Algebraic uniqueness of Kahler-Ricci flow limits and optimal degenerations of Fano varieties

We prove that for any Fano manifold X, the special R-test configuration that minimizes the H-NA functional is unique and has a K-semistable Q-Fano central fiber (W; epsilon). Moreover there is a unique K-polystable degeneration of (W; epsilon). As an application, we confirm the conjecture of Chen, Sun and Wang about the algebraic uniqueness for Kahler-Ricci flow limits on Fano manifolds, which implies that the Gromov-Hausdorff limit of the flow does not depend on the choice of initial Kahler metrics. The results are achieved by studying algebraic optimal degeneration problems via new functionals for real valuations over Q-Fano varieties, which are analogous to the minimization problem for normalized volumes.