Six-cycle weak saturation in rectangular grids and cycle weak saturation in ladders

We determine six-cycle weak saturation in every rectangular grid with both sides at least two: the minimum number of seed edges is the number of vertices. We also determine weak saturation by every even cycle in a two-row ladder, including the absent-cycle cases. Explicit optimal seeds and complete restoration orders attain binary cycle-span lower bounds, and the full target-cycle spans are computed exactly. Both ordinary and initially target-free minima are treated.