FINAL FANTASY XIII-2

Achieve a 5-Star Battle Rating on The Proto Fal'Cie Adam at the top of the Augustus Tower in 200 AF at the end of Episode 4. In order to get off the first floor of this tower you must go to the Augustus Tower in 300 AF and run through that level and get the Security Key, so that you can unlock the first elevator in the tower in order to get to the top.

Once you're up there the battle is pretty easy, though I did have my key roles maxed out to 99(Noel/COM, Serah/RAV)with the best weapons at that moment the 4.5k Gil weapons from Chocolina, and my monsters were at least level 2 almost 3. I was playing on Normal but you can do it even easier on Easy of course. All you really need to do is take out the two "arms" he sends out than start attacking him. The two "arms" and himself are easily staggered.

I used COM/RAV/RAV(Relentless Assault) on the arms, and easily dealt with them.
Then switched to RAV/RAV/RAV(Tri-Disaster) and staggered Adam.
Then back to Relentless Assault(because that was my more preferred and more powerful paradigm) and easily took half of his health before him becoming unstaggered.
So just rinse and repeat, minus the arms because they don't respawn.
As for healing I used, SEN/MED/MED(Combat Clinic) Though I really didn't need to, this boss is quite easy.

*If you somehow mess this up you can always use the Gate Seal you get from this mission to use Time Reversal to re-do this level and eventually this Boss.

If you have any questions, message or comment below.

Video Courtesy of PowerPyx.

Another method to get this is to have three simple Paradigms: Serenity (MED/MED/MED), Tri-Disaster (RAV/RAV/RAV) and Cerberus (COM/COM/COM). At this point, I had the best weapons stat-wise at the time (the ones that cost 4.5k each), and I had Cactrot as the Medic after infusing Cait Sith, who was well raised), Zwerg Scandroid for RAV (infused to have multiple elements to attack with and Dragoon as the Commando (I infused several Commandos I had raised and gave it more defense and nicely boosted Strength).

The key is to get Dragoon (which you can find in the area before you fight the Proto Fal'Cie) and raise it as much as possible, particularly with Power Slivers, though boosting its Magic won't hurt either. Then, when the fighting starts, switch to Tri-Disaster if you aren't already and Stagger the Manipulators, then Switch to Commando and knock off each one, switching when necessary. Then, in Tri-Disaster, stagger the Fal'Cie, then switch to Cerberus and let it rip. The Dragoon, with a little over 500 Strength, can hit for 2500 when the enemy is Staggered, 5000+ with Blindside, which will murder the Fal'Cie quickly. If you go for a pure Strength, it can hit well over this.

If you're in trouble, switch to Commando to get a hit in and slow the Chain drain if it isn't staggered, then to Serenity to quickly get your team back on its feet (by this time both Noel and Serah learned Raise, and my Cactrot had Raise too. They all had Ensuna and Curasa, which sped up healing), then switch back to the Paradigm you were in before.