= Solution
Sum the <spectral filtering of Hamiltonian terms> over the original finite decomposition. The total <Hamiltonian operator> commutes with its own evolution, so
$$
\sum_ZA^{(Z)}=\int w(t)e^{itH}\left(\sum_Zh_Z\right)e^{-itH}dt=H\int w(t)dt=H.
$$
The absence of all ground-to-excited <matrix elements>, proved in part (a), makes each filtered term block diagonal relative to $P_0=|\phi_0\rangle\langle\phi_0|$ and its orthogonal complement. Therefore
$$
\boxed{H=\sum_ZA^{(Z)},\qquad[A^{(Z)},P_0]=0.}
$$
A filtered term may act on the whole system, since <unitary time evolution> spreads its original support. Neither commutation with $P_0$ nor preservation of its ground-state expectation asserts that the global <ground state> minimizes each individual term. In particular, this construction does not generally turn a frustrated decomposition into a <frustration-free Hamiltonian>.
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