Energy conservation (which, as mentioned above, is violated cosmologically) only places constraints upon the flux of causal edges associated with updating events which maintain baryonic matter in the hypergraph, which in turn only places constraints on the classes of topological obstructions in the hypergraph that can be associated with elementary particles. As far as I know, this does not place any direct restrictions on the structure of the rules. Have I missed something?

I find you analysis very clear and useful but there is one confounding factor that we need to take into account, energy is not conservatived insofar as new space is constantly being generated in the real universe. There are constantly vertices being created as the universe expands so conditioning on a fixed amount of causal edges would not work. Particularly if inflation is real then we need a rule that can vary in its creation of new verticies. You conditioning proposal would work however if we had a way to delineate changes in causal edge flux due to spatial expansion versus the causal edge flux associated with particles, a kind self sustaining edge configurations.

This makes me think that we need a rule that has a fixed quantity of self sustaining energy(particle energy) from the beginning that gets diluted as more space is created, this would mimic the high energy at the time of the big bang. Perhaps the rule would in the beginning have a structure that creates a large number of self sustaining constructs in the graph but as the number if vertices increase this behavior ends. One possibility could be whatever process creates virtual particle pairs that anhiliate in our current vacume would create non-anhilating particles in the early universe.