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In orthodox electromagnetic theory, it is considered to be a $U(1)$ gauge theory.  In particular local gauge invariance is couched in terms of a $U(1)$ principal bundle.  In the S4 theory, it is clear that the right ‘circle bundle’ for electromagnetism are the normal geodesic circles over any point of the three dimensional submanifold of $S^4(1/h)$.  The complexity of the extrinsic geometry of $M$ leads to many possibilities that these geodesics corresponding to multiple points intersect each other.  In particular, the natural ‘invariance’ that one might expect should be based on ‘potentials’ $\phi(z)$ for $z \in S^4(1/h)$ rather than a proper $U(1)$-bundle which would take intersections into account.  This is an issue which would be very interesting to check by empirical tests.