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The Bounded Canopy Conjecture is that all GF(q)-representable matroids M that have an N-minor and are N-fragile have branch width bounded by a constant depending only on q and N.\n  A matroid N stabilizes a class of matroids over a field F if, for every matroid M in the class with an N-minor, every F-representation of N extends to at most one F-representation of M.\n  We prove that, if Rota's conjecture is false for GF(q), then either the Bounded Canopy Conjecture is false for GF(q)"},"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":"1006.1418","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-08T00:48:48Z","cross_cats_sorted":[],"title_canon_sha256":"81c76ef5a0eb13c589de6d9bd756a4710cd13e01ddb696364dfb6addd2ddbe73","abstract_canon_sha256":"f0af6e0af990319d25cf86ee814e068e4ba3d2ddb039dbbf543fb851020264d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:32.227338Z","signature_b64":"NKZ1bcAjsBoDJ1tHKbyhAOTzGZ/SQ9ITuaJSlmR7Re0boG0CX4nQiO1e6Y+2e2Ts3p9Bxg3gB8jNTcGdBgi8AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88424e4149f41962861ebb9a51f4a5b961a37f33870f7a688cd881a3bd5e02db","last_reissued_at":"2026-05-18T04:16:32.226948Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:32.226948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stability, fragility, and Rota's Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dillon Mayhew, Geoff Whittle, Stefan H.M. van Zwam","submitted_at":"2010-06-08T00:48:48Z","abstract_excerpt":"Fix a matroid N. A matroid M is N-fragile if, for each element e of M, at least one of M\\e and M/e has no N-minor. The Bounded Canopy Conjecture is that all GF(q)-representable matroids M that have an N-minor and are N-fragile have branch width bounded by a constant depending only on q and N.\n  A matroid N stabilizes a class of matroids over a field F if, for every matroid M in the class with an N-minor, every F-representation of N extends to at most one F-representation of M.\n  We prove that, if Rota's conjecture is false for GF(q), then either the Bounded Canopy Conjecture is false for GF(q)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.1418","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1006.1418","created_at":"2026-05-18T04:16:32.227008+00:00"},{"alias_kind":"arxiv_version","alias_value":"1006.1418v2","created_at":"2026-05-18T04:16:32.227008+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.1418","created_at":"2026-05-18T04:16:32.227008+00:00"},{"alias_kind":"pith_short_12","alias_value":"RBBE4QKJ6QMW","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_16","alias_value":"RBBE4QKJ6QMWFBQ6","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_8","alias_value":"RBBE4QKJ","created_at":"2026-05-18T12:26:13.927090+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/RBBE4QKJ6QMWFBQ6XONFD5FFXF","json":"https://pith.science/pith/RBBE4QKJ6QMWFBQ6XONFD5FFXF.json","graph_json":"https://pith.science/api/pith-number/RBBE4QKJ6QMWFBQ6XONFD5FFXF/graph.json","events_json":"https://pith.science/api/pith-number/RBBE4QKJ6QMWFBQ6XONFD5FFXF/events.json","paper":"https://pith.science/paper/RBBE4QKJ"},"agent_actions":{"view_html":"https://pith.science/pith/RBBE4QKJ6QMWFBQ6XONFD5FFXF","download_json":"https://pith.science/pith/RBBE4QKJ6QMWFBQ6XONFD5FFXF.json","view_paper":"https://pith.science/paper/RBBE4QKJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1006.1418&json=true","fetch_graph":"https://pith.science/api/pith-number/RBBE4QKJ6QMWFBQ6XONFD5FFXF/graph.json","fetch_events":"https://pith.science/api/pith-number/RBBE4QKJ6QMWFBQ6XONFD5FFXF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RBBE4QKJ6QMWFBQ6XONFD5FFXF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RBBE4QKJ6QMWFBQ6XONFD5FFXF/action/storage_attestation","attest_author":"https://pith.science/pith/RBBE4QKJ6QMWFBQ6XONFD5FFXF/action/author_attestation","sign_citation":"https://pith.science/pith/RBBE4QKJ6QMWFBQ6XONFD5FFXF/action/citation_signature","submit_replication":"https://pith.science/pith/RBBE4QKJ6QMWFBQ6XONFD5FFXF/action/replication_record"}},"created_at":"2026-05-18T04:16:32.227008+00:00","updated_at":"2026-05-18T04:16:32.227008+00:00"}