Topological dynamics beyond Polish groups

The paper proves the equivalence of (1) CAP, (2) strongly CAP, (3) closedness of AP(Lip(d)) for all continuous right-invariant diameter-1 pseudometrics d, and (4) UEB on M(G), via (2)⇒(1)⇒(3), (4)⇒(2), and the contrapositive ¬(4)⇒¬(3); it also identifies the UEB uniformity with the one generated by the entourages [Lip(d), ε] on S(G) (Proposition 4.19), thereby giving several interchangeable descriptions of the same uniform structure . The candidate solution directly supplies a clean new argument for (3)⇒(1) using the Lipschitz test X→Lip(dψ) and then appeals to the paper’s Theorem 6.1 for the remaining implications, matching the paper’s main theorem. One minor technical slip is the claim that, for a UEB set H, the gauge pseudometric dH(g,h):=sup_{f∈H}|f(g)−f(h)| is right-invariant; without additional H-invariance assumptions this is not automatic. The uniformity identification is nevertheless correct as stated in the paper (Proposition 4.19), which the solution also cites. Net: the paper’s argument is correct and complete; the model’s solution is correct in substance, offers a different proof of (3)⇒(1), and has only a minor fixable detail in Step B.