So this question is inspired by the following thread: https://forums.factorio.com/viewtopic.php?f=5&t=25008
In it, the poster is examining an $8$-belt balancer (more on that to come) which he shows fails to satisfy a desirable property, which he called universally throughput unlimited.
So what is a $n$-belt balancer? It is a configuration of belts (which move items around), and splitters (which take two belts in and balance their items on the two belts on the output side), which will balance the input of all $n$ input belts across all $n$ output belts. They are frequently used in large factories to move large amounts of items to a variety of different areas in a manner where no one belt worth of items getting backlogged (more items coming at it than it can use) results in other projects not getting full throughput (or at least as much as they can use).
The desired property called universally throughput unlimited is the following: Suppose only $k$ of the $n$ input belts are getting input (assume full input; aka, input belts are assumed saturated), and that all but $k$ of the output belts are backlogged and have no throughput (already full of items and nothing is moving on those belts). Then the full input on those $k$ input belts can be provided across the $k$ output belts (which have the same maximum throughput, hence no one output belt can handle more than one input belt's worth of throughput). This basically means that the $n$-belt balancer is never a bottleneck no matter the current input or output limitations (which lanes are getting input/available for output).
The question I have is the following: It always possible to create an $n$-belt balancer satisfying the universally throughput unlimited condition for any $n$? It not, for which $n$'s is it possible? (clearly, $n=2$ works because of how splitters behave)
I have some ideas on how to approach this problem, but am nowhere near having it solved. The first idea is about how to represent the problem: We can represent the input belts and output belts as vertices of a directed graph. The inputs being sources (in-degree=0) and the outputs being sinks (out-degree=0). The balancer is the input and output vertices together with a set of intermediate vertices which represent splitters which have $1\leq$in-degree,out-degree$\leq$2 (one or two directed edges point to them and one or two coming from them) and the associated directed edges. Looking at the problem this way, it is easy to see that a necessary condition is that input on any belt can reach any output belt (it is necessary because if not, then consider the case of all input on one input belt and all but one output belt backlogged with 0 throughput; in such a case, if you can't route that input belt's input to the output belt you won't get any throughput), but this condition is not sufficient (multiple examples that satisfy this condition have been shown both theoretically and experimentally to fail to have the desired universally throughput unlimited property).
An important thing to note is that belts can be routed under other belts via underground belts, hence planarity of the above described graph is not necessary. The fact that splitters have some very specific behaviors is important to this problem: They will always try to balance outputs provided there is no backlog, hence, in a no backlog scenario the output on each belt leaving a splitter is half of the total input on both of its input belts. If however, one of the output belts is backlogged with no throughput, then all of the throughput will be merged onto the 'free' belt up to it's throughput limit. If more than one belt worth of throughput is coming into a splitter in this case, then both input belts will start to bottleneck (each belt's effective throughput will be half of the maximum because that's how much of the saturated output belt is coming from the given input belt). Sometimes a backlog is only a reduction in throughput (do to bottlenecking down the line somewhere) in such a case, a splitter will still split input equally up to that the reduced throughput on the lowest throughput belt, after that, all remaining throughput is thrown at the belt with additional capacity until that two is saturated, and if there is any more input coming at the given splitter then both of its input belts will start to backlog.
This backlog phenomenon can result in some very subtle behaviors. Which makes simply assigning weights to the directed edges in the above described graph (constrained to a value of $[0,1]$ where $1$ is saturated and $0$ is no throughput) inadequate to describe the problem. For instance a splitter causing a backlog with some throughput but not enough to avoid backlog, can lead to a reduction in throughput for another splitter's output belt, shifting more of it's input onto the other output belt (which might cause a splitter further down that belt to suddenly become a bottleneck and backlog, etc.)
My suspicion from experimenting a tad as well as some theoretical work looking at how splitters are dividing inputs leads me to conjecture that it is not possible for all $n$, and that the most likely candidates are powers of $2$. Even then, for powers higher than $1$ it still might be impossible because of odd #s of belts having input needed to get to same number of output belts (and if balancing odd #s of belts isn't possible, then the universally throughput unlimited condition might not be satisfiable because of these cases).