Chief wakes up where he left off in Halo 3, floating in the broken remains of the Forward Unto Dawn. Cortana wakes Chief up because they've been boarded by some Covenant who were presumably guarding the planet you see in the Halo 3 Legendary ending.
This planet turns out to be a dyson sphere shield world.
You go inside, discover the Prometheans and some tid bits about the Forerunners.
You start picking up a signal, it's the UNSC Infinity responding to the distress beacon Cortana had been playing from the Dawn.
You realize you need to warn the Dawn away because there was no discernible way out of the dyson sphere.
You realize some glowing tower energy things are causing interference, you work on disabling them but instead you awaken a Forrunner surviver. He was basically the military leader of the Forrunners. He's pretty damn powerful.
The Coveneant ally themselves with him.
Infinity enters the sphere and you meet up with Latski, Cheif is placed under arrest by the Captain of the Infinity, Chief doesn't give a fuck since he's the Chief and has better things to do like stop the Forrunner douche bag from activating the Halo's on a quest for vengence against the Flood.
There was also some point where the Forrunner dude talked about how the Forrunners encountered the Human race long ago, and that they went to war with them to protect their area of space. The Forrunners then found out that we weren't expanding agressively, but running from the original Flood. Flood destroyed both civilizations since the original humans weakened the Forunners.
Anyway, Chief goes after the Forrunner douche and ends up killing him.
That's my synopsis of Halo 4.
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