Abstract: I will report on ongoing joint work with Devalapurkar and Schlank. We construct a telescopic analog of the Beilinson fiber square, providing a simple formula for the difference between the T(n)-local K-theory of a ring spectrum R and that of its T(n-1)-localization. In fact, there is a highly structured explanation for this square in terms of the T(n)-localization of the category of noncommutative motives, which we explain. There are many applications to our understanding of redshift, of which I will discuss some. In particular, we find explicit formulas for the K(n)-local K-theory of many K(n-1)-local ring spectra in terms of Galois cohomology.