{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:OFA4JWAEUDWOSTGZT4XZXWWLAC","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":"5a38c4ed18cc37ae6c39c57812b8332bdc43c352eba0b473d59ae3edfb627db6","cross_cats_sorted":["hep-th","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-07-31T18:37:08Z","title_canon_sha256":"7c105a22c7735e7a1950a782e4dd0be033f0acc2786f0e368b6c00ad5f4150a4"},"schema_version":"1.0","source":{"id":"1207.7339","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.7339","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"arxiv_version","alias_value":"1207.7339v2","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.7339","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"pith_short_12","alias_value":"OFA4JWAEUDWO","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"OFA4JWAEUDWOSTGZ","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"OFA4JWAE","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:c8423d27f0c030d18eee9f962867b3f4503e1e1131f6e8cc624e61889133b334","target":"graph","created_at":"2026-05-18T01:20:22Z","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"},"paper":{"abstract_excerpt":"In this paper, we show that via a novel construction every rank-3 root system induces a root system of rank 4. Via the Cartan-Dieudonn\\'e theorem, an even number of successive Coxeter reflections yields rotations that in a Clifford algebra framework are described by spinors. In three dimensions these spinors themselves have a natural four-dimensional Euclidean structure, and discrete spinor groups can therefore be interpreted as 4D polytopes. In fact, we show that these polytopes have to be root systems, thereby inducing Coxeter groups of rank 4, and that their automorphism groups include two ","authors_text":"Pierre-Philippe Dechant","cross_cats":["hep-th","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-07-31T18:37:08Z","title":"Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.7339","kind":"arxiv","version":2},"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":"sha256:8d17ed5e7170b60132d949cb6461dbf34452969dddd9afa09df398093eb8657d","target":"record","created_at":"2026-05-18T01:20:22Z","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":"5a38c4ed18cc37ae6c39c57812b8332bdc43c352eba0b473d59ae3edfb627db6","cross_cats_sorted":["hep-th","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-07-31T18:37:08Z","title_canon_sha256":"7c105a22c7735e7a1950a782e4dd0be033f0acc2786f0e368b6c00ad5f4150a4"},"schema_version":"1.0","source":{"id":"1207.7339","kind":"arxiv","version":2}},"canonical_sha256":"7141c4d804a0ece94cd99f2f9bdacb00849c267e42372ae826fc70a5aafce2c8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7141c4d804a0ece94cd99f2f9bdacb00849c267e42372ae826fc70a5aafce2c8","first_computed_at":"2026-05-18T01:20:22.506747Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:22.506747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1Oeq2+z0AK7LPckMVjBx+Ak+RDKwKPzI+Lr6RYSslnhpC16OwV0U7YL5dro1cwa760ScQlUSX+JooB/6G+l2DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:22.507328Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.7339","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8d17ed5e7170b60132d949cb6461dbf34452969dddd9afa09df398093eb8657d","sha256:c8423d27f0c030d18eee9f962867b3f4503e1e1131f6e8cc624e61889133b334"],"state_sha256":"816e03f9aff473f6b49aa13f7a56d9a30089d90d384e65be68b81f3340b9e347"}