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Watch Video: Halo 5 Speedrunning Tutorial - Part 13: Genesis
During the beginning sequence, just focus on jumping and thrusting as you go faster while in the air. Once you land on the ground run to the elevator There is a trigger inside the hallway leading up to the elevator that if reached too fast, the door at the bottom will take an extra 10-15 seconds to open. It might help to wait for master chief to say "how the hell are you here" before entering the hallway. This skips Cortana's dialogue and opens the door almost immediately. Run through the door and grab one of the ghosts to the left. The big door in the next section will begin to open once you drive in front of it and trigger the spawn of a knight and 3 crawlers right in front of the door. Nothing here needs to die so while your waiting it is a good idea to grab the ultra ghost to help kill the wardens. Proceed until you reach the gap where the bridge eventually spawns. Waiting for the bridge is slow so you want to ramp over the right side of the gap and proceed to the final section.
You will notice a ramp by a crashed pelican. Drive up this ramp slowly. If you drive over it too fast you will miss a load and the level will not end until you hit it. Now just use the ultra ghost to kill the 2 wardens and drive to the door to end the mission.