= Solution
Fix any configurations $s$ and $s'$. 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 $w_i(\pm1)$ are finite and strictly positive. The finite prescribed sequence therefore has positive probability and reaches $s'$.
Thus \b[every configuration is accessible from every other configuration], so the sampler is an <irreducible Markov chain>, or equivalently <phi-irreducible> with counting measure. If $s=s'$, 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.
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