OK, so here’s an idea that came to me late last night.
What if odd numbers represent bosons (energy particles) and even numbers represent fermions (matter particles)? Or, more precisely, each natural number represents one phase of existence (between interactions) of a boson or a fermion according to whether it is odd or even. Interactions between these particles are represented by Pythagorean triples. In other words, if a²+b²=c², then a and b are particles that interact at a temporo-spatial position given by c. And, of course, vice versa: two particles only interact if they are the junior members of a Pythagorean triple, and the senior member determines where and when their interaction takes place.
What sort of a Universe could a rule like this define? How would space and time be formulated? Just an idea, but maybe worth pursuing.