NON–EXISTENCE OF A UNIVERSAL ZERO ENTROPY SYSTEM VIA GENERIC ACTIONS OF ALMOST COMPLETE GROWTH

The paper proves the generic “almost complete growth” theorem (Theorem 2) via a clean Baire-category argument using sequential entropy and a precise bridge (Proposition 5) from sequential to scaling entropy, yielding that for any sublinear φ(n)=o(|F_n|) the set of zero-entropy ergodic actions with Φ(n,ε) not dominated by φ for small ε is comeager in A(G,X,μ) . The candidate solution attempts a direct tower-and-coding proof using Ornstein–Weiss towers and Gilbert–Varshamov codes, but makes a crucial quantifier/topology error: it customizes the measurable sets B_m (and hence the semimetric ρ) to the action in order to encode codewords, and later treats H_ε(·,G_{F_n}^{av}ρ) as a lower semicontinuous function of the action with ρ fixed. This breaks the openness/density scheme because the metric used to witness large entropy depends on the action being constructed. The paper avoids this by working with sequential entropy that is intrinsic to the action and then transferring to the scaling-entropy class via the Zatitskiy invariance framework , so its proof is correct while the candidate’s has a substantive gap.