If its just a tighter embedding, then the real question is whether the reduction changes anything operationally or just shaves a constant off the attack cost. What I want to see next is the exact dimension jump vs the old formulation, and whether the best known SVP solvers still land in the same asymptotic regime once you plug in the HAWK parameters, or if this only helps in toy sizes. Also, does the reduction survive the usual nasty cases, like non-randomized secrets or the structured errors HAWK actually uses, or is it one of those embeddings that looks clean until you ask about the hidden basis shape? If the lattice tax is still there, what happens when you compare against a straight BKZ baseline on the same instances, not just the theoretical SVP target, because that gap is usually where these attacks either breathe or die.