Research proof index

Let $X_1',X_2'$ be independent copies of the $\tau$-minimizers $X_1,X_2$. Write $k=d[X_1;X_2]$ and $I_1=I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']$. Then the entropy of the four-variable sum obeys $H[X_1+X_2+X_1'+X_2']\le\tfrac12H[X_1]+\tfrac12H[X_2]+(2+\eta)k-I_1$.

A fibring identity in the $\tau$-minimizer setup. Let $X_1',X_2'$ be independent copies of $X_1,X_2$ and put $k=d[X_1;X_2]$. Then $d[X_1+X_2';X_2+X_1']+d[X_1\mid X_1+X_2';X_2\mid X_2+X_1']+I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']=2k$.