Hamilton-cycle weak saturation in three families of rectangular strips

How few starting edges of a rectangular grid suffice to add every remaining edge, one at a time, with each addition completing a spanning Hamilton cycle? We give optimal constructions with rc+3 starting edges for 8×n grids with n≥8, R×11 grids with even R≥12, and R×13 grids with even R≥26. Each seed is connected and Hamilton-free. Its ordered Hamilton witnesses form a basis of the full binary Hamilton-cycle span. The constructions replace cut edges of finite reference cycles by disjoint spanning paths in repeated strips. Exact graph identities join the blocks; an explicit path argument closes the remaining boundary at width eleven. For eight rows, affine inequalities and a parity-lattice argument cover the whole odd-width schedule. The general span dimensions are not claimed as new; the contribution is the complete optimal orders and their witness bases. Finite catalogues, exact checks and supporting formal proofs accompany the article.