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Let $d\\in\\mathbb N\\cup\\{\\infty\\}$ be the conditional topomorphic degree; when $d<\\infty$, it is the least integer such that $\\pi$ is an at most $d$-to-one topomorphic extension. We prove that, for every $r\\ge2$, the following are equivalent: the system is Weyl mean $r$-equicontinuous; it is mean $r$-equicontinuous along some F{\\o}lner sequence; and $d\\le r-1$. For minimal $\\mathbb Z$-systems, this resolves a conject"},"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":"2607.27400","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-29T19:13:23Z","cross_cats_sorted":[],"title_canon_sha256":"dbe91b66e344ada48bbcb8315874163071df8dcc4cfd4f1b2b113621cca5bbdb","abstract_canon_sha256":"9ba0154ab359a83f6c9cdf120174a413a6c5c8438a438d993f6af817b3c36b5d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5dbbc850160918b3d60427eab91ef28f39e4ddbe30e9110ca5e0a18acedcfa42","last_reissued_at":"2026-07-31T00:11:41.110776Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-31T00:11:41.110776Z"},"graph_snapshot":{"paper":{"title":"Conditional Topomorphic Degree and Multivariate Mean Equicontinuity for Minimal Amenable Group Actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Chunlin Liu","submitted_at":"2026-07-29T19:13:23Z","abstract_excerpt":"Let $G$ be a countably infinite discrete amenable group acting minimally on a compact metric space $X$, and let $\\pi:X\\to X_{\\mathrm{eq}}$ be the maximal equicontinuous factor map. Let $d\\in\\mathbb N\\cup\\{\\infty\\}$ be the conditional topomorphic degree; when $d<\\infty$, it is the least integer such that $\\pi$ is an at most $d$-to-one topomorphic extension. We prove that, for every $r\\ge2$, the following are equivalent: the system is Weyl mean $r$-equicontinuous; it is mean $r$-equicontinuous along some F{\\o}lner sequence; and $d\\le r-1$. 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