It depends on granularity. If you are talking about your game of life world on the level of the rules of the game, that is equivalent to talking about our Universe on the level of the universal wave function. In both cases there are no more agents with actuators and no more do(.), as a result. That is, it’s not that your factorization will be causal, it’s that there is no causality.
But if you are taking a more granular view of your game of life world, similar to the macroscopic view of our Universe, where there are agents that can push and prod their environment, then suddenly talking about do(.) becomes useful for getting things done (just like it is useful to talk about addition or derivatives). On this macroscopic level, there is causality, but then your statement about all factorizations being causal is false (due to obvious examples involving reversing causal chains, for example).
It depends on granularity. If you are talking about your game of life world on the level of the rules of the game, that is equivalent to talking about our Universe on the level of the universal wave function. In both cases there are no more agents with actuators and no more do(.), as a result. That is, it’s not that your factorization will be causal, it’s that there is no causality.
But if you are taking a more granular view of your game of life world, similar to the macroscopic view of our Universe, where there are agents that can push and prod their environment, then suddenly talking about do(.) becomes useful for getting things done (just like it is useful to talk about addition or derivatives). On this macroscopic level, there is causality, but then your statement about all factorizations being causal is false (due to obvious examples involving reversing causal chains, for example).