Academia 4XX AF lacks any real side quests or monster battles. The only notable things here, other that the quizzes, which will be covered in a separate guide, are the Chaos Crystal Weapons and a Paradox Ending. You may remember from earlier in the story when Hope and Alyssa had you running around the various areas of the Historia Crux collecting Graviton Cores. Upon delivering the fourth of the Graviton Cores to Hope, he mentioned that he would be able to craft you a very powerful weapon, but only if he had something called the Chaos Crystal. This is a very rare Fragment that you will have to collect anyway if you are intent on getting every Fragment in the game. Luckily, you do not really have to rely on random chance or running around in the world to find it. The Chaos Crystal is a reward in the Serendipity Casino.
You can collect the Chaos Crystal from the Casino Vendors in Serendipity for 10,000 Casino Coins. Sadly, this rounds out to about 100,000 gil that you would have to spend on Casino Coins in order to get it. You can lower this amount by a pretty significant sum (about 7,500 Casino Coins lower, to a total of 75,000 gil) if you get the Bargain Hunter Fragment Skill, which you should have already. You may still need to participate in the Casino’s games in order to raise such a large amount of Casino Coins. In this case, the Chocobo races are the way to go, as they are the most consistent and reliable way to raise Casino Coins quickly. Head over to Serendipity and refer to our earlier guide to the Serendipity Casino for strategies on how to win the Chocobo races with ease.
Once you obtain the Chaos Crystal, return to Academia 4XX AF and speak with the receptionist in the Entrance area. The receptionist will give you two choices, and you can only go with one so think carefully. You can either trade your Chaos Crystal for the Odinbolt for Serah, or the Odinblade for Noel. Both of these weapons will grant the user +40 Strength, +40 Magic, and the Fragment Energy Passive Ability. This ability makes these weapons all the more deadly, as it will increase the power of the weapon as you collect more Fragments. As long as you follow this guide and grab all of the Fragments in the game, you will be virtually unstoppable. Pick one of the weapons, and know that after you do you can obtain the other from the Serendipity Casino later on.
Now for the Paradox Ending. Reset the story progress for the area in the Historia Crux using the Paradox Scope item. Return the Graviton Cores to Alyssa as you did before. However, when you hand her the fifth Core, she will offer you the Vagabond Artefact in a Live Trigger Event. Deny the Vagabond Artefact to view the Paradox Ending.