Hamilton-cycle weak saturation on even rectangular grids and width-five obstructions

We determine the Hamilton-cycle weak saturation number on every even-by-even rectangular grid. If a=min(r,c) and b=max(r,c), the value is 3b-3 for a=2 and rc+3 for a>=4. Complete constructions also give bases of the full binary Hamilton-cycle span. For width five we give connected Hamilton-free seeds of size 5R+4 for every even R>=6, establish exactness at R=6,8,10,12, and construct infinite families that separate end-cap stopping tests from full weak saturation. The accompanying support contains the mathematical details, exact certificates and checkers, claim/dependency records, pinned formal sources and verification disclosures. Exactness at width five for even heights at least14 remains open.