I stumbled upon the same dilema. You have to do the following, leave Sydney, or let her go with you if you want, and inspect the whole museum, you should find a door leading to the basement, which needs no key. Once you enter there, you will face many Protectrons, and other robots. You have to explore the entire place, I think its 2 dungeons to be precise. Once you are in the second dungeon, find your way until you find a door to another place, enter and you should be greeted by a robot with an identity problem, here you can either destroy it (bad karma) or help him (good karma).
I must tell you that if you decide to help him, its going to be a little bit longer to get the declaration, as you have to go to a rather far location if you dont have nearby knowned areas.
Anyway, you can get quickly to that robot by a secret elevator in the room where you first meet Sydney, you will see a circle in the floor, more or less in the middle of the room. Just click on it, and you will descend to a place close to this robot, though I have to warn you, bring a shotgun if you can as if can kill most robots (if you have the small guns skill) in 2 shots during VAATS.
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