It's mostly a question for philosophy of mind, I think specifically a question about intentionality. I think the closest you'll get to a mathematical framework is control theory; controllers are a weird edge case between tools and very simple agents. Control theory is mathematically related to Bayesian optimization, which I think Eliezer believes is fundamental to intelligence: thus identifying cases where a controller is a tool or an agent would be directly relevant. But I don't see how the mathematics, or any mathematics really, could help you. It's possible that someone has mathematized arguments about intentionality by using information theory or some such, you could Google that. Even so I think that at this point the ideas are imprecise enough such that plain ol' philosophy is what we have to work with. Unfortunately AFAIK very few people on LW are familiar with the relevant parts of philosophy of mind.