A transcendental non-Archimedean Calabi--Yau Theorem with applications to the cscK problem

Let $X$ be a compact K\"ahler manifold and $\alpha$ a K\"ahler class on $X$. We prove that if $(X,\alpha)$ is uniformly K-stable for models, then there is a unique cscK metric in $\alpha$. This was first proved in the algebraic case by Chi Li, and it strengthens a related result in an article of Mesquita-Piccione. K-stability for models is defined in terms of big test configurations, but we also give a valuative criterion as in the work of Boucksom--Jonsson together with an explicit formula for the associated $\beta$-invariant. To accomplish this we further develop the non-Archimedean pluripotential theory in the transcendental setting, as initiated in the works of Darvas--Xia--Zhang and Mesquita-Piccione. In particular we prove the continuity of envelopes and orthogonality properties, and using that, we are able to extend the non-Archimedean Calabi-Yau Theorem found in an article of Boucksom--Jonsson to the general K\"ahler setting.