Sums of products of Kloosterman sums to prime power moduli

We prove new bounds on complete sums of products of k additively shifted Kloosterman sums to odd high prime power moduli q=p^n, which feature substantially stronger power savings (about q^(-1/ceil(k/2)) in generic configurations) and a novel quantification of the alignment among the shifts. We prove our bounds by developing a method to estimate complete exponential sums with a broad class of phases well-controlled by a specified number of initial terms in a property reflecting p-adic differentiability.