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"Chemicals ... respond to other chemicals in a determined way, and that is quite simply the whole story of a material brain"

Yes, chemicals "don't get to pick how they react". But they act and respond in a stochastic way, not a "determined" one (see From random motion to order: diffusion and some of its implications). And the brain is not "just" chemicals, but rather an ordered assembly of them, with an organization that in turn can make use of randomness to yield ... personal choice/free will? Perhaps the brain itself is the source of the unpredictability that we expect of the immaterial?

"the immaterial self is a "posit" that we can't prove by pointing to some material thingy but it is a necessary one to explain our experience"

Yep, the immaterial self is a "posit." So too is its absence. And there are a variety of reasons to prefer, at any given time, one "posit" over another. Its not clear to me though that there is anything about "free will" in particular that makes "necessary" the posit of an "immaterial self".