You return to the office after SongBird slammed you flying into a wall. I reckon a Booker died there.
Not to mention it explains why the Lucetes' would want to warn you about taking the number 77 ball.
The Songbird didn't slam you flying into a wall. You dropped into the ocean and blacked out before washing up on the shore. You passed out after getting baptized by the preacher. They (third parties and yourself) address both these things immediately after you wake up.
The Luteces warn you about the ball so you'd be on alert for a fight. What does that have to do with the office?
What do you mean by experimenting with constants?
For me it was because a Booker died after taking the #77 ball, so they decided to warn you against taking it.
They weren't warning you about taking it. You get #77 every time, it's a constant. They were just letting you know so you'd be ready for a fight.