  We study the relation between the coregularity, the index of log Calabi-Yau
pairs, and the complements of Fano varieties. We show that the index of a log
Calabi-Yau pair $(X,B)$ of coregularity $1$ is at most $120\lambda^2$, where
$\lambda$ is the Weil index of $K_X+B$. This extends a recent result due to
Filipazzi, Mauri, and Moraga. We prove that a Fano variety of absolute
coregularity $0$ admits either a $1$-complement or a $2$-complement. In the
case of Fano varieties of absolute coregularity $1$, we show that they admit an
$N$-complement with $N$ at most 6. Applying the previous results, we prove that
a klt singularity of absolute coregularity $0$ admits either a $1$-complement
or $2$-complement. Furthermore, a klt singularity of absolute coregularity $1$
admits an $N$-complement with $N$ at most 6. This extends the classic
classification of $A,D,E$-type klt surface singularities to arbitrary
dimensions. Similar results are proved in the case of coregularity $2$. In the
course of the proof, we prove a novel canonical bundle formula for pairs with
bounded relative coregularity. In the case of coregularity at least $3$, we
establish analogous statements under the assumption of the index conjecture and
the boundedness of B-representations.
