Double variational principle for mean dimensions with sub-additive potentials

The paper’s main theorem (a double variational principle for mean dimension with sub-additive potentials) is stated and proved via three steps: (i) fixed-metric bounds linking rate-distortion dimension plus F*(µ) to metric mean dimension, (ii) a dynamical Frostman lemma giving mdim_H ≤ sup_µ(rdim ± + F*(µ)), and (iii) a marker-property metric realizing mdim_M = mdim, from which the minimax equalities follow. In contrast, the model asserts a stronger fixed-metric equality sup_µ(rdim_lower/upper + F*(µ)) = mdim_M^−/+, which the paper does not claim. The paper shows only the inequality chain sup_µ(rdim ± + F*(µ)) ≤ mdim_M and mdim_H ≤ sup_µ(rdim ± + F*(µ)); equality at a fixed metric is obtained only after choosing the realization metric and combining with mdim ≤ mdim_H. The model also omits the paper’s tame-growth assumption used in Step (ii). Hence, while the final conclusion matches, the model’s Step (1) argument is flawed relative to the paper.