I disagree with you. How can you determine if it's not paying for itself? If you're talking in absolute terms, then we spend 10,000 and it pays for itself on the 6th day. After that, it's all profit.
If you're talking in relative terms, i.e., roughly speaking, there's a choice on the 7th day to build either unique units or a Capitol, then I apologize.
But in general, you definitely need to build a Capitol. It's just important to choose the right moment and understand in WHICH castle.
2 TS
I don't understand, is your topic about which castle is more profitable to play, or about whether it's necessary (and when) to build a Capitol? :)