A NONLINEAR VERSION OF THE NEWHOUSE THICKNESS THEOREM

The paper’s Theorem 1.1 states that if the partial-derivative ratios satisfy (τ(Ki))^{-1} ≤ |∂xi f/∂z f| ≤ τ(Kd) on [0,1]^d, then f(K1,…,Kd) is exactly the interval between its min and max on K1×…×Kd. The proof covers each forbidden level set by finitely many axis-parallel “gap strips/cubes,” selects a minimal-width one, and derives a contradiction via slope bounds and thickness, first in d=2 (Lemmas 2.1–2.2) and then by slicing for general d; see the statements of Theorem 1.1 and Corollary 1.4 and their proofs in Sections 2.1–2.2 of the PDF . The candidate solution mimics this “minimal strip” approach but introduces two critical mistakes: (i) it asserts each level set f= y is a global graph z=g_y(x′) over all x′∈[0,1]^{d−1}, which need not hold (monotonicity in z gives uniqueness if a solution exists for a fixed x′, not existence for every x′); and (ii) it relies on equality cases in mean-value-type inequalities to force constant slopes, which is unjustified. These gaps undermine the correctness of the candidate’s argument, whereas the paper’s method avoids these pitfalls by covering and slicing directly.