{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:NRC42ZMNE4WWZM5UW6W6DWJINS","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":"695638926c7820a4607a43a2b5239146c32dfab0c29a518eb71e2505b1bc500b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-01-15T19:12:02Z","title_canon_sha256":"aaa121a65e39e5c9eac1c8b33f187318ce8a88adbe34a2e157371f7d90d02973"},"schema_version":"1.0","source":{"id":"1401.3708","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.3708","created_at":"2026-05-18T03:02:07Z"},{"alias_kind":"arxiv_version","alias_value":"1401.3708v1","created_at":"2026-05-18T03:02:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.3708","created_at":"2026-05-18T03:02:07Z"},{"alias_kind":"pith_short_12","alias_value":"NRC42ZMNE4WW","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"NRC42ZMNE4WWZM5U","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"NRC42ZMN","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:30f313a96ce3a49c08d22f5cf10ddbb32d43db4aa96098563793deb9ff5ba548","target":"graph","created_at":"2026-05-18T03:02:07Z","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 $\\mathcal G_2$ denote the affine group $GL(2,\\mathbb Z) \\ltimes \\mathbb Z^{2}$. For every point $x=(x_1,x_2) \\in \\R2$ let $\\orb(x)=\\{y\\in\\R2\\mid y=\\gamma(x)$ for some $\\gamma \\in \\mathcal{G}_2 \\}$. Let $G_{x}$ be the subgroup of the additive group $\\mathbb R$ generated by $x_1,x_2, 1$. If $\\rank(G_x)\\in \\{1,3\\}$ then $\\orb(x)=\\{y\\in\\R2\\mid G_y=G_x\\}$. If $\\rank(G_x)=2$, knowledge of $G_x$ is not sufficient in general to uniquely recover $\\orb(x)$: rather, $G_x$ classifies precisely $\\max(1,\\phi(d)/2)$ different orbits, where $d$ is the denominator of the smallest positive nonzero rational ","authors_text":"Daniele Mundici","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-01-15T19:12:02Z","title":"Classifying $GL(2,\\mathbb Z) \\ltimes \\mathbb Z^{2}$-orbits by subgroups of $\\mathbb R$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.3708","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:d6a2ad8bf84ff101b0e7a739c2fdd1193bdbed2158fa0758f11049cf675e01f4","target":"record","created_at":"2026-05-18T03:02:07Z","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":"695638926c7820a4607a43a2b5239146c32dfab0c29a518eb71e2505b1bc500b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-01-15T19:12:02Z","title_canon_sha256":"aaa121a65e39e5c9eac1c8b33f187318ce8a88adbe34a2e157371f7d90d02973"},"schema_version":"1.0","source":{"id":"1401.3708","kind":"arxiv","version":1}},"canonical_sha256":"6c45cd658d272d6cb3b4b7ade1d9286c8237647550c868bcc4234f467514ca17","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6c45cd658d272d6cb3b4b7ade1d9286c8237647550c868bcc4234f467514ca17","first_computed_at":"2026-05-18T03:02:07.893375Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:02:07.893375Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8QQEo0dSdo2Fnu5obdfoOz8Y00pqCyf0MUWF3qYOx+W4/XXt9kTS3qDUNzvmVAAMFz+SGYWG7uZIOAl1uoeCDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:02:07.894091Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.3708","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d6a2ad8bf84ff101b0e7a739c2fdd1193bdbed2158fa0758f11049cf675e01f4","sha256:30f313a96ce3a49c08d22f5cf10ddbb32d43db4aa96098563793deb9ff5ba548"],"state_sha256":"9bd41b31c1660af7f67ab9580caccea4a3926b299f674e1bd81330cea02d777f"}