Excess-cost factorisation: with B = (Σ√A)² and r* ∝ √A, Σ_j A_j/r_j = B·Σ_j r*_j²/r_j = B·(1 + χ²(r*‖r)) for every proposal r

For all $\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}$ and $\htmlData{mx=3}{A, r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, if $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{A}(\htmlData{mx=1}{j})}}$, $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ and $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{J}}$), then