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2020-11-28T23:23:20Z","title_canon_sha256":"25c36b2ea2fb2d52b85ef7a60905de0fea2f218e358a7b97967109aa8aac76b1"},"schema_version":"1.0","source":{"id":"2011.14232","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.14232","created_at":"2026-07-05T01:55:18Z"},{"alias_kind":"arxiv_version","alias_value":"2011.14232v1","created_at":"2026-07-05T01:55:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.14232","created_at":"2026-07-05T01:55:18Z"},{"alias_kind":"pith_short_12","alias_value":"UULPBTSKZWVB","created_at":"2026-07-05T01:55:18Z"},{"alias_kind":"pith_short_16","alias_value":"UULPBTSKZWVBGEAB","created_at":"2026-07-05T01:55:18Z"},{"alias_kind":"pith_short_8","alias_value":"UULPBTSK","created_at":"2026-07-05T01:55:18Z"}],"graph_snapshots":[{"event_id":"sha256:3a067e51682bf9638ad835494088f4784223dc2fbc4815647fca8a0f62f74ea9","target":"graph","created_at":"2026-07-05T01:55:18Z","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/2011.14232/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We classify all sets of nonzero vectors in $\\mathbb{R}^3$ such that the angle formed by each pair is a rational multiple of $\\pi$. The special case of four-element subsets lets us classify all tetrahedra whose dihedral angles are multiples of $\\pi$, solving a 1976 problem of Conway and Jones: there are $2$ one-parameter families and $59$ sporadic tetrahedra, all but three of which are related to either the icosidodecahedron or the $B_3$ root lattice. The proof requires the solution in roots of unity of a $W(D_6)$-symmetric polynomial equation with $105$ monomials (the previous record was $12$ ","authors_text":"Alexander Kolpakov, Bjorn Poonen, Kiran S. Kedlaya, Michael Rubinstein","cross_cats":["math.AG","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2020-11-28T23:23:20Z","title":"Space vectors forming rational angles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.14232","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":"sha256:88e4146bfe1ff562cee0ff998e23149e7ff15fb62bc6dc9a670a10b2164026ee","target":"record","created_at":"2026-07-05T01:55:18Z","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":"88a24e29fd367873e62ed59b083835900db3b48eaed286808ca8466b107ed979","cross_cats_sorted":["math.AG","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2020-11-28T23:23:20Z","title_canon_sha256":"25c36b2ea2fb2d52b85ef7a60905de0fea2f218e358a7b97967109aa8aac76b1"},"schema_version":"1.0","source":{"id":"2011.14232","kind":"arxiv","version":1}},"canonical_sha256":"a516f0ce4acdaa1310016137fbc9b1dd5684def45d762cd308e378d2618158dc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a516f0ce4acdaa1310016137fbc9b1dd5684def45d762cd308e378d2618158dc","first_computed_at":"2026-07-05T01:55:18.770640Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:55:18.770640Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RzIF2aTJauhW6FUlXwOvbm5Tn2RDktRspqQLKvDhIHlqMOBrNUttcxGn1DBLwJ9RwB5MMSZXdx+UM3wCAjmcCg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:55:18.771049Z","signed_message":"canonical_sha256_bytes"},"source_id":"2011.14232","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88e4146bfe1ff562cee0ff998e23149e7ff15fb62bc6dc9a670a10b2164026ee","sha256:3a067e51682bf9638ad835494088f4784223dc2fbc4815647fca8a0f62f74ea9"],"state_sha256":"f081715f2f4239d2a1dfb8ae53bfa549f8e60d8d095dbc69c9d183fc8014b3dd"}