Gemini Core Booster
Gemini Core is an Boss artifact that gives the Gemini's left (red) cannon +1 damage, and its right (blue) cannon -1 damage and stun.
Clarification
"Red" and "blue" cannon are the more technically correct terms here in case you had your ship shuffled by Ship Shuffler - the red cannon will always get +1 damage, and the blue cannon will always get -1 damage and stun, regardless of which is actually the "left" and "right" cannon post-shuffle.
The power of an attack cannot go below 0, so for example, any shot that his a Weak point will still deal 1 damage. But all modifiers, positive and negative, are factored in simultaneously when deciding the power of a shot. So for example if you have 1 Overdrive, are shooting from the blue cannon with this artifact, and use Buckshot, each shot will deal 0 damage (base 0, -1 for Power Diversion, +1 for Overdrive). This could surprise you if you gain 1 Overdrive expecting it to increase the attack power of Buckshot.
Strategy
A powerful boost to both your offense and defense, depending on what the situation calls for, provided you plan out your turn skillfully and have ways to toggle your cannons easily (like Crystal Shard, Bolt, and other cheap cards).
Stun is a very valuable defensive resource, especially against The Cobalt. You'll usually want to be dodging to your right with this artifact to move into the space cleared by your stuns, making many of Peri's rightward movement valuable.
Multi-hit attacks like Multi Blast benefit very well from the left cannon's damage boost, although they'd rather not get used from the right cannon.