Proposition 1 (Secret Key Capacity); precise result selected for the p.1 paper seed

For finite types $S,X,Y,Z$, let $W$ supply their nonemptiness and a probability law $W_s(x,y,z)$ for each $s\in S$. Let $c:S\to\mathbb R$ and $\Gamma\in\mathbb R$ satisfy $\Gamma>0$, the existence of a real $L$ with $0\le c(s)\le L$ for every $s$, and the existence of a probability law $p$ with $\sum_s p(s)c(s)\le\Gamma$. There exist positive natural numbers $u,w$, a feasible probability law $p$, and stochastic kernels $q(a\mid x,s)$, $r(b\mid a)$, for $a<u,b<w$, whose value

is a member of, and at least every member of, the set obtained by allowing all positive finite auxiliary sizes and all such feasible laws and kernels; furthermore, the sender-excited capacity equals $J$, embedded in the extended real numbers. Here $U=a,V=b$, and conditional mutual information means $I(A;B\mid C)=H(A,C)-H(C)-H(A,B,C)+H(B,C)$ with Shannon entropies in bits. The capacity in this equality is the extended-real supremum of nonnegative real rates $R$ with the following property: for every positive block length $\ell$, there is a code with positive message, public-message, and key counts $M_\ell,P_\ell,K_\ell$, a probability law $e_\ell(s^\ell\mid m)$ on $S^\ell$ for each message, maps $f_\ell(m,x^\ell)$ to the public alphabet and $a_\ell(m,x^\ell)$ to the key alphabet, and a map $b_\ell(y^\ell,t)$ to the same key alphabet; there are fixed nonnegative real constants $r_M,r_P,r_K$ such that $M_\ell=2^{\lceil\ell r_M\rceil}$, $P_\ell=2^{\lceil\ell r_P\rceil}$, and $K_\ell\le2^{\ell r_K}$; every $s^\ell$ with $e_\ell(s^\ell\mid m)>0$ satisfies $\sum_i c(s_i)\le\ell\Gamma$; and under the mass function $M_\ell^{-1}e_\ell(s^\ell\mid m)\prod_iW_{s_i}(x_i,y_i,z_i)$, the disagreement probability $\Pr[a_\ell(m,x^\ell)\ne b_\ell(y^\ell,f_\ell(m,x^\ell))]$ tends to zero, $I(a_\ell(m,x^\ell);(z^\ell,f_\ell(m,x^\ell)))/\ell$ tends to zero, and for every $\varepsilon>0$, eventually $H(a_\ell(m,x^\ell))/\ell\ge R-\varepsilon$.