{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:NRC42ZMNE4WWZM5UW6W6DWJINS","short_pith_number":"pith:NRC42ZMN","schema_version":"1.0","canonical_sha256":"6c45cd658d272d6cb3b4b7ade1d9286c8237647550c868bcc4234f467514ca17","source":{"kind":"arxiv","id":"1401.3708","version":1},"attestation_state":"computed","paper":{"title":"Classifying $GL(2,\\mathbb Z) \\ltimes \\mathbb Z^{2}$-orbits by subgroups of $\\mathbb R$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Daniele Mundici","submitted_at":"2014-01-15T19:12:02Z","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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1401.3708","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-01-15T19:12:02Z","cross_cats_sorted":[],"title_canon_sha256":"aaa121a65e39e5c9eac1c8b33f187318ce8a88adbe34a2e157371f7d90d02973","abstract_canon_sha256":"695638926c7820a4607a43a2b5239146c32dfab0c29a518eb71e2505b1bc500b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:07.894091Z","signature_b64":"8QQEo0dSdo2Fnu5obdfoOz8Y00pqCyf0MUWF3qYOx+W4/XXt9kTS3qDUNzvmVAAMFz+SGYWG7uZIOAl1uoeCDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c45cd658d272d6cb3b4b7ade1d9286c8237647550c868bcc4234f467514ca17","last_reissued_at":"2026-05-18T03:02:07.893375Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:07.893375Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classifying $GL(2,\\mathbb Z) \\ltimes \\mathbb Z^{2}$-orbits by subgroups of $\\mathbb R$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Daniele Mundici","submitted_at":"2014-01-15T19:12:02Z","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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.3708","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1401.3708","created_at":"2026-05-18T03:02:07.893492+00:00"},{"alias_kind":"arxiv_version","alias_value":"1401.3708v1","created_at":"2026-05-18T03:02:07.893492+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.3708","created_at":"2026-05-18T03:02:07.893492+00:00"},{"alias_kind":"pith_short_12","alias_value":"NRC42ZMNE4WW","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_16","alias_value":"NRC42ZMNE4WWZM5U","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_8","alias_value":"NRC42ZMN","created_at":"2026-05-18T12:28:41.024544+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NRC42ZMNE4WWZM5UW6W6DWJINS","json":"https://pith.science/pith/NRC42ZMNE4WWZM5UW6W6DWJINS.json","graph_json":"https://pith.science/api/pith-number/NRC42ZMNE4WWZM5UW6W6DWJINS/graph.json","events_json":"https://pith.science/api/pith-number/NRC42ZMNE4WWZM5UW6W6DWJINS/events.json","paper":"https://pith.science/paper/NRC42ZMN"},"agent_actions":{"view_html":"https://pith.science/pith/NRC42ZMNE4WWZM5UW6W6DWJINS","download_json":"https://pith.science/pith/NRC42ZMNE4WWZM5UW6W6DWJINS.json","view_paper":"https://pith.science/paper/NRC42ZMN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1401.3708&json=true","fetch_graph":"https://pith.science/api/pith-number/NRC42ZMNE4WWZM5UW6W6DWJINS/graph.json","fetch_events":"https://pith.science/api/pith-number/NRC42ZMNE4WWZM5UW6W6DWJINS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NRC42ZMNE4WWZM5UW6W6DWJINS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NRC42ZMNE4WWZM5UW6W6DWJINS/action/storage_attestation","attest_author":"https://pith.science/pith/NRC42ZMNE4WWZM5UW6W6DWJINS/action/author_attestation","sign_citation":"https://pith.science/pith/NRC42ZMNE4WWZM5UW6W6DWJINS/action/citation_signature","submit_replication":"https://pith.science/pith/NRC42ZMNE4WWZM5UW6W6DWJINS/action/replication_record"}},"created_at":"2026-05-18T03:02:07.893492+00:00","updated_at":"2026-05-18T03:02:07.893492+00:00"}