= Exponential clustering from layered projectors
{title2=$\alpha=\tfrac13\ln(1+\Delta/2)$}
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 $\alpha=\frac13\ln(1+\Delta/2)$ for the stated detectability contraction.
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