The Oath Sword is a quest in Xenoblade Chronicles. It can be received from Karlos near the entrance to the Hidden Machina Village on the Fallen Arm. This quest is only triggered if Fiora is in the lead. Successfully completing this quest will unlock Fiora's fourth skill branch, Rashness.
To reach the Power Pipe Ruins, travel to the Ulna Passage landmark and activate the Pressure Control Device on the west (run past the walkway to the 5th Pulse Zone and look for a red dot or just run along the walls to find it, it will not show up on the mini map). A wall will open, then climb up the wall to reach the upper level. From there head south and climb down the wall. The item will immediately appear on the map.
"You have returned the sword to Karlos and he is even more determined to keep his promise."
The Affinity Chart will change after the quest has been completed: