{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:XJ6CELGVS6Y6IP6TSXD6RPG5PL","short_pith_number":"pith:XJ6CELGV","canonical_record":{"source":{"id":"1309.5727","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-23T08:43:13Z","cross_cats_sorted":[],"title_canon_sha256":"abcccfad46e00c10ff985b6bcf9f15d36168696a61c4c62f2f9871e3d7f586cd","abstract_canon_sha256":"f4a573c43cece06ec183a23736728d8d28874934ee51b38b30cb1243cac9655d"},"schema_version":"1.0"},"canonical_sha256":"ba7c222cd597b1e43fd395c7e8bcdd7ad54f6e45b1cdd61288786c45d9df7c5a","source":{"kind":"arxiv","id":"1309.5727","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5727","created_at":"2026-05-18T02:03:38Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5727v2","created_at":"2026-05-18T02:03:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5727","created_at":"2026-05-18T02:03:38Z"},{"alias_kind":"pith_short_12","alias_value":"XJ6CELGVS6Y6","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XJ6CELGVS6Y6IP6T","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XJ6CELGV","created_at":"2026-05-18T12:28:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:XJ6CELGVS6Y6IP6TSXD6RPG5PL","target":"record","payload":{"canonical_record":{"source":{"id":"1309.5727","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-23T08:43:13Z","cross_cats_sorted":[],"title_canon_sha256":"abcccfad46e00c10ff985b6bcf9f15d36168696a61c4c62f2f9871e3d7f586cd","abstract_canon_sha256":"f4a573c43cece06ec183a23736728d8d28874934ee51b38b30cb1243cac9655d"},"schema_version":"1.0"},"canonical_sha256":"ba7c222cd597b1e43fd395c7e8bcdd7ad54f6e45b1cdd61288786c45d9df7c5a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:38.090384Z","signature_b64":"ztLj8akxNrh4fre0NBnzGXME1FjaiOleTQTRybqk1cdByRQPZbNMWY1p9UcGEwV/mhsGMNTbXPHpzvh408y7BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba7c222cd597b1e43fd395c7e8bcdd7ad54f6e45b1cdd61288786c45d9df7c5a","last_reissued_at":"2026-05-18T02:03:38.089946Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:38.089946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.5727","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":"2026-05-18T02:03:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9qnN1I7f9x/xR5Syyo92ooUYA/K4CAWWYGx9QTUrmpcI3UCb7vIy3diH/lkCCdHucwsIXIta1xBnP0fXfJSRAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T12:59:44.908292Z"},"content_sha256":"f96a00b787b68ecba3c7c329a9f9d9f3d8efbdaebf1c709a0772273c2edef337","schema_version":"1.0","event_id":"sha256:f96a00b787b68ecba3c7c329a9f9d9f3d8efbdaebf1c709a0772273c2edef337"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:XJ6CELGVS6Y6IP6TSXD6RPG5PL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the (non)existence of best low-rank approximations of generic IxJx2 arrays","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alwin Stegeman","submitted_at":"2013-09-23T08:43:13Z","abstract_excerpt":"Several conjectures and partial proofs have been formulated on the (non)existence of a best low-rank approximation of real-valued IxJx2 arrays. We analyze this problem using the Generalized Schur Decomposition and prove (non)existence of a best rank-R approximation for generic IxJx2 arrays, for all values of I,J,R. Moreover, for cases where a best rank-R approximation exists on a set of positive volume only, we provide easy-to-check necessary and sufficient conditions for the existence of a best rank-R approximation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5727","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":""},"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":"2026-05-18T02:03:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"94gw/jAZ/M2574+B0Sk0+FtH19Um46ysA1MxEvnVptrsZDTE0I3JOJZtZSLPGmPAnA1iUDvdk2edpRJzW332Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T12:59:44.908653Z"},"content_sha256":"eec28077662ed2a6f6970360837a1932e41be35f19915e378c75dc3d2b5f51b2","schema_version":"1.0","event_id":"sha256:eec28077662ed2a6f6970360837a1932e41be35f19915e378c75dc3d2b5f51b2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XJ6CELGVS6Y6IP6TSXD6RPG5PL/bundle.json","state_url":"https://pith.science/pith/XJ6CELGVS6Y6IP6TSXD6RPG5PL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XJ6CELGVS6Y6IP6TSXD6RPG5PL/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":"2026-06-20T12:59:44Z","links":{"resolver":"https://pith.science/pith/XJ6CELGVS6Y6IP6TSXD6RPG5PL","bundle":"https://pith.science/pith/XJ6CELGVS6Y6IP6TSXD6RPG5PL/bundle.json","state":"https://pith.science/pith/XJ6CELGVS6Y6IP6TSXD6RPG5PL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XJ6CELGVS6Y6IP6TSXD6RPG5PL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:XJ6CELGVS6Y6IP6TSXD6RPG5PL","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":"f4a573c43cece06ec183a23736728d8d28874934ee51b38b30cb1243cac9655d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-23T08:43:13Z","title_canon_sha256":"abcccfad46e00c10ff985b6bcf9f15d36168696a61c4c62f2f9871e3d7f586cd"},"schema_version":"1.0","source":{"id":"1309.5727","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5727","created_at":"2026-05-18T02:03:38Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5727v2","created_at":"2026-05-18T02:03:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5727","created_at":"2026-05-18T02:03:38Z"},{"alias_kind":"pith_short_12","alias_value":"XJ6CELGVS6Y6","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XJ6CELGVS6Y6IP6T","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XJ6CELGV","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:eec28077662ed2a6f6970360837a1932e41be35f19915e378c75dc3d2b5f51b2","target":"graph","created_at":"2026-05-18T02:03:38Z","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":"Several conjectures and partial proofs have been formulated on the (non)existence of a best low-rank approximation of real-valued IxJx2 arrays. We analyze this problem using the Generalized Schur Decomposition and prove (non)existence of a best rank-R approximation for generic IxJx2 arrays, for all values of I,J,R. Moreover, for cases where a best rank-R approximation exists on a set of positive volume only, we provide easy-to-check necessary and sufficient conditions for the existence of a best rank-R approximation.","authors_text":"Alwin Stegeman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-23T08:43:13Z","title":"On the (non)existence of best low-rank approximations of generic IxJx2 arrays"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5727","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:f96a00b787b68ecba3c7c329a9f9d9f3d8efbdaebf1c709a0772273c2edef337","target":"record","created_at":"2026-05-18T02:03:38Z","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":"f4a573c43cece06ec183a23736728d8d28874934ee51b38b30cb1243cac9655d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-23T08:43:13Z","title_canon_sha256":"abcccfad46e00c10ff985b6bcf9f15d36168696a61c4c62f2f9871e3d7f586cd"},"schema_version":"1.0","source":{"id":"1309.5727","kind":"arxiv","version":2}},"canonical_sha256":"ba7c222cd597b1e43fd395c7e8bcdd7ad54f6e45b1cdd61288786c45d9df7c5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ba7c222cd597b1e43fd395c7e8bcdd7ad54f6e45b1cdd61288786c45d9df7c5a","first_computed_at":"2026-05-18T02:03:38.089946Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:03:38.089946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ztLj8akxNrh4fre0NBnzGXME1FjaiOleTQTRybqk1cdByRQPZbNMWY1p9UcGEwV/mhsGMNTbXPHpzvh408y7BA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:03:38.090384Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.5727","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f96a00b787b68ecba3c7c329a9f9d9f3d8efbdaebf1c709a0772273c2edef337","sha256:eec28077662ed2a6f6970360837a1932e41be35f19915e378c75dc3d2b5f51b2"],"state_sha256":"e5098632651eeb2f46964da4195f6c42ec54972af96afe14bd3246e16eff0528"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VIJ8exve6rzbMcgmX52/8jPpXZ5MQDnrnsxltWDAc0XiB5Jjdq1E5OMKZbOYQNLCnQaHOdI/LbkcT1e5qeuyCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T12:59:44.910491Z","bundle_sha256":"0cbbdac26ca7622ea195048f6aff8b7dbf119d703efc423cfc570354772469f4"}}