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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 $h_{12}=|1⟩⟨1{|}_1+|0⟩⟨0{|}_2$ and $h_{23}=|1⟩⟨1{|}_3+2|1⟩⟨1{|}_2$ commute with their sum. Their sum is uniquely minimized by $|000⟩$, yet $h_{12}$ is lower on $|010⟩$. Filtering leaves these commuting terms unchanged, so it cannot force frustration freeness.

If each local term is already minimized by the global ground state, subtract its minimum $a_Z$ to obtain a positive operator. Nonnegative normalized spectral filtering of Hamiltonian terms preserves positivity and leaves the ground eigenvalue $a_Z$ unchanged. Thus the same state minimizes every filtered term. This preserves frustration freeness but does not create it for an initially frustrated decomposition.

A decomposition $H={\sum }_Z{h}_Z$ has frustration freeness when a global ground state minimizes every term simultaneously. For finite-dimensional Hermitian terms this implies $E_0(H)={\sum }_Z{E}_0(h_Z)$. Shifting each term by its minimum produces positive operators with a common null vector. The property depends on the decomposition, not only the total operator.

Fix an integer radius $m\ge 0$ and put $r=\lfloor m/2\rfloor$. Starting with $A_x$, scan the $2r$ layers of $K^r$ from right to left. In each layer retain precisely those bond orthogonal projections whose supports meet the current support; multiply them onto the current operator and enlarge that support to include their bonds. Discard every other projection: it commutes through the current operator and acts as the identity on $|{\psi }_0⟩$, by frustration freeness.

If $L_r$ denotes the ordered product of the retained projections, defineEach layer expands the support by at most one lattice spacing. Therefore the local projector cone in a frustration-free chain lies within radius $2r\le m$ of $x$, andThe last equality holds because every retained projection also fixes the ground-state bra.

For $r\ge 1$, the stronger estimate from (i) givesFor $r=0$, choose $A_{x(m)}=A_x$; the error is at most $\Vert A_x\Vert$, so the same bound holds. HenceHere $\alpha$ is exactly the value in (i), and the prefactor is independent of $m$ and chain length. Counting $K^r$ as $r$ rather than $2r$ layers would give an incorrect support claim; the powers of a product of two orthogonal projections are what avoid a lost factor of two in the decay rate.

Write $P_0=|{\psi }_0⟩⟨{\psi }_0|$ and $Q_0=I-P_0$. Frustration freeness implies that every local orthogonal projection fixes $|{\psi }_0⟩$ on both the left and right. HenceThe spectral gap makes $\mathrm{supp}H=Q_0\mathcal{H}$. The supplied detectability lemma givesConsequentlyNo assumption that $K$ is Hermitian or normal is needed.

A stronger estimate will keep this same $\alpha$ when we count the two projection layers in (ii). On $Q_0\mathcal{H}$, put $P=P_{\mathrm{odd}}-P_0$ and $Q=P_{\mathrm{even}}-P_0$. They are orthogonal projections, with $\Vert PQ\Vert \le q$. For $m\ge 1$,so the powers of a product of two orthogonal projections satisfyThis extra estimate uses the projection structure, rather than generic submultiplicativity alone.