What's special about a fantastic outcome? Suggestions wanted.

Based on a certain SSC post, a matrix-like scenario where a non-friendly AI decides to spare a few cubic meters of computronium to faithfully simulate humanity, rather than to simply get rid of it. For sentimental reasons and because human-boxing is ironic.

It has near maximal computational capacity, but that capacity isn't being "used" for anything in particular that is easy to determine.

This is actually a very powerful criteria, in terms of number of false positive and negatives. Sadly, the false positives it DOES have still far outweigh the genuine positives, and includes all the WORST outcomes (aka, virtual hells) as well.

I presume the answer you're looking for isn't "fun theory", but I can't tell from OP whether you're looking for distinguishers from our perspective or from an AI's perspective.

Have the FAI create James+, who is smarter than me but shares my values. In a simulation in which I spend a long time living with James+ I agree that he is an improved me. Let James++ be similarly defined with respect to James+. Continue this process until the FAI isn't capable of making improvements. Next, let this ultimate-James spend a lot of time in the world created by the FAI and have him evaluate it compared with possible alternatives. Finally, do the same with everyone else alive at the creation of the FAI and if they mostly think that the world created by the FAI is about the best it could do given whatever constraints it faces, then the FAI has achieved an excellent outcome.

Unfortunately our brains lack capacity of thinking about superior intelligence. As I understood you want to describe particular examples of what lies between scenarios 0 (stands for human extinction) and 1 (mutual cooperation and new better level of everything)

First, there are scenarios where human race is standing on the edge of extinction, but somehow ables to fight back and surive, call that Skynet scenario. Analogously, you can think of a scenario where emergence of FAI don't do any great harm, but also don't provide too much new insights and never really get far beyond Human-level intelligence

I think if there was an FAI, and it successfully built utopia, it would have to solve this problem of convincing us that it did (?using something like the Arthur/Merlin protocol?)

edit: I realize that there is a nearly limitless room for trickery here, I am just saying that if I had to define FAI success it would have to be in terms of some sort of convincing game with asymmetric computing power setup, how else would we even do it? Importantly, Merlin is allowed to lie.