  We advance the multipoint lightcone bootstrap and compute anomalous
dimensions of triple-twist operators at large spin. In contrast to the
well-studied double-twist operators, triple-twist primaries are highly
degenerate so that their anomalous dimension is encoded in a matrix. At large
spin, the degeneracy becomes infinite and the matrix becomes an integral
operator. We compute this integral operator by studying a particular non-planar
crossing equation for six-point functions of scalar operators in a lightcone
limit. The bootstrap analysis is based on new formulas for six-point lightcone
blocks in the comb-channel. For a consistency check of our results, we compare
them to perturbative computations in the epsilon expansion of $\phi^3$ and
$\phi^4$ theory. In both cases, we find perfect agreement between perturbative
results and bootstrap predictions. As a byproduct of our studies, we extend
earlier work of Derkachov and Manashov to compute the anomalous dimension
matrices of all triple-twist primaries in scalar $\phi^3$ and $\phi^4$ theory
to first and second order in epsilon, respectively.
