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So, Psychic Asylum is a spell that lets you spend 15 "mind minutes" in a mind palace, which takes about a swift action-worth of real time. In the spell it explicitly says "If you're able to prepare spells, you can use the time to prepare a single spell" which matches up with the time it takes to prepare a single spell as per the rules on magic.
Now, what about if the caster is a Wizard with the Fast Study discovery? Fast Study would let him prep his entire spell list in 15 minutes, which is well within the bounds of the spell's duration. Do those abilities interact with each other in that way? What about if the Wizard took the Brilliant Spell Preparation feat (twice), could they prepare two spells in that time or are they limited to one? What about an Exploiter Wizard with the Quick Study exploit? Can they swap out multiple spells during their mind time, or is it still limited to just the one?
Fifteen minutes is the time to prepare up to a quarter of your spells, but the spell only allows one. Increasing the speed won't do anything, because you aren't allowed to prepare as many spells as you normally could, just one. Quick Study is preparing a spell, and you can only prepare a single spell. Ditto for Brilliant Spell Preparation. Those abilities will leave you with a lot of free time to do other things in the fifteen minutes, though.
Fifteen minutes is the time to prepare up to a quarter of your spells, but the spell only allows one.
Preparing some smaller portion of his daily capacity takes a proportionally smaller amount of time, but always at least 15 minutes, the minimum time required to achieve the proper mental state.
So more accurately it takes 45 minutes to prepare all your spells, but you have to spend 15 minutes beforehand to "achieve the proper mental state". So really, the fact that it lets you prep a spell at all is breaking the prep time rule.
It's a proportionally smaller time relative to the normal hour, so you can normally prepare a quarter of your spells in fifteen minutes, but 1/4 is the smallest fraction of time allowed. In any case, Psychic Asylum says a single spell and we agree that (whatever the norm is), that is not the norm.