Ariadne Society
The Ariadne Society is an organization in Icathiria devoted to trying to solve the puzzle they believe the Stagehands have posed for them. According to members of the Society, all the mysteries of Icathiria are an enigma meant to be solved; the universe is riddled with clues leading to some solution, hinting at something they're expected to do. The patterns of the warphole connections, the seemingly random effects of the Clarke chambers... all of these are parts of some encrypted messages that the Stagehands hope the inhabitants of Icathiria can decode. And the Ariadne Society is dedicated toward doing that.
As to what happens when they break the code and follow the hidden instructions, most members of the Ariadne Society don't claim to know that. Clearly the Stagehands have awesome power, though, and any prize they arrange should be truly worthwhile. Perhaps those who solve their riddle get immortality, or godlike powers; perhaps they find a way into a land of bliss; perhaps they're even invited to join the Stagehands themselves. Certainly the reward is sure to be worth the trouble—though to many members of the Ariadne Society, the search for the pattern is engrossing and entertaining enough that they don't even give much thought to what happens afterward.
Naturally, the views of the Ariadne Society are not majority views among the inhabitants of Icathiria as a whole; most people think their idea that all of Icathiria is an enormous riddle meant to be solved is rather far-fetched. Truthfully, not even all the members of the Ariadne Society really believe it—the Society was originally founded on the view that the Stagehands fashioned a giant puzzle, and that continues to be the organization's initial position, but many of its members join simply because they enjoy puzzles and problem-solving and want to have access to—and possibly contribute to—the Society's work so far determining patterns in the Clarke chambers' effects and similar matters, but don't really believe there's any overarching riddle to be solved.