LOWERING MEAN TOPOLOGICAL DIMENSION

The paper proves Theorem 1.2 via Proposition 5.1: for fixed ε,δ it constructs a factor with mdim(Y)<δ and per-block relative width-dimension bounded by <δ, and then passes to an infinite product to achieve mdim(π,T)=0 (see Theorem 1.2 and the reduction via Proposition 5.1) . The candidate solution matches the paper through the tiling construction (Proposition 3.1), the width-dimension lemma (Lemma 4.3), the definition of g, and the small mean-dimension bound of Y_g (Steps 2–6 align with the text) . However, Step 7 of the candidate solution incorrectly concludes that equality of outputs along all times forces ε-closeness on every length-m orbit block for the constructed single factor; this misuses the Lemma 4.3 fiber-embedding (E_p) because the factor encodes only a coordinate of F, not the values of E_p on the fiber, and also overinterprets property (3) in Proposition 3.1 regarding where 0 sits relative to a tile’s interior. The paper does not claim such pointwise ε-diameter control; it only obtains a quantitative width-dimension bound and then takes a product over scales to deduce mdim(π,T)=0 .