{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:I7DXBJXFAXNLSCRTTG5BEJ5GO7","short_pith_number":"pith:I7DXBJXF","canonical_record":{"source":{"id":"2602.19923","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NA","submitted_at":"2026-02-23T15:02:17Z","cross_cats_sorted":["cs.NA","math.DG"],"title_canon_sha256":"9a16fa9c97070af60ff9247f18f09af96eda9c890c894e75ef5c1c278ba499c0","abstract_canon_sha256":"0d4934f6de6605564b557a14a342c0cc05eabf4a80cfa6ed2ffc741204e1f5c5"},"schema_version":"1.0"},"canonical_sha256":"47c770a6e505dab90a3399ba1227a677fa9358c0cb7858e6eedc78d8e6320d5c","source":{"kind":"arxiv","id":"2602.19923","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.19923","created_at":"2026-07-02T01:18:06Z"},{"alias_kind":"arxiv_version","alias_value":"2602.19923v2","created_at":"2026-07-02T01:18:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.19923","created_at":"2026-07-02T01:18:06Z"},{"alias_kind":"pith_short_12","alias_value":"I7DXBJXFAXNL","created_at":"2026-07-02T01:18:06Z"},{"alias_kind":"pith_short_16","alias_value":"I7DXBJXFAXNLSCRT","created_at":"2026-07-02T01:18:06Z"},{"alias_kind":"pith_short_8","alias_value":"I7DXBJXF","created_at":"2026-07-02T01:18:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:I7DXBJXFAXNLSCRTTG5BEJ5GO7","target":"record","payload":{"canonical_record":{"source":{"id":"2602.19923","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NA","submitted_at":"2026-02-23T15:02:17Z","cross_cats_sorted":["cs.NA","math.DG"],"title_canon_sha256":"9a16fa9c97070af60ff9247f18f09af96eda9c890c894e75ef5c1c278ba499c0","abstract_canon_sha256":"0d4934f6de6605564b557a14a342c0cc05eabf4a80cfa6ed2ffc741204e1f5c5"},"schema_version":"1.0"},"canonical_sha256":"47c770a6e505dab90a3399ba1227a677fa9358c0cb7858e6eedc78d8e6320d5c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-02T01:18:06.892236Z","signature_b64":"Ujb/SJn2HbSBpx6dbykNJi+TwqaTd2Va9ut5xkF7oaisruZ8gNF2wJBKE/JWkxl/Gejm09NtWnPEB6ptdS9PAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47c770a6e505dab90a3399ba1227a677fa9358c0cb7858e6eedc78d8e6320d5c","last_reissued_at":"2026-07-02T01:18:06.891827Z","signature_status":"signed_v1","first_computed_at":"2026-07-02T01:18:06.891827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2602.19923","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-02T01:18:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NxEgG1oO9dniA/7+Kz4o73TAPwKIaZ2DSxGNfeJ9Dp+FN29Kaumrh1XSdJE9otaB4ObKYFwsyoY0jflcuFXrBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T11:21:54.932447Z"},"content_sha256":"3e7274b8811e2f6ea98bdcbd282cf8175594c92f8b96382734e26fc14124d1e9","schema_version":"1.0","event_id":"sha256:3e7274b8811e2f6ea98bdcbd282cf8175594c92f8b96382734e26fc14124d1e9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:I7DXBJXFAXNLSCRTTG5BEJ5GO7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An new polar factor retraction on the Stiefel manifold with closed-form inverse","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["cs.NA","math.DG"],"primary_cat":"math.NA","authors_text":"Ralf Zimmermann, Rasmus Jensen","submitted_at":"2026-02-23T15:02:17Z","abstract_excerpt":"Retractions are the workhorses in Riemannian computing applications, where computational efficiency is of the essence. This work introduces a new retraction on the compact Stiefel manifold of orthogonal frames. The retraction is second-order accurate under the Euclidean metric and features a closed-form inverse that can be efficiently computed.\n  A variety of retractions is known on the Stiefel manifold, including the Riemannian exponential map, the polar factor retraction, the QR-retraction, quasi--geodesics and the Cayley retraction. The Cayley retraction is second--order accurate under the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.19923","kind":"arxiv","version":2},"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/2602.19923/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-02T01:18:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lV6qDNJaasB2IhieCao/d7+cFKunKvPj0tQINOKh8BFp00zeBeAlwDdozRqhF5gyp0jSEYQ0L1Dp0xG5MkJlCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T11:21:54.932946Z"},"content_sha256":"81014a8e6654939a117e4adbaf8adce0ca96c4af4b4402cc3a6bba8c07a1b67a","schema_version":"1.0","event_id":"sha256:81014a8e6654939a117e4adbaf8adce0ca96c4af4b4402cc3a6bba8c07a1b67a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I7DXBJXFAXNLSCRTTG5BEJ5GO7/bundle.json","state_url":"https://pith.science/pith/I7DXBJXFAXNLSCRTTG5BEJ5GO7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I7DXBJXFAXNLSCRTTG5BEJ5GO7/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":"2026-08-08T11:21:54Z","links":{"resolver":"https://pith.science/pith/I7DXBJXFAXNLSCRTTG5BEJ5GO7","bundle":"https://pith.science/pith/I7DXBJXFAXNLSCRTTG5BEJ5GO7/bundle.json","state":"https://pith.science/pith/I7DXBJXFAXNLSCRTTG5BEJ5GO7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I7DXBJXFAXNLSCRTTG5BEJ5GO7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:I7DXBJXFAXNLSCRTTG5BEJ5GO7","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":"0d4934f6de6605564b557a14a342c0cc05eabf4a80cfa6ed2ffc741204e1f5c5","cross_cats_sorted":["cs.NA","math.DG"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NA","submitted_at":"2026-02-23T15:02:17Z","title_canon_sha256":"9a16fa9c97070af60ff9247f18f09af96eda9c890c894e75ef5c1c278ba499c0"},"schema_version":"1.0","source":{"id":"2602.19923","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.19923","created_at":"2026-07-02T01:18:06Z"},{"alias_kind":"arxiv_version","alias_value":"2602.19923v2","created_at":"2026-07-02T01:18:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.19923","created_at":"2026-07-02T01:18:06Z"},{"alias_kind":"pith_short_12","alias_value":"I7DXBJXFAXNL","created_at":"2026-07-02T01:18:06Z"},{"alias_kind":"pith_short_16","alias_value":"I7DXBJXFAXNLSCRT","created_at":"2026-07-02T01:18:06Z"},{"alias_kind":"pith_short_8","alias_value":"I7DXBJXF","created_at":"2026-07-02T01:18:06Z"}],"graph_snapshots":[{"event_id":"sha256:81014a8e6654939a117e4adbaf8adce0ca96c4af4b4402cc3a6bba8c07a1b67a","target":"graph","created_at":"2026-07-02T01:18:06Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2602.19923/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Retractions are the workhorses in Riemannian computing applications, where computational efficiency is of the essence. This work introduces a new retraction on the compact Stiefel manifold of orthogonal frames. The retraction is second-order accurate under the Euclidean metric and features a closed-form inverse that can be efficiently computed.\n  A variety of retractions is known on the Stiefel manifold, including the Riemannian exponential map, the polar factor retraction, the QR-retraction, quasi--geodesics and the Cayley retraction. The Cayley retraction is second--order accurate under the ","authors_text":"Ralf Zimmermann, Rasmus Jensen","cross_cats":["cs.NA","math.DG"],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NA","submitted_at":"2026-02-23T15:02:17Z","title":"An new polar factor retraction on the Stiefel manifold with closed-form inverse"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.19923","kind":"arxiv","version":2},"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":"sha256:3e7274b8811e2f6ea98bdcbd282cf8175594c92f8b96382734e26fc14124d1e9","target":"record","created_at":"2026-07-02T01:18:06Z","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":"0d4934f6de6605564b557a14a342c0cc05eabf4a80cfa6ed2ffc741204e1f5c5","cross_cats_sorted":["cs.NA","math.DG"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NA","submitted_at":"2026-02-23T15:02:17Z","title_canon_sha256":"9a16fa9c97070af60ff9247f18f09af96eda9c890c894e75ef5c1c278ba499c0"},"schema_version":"1.0","source":{"id":"2602.19923","kind":"arxiv","version":2}},"canonical_sha256":"47c770a6e505dab90a3399ba1227a677fa9358c0cb7858e6eedc78d8e6320d5c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47c770a6e505dab90a3399ba1227a677fa9358c0cb7858e6eedc78d8e6320d5c","first_computed_at":"2026-07-02T01:18:06.891827Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-02T01:18:06.891827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ujb/SJn2HbSBpx6dbykNJi+TwqaTd2Va9ut5xkF7oaisruZ8gNF2wJBKE/JWkxl/Gejm09NtWnPEB6ptdS9PAg==","signature_status":"signed_v1","signed_at":"2026-07-02T01:18:06.892236Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.19923","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e7274b8811e2f6ea98bdcbd282cf8175594c92f8b96382734e26fc14124d1e9","sha256:81014a8e6654939a117e4adbaf8adce0ca96c4af4b4402cc3a6bba8c07a1b67a"],"state_sha256":"d74c61a5ae0fa3f836369bd95976a1268b11dadc3b8dd943f18a87c5cef2ec72"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gG+tMc4gHrdeCCfHN8y33ehtiLPHLi1TyT9XkGOV/cs4bd17Qy2zvbMBgVaIJbklw+JoeMbOAaYfEQagm1oFCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T11:21:54.936441Z","bundle_sha256":"08248a8c1f44aa33b4ad3f731aebc4274aeca7a986d66a316143e276dcc8cca4"}}