OKAY OKAY SO THIS THEORY MIGHT BE A BIT FAR FETCHED.
But I think either Patrick Rothfuss got a bit lazy with the name of the four plate door OR, it's majorly plot relevant.
Firstly, the name Valaritas sounds awfully similar to truth in latin; veritas
Could it mean that 'the truth' is what the door is hiding? It could also possibly be where all the Amyr books Kvothe is desperately looking for are being hidden. (Or they were destroyed). The Amyr part probably isn't right, but it isn't too impossible. I'm assuming that the door IS hiding the truth though. the truth of what? heck if I know. But it would make sense. It'd explain puppets reaction, why he was so serious about it and all. It'd also explain Elodin too! It's probably the reason why he was so chill about it. Elodin already knows the truth, probably because he's a namer. Puppet isn't. not officially. And that's the reason he said it isn't of any concern of a student either.
HOPEFULLY THIS MAKES SENSE???