Dimensions of harmonic measures on non-autonomous Cantor sets

The paper establishes both the Manning-type formulas for the dimensions of harmonic measure (Theorem 1.4) and their continuity under NACIFS perturbations (Theorem 1.3) using uniform Koebe distortion (Prop. 4.1), symbolic ASI estimates, and a careful comparison of cylinder diameters and harmonic masses, culminating in robust pressure/entropy–Lyapunov machinery; see Theorem 1.4 and its proof outline as well as the continuity argument back to the complex plane (Theorems 1.3–1.4, Prop. 4.1, ASI lemmas, and the harmonic-measure comparison estimates) . By contrast, the candidate solution’s key Step 6 incorrectly asserts exact parent-independence of the one-step ‘child probabilities’ and a resulting product structure ω(Xα)=∏p_{t,i_t}, which is not true in general; the paper instead proves only asymptotic siblings invariance (ASI) with quantitative decay and uses that to control entropies and pressures. The candidate also assumes a strong ball–cylinder comparability for harmonic measure that the paper avoids by working symbolically. Hence the paper’s argument is sound, while the model’s proof is flawed at a crucial step.