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.