Lutece, as noted. its signed. Now this does have some subtle implications. 1. As per the coin flip, booker has done this before. 123rd trip at least. keep in mind, there might have even been more trips prior to them starting to do the coin flips. 2. Booker picked #77 the first time, and died or was captured by the cops, and died.
3. Constants and variables. Booker having died at the raffle. The Lutece's introduce variables. Random tears opening up as noted? that could be the Lutece's introducing variables for example. the statue changing from male to female just moments before you arrive, causing you to stop and look at it very likely. While Booker picking #77 is a constant. But warning Booker about picking #77 is not. But them telling Booker to not pick #77, his reaction to picking it changes. He is now on edge, he wasnt supposed to pick it, and thus is ready for the cops.
Its a small event that tells you alot of what is going on behind the scenes.