ON RELATIVE METRIC MEAN DIMENSION WITH POTENTIAL AND VARIATIONAL PRINCIPLES

The paper proves a relative variational principle for metric mean dimension with potential via entropy of partitions (Theorem 3.4) using a local relative variational principle for pressure at a fixed open cover (Theorem 3.3), together with standard separated/spanning/cover inequalities (Lemma 2.2), and then passes to the ε→0 limit to get equality with a “max” over invariant (even ergodic) measures. The statement and proof are coherent and complete in the paper . In contrast, the model’s Step 1 incorrectly asserts, for each fixed ε, the exact equality P(d,T,ψ,ε|Y) = sup_{mesh(U)≤ε} P(T,ψ,U|Y). The paper does not claim this fixed-scale equality (it only relates them via inequalities and limits), and the model’s proof sketch for the reverse inequality glosses over multiplicity issues in the cover-to-separated transition and misuses the modulus of continuity. The model also misattributes the needed local (cover-level) variational principle to Ledrappier–Walters; the paper correctly cites a relative local principle (Theorem 3.3) rather than the global Ledrappier–Walters formula . These errors make the model’s proof formally incorrect, even though its overall strategy aligns with the paper and could be repaired by following the paper’s argument.