Okay, setting aside the silly point of whether I'm a determinist or not (and I'm clearly not
arguing for determinism, as you can see by a brief inspection of my previous posts)... Moreover, lawful with a (subjective) random element describes the actual world we live in, so it'd be foolish for me to rule it out.
I've noticed that and that's a fun thing for compatibilists to claim because it makes incompatibilists look foolish.
... well... no, but yes? I mean, it does make them look foolish, but the reason I said it was because that's what I actually did, because that's the answer to the proximate problem.
And the reason it makes them foolish is because I'm right.
You want a general proof? It lies in the definition of 'random': what you've got left after you've squeezed all the predictability out of a system. Any tendency, any memory that comes back and does something... anything that actually carries state and uses it for anything other than a high-grade PRNG... all that is the kind of stuff that you can build these predictions out of. That's what's not random.
If it's entirely new every time without any bias? You can't. That's what we call random.
These two span the space of explanations by construction. That's the whole point of constructing it. If you think you've broken it, all you've done is muddy the waters so it's not so easy to see why something happened the way it did, so you vacillate between the two and think you've found something else. Generally by abusing the probability of the random element.
Like, Alice can consider any arbitrary point of Hilbert space - its amplitude is probably nonzero - and say, "Aha! This describes a world where X stupendously unlikely event occurred! But it is high complexity, and no deterministic system forced this outcome!" But Alice did the work of picking this [s]world[/s] huge subspace of worlds out, and it is not special except in her mind. The processes that produced it are deterministic, and we randomly ended up not experiencing that branch of the universal wavefunction. There's nothing there.
The car that reliably goes for no reason? The critters that have conversations for no reason? Either something made them happen or it didn't. Denying it is an attempt to escape the connotations of randomness while accepting the defining characteristic, that it isn't predictable even in principle and with universal knowledge.