A VARIATIONAL PRINCIPLE FOR THE METRIC MEAN DIMENSION OF LEVEL SETS

The paper establishes the equality mdim_M(K_α,f,d) = Λ_φ^{mdim_M}(f,α,d) = H_φ^{mdim_M}(f,α,d) under specification via three rigorously proved inequalities: mdim_M(K_α) ≤ Λ_φ^{mdim_M} (Proposition 3.1), H_φ^{mdim_M} ≤ mdim_M(K_α) (Proposition 3.2), and H_φ^{mdim_M} ≥ Λ_φ^{mdim_M} (Proposition 3.10), which force all three quantities to coincide. The candidate solution’s Step 1 mirrors the paper’s upper bound correctly. However, its Step 2 claims a direct lower bound h(K_α,f,·) ≥ Λ_φ(α,·) from a one-shot specification concatenation, but it does not ensure that the constructed points lie in K_α (only that a single long finite-time Birkhoff average is close to α). The paper overcomes this with a Cantor-like construction plus an entropy-distribution argument. The candidate’s Step 3 also attempts to bound H_φ from above by h(K_α,f,ε) using Birkhoff for ergodic measures, but the supremum in H_φ runs over all invariant measures and the concavity/inf-over-partitions subtlety prevents the reduction to ergodic measures in this way. Hence the model’s proof is incomplete, while the paper’s argument is complete and correct.