Gradient partitioning delivers the replicated sums bit for bit

For all $\htmlData{mx=1}{\alpha \in \htmlData{mx=0}{\operatorname{Type}}}$, $\htmlData{mx=5}{\mathit{op} : \htmlData{mx=0}{\alpha} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{\alpha} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\alpha}}$, $\htmlData{mx=1}{\mathit{owners}, r \in \htmlData{mx=0}{\mathbb{N}}}$, $\htmlData{mx=5}{v : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{\mathbb{N}} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\alpha}}$, $\htmlData{mx=1}{t \in \htmlData{mx=0}{\operatorname{CombineTree}}}$ and $\htmlData{mx=1}{\mathit{idx} \in \htmlData{mx=0}{\mathbb{N}}}$: