Fixed, Thanks.

1. Given a finite set $\Omega$ of cardinality $n$, find a computable upper bound on the largest finite factored set model that is combinatorially different from all smaller finite factored set models. (We say that two FFS models are combinatorially different if they say the same thing about the emptiness of all boolean combinations of histories and conditional histories of partitions of $\Omega$.) (Such an upper bound must exist because there are only finitely many combinatorially distinct FFS models, but a computable upper bound, would tell us that temporal inference is computable.)
2. Prove the fundamental theorem for finite dimensional factored sets. (Seems likely combinatorial-ish, but I am not sure.)
3. Figure out how to write a program to do efficient temporal inference on small examples. (I suspect this requires a bunch of combinatorial insights. By default this is very intractable, but we might be able to use math to make it easier.)
4. Axiomatize complete consistent orthogonality databases (consistent means consistent with some model, complete means has an opinion on every possible conditional orthogonality) (To start, is it the case that compositional semigraphoid axioms already work?)

If by "pure" you mean "not related to history/orthogonality/time," then no, the structure is simple, and I don't have much to ask about it.

Yeah, this is the point that orthogonality is a stronger notion than just all values being mutually compatible. Any x1 and x2 values are mutually compatible, but we don't call them orthogonal. This is similar to how we don't want to say that x1 and (the level sets of) x1+x2 are compatible. 

The coordinate system has a collection of surgeries, you can take a point and change the x1 value without changing the other values. When you condition on E, that surgery is no longer well defined. However the surgery of only changing the x4 value is still well defined, and the surgery of changing x1 x2 and x3 simultaneously is still well defined (provided you change them to something compatible with E).

We could define a surgery that says that when you increase x1, you decrease x2 by the same amount, but that is a new surgery that we invented, not one that comes from the original coordinate system.

Thanks for writing this.

On the finiteness point, I conjecture that "finite dimensional" (|B| is finite) is sufficient for all of my results so far, although some of my proofs actually use "finite" (|S| is finite). The example with real numbers is still finite dimensional, so I don't expect any problems. 

Fixed, Thanks.

Looks like you copied it wrong. Your B only has one 4.

I have not thought much about applying to things other than finite sets. (I looked at infinite sets enough to know there is nontrivial work to be done.) I do think it is good that you are thinking about it, but I don't have any promises that it will work out.

What I meant when I think that this can be done in a categorical way is that I think I can define a nice symmetric monodical category of finite factored sets such that things like orthogonality can be given nice categorical definitions. (I see why this was a confusing thing to say.)

If I understand correctly, that definition is not the same. In particular, it would say that you can get nontrivial factorizations of a 5 element set: {{{0,1},{2,3,4}},{{0,2,4},{1,3}}}.

When I prove it, I prove and use (a slight notational variation on) these two lemmas.

1. If $h^F\left(X|E\right)\cap h^F\left(Y|E\right)=\left\{\right\}$, then $h^F\left(X|E\right)=h^F\left(X|\right(y\cap E\left)\right)$ for all $y\in Y$.
2. $h^F\left(\right(X{\vee }_{S}Y\left)|E\right)=h^F\left(X|E\right)\cup {\bigcup }_{x\in X}{h}^F(Y|x\cap E)$.

(These are also the two lemmas that I have said elsewhere in the comments look suspiciously like entropy.)

These are not trivial to prove, but they might help.

I think that you are pointing out that you might get a bunch of false positives in your step 1 after you let a thermodynamical system run for a long time, but they are are only approximate false positives.