Thank You! You are correct. Oops!

I deleted that claim from the post. That was quite a mistake. Luckily it seems self-contained. This is the danger of "Trivial" proofs.

Initial versions of the theory only talked about ensurables, and the addition of (and shift of attention to) controllables came much later.

Looks like a pretty good summary to me.

I think $B=\left\{\right\{u,n\left\}\right\}$.

This is not correct. It is true, however that $V$ is observable in ${\text{External}}_V\left({\text{Internal}}_V\right(C\left)\right)$.

A counter example is the 2 by 2 matrix where $A$ chooses whether to carry and umbrella and $E$ chooses whether or not it rains. Externalizing whether or not it rains has no effect on the frame, but the agent still cannot observe the rain.

Fixed, thanks.

Category-theory-first approaches

I am in general not especially proficient in category theory, and I think that the whole framework could be rewritten from the ground up by someone who is more proficient in category theory than me, and be made much better in the process.

Time and coarse world models

I feel like the partial observability I get from taking a coarsening of the world and saying an agent has observations in that coarsening is similar to the partial observability I get when saying an agent learns something at a specific time. In particular, these two things seem similar enough to me that one might be able to unify the two definitions, and in the process reveal new things about them.

Computational complexity

A random open question I am curious about, but doesn't seem that important: Is the existence of a morphism between Cartesian frames NP-complete?

Formalizing time

I think that much of the meat of what I want Cartesian frames to do is connected to time, and I have only really touched the surface of that. I think that there is a lot more to say about time, and I think there are options we have about how to think about time in Cartesian frames. The one I presented is my favorite at the moment, but I am uncertain.

For example, one might want to think about an agent, and the collection of pairs of partitions $U$ and $V$ of $W$, such that the agent has a (multiplicative?) subagent that could choose $U$, while observing $V$. This collection of pairs is closed under coarsening in both arguments, and so one could talk about a sort of Pareto frontier of how refined you can make $U$ given $V$ or vice versa. I think this Pareto frontier looks a lot like time.

Logical uncertainty

There is a sense in which Cartesian frames is a very updateless ontology, and thus I am concerned about how to make it play nicely with logical uncertainty. Indeed, Cartesian frames are basically assuming that we have a set of possible worlds, which is assuming that we have objects that are the possible world that are not realized. Logical uncertainty does not do well with this assumption. Extending Cartesian frames to connect up with logical uncertainty is a major open problem.