Each gem is tucked into a specific map zone. The Collector badge only fires when all five are in your inventory, so this list also functions as the canonical hunting order.
The Yellow Gem is the easiest of the five. Look up when entering the hotel, climb the shelves into the vent duct network, and follow it to a dead-end pipe.
The Blue Gem sits on a narrow shelf jutting out from the southern cliff. Walk to the midpoint of the bridge, then drop straight down.
The Purple Gem is tucked between the secondary mainframe computers and the support columns on the third floor of the Research Tower.
The Green Gem sits under the central metal catwalks on a copper manifold pipe that carries the water supply. Drop down and walk the pipe to the gem.
The Red Gem is the most commonly missed. It sits on an ash block behind the exhaust valve along the stone retaining slabs around the primary lava furnace.
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