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Let us accept that the universe is a scaled 4-sphere, and since this is unorthodox, recall that observed crystal rotational symmetries of orders 5, 8, 10 and 12 imply at least four spatial dimensions.  I have produced heat equation arguments for compactness — briefly the cosmic background radiation has a uniform lower bound of 2.7 K > 0 which is impossible in a noncompact manifold with lower bound on the Ricci curvature because there the heat kernel has a Gaussian upper bound.  But more precisely a scaled 4-sphere of appropriate radius produces a sharp explanation of the redshift slope.  So the physical universe is a compact three dimensional hypersurface.  Now there is a natural circle-bundle like structure on a 4-sphere, which is that normal geodesics along the hypersurface are all perfect circles.  The orthodox theories suggest from the Lagrangian of the electromagnetic field that there $U(1)$-invariance.  But the natural geometric expectation is that the electromagnetic potential be a 1-form on S4(1/h) that is normal along the manifold in the sense that the 1-form produces a nonzero value evaluated at the normal vector field.  This is similar to a $U(1)$ connection but S4(1/h) is not the total space of an actual $U(1)$-bundle.  The similarity is that the kernel of the 1-form at each point on the manifold is a choice of a ‘horizontal subspace’.  Let $\alpha$ be the one-form.  Then the analogue of the curvature form is simply $F=d\alpha$.  This electromagnetism is not $U(1)$-invariant because one is not free to translate around the circle at will.  I will make more precise this version of electromagnetism which should give some testable hypotheses.