Since there is no Episode 13 tonight, I will post here my theory since the preview for Ep 13 has been released and it seems that I am in the right track in unfolding the mysteries behind #TheKingEternalMonarch . Well, these are just my thoughts, they could be wrong or right.

Imagine that the present time is not the year 2020. 2020 is already in the past… Present Lee Gon (the one not in the year 2019~2020) is trapped in a time loop. His chaos. He made a mistake & froze time forever between him & Tae Eul. #TheKingEternalMonarch

(As to how they first met, it's not covered here.) He has tried several attempts to get out of the loop, and yet couldn't arrive to a beautiful solution. (i.e. Lady Noh brought to KOC could be his doing, Refer to Ep 12 last scene)
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And so he made another attempt, this time adding Tae Eul into the equation. Past Lee Gon considered Tae Eul a ZERO, his favorite number. According to Lee Gon, Tae-Eul has the nature of that number. Here is the explanation of Lee Gon: ZERO is actually a number of absolute power.

It can nullify any number or take everything away from it. A number trapped in a root symbol can escape under only two circumstances. They need to have a square root or meet the powerful number, zero.
#TheKingEternalMonarch

Past Lee Gon further elaborated: You're always busy and you don't care much for me. Because of you, I am powerless here, but it's fine. You are much more amazing than I have imagined. And you are standing in front of the root symbol I'm trapped in. #TheKingEternalMonarch

(Refer to Ep 3 the proposal scene outside The Alley). How did the Present Lee Gon add Tae Eul into the equation? By introducing her earlier into his life in the winter of 1994 through the ID the child Lee Gon retrieved from the Present Lee Gon. #TheKingEternalMonarch

Thereby inducing a "Butterfly Effect". According to Google: The butterfly effect is an idea that is more commonly used in chaos theory. A small change can make much bigger changes happen; one small incident can have a big impact on the future. #TheKingEternalMonarch

This is what the Present Lee Gon wants. To escape in his chaos a small change, like an ID, is what all he needs. Do we want a beautiful and happy ending for Lee Gon and Tae Eul? Do we want Lee Gon to come up with his beautiful solution? #TheKingEternalMonarch

The answer lies with Tae Eul. What we are witnessing now is a timeline of the Past Lee Gon and the Past Tae Eul.
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But this time, every decision of Tae Eul is significant as it is the key which will lead to that small incident which will change the future of the Present Lee Gon and prevent the eternal time freeze. #TheKingEternalMonarch