= Solution
Write $P=\sum_{j\leq\alpha}p_j(x)\partial^j$, setting unused <coefficients> to zero. Split the <Adler trace> into its fixed leading part and variable parts:
$$
l_P(L)=\operatorname{Tr}(P\partial^m)
+\sum_{l=0}^{m-1}\operatorname{Tr}(P u_l\partial^l).
$$
Cyclicity from part (i) gives
$$
\operatorname{Tr}(P u_l\partial^l)
=\operatorname{Tr}(u_l\partial^lP)
=\int u_l\operatorname{res}(\partial^lP)\,dx.
$$
Thus this <Hamiltonian trace functional on monic differential operators> has the required form
$$
\boxed{l_P(L)=c+\sum_{l=0}^{m-1}\int a_l(x)u_l(x)\,dx,\qquad
c=\int p_{-m-1}(x)\,dx,\quad a_l=\operatorname{res}(\partial^lP).}
$$
To make each <coefficient> explicit, expand $\partial^lp_j$ by the <Leibniz rule>. A term with $k$ <derivatives> of $p_j$ contributes to the residue exactly when $l-k+j=-1$. Consequently,
$$
\boxed{a_l(x)=\sum_{k=0}^l\binom lk p_{k-l-1}^{(k)}(x).}
$$
These are <smooth functions> depending on $P$, not on the variable <coefficients> of $L$. In particular $a_0=p_{-1}$ and $a_1=p_{-2}+p_{-1}'$. The <coefficient> $a_0$ must be included: it appears in the displayed sum even though the printed <coefficient> list starts at $a_1$.
The monic operators form an <affine space>, not a vector space. Accordingly $l_P$ is affine in the coordinates $u_l$, with constant $c$, and its differential is linear in variations. Its <variational derivatives> are $\delta l_P/\delta u_l=a_l$. It is the restriction of a linear trace pairing on the full operator space, which explains the terminology without discarding the constant term.
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