Complex fractal trainability boundary can arise from trivial non-convexity

The paper empirically demonstrates renormalization invariance for the additive and multiplicative cosine-perturbed quadratics, deriving ε̃=ε/b^2, λ̃=λ/b for the additive case and ε̃=ε, λ̃=λ/b for the multiplicative case, and concludes that α depends only on θ+ = ε/λ^2 and θ× = ε, respectively, with a sharp empirical transition to non-zero α near θ+ = 1/(2π^2) where f+ becomes non-convex. These steps are supported by their Methods and figures but are not presented as rigorous theorems; the paper explicitly frames results as numerical/heuristic and discusses limitations and assumptions (e.g., approximate independence of α from initial conditions) . The candidate solution gives exact scaling conjugacies and the convexity threshold, and derives standard L-smooth descent guarantees and an explicit escape criterion, but (correctly) stops short of a general proof that α=0 throughout the entire convex additive regime or that α>0 always holds in the multiplicative case. Since the paper does not offer a complete proof of these global fractality claims either—and indeed acknowledges methodological limits of box-dimension estimation and classification heuristics—the overall status of the fully rigorous statements appears open as of the cutoff .