{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:L7BGLHAWNT4ZHWYKFYUP4VXNLZ","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":"39f6cb48f025fc1423c017464bc480c2c1dff8d22492715263d357ab61ac381c","cross_cats_sorted":["math.RA"],"license":"","primary_cat":"math.CO","submitted_at":"2006-11-18T00:27:24Z","title_canon_sha256":"a32cba68921dd4f454cc9eeeaf08240d3a5f05ca53d1809d856f63611d6fd9eb"},"schema_version":"1.0","source":{"id":"math/0611551","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0611551","created_at":"2026-05-18T04:27:15Z"},{"alias_kind":"arxiv_version","alias_value":"math/0611551v1","created_at":"2026-05-18T04:27:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611551","created_at":"2026-05-18T04:27:15Z"},{"alias_kind":"pith_short_12","alias_value":"L7BGLHAWNT4Z","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"L7BGLHAWNT4ZHWYK","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"L7BGLHAW","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:8196eba10288787d6407b3fea2fb749e231d6ff75e24e1dad0990081cd1769dc","target":"graph","created_at":"2026-05-18T04:27:15Z","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":"Let the columns of a $p \\times q$ matrix $M$ over any ring be partitioned into $n$ blocks, $M = [M_1, ..., M_n]$. If no $p \\times p$ submatrix of $M$ with columns from distinct blocks $M_i$ is invertible, then there is an invertible $p \\times p$ matrix $Q$ and a positive integer $m \\leq p$ such that $QM = [QM_1, ..., QM_n]$ is in reduced echelon form and in all but at most $m-1$ blocks $QM_i$ the last $m$ entries of each column are either all zero or they include a non-zero non-unit.","authors_text":"Erkko Lehtonen, Stephan Foldes","cross_cats":["math.RA"],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2006-11-18T00:27:24Z","title":"Column-partitioned matrices over rings without invertible transversal submatrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611551","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":"sha256:9efdd0e3b7ffd7ff9490fcc4e1a6144cd034b57cc1a91dffc240dd391ed0f291","target":"record","created_at":"2026-05-18T04:27:15Z","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":"39f6cb48f025fc1423c017464bc480c2c1dff8d22492715263d357ab61ac381c","cross_cats_sorted":["math.RA"],"license":"","primary_cat":"math.CO","submitted_at":"2006-11-18T00:27:24Z","title_canon_sha256":"a32cba68921dd4f454cc9eeeaf08240d3a5f05ca53d1809d856f63611d6fd9eb"},"schema_version":"1.0","source":{"id":"math/0611551","kind":"arxiv","version":1}},"canonical_sha256":"5fc2659c166cf993db0a2e28fe56ed5e5b4df67b668f58d711207c21bc6a9a90","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5fc2659c166cf993db0a2e28fe56ed5e5b4df67b668f58d711207c21bc6a9a90","first_computed_at":"2026-05-18T04:27:15.464144Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:15.464144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Zcl9NVRHoew2RX6x0oQDY1ErISf1bccCPohfTfDdKZpHZyz3UZTAESkxIr7oy5Z+4O0pIDIfXv0VaDyG5hZAAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:15.464705Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0611551","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9efdd0e3b7ffd7ff9490fcc4e1a6144cd034b57cc1a91dffc240dd391ed0f291","sha256:8196eba10288787d6407b3fea2fb749e231d6ff75e24e1dad0990081cd1769dc"],"state_sha256":"26c0ea4bedc55641376baa8f16abd3e9638e2e4bdebc4e3b131fbadabeabea1e"}