Within this plane we have partitioned a group of G gates with T terminals. If Rent's rule applies and if the Rent exponent equals 0.6 and suppose we observe circuits of say 1000 gates. There is only one Rent exponent that follows from the netlist. Maybe for every given netlist there is a rent parameter that cannot be beaten. There will be a straightforward derivation from the divergence theory to the Rent's rule.
Potential function is derived by the connection of two gates.
Perhaps the vector field theory can be used for Rent's rule. Try to build a vector field for a given placement to find the singularity. We don't care about where we embed it. Rent's rule is just the minimum possible info. Almost impossible to avoid Rent's rule.