GEOTHERMAL VALLEY - PIT OF JUDGMENT CHALLENGE TOMB
IMPORTANT: There is a potential bug in this area that so far has not been patched. It may occur in the Xbox or PC versions. I don't know what causes it, so I'm not sure how to avoid triggering it. So, just in case, I strongly recommend changing save slots at the Ruins Encampment Base Camp and not overwriting your previous save until you've completed the tomb. If you encounter the bug (detailed below), you can reload your earlier save, change slots again, and retry the puzzle. If it still doesn't work, reload your earlier save and continue with the game. Hopefully the developers will fix this at some point and you can then return to this area later complete the tomb.
CHALLENGE TOMB ENTRANCE: Head north from the Ruins Encampment Base Camp, past the spot where Sofia gave her speech to the villagers earlier. Follow the path behind the waterfall and up a set of wooden stairs to the Challenge Tomb entrance. If you haven't yet dived into the pool here for the High-Dive Challenge (details), here's your chance. (screenshots)
Follow the tunnel to a shallow pit with metal rings above. Pick some mushrooms growing on the left and then use your grapple axe to swing across the pit. Squeeze through a narrow gap in the rocks to discover the Hidden Ravine Base Camp. Grapple-swing across the deeper pit beyond the camp. Just ahead the tunnel is divided by a large boulder. Follow the right wall to find an Explorer Satchel (5/5), which reveals the locations of 3 survival caches. Only the first one is visible on your map so far. The others are in the unexplored area ahead. (screenshots)
Now follow the falling water and slide down the slippery slope. Jump at the end to clear the chasm and latch onto the climbable wall. Climb onto the ledge above. Walk across the fallen tree trunk and dig up the survival cache (12/20) buried ahead on the left. Continue through the low tunnel on the right. Drop into the hole ahead and slide down into the tomb. (screenshots)
MINE CART PUZZLE RIGHT SIDE: Swim across the pool to the north side to get a glimpse of the codex behind a wooden barrier. Dig up another survival cache (13/20) buried off to the left. (screenshots)
Now tackle the puzzle. Turn around and scope out the two long mine cart tracks. The tracks on the left are blocked by debris, so run up the ones on the right. At the top, on the left, just before the octagonal platform with the mine cart, is a box containing a relic (8/13), a Rope Ring from the Time of Adaptation set. The mine cart moves, but a wooden bumper prevents you from pushing it along the track. Use your axe as a handle to turn the wheel, rotating the platform beneath the cart 180 degrees. Now you can push the cart forward along the track until gravity takes over and the cart plummets down the slope and crashes into the wooden barrier below. One down; one to go! (screenshots)
Return to the top of the track. There's magnesite embedded in the wall ahead on the left. In the dark alcove at the back, you'll find a document (17/33), Unrepentant from the Pit of Judgment series (Greek). Follow the tunnel to the right and jump across the pit. In this dead-end tunnel, there's a container with cloth and 2 Byzantine coins, a lump of magnesite, and the Archivist Map (6/9) for this area, which reveals the location of any unfound documents and relics on your map. (screenshots)
Return to the movable platform where the mine cart was and continue straight through to the ledge with the rope-wrapped post. Shoot an arrow into the beam on the other side of the cavern to create a rope line. Then slide down to the base of the wooden structure supporting the second set of tracks. (screenshot)
MINE CART PUZZLE LEFT SIDE: There's a second mine cart here, but you can't do anything useful with it yet. Mine some more magnesite in the alcove on the right. Pick up some cloth and a relic (9/13), a pair of Dice from The Athanatoi set, on the left. Then scramble up the wooden wall and climb the stairs to the top. There's more magnesite on the outside ledge and a survival cache (14/20) at the back of the alcove with the movable platform. (screenshots)
Crank the wheel to lower the platform to the level below. Notice that as it descends the platform also rotates a quarter turn counterclockwise. Cross the platform and turn the second wheel to rotate the platform a quarter turn clockwise. Now you should be able to push the mine cart onto the platform. Before raising the platform, turn the second wheel again to rotate the platform a quarter turn counterclockwise. Now turn the first wheel again to raise the platform to the level above, while simultaneously rotating it a quarter turn clockwise. When you get the cart to the upper level, you should now be able to push it forward down the tracks. (screenshots)
NOTE: If you turned the lower crank too far, or not far enough, and the platform somehow became misaligned, preventing you from pushing the cart onto the tracks, just return the cart to the lower level and start again. Notice there are 4 wooden bumpers, one on each side of the platform. Three of these remain fixed at all times. The fourth retracts into the floor to allow the cart to pass over it. Rotate the platform until the retractable bumper is farthest from the crank that controls the rotation. Then turn the other crank to raise the platform. It should then be in the correct position once you get it to the upper level. (screenshot)
When push the cart forward, it rolls on down the tracks, crashing through the pile of debris and destroying the wooden wall in front of the codex room. (screenshot)
NOTE: This is where the potential bug may occur. Some players have reported that after completing the puzzle, they are still unable to push the second mine cart more than a few feet down the track. If this happens to you and you have switched save slots, as recommended above, you can try reloading your save, changing slots once more, and then trying the puzzle again. If it still doesn't work, reload your earlier save and continue with the game. Hopefully the developers will fix this at some point and you can then return to this area later complete the tomb. If you did not change save slots, you can try restoring a backup save on Xbox or reloading the "Last Campsite" save on PC. I also encourage you to add to the bug report thread on the Square Enix forum. Be sure to mention which version of the game you're playing.
THE CODEX AND OTHER TREASURE: Swim back across the pool and enter the codex room. Here you'll find 2 clumps of mushrooms, a container of cloth, a box of supplies, and the second document (18/33) in The Pit of Judgment series, titled The Overseer (Greek). (screenshots)
Of course there's also the codex itself, which unlocks the Geologist skill (plus XP and 10,000 Expedition Credits). This skill allows you to find chromite ore, which can be used for certain advanced weapon upgrades and outfits that involve armor plating. Now whenever you mine magnesite, there's a chance you'll also get some chromite. There's a vein of ore on one side of the treasure room where you can try this now, plus 2 lumps of ore in the doorway leading outside. Chromite is listed under Exotic Resources in the Resources section of your Map/Inventory screen. (screenshots)
Swim back across the pool toward the entrance. To bypass the slippery slope where you came in, climb the rough wall on the left to the ledge above. Latch onto the next section of climbable wall and climb up and to the right. Jump across to the right and continue climbing onto the ledge above. Go through the low tunnel with the dangling roots. Return across the fallen tree. Then jump across the chasm to the climbing wall on the right. Climb to the left and upward until you reach a pair of thin, stone handholds. Continue climbing around to the left until you can drop down into the tunnel that leads back to the Hidden Ravine Base Camp. (screenshots)
After completing this tomb you could fast travel back to the Cliffside Lookout Base Camp (or wherever you left off), but you may want to revisit the Remnant herbalist who gave you the Ancient Secrets Mission. (details) Her workshop is southeast of the Ruins Encampment Base camp, near the third monolith. She'll give you 10 gold coins for each Challenge Tomb you've raided since you last saw her.
†UPDATE HISTORY: 1/16/16 - First draft of walkthrough posted online.
3/15/16 - Added note about mine cart bug in Pit of Judgment Challenge Tomb.
6/6/16 - Made a few tiny tweaks while revising the entire Geothermal Valley walkthrough, nothing major as far as strategy, though.