The High-Dimensional Weierstrass Functions

The paper’s Main Theorem 1 exactly states the dimension formula dimB graph W = dimH graph W = min{log_{λ−1} b, 1 + (d − q)(1 + log_b λ)} for real-analytic φ, with q defined as the maximal dimension of V for which π_V ∘ W is Lipschitz; this matches the model’s Part (a) claim, including the equality of Hausdorff and box dimensions . The proof architecture (Ledrappier–Young formula for μ and a reduction that peels off q Lipschitz directions) is consistent with the paper’s Theorem 3.1 and the proof of Main Theorem 1 via the coordinate transform F that isolates the (d−q) non-Lipschitz directions . However, the model’s Part (b) incorrectly asserts that W^{λ,b}_φ is never Lipschitz for any non-constant φ, which contradicts the paper: there are non-constant φ with a cohomological degeneracy (H*) for which W is smooth (e.g., φ(x)=W0(x)−λW0(bx) gives W=W0 and hence q=d) . The paper’s Main Theorem 2 (p(φ,b)<d and only finitely many λ with q(φ,λ,b)>p(φ,b)) is correct and is proved via an analytic/Fourier characterization, together with an equivalence q=q′ derived from a 1D dichotomy theorem; the model’s high-level statement about finiteness of exceptional λ is consistent with the paper, but its blanket “never Lipschitz” claim is false .