A Bernoulli count is a sufficient summary for a finite-sample posterior

Give each finite set its discrete metric $d(x,x^{\mathrm{\prime }})=0$ for equality and $1$ otherwise. Since Cauchy means that tail distances are eventually below every positive radius, a Cauchy sequence is eventually constant at radius $1\mathrm{/}2$; the metric is complete. Each finite set is countable and dense in itself, and its singletons are open, so its Borel sigma-algebra is the power set. The identity is a Borel isomorphism to this Polish presentation, so [F4] gives that $\mathrm{\Theta }$ and $\mathsf{Y}$ are standard Borel. For $\mathbb{R}$, [F5] gives a complete usual metric and countable dense subset $\mathbb{Q}$; each metric ball is an ordinary open interval, so this metric induces the usual topology. Thus $\mathbb{R}$ is Polish and its usual Borel space is standard Borel by [F4].

The prior is proper. For each $\theta$, summing the likelihood over the eight binary triples gives $(\theta +(1-\theta ){)}^3=1$, so $K$ is a probability kernel by [F1]. The count $S$ is measurable because its domain is finite.

For $k\in \{0,1,2,3\}$ let $F_k=\{y\in \mathsf{Y}:S(y)=k\}$ and define $Q(k,D)=\mathrm{\mid }D\cap F_k\mathrm{\mid }\mathrm{/}\left(\genfrac{}{}{0px}{}{3}{k}\right)$. For $t\mathrm{\notin }\{0,1,2,3\}$ set $Q(t,\cdot )={\delta }_{(0,0,0)}$. Each value is a probability measure by [F7], and for fixed $D$ the map $t\mapsto Q(t,D)$ is Borel, so $Q$ is a probability kernel. For every $\theta$, $D\subseteq \mathsf{Y}$, and Borel $C\subseteq \mathbb{R}$, $K(\theta ,D\cap S^{-1}(C))={\sum }_{k\in C\cap \{0,1,2,3\}}\mathrm{\mid }D\cap F_k\mathrm{\mid }{\theta }^k(1-\theta {)}^{3-k}$ $={\sum }_{k\in C\cap \{0,1,2,3\}}Q(k,D)\left(\genfrac{}{}{0px}{}{3}{k}\right){\theta }^k(1-\theta {)}^{3-k}={\int }_{C}Q(t,D)K_S(\theta ,dt)\mathrm{.}$ This is the common-kernel criterion [F3], so $S$ is sufficient.

The full-data likelihood at $y_0=(1,1,0)$ is ${\theta }^2(1-\theta )$. The prior-weighted likelihoods at $1\mathrm{/}3$ and $2\mathrm{/}3$ are respectively $1\mathrm{/}27$ and $2\mathrm{/}27$, so their normalization gives posterior masses $1\mathrm{/}3$ and $2\mathrm{/}3$. For $0\le ϵ<1$, the condition $\mathrm{\mid }S(Y)-2\mathrm{\mid }\le ϵ$ is exactly $S(Y)=2$. Its likelihood is $3{\theta }^2(1-\theta )$, three times the full-data likelihood at either parameter; the prior-weighted masses are $1\mathrm{/}9$ and $2\mathrm{/}9$, giving the same normalized posterior.

The event in step 3.2 has positive prior-predictive probability $1\mathrm{/}3$, so the ABC target is defined. By [F6], the rejection algorithm's accepted parameter has that target law; hence it equals the full-data posterior just computed. Explicitly, the summary likelihood differs from the full-data likelihood by the same positive factor $3$ at both parameters, which cancels from the finite posterior normalization in step 3.2; sufficiency itself was proved independently in step 3.1.