K-moduli with real coefficients

In this paper, we develop an algebraic K-stability theory (e.g. special test configuration theory and optimal destabilization theory) for log Fano $\mathbb R$-pairs, and construct a proper K-moduli space to parametrize K-polystable log Fano $\mathbb R$-pairs with some fixed invariants (e.g. dimension, volume, coefficients). All of these are well-known for log Fano $\mathbb Q$-pairs, and the strategy in this paper is trying to reduce the problems (in many cases) to $\mathbb Q$-coefficients case rather than rebuilding the whole program as in $\mathbb Q$-coefficients case.