= Solution
Put $X=\sum_s a_s^\dagger b_s$ and $Y=X^\dagger$, and take real hopping amplitude $t$ and real $\mathcal E$. Write the <anti-Hermitian> ansatz with its ordering visible:
$$
S=f(\Delta)X-Yf(\Delta).
$$
Using $Yf(\Delta)=f(\Delta+2)Y$, its <commutator> with $\mathcal E$ is
$$
[S,\mathcal E]
=f(\Delta)[\mathcal E(\Delta-2)-\mathcal E(\Delta)]X
-f(\Delta+2)[\mathcal E(\Delta+2)-\mathcal E(\Delta)]Y.
$$
Both terms equal their required hopping coefficients $-t$ if
$$
\boxed{f(\Delta)=\frac{t}{\mathcal E(\Delta)-\mathcal E(\Delta-2)}.}
$$
The shifted expression is then $f(\Delta+2)=t/[\mathcal E(\Delta+2)-\mathcal E(\Delta)]$. For a transition from difference $d$ to $d+2$, the left-ordered factor evaluates as $f(d+2)=t/[\mathcal E(d+2)-\mathcal E(d)]$, the final-minus-initial energy denominator.
The formula describes <occupation-dependent virtual hopping> on actual transitions with nonzero energy gaps. <Perturbation theory> also requires the hopping <matrix> element, including its bosonic occupation factors, to be small compared with that gap. The <resonance obstruction to perturbative elimination> at a zero denominator on a nonzero hop requires <degenerate perturbation theory>. Values on unreachable states whose hopping amplitude vanishes need not determine the generator.
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