The original reference to liquid is of course correct, e.g. ibi Isara Rhodanusque amnes ... confluunt in unum - where the rivers Rhodanus and Isere flow together [merge], but it can also be used figuratively, for example for crowds of people "flowing" together: undique ad hoc spectaculum confluere. In the context of calling the hat, I was thinking of "flow of magic" mainly to have a more nicely sounding phrase.

I agree that pileus is not a good choice, because it is actually dome-shaped and the basis for the pileolus that is worn by roman-catholic dignitarys, but then the Romans never had the concept of a pointy wizards hat. Also it was not technically the sign of an ex-slave - it might gained recognition, because the were allowed to wear the pileus - but usually worn by fishermen and workmen.

Anyway, when I realized that Eliezer is after a phrase that emphasizes the deciding/choosing trait of the hat, I remembered a description of a football (soccer) game in latin, and the term it used to describe the referee, so here's my new proposition:

disceptator, accede!

the one who decides/arbitrates, step up / step here! (imperative form)

or one could also use

disceptator, appare!

which again is the imperative of apparere - appear/show yourself

Edit: You might have confused the pileus with the phrygian cap, which is sort of pointy and looks like a smurf cap. Funny factoid, the french revolutioners mixed it up as well and chose the phrygian as a symbol of liberty.

pileum dictionis, affluere!

It would make Vergilius turn over in his grave, but it roughly means "(felt)hat (of the) talking, flow (to me) (i.e. appear)"

took the liberty to base it on the german translation of the sorting hat, which is "talking hat"

Edit: Adtendite Ordinarium means something like "mind the order". google really sucks at latin ;)