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DanielLC comments on Completeness of simulations - Less Wrong Discussion
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A sufficiently nice function is mathematically guaranteed to have at least one fixed point where f(x) = x. We need to make some assumptions to make it nice enough, but once we do that, we just set x as the hypothetical GLUT output, and f(x) to the GLUT output of the subject's reaction to x, and we know there's some value of x where the GLUT output of the reaction is the same as what the subject is reacting to.
f is the situation, and the fixed point is what you're calling X. Also, I'm not sure if there's a method to construct X. We just know it exists. You can't even just check every value of X, because that only works if it's discrete, which means it's not sufficiently nice.
Thanks for the elaboration; this is a very interesting point that I wasn't aware of. But it does seem to rely on the function having the same domain as its range, which presumably is one of the assumptions going into the niceness. It is not clear to me, although perhaps I'm just not thinking it through, that "future movements of quarks" is the same as "symbols to be interpreted as future movements of quarks".
You could think of it as x is the GLUT output, f(x) is the subject's response, and g(f(x)) is the GLUT's interpretation of the subject's response. f maps from GLUT output to subject response, and g maps from subject response to GLUT output. f and g don't have fixed points, because they don't have the same domain and range. f∘g, however, maps from GLUT output to GLUT output, so it has the same domain and range. I was just calling it f, but this way it might be less confusing.