  One of the most striking successes of the lightcone bootstrap has been the
perturbative computation of the anomalous dimensions and OPE coefficients of
double-twist operators with large spin. It is expected that similar results for
multiple-twist families can be obtained by extending the lightcone bootstrap to
multipoint correlators. However, very little was known about multipoint
lightcone blocks until now, in particular for OPE channels of comb topology.
Here, we develop a systematic theory of lightcone blocks for arbitrary OPE
channels based on the analysis of Casimir and vertex differential equations.
Most of the novel technology is developed in the context of five- and six-point
functions. Equipped with new expressions for lightcone blocks, we analyze
crossing symmetry equations and compute OPE coefficients involving two
double-twist operators that were not known before. In particular, for the first
time, we are able to resolve a discrete dependence on tensor structures at
large spin. The computation of anomalous dimensions for triple-twist families
from six-point crossing equations will be addressed in a sequel to this work.
