{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:D3YT6GI2Z63U6OLIPKGWXUNW2Y","short_pith_number":"pith:D3YT6GI2","schema_version":"1.0","canonical_sha256":"1ef13f191acfb74f39687a8d6bd1b6d62564224e186443afd517c5acfbc9abbf","source":{"kind":"arxiv","id":"2606.29317","version":1},"attestation_state":"computed","paper":{"title":"Three-Dimensional Real Affine Lie Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.SG","authors_text":"M. W. Mansouri, S. El Bourkadi, T. A\\\"it Aissa","submitted_at":"2026-06-28T10:23:59Z","abstract_excerpt":"We classify all left-invariant real affine connections in dimension three. Our approach reduces the three-dimensional problem to a two-dimensional one by decomposing each left-invariant affine connection into a two-dimensional part and an additional one-dimensional component. After characterizing all possible two-dimensional left-invariant affine connections, we return to the three-dimensional setting to obtain a simplified description of all three-dimensional left-invariant affine connections. We then explicitly solve the resulting simplified quadratic equations and perform a refined analysis"},"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":"2606.29317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2026-06-28T10:23:59Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"55df75c27495f214462705466d4d7b5494437348b67e02af629361d1c586844a","abstract_canon_sha256":"99bcf031bb11a023b6d1f4a733fa9f6561f13f64e485eb21f145be668c3f52cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T01:18:01.266408Z","signature_b64":"w87r6MmDSxFsRIGvGbciHd9Mu5x0/PX+QTTXkzJvyZOU2NGMslSGEtYIMTZYRTwz87/n8WbXh9AHBRN0zsSyAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ef13f191acfb74f39687a8d6bd1b6d62564224e186443afd517c5acfbc9abbf","last_reissued_at":"2026-06-30T01:18:01.265877Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T01:18:01.265877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Three-Dimensional Real Affine Lie Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.SG","authors_text":"M. W. Mansouri, S. El Bourkadi, T. A\\\"it Aissa","submitted_at":"2026-06-28T10:23:59Z","abstract_excerpt":"We classify all left-invariant real affine connections in dimension three. Our approach reduces the three-dimensional problem to a two-dimensional one by decomposing each left-invariant affine connection into a two-dimensional part and an additional one-dimensional component. After characterizing all possible two-dimensional left-invariant affine connections, we return to the three-dimensional setting to obtain a simplified description of all three-dimensional left-invariant affine connections. We then explicitly solve the resulting simplified quadratic equations and perform a refined analysis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29317","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.29317/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2606.29317","created_at":"2026-06-30T01:18:01.265947+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.29317v1","created_at":"2026-06-30T01:18:01.265947+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.29317","created_at":"2026-06-30T01:18:01.265947+00:00"},{"alias_kind":"pith_short_12","alias_value":"D3YT6GI2Z63U","created_at":"2026-06-30T01:18:01.265947+00:00"},{"alias_kind":"pith_short_16","alias_value":"D3YT6GI2Z63U6OLI","created_at":"2026-06-30T01:18:01.265947+00:00"},{"alias_kind":"pith_short_8","alias_value":"D3YT6GI2","created_at":"2026-06-30T01:18:01.265947+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/D3YT6GI2Z63U6OLIPKGWXUNW2Y","json":"https://pith.science/pith/D3YT6GI2Z63U6OLIPKGWXUNW2Y.json","graph_json":"https://pith.science/api/pith-number/D3YT6GI2Z63U6OLIPKGWXUNW2Y/graph.json","events_json":"https://pith.science/api/pith-number/D3YT6GI2Z63U6OLIPKGWXUNW2Y/events.json","paper":"https://pith.science/paper/D3YT6GI2"},"agent_actions":{"view_html":"https://pith.science/pith/D3YT6GI2Z63U6OLIPKGWXUNW2Y","download_json":"https://pith.science/pith/D3YT6GI2Z63U6OLIPKGWXUNW2Y.json","view_paper":"https://pith.science/paper/D3YT6GI2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.29317&json=true","fetch_graph":"https://pith.science/api/pith-number/D3YT6GI2Z63U6OLIPKGWXUNW2Y/graph.json","fetch_events":"https://pith.science/api/pith-number/D3YT6GI2Z63U6OLIPKGWXUNW2Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D3YT6GI2Z63U6OLIPKGWXUNW2Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D3YT6GI2Z63U6OLIPKGWXUNW2Y/action/storage_attestation","attest_author":"https://pith.science/pith/D3YT6GI2Z63U6OLIPKGWXUNW2Y/action/author_attestation","sign_citation":"https://pith.science/pith/D3YT6GI2Z63U6OLIPKGWXUNW2Y/action/citation_signature","submit_replication":"https://pith.science/pith/D3YT6GI2Z63U6OLIPKGWXUNW2Y/action/replication_record"}},"created_at":"2026-06-30T01:18:01.265947+00:00","updated_at":"2026-06-30T01:18:01.265947+00:00"}