Three-way equal-sum partition dynamic programming ($D_i(x,y)$)

For positive integer weights of total $S=3h$, store reachable pairs $(x,y)$ of sums assigned to the first two groups after each input list prefix. The third sum is determined by the processed total minus $x+y$. Assign each new occurrence to one of the three groups using a fresh layer; accept exactly final state $(h,h)$. The transitions are sound by construction and complete by deleting the last occurrence from a valid assignment. The state range is $0\le x,y\le h$, giving $O(n(h+1{)}^2)$ time and $O((h+1{)}^2)$ rolling space.