Monochromatic arm exponents for critical planar percolation

We determine the monochromatic three-arm exponent for critical planar percolation through the exact spectral threshold of an explicit positive integral operator family with elementary kernels, obtaining $ξ_3=0.76928375\cdots$. Unlike the monochromatic two-arm (backbone) event, the three-arm event can be obstructed by two distinct white clusters, so a Markovian exploration must retain the past connections. We introduce a two-point boundary observable that records these connections. Using the coupling between conformal loop ensemble (CLE) and Liouville quantum gravity (LQG), this observable satisfies a renewal equation in terms of quantum boundary lengths, and yields the explicit operator whose spectral information determines $ξ_3$. Our approach also gives an explicit spectral characterization of monochromatic $n$-arm exponents for all $n\ge4$.