= Solution
A <local operator> is an inflationary, idempotent, finite-meet-preserving map $j:\Omega\to\Omega$ with $j(\top)=\top$, expressed in the <internal logic of a topos>. For a <subterminal object> $U$ with truth value $u$, the <closed local operator> and <quasi-closed local operator> are respectively
$$
c(u)(p)=u\vee p,\qquad q(u)(p)=((p\Rightarrow u)\Rightarrow u).
$$
The first is a <local operator> by distributivity. For the second put $n(p)=p\Rightarrow u$. Then $p\le n^2p$, $n$ is order-reversing, and $n^3p=np$: the first inequality gives $n^3p\le np$, while its application to $np$ gives the reverse inequality. Thus $q=n^2$ is inflationary and idempotent, and $q(\top)=\top$.
For meet preservation, monotonicity gives $q(p\wedge r)\le qp\wedge qr$. Conversely put $t=(p\wedge r)\Rightarrow u$. From $t\wedge p\le r\Rightarrow u$ and $qr\wedge(r\Rightarrow u)\le u$, we obtain $t\wedge qr\le p\Rightarrow u$. Intersecting with $qp$ gives $qp\wedge qr\wedge t\le u$, hence $qp\wedge qr\le q(p\wedge r)$. This proves all local-operator axioms. Also $q(u)(0)=u$ and $q(u)(u)=u$. Equivalently, it is the largest <local operator> leaving $u$ fixed: if $k(u)=u$, then
$$
kp\wedge k(p\Rightarrow u)\le u\quad\Longrightarrow\quad kp\le k(p\Rightarrow u)\Rightarrow u\le(p\Rightarrow u)\Rightarrow u.
$$
This explains the quasi-closed terminology as relative double negation above the closed bottom $u$.
The <subobject classifier> of $\mathbf{sh}_j(\mathcal E)$ is the <closed-subobject classifier> $\Omega_j=\{p:jp=p\}$. Its bottom is $j0$, its meet is inherited, its join is $j(p\vee r)$, and implication between closed truth values is the ambient implication. To verify the last claim, if $jv=v$, then
$$
j(p\Rightarrow v)\wedge jp=j((p\Rightarrow v)\wedge p)\le v.
$$
It follows that $j(p\Rightarrow v)\le jp\Rightarrow v\le p\Rightarrow v$; inflationarity gives equality and also $jp\Rightarrow v=p\Rightarrow v$.
Suppose $j=q(u)$. Every closed $p$ lies above $u$, and its complement in $\Omega_j$ is $np=p\Rightarrow u$, also closed because $n^3=n$. We have $p\wedge np=u$. Moreover $n(p\vee np)=np\wedge n^2p=np\wedge p=u$, so $q(p\vee np)=\top$. Thus the fixed truth values form a <Boolean algebra>, and the sheaf topos is a <Boolean topos>.
Conversely suppose the sheaf topos is Boolean and put $u=j0$. Complementation in its classifier is $p\mapsto p\Rightarrow u$. Applying the double-complement identity to $jp$, and using the implication calculation above, gives
$$
jp=((jp\Rightarrow u)\Rightarrow u)=((p\Rightarrow u)\Rightarrow u).
$$
Therefore
$$
\boxed{\mathbf{sh}_j(\mathcal E)\text{ is Boolean}\iff j=q(j0).}
$$
For the cover, work in the <slice category> $\mathcal E/\Omega$. Its generic <subterminal object> is $T=(\top:1\hookrightarrow\Omega)$. Form its quasi-closed Boolean subtopos $\mathcal B=\mathbf{sh}_{q(T)}(\mathcal E/\Omega)$, and compose its embedding with the slice projection to obtain $b:\mathcal B\to\mathcal E$. The slice projection has inverse image $X\mapsto X\times\Omega$, and sheafification is left exact, so this is a <geometric morphism>.
To prove surjectivity, take a <monomorphism> $S\hookrightarrow X$ with characteristic predicate $p(x)$, and suppose $b^*$ makes it invertible. Pullback to the slice followed by sheafification makes this exactly the assertion that it is $q(T)$-dense. Internally on $X\times\Omega$, therefore,
$$
((p(x)\Rightarrow u)\Rightarrow u)=\top
$$
for the generic truth value $u$. Substitute $u=p(x)$ by pulling back along the graph $X\to X\times\Omega$. The left side becomes $(\top\Rightarrow p(x))=p(x)$, so $p(x)=\top$ and the original mono is invertible. Thus $b^*$ reflects invertible monos. If it identifies two parallel arrows, their <equalizer> becomes invertible; reflection then shows the arrows were already equal. Hence $b^*$ is faithful, and
$$
\boxed{\mathbf{sh}_{q(T)}(\mathcal E/\Omega)\longrightarrow\mathcal E\text{ is a surjection from a Boolean topos}.}
$$
The generic truth value is essential: substituting only a fixed global value such as $0$ would give a Boolean subtopos, but need not give a surjection onto the original topos.
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