One way to define the prediction space would be to have it predict the state of the universe immediately after the prediction is stated. Since anything after that is a function of that point in time, it’s sufficient. Every particle is within the distance light would have moved in that time. Each particle has at most the energy of the entire universe plus the maximum energy output of the GLUT response (I’m assuming it’s somehow working from outside this universe). This includes some impossible predictions, but that just means that they’re not in the range. They’re still in the domain. Just take the Cartesian products of the positions and momentums of the particles, and you end up in a Euclidean space.
If you want to take quantum physics into account, you’d need to use the quantum configuration space for each prediction. You only need to worry about everything in the light cone of the prediction again. You define the distance between each configuration space as the integral of the square of the magnitude of the difference of the quantum waveform at each point. Since the integral of the square of the magnitude of each waveform adds to one, it’s bounded. Since it’s always exactly one, you get a sphere, which isn’t convex, but that’s just the range. You just take the convex hull, a ball, as the domain.
The actual output of the GLUT may or may not be compact depending on how you display the output.
For those interested: Brouwer fixed-point theorem
I don’t see compactness of the set of possible predictions.
I’d use the Schauder fixed-point theorem so that you don’t have to worry as much about what space you use.
One way to define the prediction space would be to have it predict the state of the universe immediately after the prediction is stated. Since anything after that is a function of that point in time, it’s sufficient. Every particle is within the distance light would have moved in that time. Each particle has at most the energy of the entire universe plus the maximum energy output of the GLUT response (I’m assuming it’s somehow working from outside this universe). This includes some impossible predictions, but that just means that they’re not in the range. They’re still in the domain. Just take the Cartesian products of the positions and momentums of the particles, and you end up in a Euclidean space.
If you want to take quantum physics into account, you’d need to use the quantum configuration space for each prediction. You only need to worry about everything in the light cone of the prediction again. You define the distance between each configuration space as the integral of the square of the magnitude of the difference of the quantum waveform at each point. Since the integral of the square of the magnitude of each waveform adds to one, it’s bounded. Since it’s always exactly one, you get a sphere, which isn’t convex, but that’s just the range. You just take the convex hull, a ball, as the domain.
The actual output of the GLUT may or may not be compact depending on how you display the output.