Average exceptional Lie and Coxeter group hierarchies with special reference to the standard model of high energy particle physics

The notions of the order of a symmetry group may be extended to that of an average, non-integer order. Building on this extension, it can be shown that the five classical exceptional Lie symmetry groups could be extended to a hierarchy, the total sum of which is four times α¯0=137+k0 of the electromagnetic field. Subsequently it can be shown that all known and conjectured physical fields may be derived by E-infinity transfinite scaling transformation. Consequently E8E8 exceptional Lie symmetry groups manifold, the SL(2,7)c holographic modular curve boundary Γ(7), Einstein-Kaluza gravity R(n=4) and R(n=5) as well as the electromagnetic field are all topological transformations of each other. It is largely a matter of mathematical taste to choose E8 or the electromagnetic field associated with α¯0 as derived or as fundamental. However since E8 has been extensively studied by the founding father of group theory and has recently been mapped completely, it seems beneficial to discuss at least high energy physics starting from the largest of the exceptional groups.