  Fix two positive integers $d$ and $k$, a positive number $v$, and a finite
set $I$ of non-negative rational numbers, we prove that the K-semistable domain
of the log pair $(X, \sum_{j=1}^kD_j)$ is a rational polytope lying in the
$k$-dimensional simplex $\overline{\Delta^k}$, where $X$ is a Fano variety of
dimension $d$, $D_j\sim_\mathbb{Q} -K_X$, $(-K_X)^d=v$, the coefficients of
$D_j$ are contained in $I$, and $(X, \sum_{j=1}^kc_jD_j)$ is a K-semistable log
Fano pair for some $c_j\in [0,1)\cap \mathbb{Q}$. Moreover, we show that there
are only finitely many polytopes which may appear as the K-semistable domains
for such log pairs. Based on this, we establish a wall crossing theory for
K-moduli with multiple boundaries.
