FILLED JULIA AND MANDELBROT SETS FOR DYNAMICS OVER THE NORMED REAL NONASSOCIATIVE ALGEBRAS

The paper proves that for a normed real nonassociative algebra whose norm satisfies a norm square inequality, both the filled Julia set KA(fc) and the Mandelbrot set MA are bounded and closed. It does so via an escape threshold λ = max{2/η, ||c||} (Theorems 3.1–3.3), plus continuity and a sequential-closure argument. The model reaches the same conclusions using a different but compatible escape radius R(c) = max{2/η, sqrt(2||c||/η)} and expresses KA(fc) and MA as intersections of preimages/sublevel sets of continuous maps. The logical steps align; the model’s escape radius is a valid (in fact, slightly sharper) sufficient condition, and its topological arguments rely on the same continuity assumptions used in the paper. No missing hypotheses undermine either argument.