{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:7NCA5PNALLC3A3UDEWTFOVB6SV","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":"1df99be37165666b259b1545e04027b3d208aa47adf36665ee326a9cc85117c9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-05-12T10:38:19Z","title_canon_sha256":"210ca27e684079981e8bc2db92584bdca6bcc1c7114f45465c9b1379beb6b77c"},"schema_version":"1.0","source":{"id":"1405.2700","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.2700","created_at":"2026-05-18T02:52:04Z"},{"alias_kind":"arxiv_version","alias_value":"1405.2700v1","created_at":"2026-05-18T02:52:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.2700","created_at":"2026-05-18T02:52:04Z"},{"alias_kind":"pith_short_12","alias_value":"7NCA5PNALLC3","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"7NCA5PNALLC3A3UD","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"7NCA5PNA","created_at":"2026-05-18T12:28:19Z"}],"graph_snapshots":[{"event_id":"sha256:afd5a0bb184e85c43031e8c2d140e56a77ed4cb7ab57db8568de4886f6ad345f","target":"graph","created_at":"2026-05-18T02:52:04Z","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":"Let $\\mathcal{W}$ be the set of strongly real elements of $W$, a Coxeter group. Then for $w \\in \\mathcal{W}$, $e(w)$, the excess of $w$, is defined by $e(w) = \\min\\{\\ell(x) + \\ell(y) - \\ell(w) \\; | \\; w=xy, x^2 = y^2 = 1\\}$. When $W$ is finite we may also define $E(w)$, the reflection excess of $w$. The main result established here is that if $W$ is finite and $X$ is a $W$-conjugacy class, then there exists $w \\in X$ such that $w$ has minimal length in $X$ and $e(w) = 0 = E(w)$.","authors_text":"Peter J. Rowley, Sarah B. Hart","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-05-12T10:38:19Z","title":"Zero Excess and Minimal Length in Finite Coxeter Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.2700","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":"sha256:7ee7b46d6191026e7d14f9b175daec34a5cd291327fe896d265637a76c2d4f92","target":"record","created_at":"2026-05-18T02:52:04Z","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":"1df99be37165666b259b1545e04027b3d208aa47adf36665ee326a9cc85117c9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-05-12T10:38:19Z","title_canon_sha256":"210ca27e684079981e8bc2db92584bdca6bcc1c7114f45465c9b1379beb6b77c"},"schema_version":"1.0","source":{"id":"1405.2700","kind":"arxiv","version":1}},"canonical_sha256":"fb440ebda05ac5b06e8325a657543e955638192f9eb59adb7eb22cd40993b037","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb440ebda05ac5b06e8325a657543e955638192f9eb59adb7eb22cd40993b037","first_computed_at":"2026-05-18T02:52:04.931105Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:52:04.931105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YTRC20JXjF3gLBNX1E8xi3aJ+dTM62bgiP7I9doZC49TAqwqGTRtAEmsp7UfWMOFXun3gIFHV5RnsjHB0tCKDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:52:04.931600Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.2700","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ee7b46d6191026e7d14f9b175daec34a5cd291327fe896d265637a76c2d4f92","sha256:afd5a0bb184e85c43031e8c2d140e56a77ed4cb7ab57db8568de4886f6ad345f"],"state_sha256":"0e6ce589dfd6df016caeaddc65c0df350d49c4f6c6c3f0068b517eda5b2d89d7"}