Exponential clustering from layered projectors ($\alpha =\frac{1}{3}\ln (1+\Delta /2)$)

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 $\alpha =\frac{1}{3}\ln (1+\Delta /2)$ for the stated detectability contraction.