Establishes an information-geometric framework in free probability where the role of a probability distribution is played by a model-theoretic type: a functional that evaluates not only non-commutative polynomials but every logical formula built from them with suprema and infima. Within it one proves lower bounds for several free-entropy quantities along Wasserstein geodesics on the type space, characterizes their semicontinuity and genericity, and proves existence and uniqueness of quasi-moment types solving a regularized version of Santambrogio's variational problem. A counterexample shows that non-commutative laws alone cannot reconcile Wasserstein distance and entropy in the large-n limit, which is precisely why types are the correct object.
A free-probability research paper (Jekel) developing Wasserstein information geometry for the large-n limit of unitarily-invariant random multi-matrix ensembles. Its central move is to represent a di…