> If numbers cannot have infinite strings of digits, then the future can never be perfectly preordained.

Why? A simple cellular automaton like Rule 30 doesn't use any infinite data but it's future is perfectly preordained.

But being preordained doesn't mean that there can't be free will. Because having the initial state, and being able to predict later state are very different. Even simplest cellular automata are computationally irreducible, meaning that there is no shortcut, and the only way to find out their future states is to run the full computation.

So if physics can be described by finite computation, and all the future states can be obtained from initial state, the only way to obtain them would be running the whole simulation, which would be equivalent to people in that simulation living and making their choices themselves.

https://en.m.wikipedia.org/wiki/Rule_30

Nobody in the simulation you've described would actually make any choices; the choices are predetermined, baked into the heap of the CA/TM being used to drive the computation. Conway's Free Will concerns situations where the choices cannot be predetermined, as a consequence of the Kochen-Specker lemma.

Situations where the choices cannot be predetermined, are not free will, they are just randomness, because they do not depend on the one making the choice.

That said, knowing that the choice is predetermined and knowing the choice, is often very different.

For simple systems if you know the initial state you can easily predict all the future states by using a formula and not doing the computation the system performs.

But for many systems like rule 30, the only way to predict the future state is to perform exactly the computation the system is performing. So if the TM describes a person, the only way to know future states of the TM from initial state is to wait for the person in the TM to make the choice.

In Conways example, free will would be if experimenters decision what to measure is based on the state of the part of the universe describing the experimenter alone, and not connected to the parts describing other experimenters performing related measurement.

Yes, I think the parent would accept that the automata in the simulation do not have free will of the "Conway" type. I believe he is saying the agents have free will of the "compatibilist" type.

https://plato.stanford.edu/entries/compatibilism/

The definition of "free will" in this proof is not the same as the one being discussed here. The kind of "free will" that Conway's proof talks about could be exercised by an automated device that randomly chose measurement settings based on some unpredictable quantum event.

contingency is necessary but not sufficient for free will, Conway's definition is philosophically weak as it equivocates 'choice' and therefore the whole issue. The definition of choice includes not only possible options but also a selection mechanism, where for particles/random there is no selection mechanism (contingency)