3.disillusionment/suspended animation
Check the bulletin board.
If you have Archer, go to your Private Room for an important conversation. Pick the 2nd option.
Talk to Rin on the Roof.
Go to 1F for an event.
Talk to Taiga to get a quest to find a magazine.
Go to the schoolyard and talk to the marked boy who wants a Cauldron.
Talk to Sakura in the Nurse's Office for your weekly item.
Enter Arena 1F.
Move forward for an event with Alice.
You're supposed to follow her, but there's no penalty for taking your time and doing other things. However, you might as well chase her because you can always explore the side areas on your way back.
Catch up with Alice right before the Trigger location and the exit. She summons her "friend." Choose to run. You get Matrix Level 1.
You can't advance beyond Alice's "friend" for now. Work your way back toward the entrance, collecting items from this mini-maze made from hidden paths along the way.
Make sure you pick up the Cauldron from the southwest area.
Talk to Alice near the 2F stairs.
Go to the Arena entrance to talk to Alice. Tell her to move her "friend."
Talk to Rin near the 1F stairs.
Go to 3F and talk to Issei.
Talk to Rani at the end of the 3F east hall.
Talk to Rin on 1F.
Go to the Commissary and check the item store for a Ruby. It's way too expensive. Exit the shop menu for a conversation and say there's no way you can afford it.
Talk to Sakura in the Nurse's Office.
Go to the Commissary cashier again to trade for the Ruby.
Give the Ruby to Rin on 1F.
Talk to Rani on 3F.
Give the Cauldron to the boy in the schoolyard.
Enter Arena 1F.
Go back to Alice's "friend" on the north side. You can beat it now. It's pretty straightforward, just a stronger-than-average regular enemy.
Collect the 1st Trigger after it falls.
If you have Caster, go to your Private Room for an important conversation.
Talk to Rani on 3F.
Talk to Rin on the Roof.
Go to 1F for an event.
Enter Arena 2F.
You only need to chase Alice—there are no enemies today.
[Note: If you are on a New Game +, do not use the Enhancement Spike to increase your speed. This event will not end if you constantly keep running faster than Alice.]
Corner her a few times at dead ends. After the third time that you corner her, you automatically leave the Arena.
If you have Saber, go to your Private Room for an important conversation.
Go to the Garden in front of the Chapel for an event with Alice.
Go to the schoolyard for an event with Alice.
Talk to Rin on the Roof.
Talk to Taiga on 1F to give her the manga and receive the quest to get crushed ice.
Enter Arena 2F.
You just have to select your name. Feel free to pick the wrong one the first time for your own amusement. It's an officially sanctioned meme that you are not penalized for. You get Matrix Level 2.
Advance through the dungeon. Make sure you get the Crushed Ice from the northwest and the 2nd Trigger near the exit.
Talk to Touko in the Chapel.
Go to the 2F Library for an event with Alice.
Return to Touko in the Chapel.
You need to find a mirror. Go to the restrooms in the east hall on any floor. You could just do the boring thing and use one of your correct-gendered restrooms. Or you can persistently try to enter an opposite-gendered restroom 3 times for an amusing scene.
Go to the Library and check the marked bookshelf. You get Matrix Level 3.
Enter Arena 2F and grind until you're satisfied.
There is a one-day-only limited-time item called Super-spicy Mabo Tofu available at the Commissary today. There isn't much point in it, but it's worth pointing out.
Talk to Taiga to give her the Crushed Ice.
Go to the Arena Entrance for an event.
Enter Arena 2F and grind until you're satisfied.
Go to your Private Room and organize your information. You get Matrix Level E.
Equip the Exorcist Blade. Just do it.
When you've finished your preparations, talk to Kotomine on 1F.
Do this:
H G G B B A
A H G B G B
G A H B B G
B B A H G G
B G B A H G
G B B G A H
'H' means you use Hack and attack.
I seriously doubt it will actually take all the way to the end of that for her to fall, but I included extra just in case. If you're using Caster, you can also try spamming Skills instead if you really want, using the appropriate Skill to stun before an incoming enemy Skill instead of Hack. She goes down really quick that way. In either case...
Easiest. Boss. Ever.
After the battle, an event will happen.
You need to choose whether to side with Rin or Rani. This is a big deal, so choose wisely.
In either case, you end up fighting Berserker. You only need to survive for 1 turn.
You are given an option before the turn begins whether or not to use a Command Seal. Just so you know, it makes absolutely no difference whatsoever whether you use it or not. It isn't even mentioned in dialogue. That being said, I would recommend using it unless you just feel like messing around. Using it shows you all of Berserker's actions, making surviving a breeze. If you decide not to use it just for the fun of it, all you need to know is that Berserker always uses an attack Skill on the 6th move and only on the 6th move. So you can survive by spamming attack Skills for the first 5 actions and either using Hack or Guard for the 6th.