>>530094434
In Japanese version its called the Adult Card, to not be confused with a credit card. Also pyroxene in Japanese is written in a way so it could be read as "miracle(kiseki)". You are collecting miracles through your bonds with students. Anyone that read volume 3 knows what the adult card does as well.

The thing that undid Hoshino's contract was sensei not signing the contract, since he's the advisor of the club, she can't quit the club without the signature of a guardian. A theme throughout the game is that students are super powerful, they can lift several tons and run super fast, but their weakness is always "Adult stuff"; contracts, beurocracy, controlling emotions, making rational decisions, legal shit, laws, all fake bullshit that only adults understand. The students are good at acting on their emotions or acting on their intuition, but they always run into problems when they run into adult things. Hoshino for example is super strong, but she gave herself up on a bad contract because no matter how strong she is, she's a stupid kid who doesn't understand contracts. Sensei's role in this game is to navigate the adult shit that the kids can't comprehend; sensei actually read the contract and realized a loophole in the contract to invalidate it. Hoshino couldn't sign off on the contract on her own without the club advisor approving it.

Showing Blacksuit his adult card was more of a threat among associates, its something black suit respected; blacksuit is saying sensei may be correct but does he have the power to see through his ideals. The Adult card shows Blacksuit that sensei isn't all talk, he has enough to back up his words with actions.