"But I didn't do anything!" - Falling to Evil without taking actions?

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That's perfectly Neutral. Plenty of very powerful Neutral archmages explicitly isolate themselves from the world at large to study. A 'Good' wizard who does that would probably become Neutral, but not Evil. The Neutral do-nothing archmage is practically it's own Archetype. It would be trivial for him to stop any number of bad things from happening; he just doesn't care about the problems of normal folks.

The idea that someone could do nothing but study magic in their tower, never engaging with society at large, and eventually become Evil is incredibly bizarre to me.

Let's take it up a notch, same exact scenario and this time the townsfolk flee to the mage tower and bang on the door and plead to be let in. All the good aligned wizard would have to do is let them into the tower not even engage the orcs and they would be saved. But he instead chooses to continue his research. I don't think that is a neutral act.

I think this one is a little bit different because the Prince's social role has some amount of built in obligation to help people. In this case the Prince isn't doing nothing, he's commanding people who probably would otherwise help to not do so.

The Prince is technically doing nothing but I get what you mean. However, I think this might also be an evil act for the soldiers who choose to do nothing at the Prince's orders.