Mixed-Variable Optimisation as a Metric Product Space: Transient Categorical Geometry and a Hierarchy of Local Optimality

This preprint develops a geometric framework for mixed-variable optimisation in which categorical variables are endowed with transient cyclic-ordering-induced metrics.

The mixed search space is formalised as a weighted Cartesian product of metric spaces, and it is shown that dynamic rotation of the categorical component preserves metric validity and induces uniform metric equivalence across epochs.

A hierarchy of local optimality (instantaneous, robust, and categorical) is introduced, together with explicit coverage bounds establishing that ⌈(k−1)/2⌉ cyclic orderings suffice for full categorical adjacency coverage for a k-level variable.

The results provide structural foundations for mixed-variable optimisation independent of any particular algorithmic paradigm.