Moments of finitary factor maps between Bernoulli processes

The paper proves that if a finitary factor ϕ between i.i.d. Zd processes with equal entropy has finite d/2-moment coding radius, then the informational variances coincide. It does so via a finite-torus modeling, a key inequality µ([x̂]) ≤ ν([ϕ̂n(x̂)]) (Lemma 2.3), and a CLT-based positive-part comparison, yielding Theorem 1.1 rigorously (see the statement and proof outline around Theorem 1.1 and Proposition 3.1) . By contrast, the candidate solution’s “radius–expansion” Doob-martingale argument incorrectly asserts that only a width-1 boundary layer contributes at expansion step t (δi,t = 0 unless dist∞(i, ∂Bn) ≤ t+1). Interior sites with large coding radius can also be affected when the newly revealed ring intersects their coding ball, which occurs when dist∞(i, ∂Bn) + t + 1 ≤ R(i). Properly accounting for this shows the expected number of affected sites at step t scales like |∂Bn|·E[(R−t)+], not |∂Bn|·P(R>t). Summing over t then typically requires E[R2] < ∞ (e.g., in d = 3), which is stronger than the paper’s hypothesis E[Rd/2] < ∞. The model’s subsequent finite-coloring and variance-additivity step is also not justified under unbounded radii. Hence the model’s proof does not establish the claimed result, while the paper’s proof is complete and correct under its stated assumptions .