(OP here) That's a good point---these mereologies tend to grow very fast with system size. Exponential is not even so bad. There is the 'redundancy' mereology for example, which scales with the Dedekind numbers. This appears quite often in information theory and neuroscience, but quickly becomes intractable.
As I see it, emergence comes in two flavours: a higher-order interaction among microscopic parts is already emergent in the sense that it is a non-atomic thing that determines the behaviour of atoms (I use atoms to refer to the 'singletons' or smallest elements of the theory, not necessarily physical atoms). But you're completely right in saying that there is another sense of emergence which only really happens for a 'thermodynamic' number of atoms. The difference seems somehow captured by the contrast between:
-- the whole is more than the sum of the parts
-- the whole is less than the sum of the parts.
Both are commonly called emergence! If it turns out that you don't need to keep track of all birds in a flock to describe its behaviour, then we call that emergent because the whole is somehow less than the sum of the parts.
Your example of genetics is interesting, because it is actually what got me interested in this problem in the first place. I spent most of my PhD struggling with calculating up to 7-point interactions among genes, and you indeed need some clever tricks to make this tractable. I used causal discovery methods to rule out most potential interactions based on conditional dependencies. This is now a piece of open-source software: https://www.embopress.org/doi/full/10.1038/s44320-024-00074-...
Author of the post here: There is quite a deep connection actually. You can assign a simplicial complex to a partial order P with a max and a min element (0 and 1). Then the Möbius function on P calculates the (reduced) Euler characteristic of that simplicial complex as µ(0, 1)=\Chi. For example, if the partial order is a power set mereology (a Boolean algebra) on 3 elements, then the associated simplicial complex is a triangle, and µ(0, 1) = µ(\emptyset, {a, b, c})=(-1)^3, which is the correct answer as a triangle (without interior) is homeomorphic to a circle.
In a way, calculating quantities q through Möbius inversion is just calculating Euler characteristics, weighted by by Q. (with some caveats)
The Möbius function actually appears quite often in statistics/probability theory. I wrote more about this in the paper (https://arxiv.org/pdf/2404.14423), but in short: if you invert moments with the powerset Möbius function then you get central moments, but if you invert moments with the partition Möbius function, then you get cumulants. In fact, you can vastly generalise this by changing the mereology from partitions to ordered partitions etc.
The pairwise description you list correspond to a different link, namely, one where each pair is actually connected. For the shown rings, the 'correct' description would be:
A is not connected to B
B is not connected to C
C is not connected to A
A, B, and C are connected
This seems paradoxical, but the paradox is resolved by the 'higher-order' linkage.
That's right! This is secretly about doing calculus on posets. You can actually generalise some notions from incidence algebras to other settings, like groupoids and categories, where you can play the same games. This is something I mostly haven't looked at yet, but I think it might be fun, and some people seem to have found it useful (for example: https://arxiv.org/abs/1809.00941 and https://arxiv.org/abs/2501.06662)