I edited this post from asking how to solve the issues of acquiring Boolean Gemini to helping give people the solution to this issue.
First, you need to finish the quest that has you kill 2 Taken Zealots on each planet.
Then after turning it in, you may have to wait a few hours up to the daily reset before Petra gives you the quest 'Lost and Found,' and definitely take that quest.
Finnally, it seems that Boolean Gemini isn't guaranteed to drop after 'Lost and Found,' but it still could, so it seems like the gun is awarded randomly from one of the Reef quests like 'Lost and Found,' 'Wolves of Mars,' etc.
If you're still having trouble with this you may need to have your Queen rank at 3 or above, but I'm not sure if this is nessisary. Let me know if you got it under rank 3 with the Queen.
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3 RepliesFrom all the threads this is the determined requirements for the drop 1.lvl3 queens rank (Bungie help posted this on another thread) 2.complete ALL wolves on Mars quests for Varkis 3.complete all Petra quest line and turn in "lost and found" I know this works because I did it this morning and got the drop Keep grinding Gaurdians