[FartyMcFarter]

Then it's impossible to trust the coin in the general case.

Proof: Imagine the extreme case of the coin containing AI that knows exactly how you use it and how to manipulate each toss result. The coin itself can decide the outcome of your procedure, so it's impossible to trust it to generate randomness.

[lisper]

It's also impossible to prove that a given coin is not being controlled by an AI. (Or a deity.)

[FartyMcFarter]

Yes, which is why you can only trust abstract coins that exist in a formal system which assumes independent tosses :)

If you require true randomness without any assumptions this is not the universe for you.

[jjk166]

Just perform the same coin toss in two universes.