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It doesn't seem to be universally true. For example, a thermostat's action is correlated with past temperature. People are similar to thermostats in some ways, for example upon touching a hot stove you'll quickly withdraw your hand. But we also differ from thermostats in other ways, because small amounts of noise in the brain (or complicated sensitive computations) can lead to large differences in actions. Maybe Carroll is talking about that?
Good point. But consider the nearest scenarios in which I don't withdraw my hand. Maybe I've made a high-stakes bet that I can stand the pain for a certain period. The brain differences between that me, and the actual me, are pretty subtle from a macroscopic perspective, and they don't change the hot stove, nor any other obvious macroscopic past fact. (Of course by CPT-symmetry they've got to change a whole slew of past microscopic facts, but never mind.) The bet could be written or oral, and against various bettors.
Let's take a Pearl-style perspective on it. Given DO:Keep.hand.there, and keeping other present macroscopic facts fixed, what varies in the macroscopic past?