 Solution

Use the sorted sparse dot product merge, processing a coordinate only when it appears in both supports of a vector:Both structures must satisfy the representation invariant; equal dimension is checked explicitly. The loop invariant says that result contains exactly the products for all matching indices already passed, and that every skipped unequal index has zero in the other vector. The latter follows from sortedness: if p.index < q.index, that index cannot appear at q or later, and all earlier entries of the other vector have already been settled. Equality contributes its product exactly once. The total number of remaining ordinary entries is a loop variant; every iteration advances at least one cursor. Once either cursor reaches its header, all remaining entries pair with zeros. Thus the exact-arithmetic result is ${\sum }_{i}vec{1}_{i}vec{2}_i$. With $k_1,k_2$ stored entries, time complexity is $O(k_1+k_2+1)$ and auxiliary space is constant. Floating-point products and additions may round or overflow even when input coordinates are finite; no exact real-arithmetic accuracy claim is made for Double.