I have done considerable retooling and clarifying of my paradox theory of Lost. See the original here. I am leaving out a lot of my notes from the original in order to make it easier to follow.
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The Problem
After the explosion of Jughead in the season 5 finale, there is only a single reset, an Oceanic flight that never crashes (despite all of the timeline jumps from the last season). Theoretical physics/math of dynamic systems argues that for each spacetime there is only one future state (that is to say, each timeline is a closed system, it may loop, but only on itself with the same variables so that no paradoxes are possible).
Let me better explain future state at this point and why LAX is key. I will use a hypothetical example.
If I wear a red dress, get on a flight and never change my outfit, then I will always be wearing a red dress when I land at my destination. This is the future state of my original timeline.
Let us refer to the original condition of spacetime before the crash on the island in season one as P (primary). Let's represent the timeline we see in LAX as S (secondary).
Without the interference of the island and with no crash having occurred P will always be P:
P (leave Australia)---------------time------------------P (arrive in LAX)
If, on the other hand, i board a plane wearing a red dress, never change my wardrobe, but reach my destination wearing a blue dress, then there has been a interference or fracture of some sort in spacetime. We saw this in last season when time on the island was different from time on the Charles' ship.
So the LAX timeline looks like this:
P (leave Australia)---------------time--------------|
| -------- S (arrive in LAX)
Now, why do I say there is a different future state (what we see in S) from the original timeline (P)? There are many issues that make this clear, here are a few:
Desmond is on the flight
Jack gets one bottle of alcohol rather than two
Shannon, Michael, Walt, and Vicent are missing
Kate recognizes Jack in E603
Because there are differences between the primary timeline (season 1) and its future state (season 6), we therefor cannot be dealing with the future state of the original Oceanic flight if it never crashed.
We are dealing with a completely different future state.
And just as important as the above point, is that there is only a single future state despite all of the time jumps in the original timeline. Let me better explain this by organizing things using a hypothetical example:
P1 (primary 1): original condition/timeline
P2 (primary with time jump) I travel back in time to 1928 and kill some random person.
S1 (secondary future state spins off) I return to my original timeline, but I have altered something (killing someone in the past) and so my original timeline is a branching off to a secondary.
I hope you are with me thus far. This can get confusing.
Since in season 5, there were multiple jumps and interactions with the past, this should have spawned multiple branches (P1 to S1, P1 to S2, and so on) of the future. But it did not. We only have a single future state after the explosion of Jughead.
If we were dealing with a multi-dimensional universe (MDU), then the jumps would not degrade the integrity of spacetime. But we appear to have a schism that is atypical to what a dimensional breach might look like.
In other words, in a single dimensional reality, the plane should have arrived with every single variable the same as when it departed without the introduction of the island. But the plane leaves Australia in a "red dress," does not change into a new outfit (island) and somehow lands in LAX in a "blue dress."
In a MDU, this would not be an issue because the variables are different. Let's consider the same red/blue dress scenario.
If I board a plane in Australia in a red dress and never change my outfit on the plane, then I should be wearing a red dress when I land in LAX. This is MDU1.
In MDU2, I board a board a plane in Australia in a blue dress and never change my outfit on the plane and as in MDU1, when I land, I should be wearing the same outfit - in this case, a blue dress.
As you can see, each dimension still operates on the same principles of time. Now, how would the MDU theory explain why I start off in a red dress and end up in a blue dress? Like this:
I board a plane in Australia wearing a red dress and never change my outfit. A dimensional breach occurs and I enter MDU2 while on the plane, where I am wearing a blue dress.
The MDU theory would account for the discrepency. But there is a problem with this theory as applied to Lost. Time changes only affect the Losties. As we have seen in Season 5, the Losties and only the Losties experience different time jumps, while the rest of the world and even the island do not.
MDU2 would have its own fixed rules of physics that it must operate on, just as MDU1. Spacetime, although in a closed system for each, remains fixed (p will always be p if nothing interferes).
So we are clearly not dealing with a MDU given the time jumps in season 5 that ONLY affected the Losties.
This leaves us back at square one.
The Paradox
If we are not dealing with an MDU, and it is clear that we are not (or the Lost writers dispensed with physics in order to create a tidy ending to the show), then we are left with the problem of how P becomes S (how a red dress turns into a blue dress).
All of this suggests a fracture of spacetime. Given the constant warnings we have seen (Eloise to Desmond, for example) that we have seen about "course correction" of the universe and the end humanity's ultimate self destruction (as expressed in the Velenzetti Equation), it is clear that we are dealing with a very fractured spacetime, that will continue to fracture until the integrity of spacetime becomes entirely degraded.
We are also told about the need of "constants" in order to keep a connection between pieces of the puzzle (fractured pieces).
Clearly a paradox has caused a fracture that continues. What that paradox is remains to be seen. Perhaps the paradox occures when Charles W leaves the island to have an affair. This is never fully explained. Perhaps the father of Clare is the breach? Perhaps Charles and Christian are both Clare's father given the time travel (in P, Christian is the father, in S, Charles is the father). It really is not clear what causes the paradox.
Just so that we understand what a paradox in spacetime is, let me use a standard explanation. What is a grandfather paradox? Consider the following scenario.
If i travel back through time into the past and kill my maternal grandfather before he meets my grandmother, then they will never meet, never have my parents, and I will never born and so I cannot travel back in time to kill my grandfather.
But what if time travel is possible? How would such a paradox be avoided? What would safeguard the integrity of spacetime?
Valenzetti Equation
We know that the Valenzetti Equation is the mathematical explanation of how humanity will ultimately destroy itself: 4 8 15 16 23 42.
Each number represents a variable of some sort and together this string of numbers expresses the entire problem I just described above.
The solution to this problem would be a "course correction."
If spacetime becomes so fractured that the fractures threaten the very fabric of the universe, how would the universe course correct in order to save itself?
Think of a puzzle. You don't know what goes where initially, but after some initial trial and error, you assemble enough to get a sense of what the puzzle is and where the pieces go. The clusters of the puzzle you have solved (constants) guide you to building the rest of the puzzle.
But what if their were countless pieces as spacetime continued to fracture, how would you ever orient yourself?
You would do so with constants - the pieces of spacetime that are fixed, around which you can insert the other pieces.
We know Desmond is a constant. If we know that Desmond is an unchanging and fixed piece of the puzzle and if his constant (the next piece that fits into his puzzle piece) is Penny, then we can build from there and reconstruct every other connection (Charles W, Charlie Jr., even Jack).
Aaron is the Other Constant
Aaron is the other constant we can be certain of at this point.
Consider all of the other variables (Losties) and you cannot escape something key after watching E603 - those key factors all lead to one - and now visible - unifying point: Aaron.
Look at what we now know:
Kate is involved in the delivery of Clare's baby for the third time now, and this time in the future state. So no matter what has happened or how many changes have taken place, Kate is always helping in the delivery of Aaron, in multiple versions of P, and also now in S.
Aaron's name is always Aaron
Ethan is still involved with Clare and her baby in a medical capacity and still having a hard time poking her with needles.
Ethan is the last baby to survive being born on the island, until Aaron
Jack, Clare, Kate, Ethan, Juliet, and Christian all have relationship to Aaron's birth.
So Aaron appears to be a constant just like Desmond, around whom a pieces of the puzzle can be rebuilt. Perhaps this is why Clare turns to the dark side on the current version of the island - because Aaron is gone?
Jacob is the Course Correction who selects the puzzle pieces to organize them (as different variables) that lead to a solution of the Velenzetti Equation. That is to say, Jacob is the personification of this.
MIB is the antithesis, he is the growing schism personified.
So in short:
The numbers of the Valenzetti Equation describe a paradox in space-time that must be course corrected or the universe will be destroyed. This paradox has already splintered spacetime and will continue to do so until the integrity of the universe collapses entirely.
The Losties, the others, the world, is how this problem and solution play out. In other words, Lost is short for a fractured spacetime.
The solution to this paradox and the Valenzetti equation (and to Lost really) is somehow related to fixed elements or constants in spacetime, two of whom we know for sure to be Aaron and Desmond.
Desmond is a constant for Jack, Eloise, Faraday, and Penny (brother and sister), and Charles Widmore (father of Faraday and Penny with Eloise)
Aaron is a constant for Clare, Jack (brother and sister), Christian (father of Clare and Jack), Ethan and Kate