Research proof index

A preliminary endgame bound. In the $\tau$-minimizer setup, let $k=d[X_1;X_2]$ and let measurable $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$. Put $\delta=I[T_1:T_2]+I[T_2:T_3]+I[T_3:T_1]$ and $c[T_1\#T_2]=d[X_1^0;T_1]-d[X_1^0;X_1]+d[X_2^0;T_2]-d[X_2^0;X_2].$ Then $k\le\delta+\eta c[T_1\#T_2]+\frac{\eta}{2}\bigl(I[T_1:T_3]+I[T_2:T_3]\bigr).$

A symmetry identity in the $\tau$-minimizer endgame. Let $X_1',X_2'$ be independent copies of $X_1,X_2$, and set $U=X_1+X_2$, $V=X_1'+X_2$, $W=X_1'+X_1$, and $S=X_1+X_2+X_1'+X_2'$. Then the conditional mutual informations agree: $I[V:W\mid S]=I[U:W\mid S]$.

In the $\tau$-minimizer endgame, let $X_1',X_2'$ be independent copies of $X_1,X_2$, set $k=d[X_1;X_2]$, $U=X_1+X_2$, $V=X_1'+X_2$, $W=X_1'+X_1$, $S=X_1+X_2+X_1'+X_2'$, and $I_1=I[U:V\mid S]$. If $c[A\mid S\#A\mid S]=\sum_{i=1}^2\bigl(d[X_i^0;A\mid S]-d[X_i^0;X_i]\bigr)$, then $c[U\mid S\#U\mid S]+c[V\mid S\#V\mid S]+c[W\mid S\#W\mid S]\le(6-3\eta)k+3(2\eta k-I_1)$.