This submission gives a full Lean proof that finite-graph twin-width and mixed minor number are functionally equivalent: each parameter is bounded by a numerical function of the other.

The concept surface has two review units. The first gives the complete definition of mixed minor number — interval divisions, mixed matrix cells, mixed minors, and vertex-ordered adjacency matrices — phrased with plain structures and Prop-valued predicates, and with the parameter an infimum of an explicit set of naturals. The second states the functional equivalence with the twin-width parameter of submission Lax1, applied directly to the two parameters through a shared uniform signature. The proof runs through the Marcus–Tardos theorem, the matrix grid-minor theorem, and the graph twin-decomposition bridge; both submitted parameters are proved pointwise equal to their source counterparts, via the black-is-complete / red-is-non-homogeneous invariant for contraction sequences and a pointwise division-and-cell translation for mixed minors.

Ported from the original formalization by Édouard Bonnet (github.com/EdouardBonnet/leaning, twin-width, MIT-licensed).