Unnumbered numerical identity: HRW squashed CMI equals nonlocality cost

For every $\delta,\eta\in\mathbb R$ satisfying $0\le\delta\le1$ and $-\tfrac14\le\eta\le\tfrac1{12}-\tfrac{\delta}{3}$, let $p$ be the Boolean array given at $x=y=0$ by $p(x,y,a,b)=(1-\delta)/2$ when $a=b$ and $\delta/2$ otherwise, and at every other input pair by $3/8-\eta/2$ when $a\oplus b=xy$ and $1/8+\eta/2$ otherwise. A valid array means a nonnegative array normalized over $a,b$ at each $x,y$, whose $b$-marginal is independent of $x$ and whose $a$-marginal is independent of $y$. A finite decomposition of $p$ consists of a natural number $n$, weights $w_i\ge0$ summing to $1$, and valid arrays $v_i$, with $\sum_iw_iv_i=p$ pointwise; its support is the set of arrays $v_i$ with $w_i>0$. Such a decomposition is minimal when every positively weighted member is a vertex and no finite decomposition of $p$ has support strictly contained in its support; a vertex is a valid array $v$ for which $v=tq+(1-t)r$, with valid $q,r$ and $0<t<1$, implies $q=r=v$. For every supplied complete extension $c$ of $p$, the hypotheses therefore include validity of $p$, positive natural numbers $Z,E$, and, for each $z<Z$, a minimal decomposition $(w_{z,e},v_{z,e})_{e<E}$ of $p$; all these decompositions use fixed member labels $v_{z,e}=v_e$, and every minimal finite decomposition $d$ of $p$ has exactly one index $z<Z$ such that, for every array $v$, $\sum_{e:\,v_e=v}w_{z,e}$ equals the total weight assigned to $v$ by $d$. Zero weights and repeated member labels are allowed. For every probability vector $q=(q_z)_{z<Z}$, meaning $q_z\ge0$ and $\sum_zq_z=1$, define $P^{x,y,q}(a,b,e)=\sum_zq_zw_{z,e}v_e(x,y,a,b)$. For a finite real-valued array $r$, put $H(r)=-\sum_i r_i\log_2 r_i$, using the total real logarithm, and define

The quantity called $\operatorname{squashedCMI}(c)$ is $\max_{x,y\in\{0,1\}}\inf_q I(P^{x,y,q})$. To define the other quantity, an array is local when it is nonnegative and normalized and admits a representation $\sum'_l\mu(l)_{\mathbb R}\alpha_l(a\mid x)\beta_l(b\mid y)$ for some arbitrary latent type $L$, probability mass function $\mu$, and nonnegative, normalized local response functions $\alpha_l,\beta_l$. Let $F(p)$ be the real infimum of $t\in[0,1]$ admitting $p=tv+(1-t)l$, where $v$ is a vertex without a local model and $l$ is local, with the total real-infimum convention giving $0$ if this set is empty. Set

The nonlocality cost at inputs false, false is, by definition, $F(p)\log_2(\min(A_0,B_0))$. The theorem asserts both equalities

The parameter bounds are inclusive, and the assertion applies to every complete extension $c$ satisfying the stated structure.