A classical Wu-manifold answer to a free-rank question

The fixed Wu manifold W = SU(3)/SO(3), with the real-matrix SO(3) inclusion, is a finite simply connected complex of rational homotopy type S^5. Its actual continuous free toral rank and free 2-rank are zero, while its free p-rank is one for every odd prime p. Every cyclic group of odd order acts freely and smoothly on this same W. This supplies an affirmative answer to the existence question printed on page 1912 of Hanke's OWR 32/2010 contribution, under its literal actual-space free-action definitions and all its homotopy hypotheses. Almost-free and rational toral ranks are different invariants; no replacement space or prime-dependent space is used. This explanatory research note assembles classical facts. Yeroshkin's 2014 preprint and 2015 circle-isotropy calculation already contain the weights (1,1,-2), whose odd cyclic subgroups act freely. Kuhn and Lloyd's 2020 preprint already explicitly discusses the no-free-Wu-involution obstruction; the 2022 version retains it. Debray and Yu's 2024 paper and author-hosted 2025 version give the classical tangent characteristic number and credit older sources. The note supplies a complete topological covering and tangent-microbundle argument covering arbitrary continuous involutions, and a two-row Borel spectral-sequence proof of the odd-primary rank-one upper bound. It claims no new manifold, circle action, classical ingredient or classification. Earlier actual free finite-versus-circle gaps are credited explicitly: Mundet i Riera's 2021 first preprint, Theorem 1.14 and Section 15 (published Theorem 1.11 and Section 13, online 2024, volume 2026), supplies regular self-covers of every odd degree of the fixed Cappell-Weinberger-Yan manifolds. Circle-cover pullback gives actual free cyclic actions for every odd prime, while nontrivial continuous circle actions are excluded. These do not meet Hanke's abelian fundamental-group hypothesis: their mapping-torus group contains a nontrivial centerless, hence nonabelian, subgroup. They are not dismissed merely as effective-action examples. Two independently positioned bounded priority audits and root primary-source checks verified no complete predecessor meeting all the selected question's hypotheses in the inspected statements. Priority remains unresolved; older books, theses, journal versions, citation coverage and unindexed work remain gaps. No firstness, certified novelty, absence theorem or current worldwide-open claim is made. The companion archive preserves all 48 frozen candidate files, including three original archives, exact statement and full selected-record identities, two independent mathematical families, two independently positioned priority families, postseal adversarial checks and reproducible finite diagnostics. Core verification uses Python's standard library. Optional primary-body custody is distinguished from core authored-source verification; full third-party PDFs, extracted texts and rendered pages are not redistributed. Finite computations support the written proofs and do not formally certify topology or establish an infinite-prime claim by sampling. AI tools were used extensively in development, mathematical and literature investigation, drafting, computation, reproduction and adversarial review. Reviewer session reuse and exposure limits are documented. This is an unrefereed preprint without external human peer review or proof-assistant certification. The note and authored companion material are licensed CC BY 4.0.