= Commuting with a ground projector does not imply frustration freeness
The condition $[g_Z,P_0]=0$ makes a unique global <ground state> an <eigenvector> of every filtered term, but its termwise eigenvalues can exceed the corresponding minima. For example $h_{12}=|1\rangle\langle1|_1+|0\rangle\langle0|_2$ and $h_{23}=|1\rangle\langle1|_3+2|1\rangle\langle1|_2$ commute with their sum. Their sum is uniquely minimized by $|000\rangle$, yet $h_{12}$ is lower on $|010\rangle$. Filtering leaves these commuting terms unchanged, so it cannot force <frustration freeness>.
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