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Combine a local projector cone in a frustration-free chain with the contraction bound from the detectability lemma. If the localized operator misses a second observable's support, their commutation and the exact ground-state bra identity convert the connected correlation function into an approximation error. The two-projection power estimate yields spatial decay rate $\alpha =\frac{1}{3}\ln (1+\Delta /2)$ for the stated detectability contraction.

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.