= Solution
\b[True for the finite bound-level Morse model with nonzero adjacent control couplings.] The <Morse oscillator> energies have the form
$$
E_n=A(n+1/2)-B(n+1/2)^2,
\qquad E_{n+1}-E_n=A-2B(n+1).
$$
Over the physical bound-level range these adjacent gaps are positive, and $B\ne0$ makes them pairwise distinct. Label the retained levels consecutively. With $H_1=\sum_j d_j(E_{j,j+1}+E_{j+1,j})$, all $d_j\ne0$, the squared adjoint action of $iH_0$ has <eigenvalue> $-(E_{j+1}-E_j)^2$ on each adjacent skew-Hermitian coupling. A polynomial chosen by <Lagrange interpolation> can therefore isolate any one adjacent coupling from $iH_1$.
Commuting an isolated coupling once with the drift produces its independent antisymmetric quadrature. The bracket between the two quadratures yields a traceless diagonal generator; brackets along the connected chain produce every nonadjacent off-diagonal generator. Thus <connected nondegenerate transition chain generates special unitary control> proves $\mathfrak{su}(N)\subseteq\mathfrak g$, which gives density-state controllability. If the physical drift has nonzero <trace>, subtracting its traceless part also produces the identity direction, giving $\mathfrak u(N)$ and full operator controllability. Shifting the energy origin may remove that extra phase direction but does not change state controllability. The assertion concerns the finite-level model in the question, not the entire bound-plus-dissociation <Hilbert space> of the untruncated Morse potential.
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