Proposition 4 ([Cylinder barriers and height tails]). On the infinite cylinder of period N , the individual critical weights of essential polygons meeting a fixed horizontal level have bounded sum. From a fixed bottom port in a …

We combine with Proposition 9 ([Essential polygons in a slab]) to control vertical span. An essential polygon with vertical height span s , or an arch from the bottom reaching up by s , must hit all short essential polygons in each slab completely traversed. Using order s/(CN) disjoint slabs and the slab mass bound gives H_H(γ;t) (c' s/N) for t≥1 , c'>0 . Consequently the unadorned mass of essential polygons meeting a given horizontal line on the infinite cylinder is O(1) : sum by spans between jN,(j+1)N , each time using a containing cylinder of height O((j+1)N) and the last integral bound. The unadorned arch mass from a fixed base midpoint reaching s in a semi-infinite cylinder is e^ -c' s/N by the arch flux bound and exhaustion. We explain also the height-tail bound for unmarked polygons. For N≥3 a polygon of large height can be opened into such an arch at the bottom of its lowest occupied triangle row by bounded-cost surgery. In that row an upward triangle center v must occur (any downward triangle there needs a tilted connection to such a center), using both tilted neighbors w_± . If one such neighbor uses its other tilted connection to another upward center u , cut that segment v,w,u (remove w ) and attach v,u to the base line directly. Otherwise both neighbors continue north and the upward centers across their other tilted connections are unoccupied (being occupied would force both slants there). Remove v and attach w_± to the base via these free centers. In either case the surgery yields a simple arch, still reaching within bounded distance of the same height, with bounded multiplicity given the local surgery site. For N=2 , every polygon occupying more than one triangular strip uses exactly one of the two upward centers in its lowest strip. Indeed both upward centers would each have to use both tilted links, forcing the polygon to be the four-edge row cycle. Thus both downward centers are used and continue into the next strip, while the other upward center is unoccupied. Replace the two-link segment through the occupied upward center by the two-link segment through the other one. This gives a simple polygon of the same length and exactly the same minimum and maximum heights. The two alternative segments together form the essential four-edge row cycle, so the replacement changes the winding number by 1 or -1 and takes a contractible polygon to an essential one. The operation is an involution, recovered from the lowest strip, and hence is injective. The already established essential-polygon tail therefore bounds these contractible polygons as well. At N=1 every polygon is a row digon and has bounded height span. These two cases complete the height-tail argument at the remaining periods. Thus polygons meeting a given level with span between s and 2s have mass (N,s)e^ -c' s/N , by summing arch bounds and surgery locations. Rotation gives the same argument in either other lattice-axis orientation.