 Solution

Fix any configurations $s$ and $s^{\prime }$. List the sites where they differ and update those sites in turn to their target signs. Every specified site has positive probability $1/K^2$ of being selected. At every intermediate configuration, both values of its full conditional distribution have positive probability, since all weights $w_i(\pm 1)$ are finite and strictly positive. The finite prescribed sequence therefore has positive probability and reaches $s^{\prime }$.

Thus every configuration is accessible from every other configuration, so the sampler is an irreducible Markov chain, or equivalently phi-irreducible with counting measure. If $s=s^{\prime }$, a single update that keeps its selected spin unchanged also has positive probability. This simultaneously proves positive self-transition probability and aperiodicity, including $K=1$. The sign of finite $J$ does not alter the accessibility argument.