This guide contains spoilers for one of the locations in Duviri. If you want to do this puzzle on your own, stop reading this guide.

Some notes (thanks to Tohirin for pointing some of these out):

• Archarbor only appears during Joy, Sorrow, and Envy Spirals.
  
• It gives rare resources as a reward, although I'm not sure exactly what. Some people have reported getting Cinta parts, but I haven't gotten any in my 5-6 attempts now at this puzzle.
  
• Unlike normal puzzles of this sort, the symbols won't light up or make a sound when you get the correct one.

Here's a map of Archarbor.
B = Bust
S = Security

Your first goals are to disable security in the southwestern part of the map by interacting with the podium (labeled "S"). I'm not sure if you have to disable the rest of the security on the island.

Along with getting the security console, you want to interact with all three of the busts of Dominus Thrax (labeled "B") so they face the center of Archarbor, the same way the podiums are facing. Once they are in the right position, they will all start glowing.

Once you disable all three security consoles and move all three busts, head to the southwest security node and head underground. There, you'll find a glowing object with a console next to it. Interact with it and look outside, and you'll see the door to your next objective being opened. Now you'll want to head underneath the island and go to that door.

Once inside, you'll find an owl sitting in the center of the room with several glowing blue objects around it. In case you haven't seen these weird tube/pot objects yet, you shoot them in order to change the symbol it shows. On one of the walls, you'll see this:
You need to match the symbols in the tubes with those on the wall. But hold on, you'll notice that some of the tubes on the wall aren't glowing and half of the symbols are obscured. Someone's already figured out the puzzle and uploaded the following image to the official Warframe Wiki, which labels each symbol and which tube it corresponds to:

To get started, there are four tubes you need to make functional, two on either side of the wall of symbols, designated by podiums in front of them. These are tubes are 8, 5, 4, and 3.

The glowing blue balls you need are in the four tubes on the other side of the room from the wall of symbols. You have to interact with each stand in a different way- one you just shoot, one stand you get close and "push", one has a glowing ball ready to be picked up, and for the last one, you shoot a little glowing bit connected to it. Once you have the glowing balls, pick them up and throw them into the nonfunctional tubes.

Each symbol corresponds to a specific tube on either side, just use the image for reference. Positions 4, 5, 6, and 7 only show the exact middle cut of a symbol- there may be a bit of confusion as to what goes where, but if you record everything on paper you'll figure it out even without this.


Here's how you interact with the tubes to change the symbols (or open them, if you haven't figured it out):

• 2: Shoot the little node attached to it to open the tube.
  
• 8: Shoot the bottom tube to open the top one.
  
• 5: Shoot the symbol, don't shoot the node.
  
• 7: Shoot the symbol.
  
• 6: Shoot the symbol.
  
• 4: Shoot the symbol, don't shoot the node (Once you input the correct symbol, remove the ball and use it elsewhere).
  
• 3: Stand on the platform to open the tube.
  
• 1: Shoot the black vines to open the tube.

Tubes 2, 5, 6, and 3 shoot damaging lasers at you as you change the symbol. If you dodge quickly, you won't take any damage. For future reference, whenever you see these puzzles out in Duviri and you see them flashing rainbow lights like here, get ready to dodge.

Find that you're missing a ball but you swear you've used every single one? Double-check tube 4. If it still has a ball, you can safely remove it and the current symbol will be saved on it. You'll know once you've set all the correct symbols as the game will immediately reward you with loot and a decree. Once you see that message, go to the owl and collect some more loot off the ground.

Just to be clear, this puzzle is randomized every time you go into Duviri. I'm sharing my puzzle solution so you get a better idea of what the right symbols might be for you in your own game.
Here's my symbol wall:
In order, going left to right:


Remember that the slit shows the middle portion of the symbol, they don't need to be in order, and it's a lot easier if you write it down. Good luck!

Thanks for checking this out. I'll be sure to update it to be more detailed once I get the chance to do the puzzle again.

[EDIT] Brozime put out a video on how to solve the puzzle, which you can watch here, if anyone wants a decent video guide on how to do it.

[EDIT] The official Warframe Wiki added an image to help people figure out the puzzle, which I used in this guide. I got the image off this page[warframe.fandom.com] in the Wiki, but be wary of spoilers.

11/27/23 Edit: Changed the wording in a few areas, and added a note about checking tube 4 to reuse the ball.

Any concerns? Clarifications? Feel free to leave a comment.