  We prove the full range of estimates for a five-linear singular integral of
Brascamp-Lieb type. The study is methodology-oriented with the goal to develop
a sufficiently general technique to estimate singular integral variants of
Brascamp-Lieb inequalities that do not obey H\"older scaling. The invented
methodology constructs localized analysis on the entire space from local
information on its subspaces of lower dimensions and combines such tensor-type
arguments with the generic localized analysis. A direct consequence of the
boundedness of the five-linear singular integral is a Leibniz rule which
captures nonlinear interactions of waves from transversal directions.
