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Under these hypotheses this establishes the planarity conjecture stated in \\cite{Pask}. The obstruction side of the argument uses only the non-planarity of $K_5$; it makes no appeal to the four-colour theorem. The engine is a monotonicity property of the set of colours emitted at a vertex (``backward propagation''), which forces, i"},"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":"2606.05727","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-06-04T05:36:44Z","cross_cats_sorted":["math.CO","math.OA"],"title_canon_sha256":"8fec2e64604e9d0ac936fc422bedb130f8c2ff405e315bcd512d4207ca821347","abstract_canon_sha256":"fdc4fc7ad3891be953a07a6a370b17d3dc60229e62c3c570ec527defd6485eff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-05T01:15:00.858254Z","signature_b64":"xEjGa97yhetu1uKWrngikFnx9PzwYAYbGwq5q8cnQ/umRc9K0YF9huTqoQWxhfsBCmCarwBxwAQGRFt3LDKmCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c8a9b746c7d1e09dee2b02ef60b9a2a1dee960f7753093d35c0ac3462bff583","last_reissued_at":"2026-06-05T01:15:00.857849Z","signature_status":"signed_v1","first_computed_at":"2026-06-05T01:15:00.857849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Planar higher-rank trees have rank at most four","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.OA"],"primary_cat":"math.CT","authors_text":"David Pask","submitted_at":"2026-06-04T05:36:44Z","abstract_excerpt":"We prove that a finite, connected, singly connected, locally convex higher-rank tree whose $1$-skeleton is planar and which is \\emph{non-degenerate}, in the sense that every edge of each colour forms a commuting square with every other colour, has rank at most four. 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