Is it handing me answers if I happen to think of a good question?

[She shrugs.]

Then I think you still count as Battler, and you came into existence while thinking about the events of Rokkenjima. So the first two are true. I don't think your own mind, even if you have a different name or whatever else, would count as someone who knows Battler - after all, if that person knows Battler, he wouldn't have to think so hard about it.

On the other hand, you could definitely say that if the last two are true, then the first can't be. But that's not the way I see it, anyway.