Weighted Contraction and Long-Time Limits for Monotone Prandtl Flow on a Finite Strip

For existing positive classical solutions of the two-dimensional Prandtl equation with unit outer flow and zero pressure gradient on a finite strip, we prove exponential contraction in a weighted reciprocal distance. Uniform power lower and Gaussian-type upper data bounds give an explicit stationary barrier. A weighted Kato estimate then supplies the contraction rate and inflow forcing term without concavity or smallness assumptions. The distance controls the physical velocity in $L^1_xL^\infty_y$ and accumulated wall-shear and outflow discrepancies. Time translations construct stationary and periodic limits for convergent and asymptotically periodic inflow, respectively. These limits satisfy the equation and boundary data in a transposition formulation without a prescribed reference solution or derivative estimate. With an additional uniform bound on sliding-time averages of first tangential reciprocal derivatives, the limits belong to the weak class of Jia, Lei, and Yuan and are smooth up to the wall for $0<x<X$; trajectories converge locally with all derivatives away from the streamwise endpoints and far field. A sufficient initial/inflow criterion is proved in a stated differentiated solution class. The explicit contraction constants are unchanged. The principal exponent range is $1\leq m\leq3/2$, with a separate a priori contraction extension to $m<2$. The deposit contains the signed manuscript PDF and a reproducible source bundle (LaTeX/Typst sources, build and check scripts, vendored dependencies, QA reports).