Yes, the whole idea is to not store state. You only need a master password and domain to generate your site-specific password, and possibly a few custom parameters describing password rules for weird sites.
I mean, it's great to "not store state" until you get to the last part of your comment: "possibly a few custom parameters describing password rules for weird sites", plus it also needs to include things like the number times you've rotated the password for each site. The point of my question was to ask how well your implementation manages state across multiple machines, but if your answer is "it doesn't need to" then I just don't see how you can satisfy my requirements.