> 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".