Exponential Laws for Mixed-Order Ultradifferentiable Mappings

We introduce the mixed-order classes C [ M ] r , s by combining mixed differentiability with the weight-matrix framework, in both the Beurling and Roumieu cases. At finite order we prove a chain rule under the Faà-di-Bruno property. The currying map is a linear injection. If r and s are finite, the class coincides with C r , s and the exponential law reduces to the mixed-order law without weights. Continuity of the inverse on the image holds when a lower weight bound is available at a single index.