Luna 4 Archon Quest and World Formula
What relation did the world formula getting completed have with columbina reawakening? Columbina needed her authroity soul and body which she got when the moon got pulled over to the false sky . But why was the world formula needed?
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Basically, Sandrone wanted to solve for X. X = Columbina's Rebirth. With the variables currently present in the World, only 90% of the Formula could be solved. To solve that final 10%, an external variable was needed: The Frost Moon.
>"I know where potential variables within the world come from... So, what about the moons? Are there any variables there?"
The Formula's progress was based on the current variables present in the world and their placement. As things progressed, more conditions of the Formula would be met, alonging it to progress. Notice how the Formula stops at 90%, then jumps up when the Moon starts to fall down. Sandrone states that the Formula could take them to 90%, and bringing the Frost Moon down was required for the final 10%.
>"Sandrone: My formula can get us 90% of the way. It will connect you to Columbina. Then, I will bring the formula's output to the Traveler's side. For the last 10%... Do whatever it takes, just bring the moon down."
Aether was a necessary variable in the formula, as was his connection to Columbina, his presence was needed for the Formula's completion. That's why Sandrone sacrificed herself so that he could keep going. Part of the Formula also required for Dottore to be stalled for a certain amount of time to allow for the perfect time for the Frost Moon to come down.
The Formula would constantly update itself with every single event that would occur, as well as take into account time. The closer to 100% it was, the closer we were to the required conditions for Columbina's revival.
I probably could've worded that better, but yeah, I'm pretty sure this is how the World-Formula works.
>Columbina: (Kuuvahki responds to me, because... it is me. My body, answering the call of my soul...)
(And the golden energy I've encountered on my way here was, in fact, that very soul — which I am about to leave here to guide me...)
The Golden Energy that leads Columbina to each portal turns out to be the energy of her soul, left by the Future Columbina to guide her past self. Realistically, Columbina could've went back in time at any point where Luonnotar was. Instead, each moment was meticulously chosen for her. I believe Future Columbina created each Portal at the ideal moment where Columbina needed to intervene.
Columbina doesn't use her own energy going into these Portals, rather she utilizes the energy of the portals in order to go back in time.
>Columbina: (Time is flowing backwards here, and I can use that familiar power to enter the world of the past...)
Perhaps how it works is this: Future Columbina sets the portals up to teleport Past Columbina to a particular time, using Luonnotar as the anchor point, perhaps partially as a hint so Past Columbina would realize Luonnotar was a fragment of her soul. For the final portal, she is simply teleported to where Luonnotar is born so that she can create Luonnotar and properly establish the cycle.
I don't think Luonnotar was necessary for Columbina to be teleported to Teyvat, as we see how Columbina is able to teleport the Three Moon Sisters to Teyvat after they give up their Lunar Authority.
Basically, those portals were created by Future Columbina to teleport her past self to times where she needed to change things while also making sure she encounters Luonnotar and realizes that Luonnotar is a part of her. Hopefully that all makes sense, I wish they had explained that part a bit more.