Dynamical characterization of central sets along filter ∗

The paper proves the filter-relative central-set characterization cleanly via Lemma 3.1 (F-proximality ⇔ equality under some p ∈ F), Theorem 3.2 (uniform recurrence ⇔ fixed by an idempotent in a minimal left ideal of F), and a standard shift coding in the proof of Theorem 1.5; it also carefully ensures U is a neighbourhood of y by showing y(e)=1 in the coding step. These steps are present and consistent in the uploaded PDF (see Theorem 1.5 statement/definitions and its proof, and supporting lemmas: Lemma 3.1, Theorem 3.2, Lemmas 3.3–3.5) . By contrast, the model’s solution contains two substantive flaws: (i) it relies on a nonstandard and generally false claim that if p is an idempotent in K(F̄) and A ∈ p, then A is F-syndetic (already false in the classical case F={S}); and (ii) in the (i)⇒(ii) coding step it declares U={w: w(1)=1} without proving that y=T_p(x) satisfies y(1)=1, hence not ensuring U is a neighbourhood of y. The paper avoids both issues via the minimal-ideal machinery and an explicit y(e)=1 verification in the proof of Theorem 1.5 .