Remark on the local nature of metric mean dimension

Tsukamoto’s Theorem 1.1 exactly states the local formula the model aims to prove and provides a complete proof using Bowen’s product lemma (Lemma 2.1) and a key covering estimate (Proposition 2.2), culminating in the bound S(X, ε) ≤ S(X, δ) + sup_x S(B_δ(x, d_Z), ε/4) and hence the desired limsup/liminf equalities for metric mean dimension . By contrast, the model’s Step 3 asserts a stronger inequality S(X, Cε) ≤ sup_x S(B_δ(x, d_ℤ), ε) (no additive term), attributing it to Bowen–Tsukamoto. This inequality does not appear in the paper and in fact omits an essential additive term S(X, δ) that arises from covering X by δ–Bowen balls; Tsukamoto’s argument explicitly carries this term through the estimates . While the model’s final conclusion (the local formula) matches the paper, the proof sketch relies on a false intermediate claim, so the model’s proof is not correct as stated.