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Just an opinion, I think maybe they decided to call her “illegitimate child” after FL’s mother died.
If I understand the translation correctly, imho the order must be something like this:
>FL’s parents fell in love and lived in the Empire
>FL’s father told the family (the Count) that FL’s mother was someone who helped him in the battlefield
>Pregnancy happened
>Met the respective King/Emperor/Leader of their countries, get accepted as a way to stop the wars
>The peace treaty made from the pregnancy supposed to be known only for both Leaders and the FL’s Parents
>But somehow, the younger princess found out, she announced the prophecy, cry cry cry me a river, and The North’s Leader forgave her
>But the older princess (FL’s mother) realized the scheme behind it tears and flee
>Go to Empire, stay there, and give birth in the estate (the Count’s house)
>This happened while the fam still didn’t know about the identity of FL’s mother
>FL’s mother back to North, pretended to lost the child and even lost her memories
>While FL staying with her father in the Empire
>The Count and fam didn’t know about this plan and thought FL’s mother abandoned her baby
>FL’s mother assassination
>The Emperor, the Duke, and FL’s father agree to do wars once more (the treaty broken)
>While in reality FL’s father only looking for the culprit that killed FL’s mother, the Duke knew and helped him by pushing him to be on the battlefield
>FL’s father meet FL’s mother’s loyal guard, some misunderstandings and get killed
>The Count blamed the Duke for pushing the wars again and again that it killed FL’s father in the end
>The Count decided to adopt the orphaned-baby-FL
>But the hair colour and facial features were too different than the fam/siblings
>Not to mention what kid could be happy with the fact her mother abandoned her and her father get killed by her mother’s brethren— or so I guessed
>The time of the Countess pregnancy was also something to think about (imo because of noblewoman usually had this tea parties, it would be hard to miss a pregnancy, no?)
>Thus the decision came to adopt her as illegitimate child
**second theory
The fam knew about the younger princess scheme. Rather than them finding out the FL’s mother indeed gave birth, and triggered another assassination, isn’t it better to let the world know she was “the Count’s mistake” while travelling to the North?
Thus, both FL’s father and the Count decided to register her as illegitimate child of the Count.
But if the second theory is true, it cannot explain why the Count hates the Duke.
While the first theory, where the Count didn’t know the plan, is more possible. The Count only know that the Duke was the one who keep pushing FL’s father to go to battlefield and hate him for that.