I have made progress in showing that minimizing the Einstein-Hilbert functional (total scalar curvature) over hypersurfaces even when the Euler-Lagrange equations are not exactly the same as the classical Einstein gravitational field equations still satisfy the classical equation with a stress-energy tensor a function of the second fundamental form. If one calculates the Ricci curvature of the hypersurface, one automatically obtains a cosmological constant term that is . Thus it is reasonable to consider gravity as an instantaneous choice of a hypersurface whose induced metric represents gravity in the physical universe immersed in . This is from the gravity end of things.

Now quantization of energy can be directly explained by the shape of the ambient 4-sphere. I have provided independent empirical evidence for this shape by a sharp (1.8% error) match to the redshift slope using this model. So the next obvious question is whether the Dirac equation can do a good job explaining the hydrogen spectrum in an S4 universe and the answer seems to be yes. I will come back to the question of explaining Lamb shift of 2S and 2P orbitals. The Dirac equation with the Coulomb potential has an exact solution in this case in terms of explicitly calculable universal constants (that depend on the eigen-level). We get automatic mass restrictions from the discrete spectrum of the Dirac operator. This is from the quantum mechanics end of things. More verification is necessary but looking at the equations on eigenspaces of eigenvalue of the Dirac operator,

we can see that in order for the mass to produce a solution to the equation, it should be quite close to a Dirac eigenvalue (integer multiples of ) and the difference has to be exactly where is the scalar curvature . In fact the only choice for is when ]$ and then we still have to check whether the difference term matches, i.e. that the above equation is satisfied. If not, then the Dirac equation for mass will have no nontrivial solutions. So assuming that we accept the explanation of quantization of energy being due to the shape of the universe, and we accept the Dirac model — this is a classical issue in the sense that there is no ‘quantization’ mechanism beyond reaching the Dirac equation — the masses must be quantized in a precise manner.

Now one of the geometric features of any hypersurface (modeling physical universe) is that the normal geodesic forms a perfect circle of common radius. This is a natural construction to consider for electromagnetism. Perhaps one explanation for why Einstein did not succeed in a unified field theory — electromagnetism and gravity do not fit in a three-dimensional manifold. But the intersting feature of these normal circles is that they do not form the total space of a circle bundle because the normal circles from different space points intersect. One can overcome this issue by mimicking how the connection formalsim is developed in gauge theory: we simply consider the electromagnetic potential as a 1-form on the total space . Joint dynamics of gravity (which is minimizing some functional over hypersurfaces every moment) and electromagnetism which has its own dynamics can be addressed in many different ways. One simple way is to define the electromagnetism Lagrangian on the total space and then add that Lagrangian to the Einstein-Hilbert action before minimizing the action.

The constructions above are not frivolous mathematical exercises not simply because of the empirical evidence presented for an S4 universe but because some serious issues can be resolved easily (in principle): working from the explanation of quantization via geometry, we can address the question of energy of the vacuum which will now be consistent with the cosmological constant. Additional gauge forces do not cause any trouble for S4 models. But it is an interesting feature of the S4 model(s) that the natural electromagnetism geometrically is not a gauge theory that might be tested experimentally with a more fleshed out model. One could even be optimistic to retain or improve QED predictions with a mathematically robust model without need for renormalization (although if one wants renormalizable quantum gauge theories, these could be addressed as well using BRST cohomology for example using a compact embedding of the complex sphere in using the machinery of geometric quantization; the reference and idea of complex spheres was suggested by Pete Morrison who I thank for it).

But there is a natural larger question that should be resolved: could it not be that not only the fundamental quantization of energy but all quantum phenomena could be reduced to the 4-sphere geometry (which comes with spin structure, Dirac operator etc.)? The above example of forced quantization of possible masses is simply a consequence of the Dirac spectrum being discrete which is a geometric property. There are many reasons why this is not merely a mathematical curiosity but a serious question. If we could produce a physics that can explain important observations without the machinery of quantum field theory (and reproduce its accuracy) then we would have accomplished a vast parsimony improvement in physics while reaching a consistency between gravity and other forces. The above solution uniting gravity and electromagnetism/quantum phenomena explains why gravity is quite different from the other forces: it is fundamentally a phenomena of three dimensions in a four (space) dimensional universe.

The above picture also tells us that the four macroscopic spatial dimensions are not quite the usual physical dimensions (which continually change) but they are accessible to electromagnetism. This is important for the empiricists who have a blanket rejection of higher dimensions based on the idea that higher space dimensions would be accessed by us in the same manner in which we access the three dimensions. Now what about conservation of charge etc.? This is an issue to consider for more detailed research. A non- gauge invariant electromagnetism would be quite a surprise if shown empirically.

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