After some pretty extensive scouring of forums, I didn't chance upon a thorough explanation of the Lutece's, those delightful embodiments of the game's "we be fuckin' with you from minute one" mantra. I've gotta agree with our illustrious Mr. Gerstmann in his assessment that they are the game's most fascinating characters. So without further ado, allow me to test your patience.....
To begin, the Lutece's aren't twins. They are in fact two differing versions of the same person from two different universes, one born with a Y chromosome, the other with an X. Robert originally existed in tandem with the Booker who refused the baptism, learned to live with sin, had Anna, etc. Rosalind originally existed in the universe in which Booker accepted the baptism, "birthing" Comstock and, soon after, Columbia.
These two Lutece's of differing gender and universe gleaned the existence of each other and the alternate universe each inhabited because they were both (being of the same mind, literally) experimenting with the atom by way of what Rosalind refers to as the Lutece field. Rosalind on a much more impressive scale via the benefit of Comstocks funding, but nevertheless, through their tinkering the Lutece's were made aware of their alternate universe counterpart, and they began to communicate via trans-dimensional Morse code. This Morse code was implemented by turning their machines that generated the Lutece field on and off in a in a sequence recognizable by the Lutece in an alternate world..This, along with Comstock's money, lead to the creation Rosalind's machine, which could not only discern alternate worlds, but tear into them. This allowed for Comstock to acquire Anna from Booker with the help of Robert Lutece, the Lutece in the non-baptized Booker's world. It is explained, via voxophone, that exposure to the Lutee's machine has resulted in Comstock's sterility, leading him to use these rather alternative methods to seek out an heir, a seed, that will "sit the throne and drown in fire the mountains of man." When you arrive at the end game scene in which Booker hurtles down the alley in an attempt to stop the "transaction" you'll see Robert heatedly explaining to Rosalind (who is on the other side of the tear, Comstock and Columbia's side) his trepidation regarding stepping into another universe, though he ultimately does when Booker attempts to intervene, along with Comstock who is carrying baby Anna. There is a sightseer video that documents the Columbia denizen's confusion at a Lutece "twin" suddenly arriving in Columbia, as no one knew Rosalind had kin.
Anna/Elizabeth's finger inhabiting one universe and the rest of her another is what allows her to tear at will when not restricted by a siphon. A Rosalind voxophone postulates that the girl's power derives from the fact that "the universe doesn't like its peas mixed with its porridge." This likewise explains the craziness regarding the reincarnated Lady Comstock, as like Elizabeth, she was a single entity split across two universes. This concept also accounts for the Lutece's brand of universe hopping witnessed throughout the game, but we'll get there. Ancillary detail: when you're tooling around Monument Tower, before you rescue Elizabeth, the third siphon containing the hand towel with menarche bloodstains (creepy) reads age thirteen, and a room or two later you are presented with a chart that documents the drastic spike in Liz's trans-dimensional potency post puberty, at which point the siphon was installed.
Anyhow, when Robert jumps from his universe to Rosalind's, he finds his mind, and I quote from the very first visual the game presents us with, "desperately struggling to create memories where none exist." And who is this quote attributed to? Why, an R. Lutece no less, from their book The Barriers of Trans-Dimensional Travel. While Robert had Rosalind to help him sort out the cognitive dissonace that crops up when someone is transposed from one world into another, it was this period of disorientation that gave Robert the confidence that Booker would develop a narrative for himself when they ripped him from his universe of drunken, depressed stupor incited by the death of his wife and his sale of Anna. You can hear the Lutece's bickering about it in the scene on the docks that occurs directly before the lighthouse approach that instigates the game. When Booker groggily begins to verbalize his "bring us the girl, wipe away the debt" falsehood as the Lutece's carry him to the rowboat, Robert tells Rosalind see, I knew it would happen, because I lived it.
But why and how did the Lutece's come to tear Booker into the 1912 of Columbia and Comstock? Because the Lutece's eventually grew uneasy with what they'd helped create in Elizabeth, and like Lady Comstock before them, were killed when they made clear they would not keep silent. Well, an attempt to kill them was made. The task was handed to Fink, for as Columbia's foremost innovator in technology (innovation gleaned through observation of future biologists, physicists, etc via tears, which accounts for those fucking fantastic anachronistic jingles, vigors, the merging of man and beast that resulted in Songbird [elaborated upon in a Fink voxophone], etc.) he would best know how to make a sabotage of the Lutece's machine whilst they were using it appear a tragic accident. However, instead of killing the Lutece's, exploding their machine in the midst of them opening tears resulted in, as speculated by Rosalind in a voxophone, the singular entity of each Lutece being spread throughout all worlds. This is how they were able to talk to the photographer who documented their funeral. This is how, like omnipotent endgame Liz, they are able to open tears and appear where they like at will. I suppose that begs the question why they didn't just handle shit themselves, and instead used Booker. Speculative: I would say that you still need Booker's cooperation for that, and offering redemption and the chance to be reunited with a long lost daughter goes down easier than hey sorry but you need to be drowned ASAP cause we can glimpse every possible permutation of the universe and some variants of you done fucked up just trust us. Also, given Ken Levine wrote this game, I feel pretty safe in assuming Rosalind and Robert are shout outs to Hamlet's Rosencrantz and Guildenstern. Basically, they're the instigators.
As far as constants and variables go, the early game coin flip that always lands heads is an example of a constant. The hundred or so tallies for heads represents the hundred or so Booker DeWitts of slightly altered branching worlds who in total have flipped a hundred or so coins that the Lutece's, with their ability to be present in all worlds, have documented. Hence, when Robert expresses that the coin flip is never as satisfying as he thinks it'll be, Rosalind replies, "Chin up, there's always next time." The PI office death screen is a representation of these slightly differing worlds, as every time you die, when you open that door to Columbia that places you 50 or so feet away from the enemies that led to your demise, you are now playing as a Booker of a differing world in which he was not killed by those same enemies. Or at least, not in the same way, as even if as the player this time you surmount that particular encounter, the rules the game lays down would dictate that there are infinite ways in which Booker perishes in that encounter alone, just as there are myriad Booker's who manage to wrest Elizabeth from the songbird via the cipher given by elderly Elizabeth, and just as there are myriad Elizabeth's who fail to decipher said cipher in time to implement it for this or that reason, and so on, and so on. The branches are cut when the root (the acceptance of the baptism) is pulled, as they say over and over throughout the course of the game. Booker also always gives Anna away, as evidenced by Elizabeth explaining that no matter how long you wait, you always give him (Robert Lutece) what he wants. Robert's adamant stance on Booker not rowing in the opener is also a constant, as all the times they've taken him to the lighthouse in other universes, he apparently hasn't rowed. The cage and bird chokers the Lutece's offer are a representation of variables, admittedly along with, as detailed above, just about everything else you do in the game.
And there you have the Lutece's place in the story. I suppose you could say that really, it's all their fault. And yet, I can't help but enjoy that even in the midst of being scattered throughout universes, even in the middle of attempting to right so many wrongs they are very much responsible for, they're still conducting experiments and sniping at each other. And despite the fact that they remain aloof to the very end, even when they know Booker is about to walk into his own drowning, they lead him there knowing that when he dies, they as a united pair will cease to exist. And while sure, it's guaranteed that in other infinite worlds they'll reenact their connection without the aid of Comstock, their current incarnations won't experience that. I love that the duo first introduced as comic relief ultimately join the narrative confluence in such a critical and, when given some thought, heartbreaking fashion.
Loved the competition between Murder of Crows, Undertow, and Bucking Bronco for most disgusting passive animation in any game ever.
Loved the aesthetic of draped red throughout the war torn streets that embodied a simultaneously sinister and gorgeous this city is bleeding bit of symbolism.
Loved, as mentioned in the quick look, that violin sting for headshots and executions.
Loved the accompanying piano note as each Elizabeth ceases to exist, particularly that final, haunting ping as it pans up and cuts to black.
Loved the at times hilariously overt foreshadowing my second time through.
Hated the tonal dissonance between grim, affecting character beats for Elizabeth and, not a minute later, her glib quip regarding a requested lockpick.
Curious how people feel about the emotional gut punch ending of 'the Booker and the Elizabeth you grew fond of throughout the game no longer exist' being dulled by the 'wait, what the fuck' aspect that pervades your (or at the very least my) first time through. I think it makes it all the richer.