I will present joint work with Istvan Kadar (to appear soon) that achieves the following: For a large class of quasilinear perturbations to the Minkowskian linear wave equation, we show that semi-global scattering solutions arising from data posed on an ingoing null cone and on past null infinity exist and obey sharp energy estimates. We then show that if the data admit a polyhomogeneous expansion up until some order, then so does the solution. Finally, we present an algorithm suitable for then computing the precise coefficients in the expansions towards future null infinity. This work is motivated by the desire to understand the asymptotic properties of gravitational radiation near future null infinity, and, in particular, solves the issue of summing the fixed-angular-mode estimates obtained in previous works from the series "The Case Against Smooth Null Infinity".