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It has been around 8 years since I first realized that space, not spacetime, should be a four-sphere of fixed radius.  I was not trained in physics with a mathematics background.  Since then I have gradually learned some of what the physicists had accomplished and looking at the issues that have animated their work it is now fairly clear that in terms of empirical successes, the quantum field theory model of particles and gauge forces of Glashow, Weinberg and Salam (including the Higgs boson, which they predicted and which was found in 2012) is correct and quite likely supersymmetry is not a feature of nature.  This is a 2009 talk of Weinberg who is without questions a great physicist Weinberg talking about HiggsWeinberg talking about Higgs.  I am still quite sure that S4 theory is correct with the Standard Model, and Big Bang cosmology is wrong.  In fact, it is pretty clear to me now that there is nothing beyond the Standard Model in particle physics and the project of quantum gravity is the wrong direction.  I don’t think there is a graviton.  The Universe is a constructed in my view as a hypersurface of a scaled four-sphere where the quantum field theory is somehow attaining a complicated (29 adjustable parameters) theory despite the fact that there is a totally wrong cosmology, which leads to the ‘cosmological constant’ problem.  The cosmological constant can be calculated in the ‘Einstein static model’ — and Einstein’s first instinct was in the right direction of an eternal three-sphere universe.  The measured cosmological constant is positive, $\Lambda=10^{-52} m^2$.  People have calculated the vacuum energy density (see energydensityeinsteinuniversein the Einstein model and it is
$\rho =\frac{1}{480 \pi^2 a^4}$
where $a$ is the radius of the universe, and the standard explanation that the Einstein-static universe is unstable is not quite true, see stabilityeinsteinuniverse)  If we take the estimated radius of the universe, $a_{est} = 4.4 \times 10^{26} m$ then we actually get an energy density $\rho = 5.63\times 10^{-111}$.  We get much closer to the measured energy density if instead we use a radius $\sqrt{a_{est}}$, then $\rho\sim 10^{-59}$.