{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TBM4LU2XW7M637UZYO3EHJEVOW","short_pith_number":"pith:TBM4LU2X","canonical_record":{"source":{"id":"2606.07407","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-06-05T15:50:36Z","cross_cats_sorted":[],"title_canon_sha256":"8a63a80772d93f8d4adf512c26f0ced36cdc056b52b38196caf619591b37a211","abstract_canon_sha256":"042ff3382c31fd0467da6f2183dfc4b7a02fbfb313164929dc777e8451d3cccd"},"schema_version":"1.0"},"canonical_sha256":"9859c5d357b7d9edfe99c3b643a49575a77f5cff75cb1af3c68d3e4607c9a884","source":{"kind":"arxiv","id":"2606.07407","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07407","created_at":"2026-06-08T01:05:25Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07407v1","created_at":"2026-06-08T01:05:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07407","created_at":"2026-06-08T01:05:25Z"},{"alias_kind":"pith_short_12","alias_value":"TBM4LU2XW7M6","created_at":"2026-06-08T01:05:25Z"},{"alias_kind":"pith_short_16","alias_value":"TBM4LU2XW7M637UZ","created_at":"2026-06-08T01:05:25Z"},{"alias_kind":"pith_short_8","alias_value":"TBM4LU2X","created_at":"2026-06-08T01:05:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TBM4LU2XW7M637UZYO3EHJEVOW","target":"record","payload":{"canonical_record":{"source":{"id":"2606.07407","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-06-05T15:50:36Z","cross_cats_sorted":[],"title_canon_sha256":"8a63a80772d93f8d4adf512c26f0ced36cdc056b52b38196caf619591b37a211","abstract_canon_sha256":"042ff3382c31fd0467da6f2183dfc4b7a02fbfb313164929dc777e8451d3cccd"},"schema_version":"1.0"},"canonical_sha256":"9859c5d357b7d9edfe99c3b643a49575a77f5cff75cb1af3c68d3e4607c9a884","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:05:25.289127Z","signature_b64":"UynuMt/P0INbwRs0Q47K4sIGNC2LMJyZHb/rDMtqapaYt1tYPnO8Sa6jAaOzEvNZV2CprFiyfAoggXqQszmnAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9859c5d357b7d9edfe99c3b643a49575a77f5cff75cb1af3c68d3e4607c9a884","last_reissued_at":"2026-06-08T01:05:25.288261Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:05:25.288261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.07407","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":"2026-06-08T01:05:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K/5WP68VlK04MAGqrNZqv4AnyLbri6CVhtL62SLA0SjA10voQecUJLqBkzwyeVO0e5+6MDjJB4WiLtyhn2UPBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T03:09:01.488430Z"},"content_sha256":"bb5ee3cb21a505df840e348f89dfe3114572def2b307efbdd97e0c46a5748e76","schema_version":"1.0","event_id":"sha256:bb5ee3cb21a505df840e348f89dfe3114572def2b307efbdd97e0c46a5748e76"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TBM4LU2XW7M637UZYO3EHJEVOW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Structured matrix factorization length","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jeong-Hoon Ju, Taehyeong Kim","submitted_at":"2026-06-05T15:50:36Z","abstract_excerpt":"Every (resp. a generic) complex $n \\times n$ matrix can be expressed as a product of $2n+5$ (resp. $\\lfloor n/2 \\rfloor +1$) Toeplitz matrices. Motivated by this result, it is natural to ask the following question: what is the minimum number of Toeplitz matrices required to factor a given matrix? We generalize this question from Toeplitz structure to more general structures. In this paper, we introduce the notion of structured matrix factorization length when the set of matrices with a given structure is an affine variety $X \\subseteq \\mathbb{C}^{n \\times n}$. Then we introduce the $r$-th $X$-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07407","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/2606.07407/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":"2026-06-08T01:05:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oi6yYN7fjznkZ8aKqeBQtLVZfoxybFN5Cv43KVL+Il4eRG15WPORc5kdatydtlBGy3S/sSw+wK8oBOkAEsN1CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T03:09:01.489170Z"},"content_sha256":"84e223af624d26554de605d005dee61d4b8876eb5b748b94a72a695e8c31ef88","schema_version":"1.0","event_id":"sha256:84e223af624d26554de605d005dee61d4b8876eb5b748b94a72a695e8c31ef88"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TBM4LU2XW7M637UZYO3EHJEVOW/bundle.json","state_url":"https://pith.science/pith/TBM4LU2XW7M637UZYO3EHJEVOW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TBM4LU2XW7M637UZYO3EHJEVOW/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":"2026-06-12T03:09:01Z","links":{"resolver":"https://pith.science/pith/TBM4LU2XW7M637UZYO3EHJEVOW","bundle":"https://pith.science/pith/TBM4LU2XW7M637UZYO3EHJEVOW/bundle.json","state":"https://pith.science/pith/TBM4LU2XW7M637UZYO3EHJEVOW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TBM4LU2XW7M637UZYO3EHJEVOW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TBM4LU2XW7M637UZYO3EHJEVOW","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":"042ff3382c31fd0467da6f2183dfc4b7a02fbfb313164929dc777e8451d3cccd","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-06-05T15:50:36Z","title_canon_sha256":"8a63a80772d93f8d4adf512c26f0ced36cdc056b52b38196caf619591b37a211"},"schema_version":"1.0","source":{"id":"2606.07407","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07407","created_at":"2026-06-08T01:05:25Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07407v1","created_at":"2026-06-08T01:05:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07407","created_at":"2026-06-08T01:05:25Z"},{"alias_kind":"pith_short_12","alias_value":"TBM4LU2XW7M6","created_at":"2026-06-08T01:05:25Z"},{"alias_kind":"pith_short_16","alias_value":"TBM4LU2XW7M637UZ","created_at":"2026-06-08T01:05:25Z"},{"alias_kind":"pith_short_8","alias_value":"TBM4LU2X","created_at":"2026-06-08T01:05:25Z"}],"graph_snapshots":[{"event_id":"sha256:84e223af624d26554de605d005dee61d4b8876eb5b748b94a72a695e8c31ef88","target":"graph","created_at":"2026-06-08T01:05:25Z","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/2606.07407/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Every (resp. a generic) complex $n \\times n$ matrix can be expressed as a product of $2n+5$ (resp. $\\lfloor n/2 \\rfloor +1$) Toeplitz matrices. Motivated by this result, it is natural to ask the following question: what is the minimum number of Toeplitz matrices required to factor a given matrix? We generalize this question from Toeplitz structure to more general structures. In this paper, we introduce the notion of structured matrix factorization length when the set of matrices with a given structure is an affine variety $X \\subseteq \\mathbb{C}^{n \\times n}$. Then we introduce the $r$-th $X$-","authors_text":"Jeong-Hoon Ju, Taehyeong Kim","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-06-05T15:50:36Z","title":"Structured matrix factorization length"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07407","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:bb5ee3cb21a505df840e348f89dfe3114572def2b307efbdd97e0c46a5748e76","target":"record","created_at":"2026-06-08T01:05:25Z","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":"042ff3382c31fd0467da6f2183dfc4b7a02fbfb313164929dc777e8451d3cccd","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-06-05T15:50:36Z","title_canon_sha256":"8a63a80772d93f8d4adf512c26f0ced36cdc056b52b38196caf619591b37a211"},"schema_version":"1.0","source":{"id":"2606.07407","kind":"arxiv","version":1}},"canonical_sha256":"9859c5d357b7d9edfe99c3b643a49575a77f5cff75cb1af3c68d3e4607c9a884","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9859c5d357b7d9edfe99c3b643a49575a77f5cff75cb1af3c68d3e4607c9a884","first_computed_at":"2026-06-08T01:05:25.288261Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T01:05:25.288261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UynuMt/P0INbwRs0Q47K4sIGNC2LMJyZHb/rDMtqapaYt1tYPnO8Sa6jAaOzEvNZV2CprFiyfAoggXqQszmnAg==","signature_status":"signed_v1","signed_at":"2026-06-08T01:05:25.289127Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.07407","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb5ee3cb21a505df840e348f89dfe3114572def2b307efbdd97e0c46a5748e76","sha256:84e223af624d26554de605d005dee61d4b8876eb5b748b94a72a695e8c31ef88"],"state_sha256":"b6cdd411a2566b78b8252dafe29d9eadd8313726670dc3d9a47be8b7765ac8fe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"24Dm9qbM+ziwzH5b46VX3ZnrEGjBygPUZMy/qWXIipHG3hc+uUEnFuWjuFt4Yw9JZVcYHcq/HfRwh7cqDFopBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T03:09:01.493348Z","bundle_sha256":"2b41ed951d5ba8ba5278418178878e4798d3e3c39314a3abaecb8e877a7e978e"}}