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When Neo is first shown about the Matrix, the real world, and how all humans are enslaved by the machines to harness their energy, Neo has a profound panic attack. While watching this, Cypher says "He's gonna pop!"

What does Cypher mean by that? Does he mean that someone can die from the mental shock of learning about the Matrix? Does he mean that Neo could experience a physical "pop" in his body, like a blood vessel bursting, and it could kill him?

No, it simply means that he's going to be sick and/or pass out from the shock of what he has learned.

Its slightly more than just he's going to puke.

Neo just went through a profound, shocking and for him completely unfathomable situation he can barely process whats going on, his mind, as the kids say,

has been blown

Cypher, the depressing pessimistic person he is, said, "he's gonna pop" probably because he sees Neo literally rejecting reality, or not reality and is probably going to have a heart attack, and die, this is probably a common occurrence to people who are ripped out of this kind of synthetic world.