> You may be able to write down some PDEs or field equations that may describe things at some base level, but even statistical mechanics, which is really what governs a huge amount of what we see and interact with, is just a pretty good approximation.

QED.

When the approximation is indistinguishable from observation over a time horizon exceeding a human lifetime, it's good enough for the purpose of "would a simulation of a human be intelligent by any definition that the real human also meets?"

Remember, this is claiming to be a mathematical proof, not a practical one, so we don't even have to bother with details like "a classical computer approximating to this degree and time horizon might collapse into a black hole if we tried to build it".

> Citation needed. If you've spent any time dynamical systems, as an example, you'd know that the computer basically only kind of crudely estimates things, and only things that are abstractly near by. You may be able to write down some PDEs or field equations that may describe things at some base level, but even statistical mechanics, which is really what governs a huge amount of what we see and interact with, is just a pretty good approximation. Computers (especially real ones) only generate approximate (to some value of alpha) answers; physics is not reducible to a computer program at all.

You're proving too much. The fact of the matter is that those crude estimations are routinely used to model systems.