I know your pack full of idea's about whats behind the four plate door. Barrow king, Jax, the moon. From my work, the list of reasonable translations given the words we see in the book is surprising small and (spoiler) not particular enlightening.
It can be split a couple interesting ways based on available words. The best is the simplest to see:
Valar-itas
"valar-" could be valor or alar. Alar we know is a type of magic, but it also might be used to express "strength of will" :
She cocked her head. “Why don’t you call it strength of will, then?” “Alar sounds better,”Wilem said. ~TWMF
or maybe just "belief".
"-itas" Has two options. The first, and best is "atas" aka story|history
“Its story?” I asked. Vashet shrugged. “In Ademic it is Atas. It is the history of your sword. Everyone who has carried it.
Right here we can stop and be fairly proud of "The story of Will" Confirming that indeed, what's really behind the door is the friends we made along the way. Poor Wilem. Haha! surprise! Bu really, does some combination of will/belief|courage and history|story tell us something? Let me know in the comments below, it certainly isn't helping me.
A translation of Ve Valora Sartane, the name of the song Kvothe sings to Nina, would be very useful. Unfortunately, If this piece is the key, the answer seems to be forth coming. So the far second choice for "-itas" is "tass" as used in:
... translating Edamete tass and grow terribly offended when I discover it means ‘the Edema Drip.’ ~ NOTW
to mean "drip". Ok. So "Will of the Drip" sounds like a 80's horror film. But "Will of the infected" might give you some pause. Think danced. Alternatively, drip as water yields "The will of the water". Which is interesting to me given I'm curious about the proximity of Auri's pool and the four plate door. From my notes, it's impossible to tell how they line up, so it's possible the door is under/alongside it.
Didn't Kvothe mention the door had no air gapes?
It had no hinges. No handle. No window or sliding panel. Its only features were four hard copper plates. They were set flush with the face of the door, which was flush with the front of the frame, which was flush with the wall surrounding it. You could run your hand from one side of the door to the next and hardly feel the lines of it at all.
Also wasn't the stairs to Puppet's room (below the four plate door) smooth as a riverbed?
“Where are we going, anyway?” I asked, mostly to stop Simmon’s henpecking. “Sub-three,” Wilem said as he turned to descend a long flight of stone steps. Centuries of use had worn down the stone, making the stairs look as bowed as heavy-laden shelves. As we started down, the shadows made the steps look smooth and dark and edgeless, like an abandoned riverbed worn from the rock. ~
So maybe just "Strong water"? Here I bet you thought it was a metaphor:
One a door that holds the flood ~ TWMF
You would think Chronicler would have mentioned something like the flooding of the archives lower levels. But you would also think he would comment on a dead king returning as well.
In the interest of fairness, we have to consider why such a door would need copper plates. Given Elodin's cell had a copper door and copper was the only thing left after master Namers naming, it's a good bet that Copper is either nameless, or rarely named. The idea is lent strength by Martin's story which gives Taborlin a copper sword, a decent weapon against namers. The other interesting bit is the number of plates. Here my only idea is to note that Lyra, Aleph, Selitos and Lax make four and while it's a good guess that Lax/Jax is sealed away, 3 is still close to four and maybe another stepped up to lend their powers to keep him down. The number four given special significance also in that it's called the four corners, possible for similar reasons (credit to u/bhaluun for the observation).