{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:WPUNJPZUJRDDQOIVF7SVQ6G44F","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":"700d1e2f81c523449e4702a41b5bbc0e5d3ec6e35d15c50fb2389597be7109bd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-07-06T05:31:24Z","title_canon_sha256":"2a31ffd73142f4275e5a44841e07a52a50557a7262c802b75c9ca2b0907ba662"},"schema_version":"1.0","source":{"id":"2607.05466","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.05466","created_at":"2026-07-08T01:18:10Z"},{"alias_kind":"arxiv_version","alias_value":"2607.05466v1","created_at":"2026-07-08T01:18:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.05466","created_at":"2026-07-08T01:18:10Z"},{"alias_kind":"pith_short_12","alias_value":"WPUNJPZUJRDD","created_at":"2026-07-08T01:18:10Z"},{"alias_kind":"pith_short_16","alias_value":"WPUNJPZUJRDDQOIV","created_at":"2026-07-08T01:18:10Z"},{"alias_kind":"pith_short_8","alias_value":"WPUNJPZU","created_at":"2026-07-08T01:18:10Z"}],"graph_snapshots":[{"event_id":"sha256:a88717d5cad0cb676637f3f4c5992f97cf8b78474e32ee1664f8796ecfe4c806","target":"graph","created_at":"2026-07-08T01:18:10Z","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/2607.05466/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The reflection subgroups of a reflection group have a natural lattice structure given by the reflections that they contain. By considering the conjugation action orbits of the reflection subgroups for a given root line, we are able to give an essentially combinatorial way to calculate the lattice of all the normal reflection subgroups of a given (finite irreducible) reflection group, and natural generators for them. Moreover, we observe that every complex reflection group is a normal subgroup of the unique maximal reflection group which shares its collineation group. Hence, we are able to pres","authors_text":"Shayne Waldron","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-07-06T05:31:24Z","title":"The lattice of normal reflection subgroups of an irreducible reflection group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05466","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":"sha256:5a8baa4f239eca9cf184871573619f098d7b2c1aff58cbf6e3fe8ee0568e8919","target":"record","created_at":"2026-07-08T01:18:10Z","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":"700d1e2f81c523449e4702a41b5bbc0e5d3ec6e35d15c50fb2389597be7109bd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-07-06T05:31:24Z","title_canon_sha256":"2a31ffd73142f4275e5a44841e07a52a50557a7262c802b75c9ca2b0907ba662"},"schema_version":"1.0","source":{"id":"2607.05466","kind":"arxiv","version":1}},"canonical_sha256":"b3e8d4bf344c463839152fe55878dce14bff964c6ddb0c30667118afa8524035","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b3e8d4bf344c463839152fe55878dce14bff964c6ddb0c30667118afa8524035","first_computed_at":"2026-07-08T01:18:10.776399Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-08T01:18:10.776399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"v+BerQ/Uh6ofr9Gbd5YYHhaib92CEKeqRo2TH968ZPEeG6KOeZ/II1gaNXm2GwypjUPE07RlSyxLv0O93xdKCA==","signature_status":"signed_v1","signed_at":"2026-07-08T01:18:10.776843Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.05466","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5a8baa4f239eca9cf184871573619f098d7b2c1aff58cbf6e3fe8ee0568e8919","sha256:a88717d5cad0cb676637f3f4c5992f97cf8b78474e32ee1664f8796ecfe4c806"],"state_sha256":"9e6ad4f55a094e59038fbd1daec2002fdd203f0fd3546c77494880743f6ed0c0"}