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

We prove polynomial upper bounds for the localization length of the Lorentz mirror model and the Manhattan model on the even cylinder.

We first show that a fixed positive lower bound for short-direction crossings of a $100n\times n$ rectangle implies localization on scale $O(n^{10})$.

The proof is genuinely cylindrical and combines winding barriers with a two-site switching and double-counting argument.

Together with a planar confinement argument proved here, this yields unconditional cylinder localization for both models.

For Lorentz mirrors, a planar escape estimate ensures the required crossing lower bound; for Manhattan mirrors, planar confinement handles any fixed scale at which the crossing lower bound fails.