Averaging principle for rough slow-fast systems of level 3

The paper proves the strong averaging principle for level-3 rough slow–fast systems under (A),(H1)–(H6), using a level-3 controlled-path stability estimate that requires the 3β-Hölder norm of the additive remainder M. It establishes non-explosion, a Khasminskii decomposition, and a key bound E[||M||_{3β}^2] ≤ C(δ^{2β} + δ^{2(1−3β)} + δ^{−6β}ε), from which convergence follows by choosing δ ≍ ε^{1/(6β)} log(1/ε). The candidate solution omits the need for 3β-regularity, works only with a β-Hölder control of H^ε(t) = ∫ R^ε ds, and derives an incorrect two-term balance δ^{β} + ε/δ. This misses the level-3 requirement and the δ^{2(1−3β)} contribution, so its stability step and optimization are not justified for level-3 RDEs.