MCMULLEN’S AND GEOMETRIC PRESSURES AND APPROXIMATING HAUSDORFF DIMENSION OF JULIA SETS FROM BELOW

The paper’s Main Theorem (Theorem 2.8) asserts: (1) P(f,t)=P_tree(f,t,z)=P^0_tree(f,t,z) for all non-exceptional z; (2) under uniformly shrinking puzzles, P(f,t)=P̂_McM(f,t)=P^0_McM(f,t) ≤ P_McM(f,t); (3) under (buas), P(f,t)=P_pullinf_tree(f,t) and the branchwise subexponential ratio (2.11) holds. All three items are explicitly stated and proved or reduced to standard results within the paper’s framework , with the definitions and constructions spelled out in Sections 2–5 . The candidate solution reconstructs essentially the same argument: (A) relies on known equivalences of pressures for rational maps; (B) codes via McMullen-type puzzles and spectral radii and handles near-critical truncation/fuzzy weights; (C) uses buas to obtain uniform shrinking and Koebe distortion to deduce (2.11). Apart from a minor, unnecessary strengthening of the required shrinking rate in (C), the model’s solution tracks the paper’s logic and conclusions accurately. Hence both are correct and substantially aligned.