Here is the full list of all 15 Apex Construct trophies - there are 13 bronze, 1 silver and 1 gold.
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Defeat the Bear Behemoth in the CARF Teleportation lab
Gain the highest security access level
Gain access to the locked room in the safe house
Defeat the Bear Behemoth in the caves
Power up the cloaking device
Electrify water & damage an enemy
Hurt/kill an enemy with a Radiance Mine
Pay Mothr a return visit
Crash Fathr's Head to the ground
Spend RP at a Vending Machine
Get to the safe house
Alert the crabs in Hello World