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In this article, we prove the matrix version of the Moreira theorem. We prove that if $A$ and $B$ are two finite image partition regular matrices of the same order, then for every finite coloring of the set of naturals, there exist two vectors $\\overrightarrow{X}, \\overrightarrow{Y}$ such that $\\{A\\overrightarr"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2501.16595","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-01-28T00:19:14Z","cross_cats_sorted":[],"title_canon_sha256":"34cf7f0feac8abc8745862c2c474bd65ffd5fccf444389ee68558d2a8ed218c9","abstract_canon_sha256":"d18b3e9ae83895f60cce754c67b4f54e9df1ccc8b67d2e8673a2047bb3fe5a68"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:06:15.583581Z","signature_b64":"vtcKttivmYlUltFyqr49Wu9KUgMLfzXht0CFavbDN1UZ+tniqKqtLSJ0obAYqVmlL3oW+94xojCm346IqXsaBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"664ae0e6ce7c20406a4934d848730233c2cb7d38c95505691a761d09b19c2207","last_reissued_at":"2026-07-05T10:06:15.583203Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:06:15.583203Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Matrix Formulation of Moreira Theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sayan Goswami","submitted_at":"2025-01-28T00:19:14Z","abstract_excerpt":"In a celebrated article, Moreira proved for every finite coloring of the set of naturals, there exists a monochromatic copy of the form $\\{x,x+y,xy\\},$ which gives a partial answer to one of the central open problems of Ramsey theory asking whether $\\{x,y,x+y,xy\\}$ is partition regular. In this article, we prove the matrix version of the Moreira theorem. We prove that if $A$ and $B$ are two finite image partition regular matrices of the same order, then for every finite coloring of the set of naturals, there exist two vectors $\\overrightarrow{X}, \\overrightarrow{Y}$ such that $\\{A\\overrightarr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.16595","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/2501.16595/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2501.16595","created_at":"2026-07-05T10:06:15.583265+00:00"},{"alias_kind":"arxiv_version","alias_value":"2501.16595v1","created_at":"2026-07-05T10:06:15.583265+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.16595","created_at":"2026-07-05T10:06:15.583265+00:00"},{"alias_kind":"pith_short_12","alias_value":"MZFOBZWOPQQE","created_at":"2026-07-05T10:06:15.583265+00:00"},{"alias_kind":"pith_short_16","alias_value":"MZFOBZWOPQQEA2SJ","created_at":"2026-07-05T10:06:15.583265+00:00"},{"alias_kind":"pith_short_8","alias_value":"MZFOBZWO","created_at":"2026-07-05T10:06:15.583265+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MZFOBZWOPQQEA2SJGTMEQ4YCGP","json":"https://pith.science/pith/MZFOBZWOPQQEA2SJGTMEQ4YCGP.json","graph_json":"https://pith.science/api/pith-number/MZFOBZWOPQQEA2SJGTMEQ4YCGP/graph.json","events_json":"https://pith.science/api/pith-number/MZFOBZWOPQQEA2SJGTMEQ4YCGP/events.json","paper":"https://pith.science/paper/MZFOBZWO"},"agent_actions":{"view_html":"https://pith.science/pith/MZFOBZWOPQQEA2SJGTMEQ4YCGP","download_json":"https://pith.science/pith/MZFOBZWOPQQEA2SJGTMEQ4YCGP.json","view_paper":"https://pith.science/paper/MZFOBZWO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2501.16595&json=true","fetch_graph":"https://pith.science/api/pith-number/MZFOBZWOPQQEA2SJGTMEQ4YCGP/graph.json","fetch_events":"https://pith.science/api/pith-number/MZFOBZWOPQQEA2SJGTMEQ4YCGP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MZFOBZWOPQQEA2SJGTMEQ4YCGP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MZFOBZWOPQQEA2SJGTMEQ4YCGP/action/storage_attestation","attest_author":"https://pith.science/pith/MZFOBZWOPQQEA2SJGTMEQ4YCGP/action/author_attestation","sign_citation":"https://pith.science/pith/MZFOBZWOPQQEA2SJGTMEQ4YCGP/action/citation_signature","submit_replication":"https://pith.science/pith/MZFOBZWOPQQEA2SJGTMEQ4YCGP/action/replication_record"}},"created_at":"2026-07-05T10:06:15.583265+00:00","updated_at":"2026-07-05T10:06:15.583265+00:00"}