{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:TGAFZCEWY6QP27RCQU4UR3YDOZ","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":"a05929716738850f462389f3b2bab563259581c11da0a446643f8a8a3c1d596c","cross_cats_sorted":["cs.SC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-26T16:50:00Z","title_canon_sha256":"bc3ccb6e44fed029f3b4491674729a50dbf87b168d0d69bba198b33ff05ccc04"},"schema_version":"1.0","source":{"id":"1401.6667","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.6667","created_at":"2026-05-18T03:00:36Z"},{"alias_kind":"arxiv_version","alias_value":"1401.6667v2","created_at":"2026-05-18T03:00:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.6667","created_at":"2026-05-18T03:00:36Z"},{"alias_kind":"pith_short_12","alias_value":"TGAFZCEWY6QP","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"TGAFZCEWY6QP27RC","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"TGAFZCEW","created_at":"2026-05-18T12:28:49Z"}],"graph_snapshots":[{"event_id":"sha256:3734dfbdc9c84b05471a033a75f893197cc531612dbf118d75f2fb0afca781c3","target":"graph","created_at":"2026-05-18T03:00:36Z","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":"For a prime $p$ and a matrix $A \\in \\mathbb{Z}^{n \\times n}$, write $A$ as $A = p (A \\,\\mathrm{quo}\\, p) + (A \\,\\mathrm{rem}\\, p)$ where the remainder and quotient operations are applied element-wise. Write the $p$-adic expansion of $A$ as $A = A^{[0]} + p A^{[1]} + p^2 A^{[2]} + \\cdots$ where each $A^{[i]} \\in \\mathbb{Z}^{n \\times n}$ has entries between $[0, p-1]$. Upper bounds are proven for the $\\mathbb{Z}$-ranks of $A \\,\\mathrm{rem}\\, p$, and $A \\,\\mathrm{quo}\\, p$. Also, upper bounds are proven for the $\\mathbb{Z}/p\\mathbb{Z}$-rank of $A^{[i]}$ for all $i \\ge 0$ when $p = 2$, and a conje","authors_text":"Andy Novocin, Mark Giesbrecht, Mustafa Elsheikh","cross_cats":["cs.SC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-26T16:50:00Z","title":"Ranks of Quotients, Remainders and $p$-Adic Digits of Matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.6667","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:b2c1a6be94bd6ac82beb3841387a3b399250cf8fb78bdf9ab1e92ab672eea410","target":"record","created_at":"2026-05-18T03:00:36Z","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":"a05929716738850f462389f3b2bab563259581c11da0a446643f8a8a3c1d596c","cross_cats_sorted":["cs.SC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-26T16:50:00Z","title_canon_sha256":"bc3ccb6e44fed029f3b4491674729a50dbf87b168d0d69bba198b33ff05ccc04"},"schema_version":"1.0","source":{"id":"1401.6667","kind":"arxiv","version":2}},"canonical_sha256":"99805c8896c7a0fd7e22853948ef037641225d2dfebe625e205d4ace1c8eacc5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99805c8896c7a0fd7e22853948ef037641225d2dfebe625e205d4ace1c8eacc5","first_computed_at":"2026-05-18T03:00:36.608880Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:00:36.608880Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Hz78SsjHf03A/XLBtZnGKLw9CesjpbxwQaEx33WB4LxQaRh254B1k/OKWGz/oBaNBMP+TrQfZg1ctSnxbUGnDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:00:36.609638Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.6667","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2c1a6be94bd6ac82beb3841387a3b399250cf8fb78bdf9ab1e92ab672eea410","sha256:3734dfbdc9c84b05471a033a75f893197cc531612dbf118d75f2fb0afca781c3"],"state_sha256":"c9163af557887d53f1aeb1d542381b68e9690051ee1582dcd393359838ae01a1"}