I'm looking for relatively interesting examples of theorems or hypotheses that could be proven (or disproven) by running at most BB(n) [S(n)] operations - on a $n-$state, two-symbol TM , where $n$ is preferably, but not necessarily $\leq 10$, optimally $4-7$.
For example, the answer here - gives as an example a ten-state machine that computes collatz conjecture (edit: only for a single input/case given on a tape, not the whole sequence, so it may be slightly less relevant to the question).
Although this example and examples such as Goldbach conjecture are still very interesting, I would like to stress that the main focus is work related to the smallest machines.
It doesn't necessarily have to be anything sophisticated, even some simple ancient greek theorem or property would do, but more complex examples are obviously preferred. In the case of scarcity or If you deem it as interesting, it's fine to give even a non-intuitive numerical sequence that is somehow nicely explained by types of non-halting machines (wombats... monkeys...) or by the behavior of BB champions.