Generic Hölder Level Sets and Fractal Conductivity

The paper proves rigorously that for F = C × C built from the explicit fat Cantor C with ln = 1/(2^{n+1}−1), there is a phase transition at α = 1/2: D*(α,F) = 0 for 0 < α < 1/2 and D*(α,F) = 1 for 1/2 < α ≤ 1 (Theorem 6.1). The α < 1/2 side follows from a general density result on (ν,ρ)-separated structures; the α > 1/2 side uses a Hausdorff capacity estimate (Lemma 6.3), an extension theorem, and a Fubini argument to force 1-dimensional fibers on a dense open set, combined with the general upper bound D* ≤ dim_B(F) − 1 = 1. All key steps are present and justified in the paper. By contrast, the candidate solution outlines a different approach via Marstrand–Mattila slicing and a Baire stability scheme for α > 1/2, and a multi-scale ‘core quantization’ construction for α < 1/2, but leaves crucial lemmas unstated and unproved, and contains inaccuracies (e.g., asserting a 1-Lipschitz projection is automatically in C_1^α(F) without normalizing constants). Hence the model’s argument is incomplete and unreliable, even though it guesses the correct final statement.