Research proof index

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 $c[U\mid S\#U\mid S]+c[V\mid S\#V\mid S]+c[W\mid S\#W\mid S]\le(6-3\eta)k+3(2\eta k-I_1)$.

Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent $G$-valued random variables. Then $d[Y_1-Y_3; Y_2-Y_4] + d[Y_1|Y_1-Y_3; Y_2|Y_2-Y_4]$ $+ I[Y_1-Y_2 : Y_2 - Y_4 | Y_1-Y_2-Y_3+Y_4] = d[Y_1; Y_2] + d[Y_3; Y_4].$

For four measurable random variables $Y_0,Y_1,Y_2,Y_3$ in a finite abelian group, the conditional mutual-information term used in the fibring identity can be reduced to $I\!\left[(Y_0-Y_1,Y_2-Y_3):(Y_0-Y_2,Y_1-Y_3)\,\middle|\,Y_0-Y_1-Y_2+Y_3\right]=I[Y_0-Y_1:Y_1-Y_3\mid Y_0-Y_1-Y_2+Y_3].$

A pair of measures minimizing $\tau$ exists.

For p.η ≤ 1/8, there exist τ-minimizers X₁, X₂ at zero Rusza distance. For p.η < 1/8, all minimizers are fine, by tau_strictly_decreases'. For p.η = 1/8, we use a limit of minimizers for η < 1/8, which exists by compactness.

If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that $\tau[X'_1;X'_2] < \tau[X_1;X_2]$. Phrased in the contrapositive form for convenience of proof.