Ax-Schanuel Conjectures for Variations of Mixed Hodge Structures in O-Minimal Expansions

We establish a functional transcendence theorem of Ax-Schanuel type for admissible variations of mixed Hodge structures (VMHS). By embedding the period domains of VMHS into a definable category within the o-minimal structure Ran, exp, we analyze the geometry of the period map : S D. We prove that if V is an algebraic subvariety of S intersecting the graph of the period map in a locus of excessive dimension, then the projection of this intersection to S is contained in a proper weak Mumford-Tate subdomain. This result generalizes previous work by Mok, Pila, and Tsimerman on Shimura varieties and Bakker and Tsimerman on pure variations of Hodge structures. Our approach leverages the definability of the mixed period map and recent volume bounds on Griffiths transverse subvarieties, providing a crucial step towards the Zilber-Pink conjectures for mixed Shimura varieties.