Candidate Names

I had a theory as to why Kate's name wasn't written on the wall. I don't know if it's said before so here it is. When Jacob meets her when she was young after the owner of the market caught her stealing and Jacob paid the lunchbox he said to her:'You are not going to steal again now are you?'. Although he said this Kate broke her promise so Jacob wouldn't want her as a candidate. Also on the wall it wrote the name: Kwon. I believe this is Jin because again when Jacob visited the two he told them not to take their love for granted but Sun had an affair. So she is not a candidate because she didn't listen to Jacob. I believe that although Jacob helped them through their lives he always let them to have free will. It seems logical to me. Give me your thoughts please :)

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