  We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums
with characters modulo a prime $q$ which, for both variables of length $M$, are
non-trivial as soon as $M\geq q^{3/8+\delta}$ for any $\delta>0$. This range,
which matches Burgess's range, is identical with the best results previously
known only for simpler exponentials of monomials. The proof combines
refinements of the analytic tools from our previous paper and new geometric
methods. The key geometric idea is a comparison statement that shows that even
when the "sum-product" sheaves that appear in the analysis fail to be
irreducible, their decomposition reflects that of the "input" sheaves, except
for parameters in a high-codimension subset. This property is proved by a
subtle interplay between \'etale cohomology in its algebraic and diophantine
incarnations. We prove a first application concerning the first moment of a
family of $L$-functions of degree $3$.
