"continuum-many" means "the same cardinality as the real numbers".

Ok, fair enough. In that case I must merely disagree that there exist this many possible arrangements of matter; it seems to me that the arrangements are actually countably infinite.

As for particles being in bound states and having finitely many degrees of freedom: I'd be surprised if it altered the 'bigger picture' whereby all possible rearrangements of the matter in your body (or in the solar system as a whole, say) get some (possibly minuscule) amplitude assigned to them.

That's true, but the question is whether that number has the cardinality of the reals or the integers. I think it's the integers, due to the quantisation phenomenon in bound states; everything is in a bound state at some level. After my last post it occurred to me that the quantised states might be so close together that they'd be effectively indistinguishable; however, there would still be a finite number of distinguishable states. Two states are not meaningfully different if a quantum number changes by less than the corresponding uncertainty, so in effect the wave-function is quantised even in a continuously-varying number. Once you quantise it's all just combinatorics and integers.