{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:Y2SX6INZXA6GJRVOSJVHJ2SYOB","short_pith_number":"pith:Y2SX6INZ","canonical_record":{"source":{"id":"2607.26863","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-29T12:46:20Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"d155b68c50f24cbf4e1527fc2235db3dad138c1794a370d1e07cd0a8d5966f5b","abstract_canon_sha256":"fbbee30551f84839780888d931b5fb082c2e7733a00612b45069af2ee4c8a9d0"},"schema_version":"1.0"},"canonical_sha256":"c6a57f21b9b83c64c6ae926a74ea58707e52060888483b953b25b1a51c25824e","source":{"kind":"arxiv","id":"2607.26863","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26863","created_at":"2026-07-30T01:22:33Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26863v1","created_at":"2026-07-30T01:22:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26863","created_at":"2026-07-30T01:22:33Z"},{"alias_kind":"pith_short_12","alias_value":"Y2SX6INZXA6G","created_at":"2026-07-30T01:22:33Z"},{"alias_kind":"pith_short_16","alias_value":"Y2SX6INZXA6GJRVO","created_at":"2026-07-30T01:22:33Z"},{"alias_kind":"pith_short_8","alias_value":"Y2SX6INZ","created_at":"2026-07-30T01:22:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:Y2SX6INZXA6GJRVOSJVHJ2SYOB","target":"record","payload":{"canonical_record":{"source":{"id":"2607.26863","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-29T12:46:20Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"d155b68c50f24cbf4e1527fc2235db3dad138c1794a370d1e07cd0a8d5966f5b","abstract_canon_sha256":"fbbee30551f84839780888d931b5fb082c2e7733a00612b45069af2ee4c8a9d0"},"schema_version":"1.0"},"canonical_sha256":"c6a57f21b9b83c64c6ae926a74ea58707e52060888483b953b25b1a51c25824e","receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6a57f21b9b83c64c6ae926a74ea58707e52060888483b953b25b1a51c25824e","last_reissued_at":"2026-07-30T01:22:33.590962Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T01:22:33.590962Z"},"source_kind":"arxiv","source_id":"2607.26863","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":"2026-07-30T01:22:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q0xpcnBQ65ZPKKtqLkQC1l671EcDWUf76tM6UG7XXyx1Q4Min5KUbBIfHr3kolk2CmA3zYuA35rl8nudswukCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T22:14:32.265662Z"},"content_sha256":"96c2b2ad6a581043e61da5289f657b2bec90e6389df1994057200e5bf536df08","schema_version":"1.0","event_id":"sha256:96c2b2ad6a581043e61da5289f657b2bec90e6389df1994057200e5bf536df08"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:Y2SX6INZXA6GJRVOSJVHJ2SYOB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Convergence rates for pivoted QR and LU","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Marc Aur\\`ele Gilles","submitted_at":"2026-07-29T12:46:20Z","abstract_excerpt":"Pivoted QR and pivoted LU decompositions are greedy algorithms used to compute low-rank approximations of matrices from selected columns, or selected rows and columns. Despite their practical robustness, general worst-case bounds comparing their errors with those of the best corresponding low-rank approximations contain exponentially growing factors and do not explain their behavior under modest singular value decay. We prove that under approximate greedy pivoting, their error is controlled by the determinant of a submatrix, which is bounded by the geometric mean of the leading singular values"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26863","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/2607.26863/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":"2026-07-30T01:22:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sGTyHp11dpweFCRHLEvzWO5kglcrptK38HEXucroDp9IyeYxnLh6QT27ViNADjNPWHhRXF3EZWe1/3VYVtEXDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T22:14:32.266186Z"},"content_sha256":"6eb6da06b557a64c0cef901ca04ec506b5eb7929d301f93aad22cc14f9fa585f","schema_version":"1.0","event_id":"sha256:6eb6da06b557a64c0cef901ca04ec506b5eb7929d301f93aad22cc14f9fa585f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y2SX6INZXA6GJRVOSJVHJ2SYOB/bundle.json","state_url":"https://pith.science/pith/Y2SX6INZXA6GJRVOSJVHJ2SYOB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y2SX6INZXA6GJRVOSJVHJ2SYOB/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":"2026-08-21T22:14:32Z","links":{"resolver":"https://pith.science/pith/Y2SX6INZXA6GJRVOSJVHJ2SYOB","bundle":"https://pith.science/pith/Y2SX6INZXA6GJRVOSJVHJ2SYOB/bundle.json","state":"https://pith.science/pith/Y2SX6INZXA6GJRVOSJVHJ2SYOB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y2SX6INZXA6GJRVOSJVHJ2SYOB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:Y2SX6INZXA6GJRVOSJVHJ2SYOB","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":"fbbee30551f84839780888d931b5fb082c2e7733a00612b45069af2ee4c8a9d0","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-29T12:46:20Z","title_canon_sha256":"d155b68c50f24cbf4e1527fc2235db3dad138c1794a370d1e07cd0a8d5966f5b"},"schema_version":"1.0","source":{"id":"2607.26863","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26863","created_at":"2026-07-30T01:22:33Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26863v1","created_at":"2026-07-30T01:22:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26863","created_at":"2026-07-30T01:22:33Z"},{"alias_kind":"pith_short_12","alias_value":"Y2SX6INZXA6G","created_at":"2026-07-30T01:22:33Z"},{"alias_kind":"pith_short_16","alias_value":"Y2SX6INZXA6GJRVO","created_at":"2026-07-30T01:22:33Z"},{"alias_kind":"pith_short_8","alias_value":"Y2SX6INZ","created_at":"2026-07-30T01:22:33Z"}],"graph_snapshots":[{"event_id":"sha256:6eb6da06b557a64c0cef901ca04ec506b5eb7929d301f93aad22cc14f9fa585f","target":"graph","created_at":"2026-07-30T01:22:33Z","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/2607.26863/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Pivoted QR and pivoted LU decompositions are greedy algorithms used to compute low-rank approximations of matrices from selected columns, or selected rows and columns. Despite their practical robustness, general worst-case bounds comparing their errors with those of the best corresponding low-rank approximations contain exponentially growing factors and do not explain their behavior under modest singular value decay. We prove that under approximate greedy pivoting, their error is controlled by the determinant of a submatrix, which is bounded by the geometric mean of the leading singular values","authors_text":"Marc Aur\\`ele Gilles","cross_cats":["cs.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-29T12:46:20Z","title":"Convergence rates for pivoted QR and LU"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26863","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":"sha256:96c2b2ad6a581043e61da5289f657b2bec90e6389df1994057200e5bf536df08","target":"record","created_at":"2026-07-30T01:22:33Z","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":"fbbee30551f84839780888d931b5fb082c2e7733a00612b45069af2ee4c8a9d0","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-07-29T12:46:20Z","title_canon_sha256":"d155b68c50f24cbf4e1527fc2235db3dad138c1794a370d1e07cd0a8d5966f5b"},"schema_version":"1.0","source":{"id":"2607.26863","kind":"arxiv","version":1}},"canonical_sha256":"c6a57f21b9b83c64c6ae926a74ea58707e52060888483b953b25b1a51c25824e","receipt":{"builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c6a57f21b9b83c64c6ae926a74ea58707e52060888483b953b25b1a51c25824e","first_computed_at":"2026-07-30T01:22:33.590962Z","kind":"pith_receipt","last_reissued_at":"2026-07-30T01:22:33.590962Z","receipt_version":"0.3","signature_status":"unsigned_v0"},"source_id":"2607.26863","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:96c2b2ad6a581043e61da5289f657b2bec90e6389df1994057200e5bf536df08","sha256:6eb6da06b557a64c0cef901ca04ec506b5eb7929d301f93aad22cc14f9fa585f"],"state_sha256":"7ce88d532c6ee1e02c3f3bc8e10718160571f03e1bc1271ffbc07eacd9e8a9f3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h6m++kjO+pCDNbZ2xx+VjAlcpH14o0W8B4YEZPGePnXlK4EjIxVrUaMAVbgSuKd3Cj+HkWzUODn3HmcKqqQzCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T22:14:32.270176Z","bundle_sha256":"de46a7f33e0c9fe15d8bc23a10f3a15ec34fae14f605864ae4d640a4ba00c400"}}