Okay. To start with, each room is denoted by a square in a grid in the mini-map. It seems likely that each room could be denoted by a pair of numbers, and possibly a third for dungeon level. Also, that Statue talked about rooms and moving them using pairs of numbers... (3,4). It seems to me that the Catalyst does things with rooms, but what? Well, since there are two sets of coordinate input boxes, it must select two rooms. This is supported by the addition of the "z" box, which denotes up or down space in Cartesian coordinates. So it does something with two rooms, and really it seems most likely that that something is switching them because for anything else, there wouldn't need to be two sets of inputs. For the finale, look at the name. The name of a Soul Star typically hints at what it does... though the Vain Crescendo still stumps me... for instance the Astral Influx letting the Target have whoever he/she chooses be guided by the Spirits, and the Illborn Exchange letting us exchange Illborn for essence, and the Ardent Circuit manipulating circuits... you get the idea. So for Asynchrony Catalyst, it acts as a spark, a catalyst, for some sort of powerful action, and for Asynchrony, it causes un synchrony, or discord... exactly the type of discord that would be created if say, a room from dungeon level 6 was switched with a room from dungeon level 3 or 4. You know the dungeon levels all have their own theme, a river of blood found in a skyship wouldn't make sense. So... although we didn't get a z box until recently, moving a room from elsewhere in a dungeon level would still be Asynchronous. Although I am nearly certain I am correct about this, I would not recommend testing it with dungeon level 7 or 8, due to possibly provoking Oric if we tried. Once we get back to the Catalyst, I recommend that we go to a dungeon level considered safe, or normal, like Two, and type in (the room next to us), (2,2,6), which should net us a room from dungeon level Six. Mini-theorycraft get!
A more concise, easier-to-read version of this would be:
-We know dungeon rooms are denoted by tiles on the grid mini-map, so it's possible to denote them as ordered pairs, ie (5,2). We've seen that creepy statue doing exactly that, with (3,4).
-The Catalyst must affect rooms, because [REASONING_NOT_FOUND]
-Since the Catalyst affects rooms and has two sets of input fields, it must do something to two rooms at once. This is supported by the addition of the Z axis, since that would allow these things to operate on different Z-levels.
-This thing is probably switching them because I have no idea what else it could be.
-The name of a Soul Star kind of hints at its purpose sometimes if you're looking at it from the right angle; for instance, Illborn Exchange lets you swap Illborn for their Essences, Ardent Circuit can manipulate circuits assuming they have menus like every other thing it can manipulate, and the Astral Influx controls who a ravening swarm of brain-damaged ghosts advises.
-Asynchrony Catalyst is a catalyst, or spark, that does the opposite of synchrony, so discord. Therefore it's a discordant spark, which means swapping dungeon rooms because... that'd be weird!
As a side benefit to being easier for everyone else to understand, detailing everything out like this can help you find issues with your own reasoning.