Monotone coupling from the past

If update maps preserve a finite partial order with a bottom and top state, it suffices to run those two extremal states from the same backward horizon. Equality of their time-zero images forces equality of all initial states by the order sandwich. For ferromagnetic Ising heat-bath updates, use the same updated vertex and uniform variable in every chain; the conditional probability of $+1$ increases with every neighbour's spin. A block visiting all vertices once with all uniforms below the least conditional probability of $+1$ maps every state to all plus, proving almost-sure termination.