The proposed pathological process is tasked basically with being unpredictable in the face of a prediction process that expects it to do so and has unlimited time to prepare.

This prediction process could for example consider many possible shapes of environments, ecosystems, knowledge and research exposure the pathological process would be exposed to.

The pathological process might for example generate new random universes, with random physical laws, and perform simulations within these in order to consider things unlikely to be predicted.

How large is this space? Does it matter?

Both systems will likely attempt to derive general descriptions of systems in which inferences can be made.

I am very off track! Is this relevent? At what point is it relevent or not?

No. Things are only relevent when they relate to challenges we might encounter in the real world.

In reality, computational systems are relatively small. They exist in tiny algorithmic systems inside our larger universe. One can be fully observed by another, and two can mutually fully observe each other.

An all-prediction program would then be tasked with sufficiently hiding its workings from pathological programs, and observing and preparing for their behavior in advance of engaging them. This is entirely possible only if it has access to all other program forms before it is tasked to work with them. The program's spec would provide it with that access.

I don't know why Turing assumes that the predicted program can observe the predicting program. Have not read the paper. Don't know what concepts are within its scope.

In reality, programs happen that are not observed by another given program prior to their execution, and such situations aren't usually considered interesting when trying to predict program behavior alone.

Concept closes to writing a program which predicts whether another halts, which means writing something similar to an optimizing compiler and an algebraic solver. Some of the spaces of algebraic solution do indeed look unreasonable to solve given how many unproved mathematical theorems there are. Regardless, such an analyser would be incredibly useful for other tasks than the pathological one.