Uniform marked-polygon estimates and sharp finite bridge moments

We prove a uniform bound for critical honeycomb polygons through two axial marks on a periodic staircase. The bound retains an explicit power of the ratio between the period and the marked separation. We also prove sharp finite strip estimates: bridge mass of order h${}^{-1/4}$, first-length mass at most $C{h}^{13/12}$, and mass at least $c{h}^{-1/4}$ on bridges with length at least $c{h}^{4/3}$.