{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:77GJ66KNMKHXLULXFUMS64H72H","short_pith_number":"pith:77GJ66KN","canonical_record":{"source":{"id":"1801.00699","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2018-01-02T16:13:32Z","cross_cats_sorted":[],"title_canon_sha256":"f4f5bbe9bff48e27b4207026372c616079dc1f1d4d6e5679bdd63cecc0abf444","abstract_canon_sha256":"4e04ff7550f380e4ae04ddb7ad7d2f63ee305d447c472ff2433e0fe4d1abd25b"},"schema_version":"1.0"},"canonical_sha256":"ffcc9f794d628f75d1772d192f70ffd1efcfa256b89b9f25622343f8f99ceb7b","source":{"kind":"arxiv","id":"1801.00699","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00699","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00699v1","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00699","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"pith_short_12","alias_value":"77GJ66KNMKHX","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"77GJ66KNMKHXLULX","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"77GJ66KN","created_at":"2026-05-18T12:32:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:77GJ66KNMKHXLULXFUMS64H72H","target":"record","payload":{"canonical_record":{"source":{"id":"1801.00699","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2018-01-02T16:13:32Z","cross_cats_sorted":[],"title_canon_sha256":"f4f5bbe9bff48e27b4207026372c616079dc1f1d4d6e5679bdd63cecc0abf444","abstract_canon_sha256":"4e04ff7550f380e4ae04ddb7ad7d2f63ee305d447c472ff2433e0fe4d1abd25b"},"schema_version":"1.0"},"canonical_sha256":"ffcc9f794d628f75d1772d192f70ffd1efcfa256b89b9f25622343f8f99ceb7b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:52.574155Z","signature_b64":"AYWPSgr9/Mm0S5I3lzpBGrVPVZimCl8h2IVPxGjXj4nD9GdZ/dGTyVMy4Q7tlhGEp0hjG4O2WzGi2DDMYAR7BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ffcc9f794d628f75d1772d192f70ffd1efcfa256b89b9f25622343f8f99ceb7b","last_reissued_at":"2026-05-18T00:26:52.573540Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:52.573540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.00699","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GAPJmIumT1FL0Vj4TNU3tuFxY1TpO9cdxhFUOq6LRmZ6tiRKzxoQxXj5dz60BGbNe+yG35ataXDjIP5HjOdSBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T07:42:59.949806Z"},"content_sha256":"d4b3a6437bfd1200746b7dcfad76d0e30ad03bc2b39bf060cc419c727d691cbc","schema_version":"1.0","event_id":"sha256:d4b3a6437bfd1200746b7dcfad76d0e30ad03bc2b39bf060cc419c727d691cbc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:77GJ66KNMKHXLULXFUMS64H72H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sandwich classification for $O_{2n+1}(R)$ and $U_{2n+1}(R,\\Delta)$ revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"Raimund Preusser","submitted_at":"2018-01-02T16:13:32Z","abstract_excerpt":"In a recent paper, the author proved that if $n\\geq 3$ is a natural number, $R$ a commutative ring and $\\sigma\\in GL_n(R)$, then $t_{kl}(\\sigma_{ij})$ where $i\\neq j$ and $k\\neq l$ can be expressed as a product of $8$ matrices of the form $^{\\epsilon}\\sigma^{\\pm 1}$ where $\\epsilon\\in E_n(R)$. In this article we prove similar results for the odd-dimensional orthogonal groups $O_{2n+1}(R)$ and the odd-dimensional unitary groups $U_{2n+1}(R,\\Delta)$ under the assumption that $R$ is commutative and $n\\geq 3$. This yields new, short proofs of the Sandwich Classification Theorems for the groups $O_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00699","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c8Y/p2LTrn19hyDo4wouxnCkvHKEfFNzdpRcbgst2mLwHkkfzLfQOmHOjWr0Exs83u+YIVhqPqS/37XLDeiBBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T07:42:59.950465Z"},"content_sha256":"2de6d58efe71bfb7db305f43af5b3b9f4cbc39246b511f031869e9c0b151ddb6","schema_version":"1.0","event_id":"sha256:2de6d58efe71bfb7db305f43af5b3b9f4cbc39246b511f031869e9c0b151ddb6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/77GJ66KNMKHXLULXFUMS64H72H/bundle.json","state_url":"https://pith.science/pith/77GJ66KNMKHXLULXFUMS64H72H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/77GJ66KNMKHXLULXFUMS64H72H/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":"2026-05-31T07:42:59Z","links":{"resolver":"https://pith.science/pith/77GJ66KNMKHXLULXFUMS64H72H","bundle":"https://pith.science/pith/77GJ66KNMKHXLULXFUMS64H72H/bundle.json","state":"https://pith.science/pith/77GJ66KNMKHXLULXFUMS64H72H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/77GJ66KNMKHXLULXFUMS64H72H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:77GJ66KNMKHXLULXFUMS64H72H","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":"4e04ff7550f380e4ae04ddb7ad7d2f63ee305d447c472ff2433e0fe4d1abd25b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2018-01-02T16:13:32Z","title_canon_sha256":"f4f5bbe9bff48e27b4207026372c616079dc1f1d4d6e5679bdd63cecc0abf444"},"schema_version":"1.0","source":{"id":"1801.00699","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00699","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00699v1","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00699","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"pith_short_12","alias_value":"77GJ66KNMKHX","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"77GJ66KNMKHXLULX","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"77GJ66KN","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:2de6d58efe71bfb7db305f43af5b3b9f4cbc39246b511f031869e9c0b151ddb6","target":"graph","created_at":"2026-05-18T00:26:52Z","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":"In a recent paper, the author proved that if $n\\geq 3$ is a natural number, $R$ a commutative ring and $\\sigma\\in GL_n(R)$, then $t_{kl}(\\sigma_{ij})$ where $i\\neq j$ and $k\\neq l$ can be expressed as a product of $8$ matrices of the form $^{\\epsilon}\\sigma^{\\pm 1}$ where $\\epsilon\\in E_n(R)$. In this article we prove similar results for the odd-dimensional orthogonal groups $O_{2n+1}(R)$ and the odd-dimensional unitary groups $U_{2n+1}(R,\\Delta)$ under the assumption that $R$ is commutative and $n\\geq 3$. This yields new, short proofs of the Sandwich Classification Theorems for the groups $O_","authors_text":"Raimund Preusser","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2018-01-02T16:13:32Z","title":"Sandwich classification for $O_{2n+1}(R)$ and $U_{2n+1}(R,\\Delta)$ revisited"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00699","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:d4b3a6437bfd1200746b7dcfad76d0e30ad03bc2b39bf060cc419c727d691cbc","target":"record","created_at":"2026-05-18T00:26:52Z","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":"4e04ff7550f380e4ae04ddb7ad7d2f63ee305d447c472ff2433e0fe4d1abd25b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2018-01-02T16:13:32Z","title_canon_sha256":"f4f5bbe9bff48e27b4207026372c616079dc1f1d4d6e5679bdd63cecc0abf444"},"schema_version":"1.0","source":{"id":"1801.00699","kind":"arxiv","version":1}},"canonical_sha256":"ffcc9f794d628f75d1772d192f70ffd1efcfa256b89b9f25622343f8f99ceb7b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ffcc9f794d628f75d1772d192f70ffd1efcfa256b89b9f25622343f8f99ceb7b","first_computed_at":"2026-05-18T00:26:52.573540Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:52.573540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AYWPSgr9/Mm0S5I3lzpBGrVPVZimCl8h2IVPxGjXj4nD9GdZ/dGTyVMy4Q7tlhGEp0hjG4O2WzGi2DDMYAR7BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:52.574155Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.00699","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d4b3a6437bfd1200746b7dcfad76d0e30ad03bc2b39bf060cc419c727d691cbc","sha256:2de6d58efe71bfb7db305f43af5b3b9f4cbc39246b511f031869e9c0b151ddb6"],"state_sha256":"5b19a2bf9c14b5885ea01b739d3e2e46ab9bd663f7c3eb9fdfa4d26b106a623f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"isCd9cGOhnscv1Pulrz/5ZfksGWK9ENKv9IU66VTQ4vzduFBVkyrL+ThUEFFvzv5Be8ecI5xCdtY4STAym4GCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T07:42:59.953455Z","bundle_sha256":"cef6fdb253816520a892d069a9d113e538a278bdf36e14b2390725e6ec01ef6e"}}