The variational cost law is Sibson's order-2 information radius: Σ_i Σ_j (λ_i π_ij)²/(λ_i r_j) = Σ_i λ_i Σ_j π_ij²/r_j, so log(min_r E[Σπ²/r]) = I₂(row; key)

For all $\htmlData{mx=1}{I, J \in \htmlData{mx=0}{\operatorname{Type}}}$, $\htmlData{mx=3}{\mathit{lam} : \htmlData{mx=0}{I} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, $\htmlData{mx=5}{\pi : \htmlData{mx=0}{I} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{J} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ and $\htmlData{mx=3}{r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$, if $\htmlData{mx=4}{\forall i,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{\mathit{lam}}(\htmlData{mx=1}{i})}}$ and $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{I}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$), then