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## So why can one produce maximally parsimonious grand unification on scaled S4

Macroscopically, the forces are electromagnetism and gravity.  In previous posts, I had noted down how an electromagnetic potential fits adapted to a hypersurface of S4 and one can see how the geometry of S4 naturally gives us the duality between electric and magnetic fields.  The Ricci curvature of the submanifold automatically contains a ‘cosmological constant’ term which happens to be quantitatively close to the measured value.  There is the essence of the unification of these two forces but additionally we can see how electromagnetism would not be a genuine $U(1)$ gauge theory which can be used for experimental tests of S4 theory.  At each point of the ‘physical hypersurface’ the normal geodesic is a circle of fixed radius but the total space is not that of a bundle as there can be intersections.

As for weak and strong nuclear forces, there are deterministic Yang-Mills theories that are worked out for S4 by many people starting with Michael Atiyah’s work in the 1970s.  In this setting we have additional simplifications such as showing that quantum mechanics is fooled by randomness.  What this amounts to is showing how Schroedinger eigenfunctions are picking out from a deterministic Hamiltonian system those orbits which are periodic; for these I have highlighted key work by Vieri Benci and James Ralston.  The unification of gravity and electromagnetism is the key point, and it should not be surprising that with four space dimensions this is possible–this was done by Theodor Kaluza and then again by Hermann Weyl.  Einstein had objected to a fourth space dimension.  Thus S4 physics is not bringing a fourth space dimension although I have highlighted the objective evidence for four space dimensions from crystal symmetries of orders 5, 8, 10 and 12.  Rather one of my key contributions is probably compactness of the universe for which I had used the CBR lower bound as experimental evidence and Gaussian upper bounds on complete noncompact manifolds with control of Ricci curvature.  Compactness has many consequences including showing existence of periodic orbits in Benci’s theorem.

The ‘maximally parsimonious’ grand unification possible for a scaled S4 boils down to the fact that the scaling factor of $1/h$ simultaneously produces the right quantization and produces the right cosmological constant and unification of electromagnetism and gravity is fairly simple.  Compare this simplicity of an almost classical grand unification to the complexities of string theories.