I don't have any research to back this up, but it's possible that framing it as being more focused on less interesting work rather than "productive" helps clarify what might cause it. When we are well rested, our brains are functioning at a higher level, which includes spending energy making micro decisions about how important things are, what the priority is, what work is related to what you are currently looking at, and so on. When over tired, our bodies exert less energy on non critical processes, which might mean not overanalyzing or being distracted away from a task you'd rather not have to do.
So getting certain work done is easier when tired, but productivity would arguably include doing the "right" thing, mapping the right information together, and so on, which you'd want to be better rested for.
Curious to hear if you agree on the productivity vs focus clarification, or if you feel like you're truly more productive in the sense that you do some of your best work (greatest ideas, executed efficiently, with minimal errors) when tired?
So getting certain work done is easier when tired, but productivity would arguably include doing the "right" thing, mapping the right information together, and so on, which you'd want to be better rested for.
Curious to hear if you agree on the productivity vs focus clarification, or if you feel like you're truly more productive in the sense that you do some of your best work (greatest ideas, executed efficiently, with minimal errors) when tired?