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We also show that if $A$ is a non-singular affine algebra of dimension $d$ over an algebraically closed field $k$ such that $d! A = A$, $d \\equiv 2 \\pmod 4$ and $I$ an ideal of $A$, then $Um_d(A, I) = e_1{Sp}_d(A, I)$. As a consequence it is proved that if $A$ is a non-singular affine algebra of dimension $d$ over an algebraically closed field $k$ such that $(d + 1)!A = A$, $"},"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":"1511.09419","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2015-11-30T18:31:21Z","cross_cats_sorted":[],"title_canon_sha256":"c712e071825a9aed2f8a5e34d2af82ce1089f61a82e5cdb6fe3278c0c92a1ed9","abstract_canon_sha256":"ccacdc8d763ddd4d302853eb997384f1e7593e7ef4773fb7decfab380d11c2db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:40.294486Z","signature_b64":"q5WWY3nKjYACjHAubfNoXUOPsLiiablFc/pFyv/GohDmo+jnwjd19sTMpItSEBQpot0ZZlWO4f5U49cyzUDqBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15d31ccc260174383ea91bba5409b5dee6f0bc80e41927b039f59a1026d3ee88","last_reissued_at":"2026-05-18T01:25:40.293857Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:40.293857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal injective stability for the symplectic $K_1Sp$ group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"Anjan Gupta","submitted_at":"2015-11-30T18:31:21Z","abstract_excerpt":"If $R$ is a commutative ring, $I$ an ideal of $R$ and $v, w \\in Um_{2n}(R, I)$ then we show that $v, w$ are in the same orbit of elementary action if and only if they are in the same orbit of elementary symplectic action. We also show that if $A$ is a non-singular affine algebra of dimension $d$ over an algebraically closed field $k$ such that $d! A = A$, $d \\equiv 2 \\pmod 4$ and $I$ an ideal of $A$, then $Um_d(A, I) = e_1{Sp}_d(A, I)$. As a consequence it is proved that if $A$ is a non-singular affine algebra of dimension $d$ over an algebraically closed field $k$ such that $(d + 1)!A = A$, $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09419","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":"1511.09419","created_at":"2026-05-18T01:25:40.293939+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.09419v1","created_at":"2026-05-18T01:25:40.293939+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.09419","created_at":"2026-05-18T01:25:40.293939+00:00"},{"alias_kind":"pith_short_12","alias_value":"CXJRZTBGAF2D","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"CXJRZTBGAF2DQPVJ","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"CXJRZTBG","created_at":"2026-05-18T12:29:17.054201+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/CXJRZTBGAF2DQPVJDO5FICNV33","json":"https://pith.science/pith/CXJRZTBGAF2DQPVJDO5FICNV33.json","graph_json":"https://pith.science/api/pith-number/CXJRZTBGAF2DQPVJDO5FICNV33/graph.json","events_json":"https://pith.science/api/pith-number/CXJRZTBGAF2DQPVJDO5FICNV33/events.json","paper":"https://pith.science/paper/CXJRZTBG"},"agent_actions":{"view_html":"https://pith.science/pith/CXJRZTBGAF2DQPVJDO5FICNV33","download_json":"https://pith.science/pith/CXJRZTBGAF2DQPVJDO5FICNV33.json","view_paper":"https://pith.science/paper/CXJRZTBG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.09419&json=true","fetch_graph":"https://pith.science/api/pith-number/CXJRZTBGAF2DQPVJDO5FICNV33/graph.json","fetch_events":"https://pith.science/api/pith-number/CXJRZTBGAF2DQPVJDO5FICNV33/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CXJRZTBGAF2DQPVJDO5FICNV33/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CXJRZTBGAF2DQPVJDO5FICNV33/action/storage_attestation","attest_author":"https://pith.science/pith/CXJRZTBGAF2DQPVJDO5FICNV33/action/author_attestation","sign_citation":"https://pith.science/pith/CXJRZTBGAF2DQPVJDO5FICNV33/action/citation_signature","submit_replication":"https://pith.science/pith/CXJRZTBGAF2DQPVJDO5FICNV33/action/replication_record"}},"created_at":"2026-05-18T01:25:40.293939+00:00","updated_at":"2026-05-18T01:25:40.293939+00:00"}