Possible uses of the binary icosahedral group in grand unified theories

There are exactly three finite subgroups of SU(2) that act irreducibly in the spin 1 representation, namely the binary tetrahedral, binary octahedral and binary icosahedral groups. In previous papers I have shown how the binary tetrahedral group gives rise to all the necessary ingredients for a non-relativistic model of quantum mechanics and elementary particles, and how a modification of the binary octahedral group extends this to the ingredients of a relativistic model. Here I investigate the possibility that the binary icosahedral group might be related in a similar way to grand unified theories such as the Georgi--Glashow model, the Pati--Salam model, various $E_8$ models and perhaps even M-theory. This analysis suggests a possible way to combine the best parts of all these models into a new model that goes further than any of them individually. The key point is to separate the Dirac spinor into two separate concepts, one of which is Lorentz-covariant, and the other Lorentz-invariant. This produces a model which does not have a CPT-symmetry, and therefore supports an asymmetry between particles and antiparticles. The Lagrangian has four terms, and there is a gauge group GL(4,R) which permits the terms to be separated in an arbitrary way, so that the model is generally covariant. Quantum gravity in this model turns out to be described by quite a different gauge group, namely SO(5), acting on a 5-dimensional space that behaves like a spin 2 representation after suitable symmetry-breaking. I then use this proposed model to make a few predictions and postdictions, and compare the results with experiment.