{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:7ZRAUB54HWY6QW3OYZ5KJ47ZMY","short_pith_number":"pith:7ZRAUB54","canonical_record":{"source":{"id":"1111.2570","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-10T19:30:41Z","cross_cats_sorted":[],"title_canon_sha256":"351b0e8d93b474e346411af5c6d876f49feade80b8bdf0e9968d574e1e896557","abstract_canon_sha256":"4a079d074f8ff67bbc5752441f0b34bd5c1509dc6590f9ff34f414a53ce4f107"},"schema_version":"1.0"},"canonical_sha256":"fe620a07bc3db1e85b6ec67aa4f3f9660fe6e7ae4dec277adede51979ac3156a","source":{"kind":"arxiv","id":"1111.2570","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2570","created_at":"2026-05-18T04:04:44Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2570v2","created_at":"2026-05-18T04:04:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2570","created_at":"2026-05-18T04:04:44Z"},{"alias_kind":"pith_short_12","alias_value":"7ZRAUB54HWY6","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7ZRAUB54HWY6QW3O","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7ZRAUB54","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:7ZRAUB54HWY6QW3OYZ5KJ47ZMY","target":"record","payload":{"canonical_record":{"source":{"id":"1111.2570","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-10T19:30:41Z","cross_cats_sorted":[],"title_canon_sha256":"351b0e8d93b474e346411af5c6d876f49feade80b8bdf0e9968d574e1e896557","abstract_canon_sha256":"4a079d074f8ff67bbc5752441f0b34bd5c1509dc6590f9ff34f414a53ce4f107"},"schema_version":"1.0"},"canonical_sha256":"fe620a07bc3db1e85b6ec67aa4f3f9660fe6e7ae4dec277adede51979ac3156a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:04:44.608099Z","signature_b64":"Xha6NOQtHYzh1Yw7k4mXYdmZI+XQZTYa3LNYvI3T6fRcd8FXBc5hy4sVQ8IO9FklG0rwb63PDE50a/fKIx2/Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe620a07bc3db1e85b6ec67aa4f3f9660fe6e7ae4dec277adede51979ac3156a","last_reissued_at":"2026-05-18T04:04:44.607608Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:04:44.607608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.2570","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:04:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Sf6cfkDekJl+Z0Ym8HqbADHRhP9qqX3mn2xJqbzFWN7WF6O2AEhNkJph7cMwoGyI698DC9lje0bcmqlbY6A7DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T11:23:44.184452Z"},"content_sha256":"10279e7142b4d8afbce42e81460da268610bf0299ba99dffed04ca4ade6c2b70","schema_version":"1.0","event_id":"sha256:10279e7142b4d8afbce42e81460da268610bf0299ba99dffed04ca4ade6c2b70"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:7ZRAUB54HWY6QW3OYZ5KJ47ZMY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On groups with Cayley graph isomorphic to a cube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Colin Hagemeyer, Richard Scott","submitted_at":"2011-11-10T19:30:41Z","abstract_excerpt":"We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is simply-transitive on the vertices and the edge stabilizers are all nontrivial. The action on the cube extends to an orthogonal linear action, which we call the geometric representation. We prove a combinatorial decomposition for cube groups into products of 2-element subgroup, and show that the geometric representation is always reducible."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2570","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:04:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IqdzmLnnHKAoZhV3yQk5eBHmuGVp7giMSkKn1bZ18eqN/ZY5QaGHZIKkhzmTOCSvBZDLzPp/gB7aqA+GhORwDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T11:23:44.184802Z"},"content_sha256":"e8706fbaa036c3a603c1a8d8f339d080943cad76c912373052f2ef7a580fa20b","schema_version":"1.0","event_id":"sha256:e8706fbaa036c3a603c1a8d8f339d080943cad76c912373052f2ef7a580fa20b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7ZRAUB54HWY6QW3OYZ5KJ47ZMY/bundle.json","state_url":"https://pith.science/pith/7ZRAUB54HWY6QW3OYZ5KJ47ZMY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7ZRAUB54HWY6QW3OYZ5KJ47ZMY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T11:23:44Z","links":{"resolver":"https://pith.science/pith/7ZRAUB54HWY6QW3OYZ5KJ47ZMY","bundle":"https://pith.science/pith/7ZRAUB54HWY6QW3OYZ5KJ47ZMY/bundle.json","state":"https://pith.science/pith/7ZRAUB54HWY6QW3OYZ5KJ47ZMY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7ZRAUB54HWY6QW3OYZ5KJ47ZMY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7ZRAUB54HWY6QW3OYZ5KJ47ZMY","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":"4a079d074f8ff67bbc5752441f0b34bd5c1509dc6590f9ff34f414a53ce4f107","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-10T19:30:41Z","title_canon_sha256":"351b0e8d93b474e346411af5c6d876f49feade80b8bdf0e9968d574e1e896557"},"schema_version":"1.0","source":{"id":"1111.2570","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2570","created_at":"2026-05-18T04:04:44Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2570v2","created_at":"2026-05-18T04:04:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2570","created_at":"2026-05-18T04:04:44Z"},{"alias_kind":"pith_short_12","alias_value":"7ZRAUB54HWY6","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7ZRAUB54HWY6QW3O","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7ZRAUB54","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:e8706fbaa036c3a603c1a8d8f339d080943cad76c912373052f2ef7a580fa20b","target":"graph","created_at":"2026-05-18T04:04:44Z","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":"We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is simply-transitive on the vertices and the edge stabilizers are all nontrivial. The action on the cube extends to an orthogonal linear action, which we call the geometric representation. We prove a combinatorial decomposition for cube groups into products of 2-element subgroup, and show that the geometric representation is always reducible.","authors_text":"Colin Hagemeyer, Richard Scott","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-10T19:30:41Z","title":"On groups with Cayley graph isomorphic to a cube"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2570","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":"sha256:10279e7142b4d8afbce42e81460da268610bf0299ba99dffed04ca4ade6c2b70","target":"record","created_at":"2026-05-18T04:04:44Z","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":"4a079d074f8ff67bbc5752441f0b34bd5c1509dc6590f9ff34f414a53ce4f107","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-10T19:30:41Z","title_canon_sha256":"351b0e8d93b474e346411af5c6d876f49feade80b8bdf0e9968d574e1e896557"},"schema_version":"1.0","source":{"id":"1111.2570","kind":"arxiv","version":2}},"canonical_sha256":"fe620a07bc3db1e85b6ec67aa4f3f9660fe6e7ae4dec277adede51979ac3156a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe620a07bc3db1e85b6ec67aa4f3f9660fe6e7ae4dec277adede51979ac3156a","first_computed_at":"2026-05-18T04:04:44.607608Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:04:44.607608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Xha6NOQtHYzh1Yw7k4mXYdmZI+XQZTYa3LNYvI3T6fRcd8FXBc5hy4sVQ8IO9FklG0rwb63PDE50a/fKIx2/Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:04:44.608099Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.2570","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:10279e7142b4d8afbce42e81460da268610bf0299ba99dffed04ca4ade6c2b70","sha256:e8706fbaa036c3a603c1a8d8f339d080943cad76c912373052f2ef7a580fa20b"],"state_sha256":"6b19034d169da4f58f66820d884ad84a842baa472827136217e869868ae152c9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lT48bZs6ePcf7Y2cPRNjM5lwTXjpWeo+9mmtSgUdERejyE2/TpGcuD1sTUW2eTI0cKkxT+uQrfbGfgetEt5HDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T11:23:44.186733Z","bundle_sha256":"2a4ed666179091a97e74ed463138d66a436cd22773b82a99588d1ad8c35446e1"}}