Dear Akimbo,
First thank you for your kind interest. This post is a tentative response to your question having in mind your very pedagogical essay about monads.
You: Monad - a fundamental unit of geometry; that of which there is no part;...
i. extended objects, not further extensible or compressible.
ii. they are fundamental and not a composite of other 'its'.
iii. they are the fundamental units of geometry, both body and space.
Me: The points of the geometries I am dealing with could perhaps be seen as monads. (e.g. the 7 points of the Fano plane in Fig. 1a. Then in Fig 1b the same points are extended as edges).
You: monads are 'it' and their change between two alternate states is the 'bit'.
Me: Agree. One edge in Fig. 2b is either black (bit 1) or white (bit 0).
You: the two-valued attribute
denoted by 0 and 1 must really occupy the deepest part of the basement!
Me: Agree, but as two elements of a triple {0,1, \infty}.
Stephen Anastasi: (above) "not only does the universe collapse to a single minimally simple omnet, all of mathematics went down the tube with it.",
Me: The translation of this sentence would be the Belyi theorem (see the step 3 in my Sec. 2 giving the definition of a child's drawing) and the property that the child's drawing D itself is the preimage of the segment [0,1], that is D=f^-1([0,1]), where the Belyi function f corresponding to D is a rational function. All black vertices of D are the roots of the equation f(x)=0, the multiplicity of each root being equal to the degree of the corresponding vertex. Similarly, all white vertices are the roots of the quation f(x)=1. Inside each face, there exits a single pole, that is a root of the equation f(x)=\infty. Besides 0, 1 and \infty, there are no other critical value of f.
Sorry about the technicalities.
You: But what about the space then?
Me: Although the model of dessins d'enfants may be applied differently, practically, in my essay, it corresponds to the (Heisenberg) space of quantum observables such as the Pauli spin matrices, or tensorial agregates of them. You would say that they cannot be monads in such a case! But they cannot be divided in the sense that the parties (let's say Alice, Bob and Charlie for the three-partite case, I used the Fano plane for this case) are linked once for all, whatever state they share, entangled or not. I don't know about Mach, I have to think more.
I am sure that it does not dissolve your question, at least it gives you a hint, hopefully, of what this kind of maths may do.
Please rate my essay if you like it.
Best wishes,
Michel