Polynomial bound for the localization length of Lorentz mirror model on the 1D cylinder

We prove a polynomial upper bound for the localization length of the Lorentz mirror model and the Manhattan model on the even cylinder. The main input is a conditional cylinder-localization theorem in the winding regime: if short-direction crossings of $100n\times n$ rectangles have probability bounded below, then many closed trajectories wind around the cylinder. These winding trajectories are topological barriers and the localization follows from a two-switch surgery and double-counting argument.