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1:44 AM (0 minutes ago)



These are universal constants c^{mn}_j which appear in the nonlinear mixing terms of the solution of the NavierStokes equations in the eigenbasis of S4. These are scale invariant in the sense that scaling the eigenvalues by k should produce:
c^{mn}_p = c^{(km)(kn)}_{kp}
This does not seem like a hard problem since scaling the sphere by (1/k) should do this. I will actually spend some time to work this out at some point — it just looks a bit messy and besides 2D spherical harmonics the 4D case is not used so it’s still research territory but this would be a breakthrough in our understanding of Nature in terms of turbulence phenomena. When Andrei Kolmogorov in 1941 came up with his 5/3 scaling law it was a breakthrough in our understanding of turbulence. That underneath the hood the source of the scaling might be the geometry of the entire universe and thus scale invariance in all the quadratic coupling problems would instantly demystify stochasticity, turbulence, and source of fractal geometry of Nature instantly.
In other words, I claim that it is the CLEBSCHGORDON COEFFICIENTS in S4 spherical HARMONICS AND THEIR SCALE INVARIANCE that is the geometric explanation of all the fractal phenomena that we observe in Nature empirically. Note that no such explanation is natural to the euclidean geometry. You need compactness of space to have any expansion at all. Therefore SPHERICAL geometry of space is crucial to explain scale invariance phenomena. Thus my S4 model radically alters human understanding of the universe in which we live by clarifying the mathematical structure behind scale invariance and fractal geometry of Nature. Benoit Mandelbrot had gone far into modeling of fractal phenomena with a long history going back to Cantor but no one before me has completely solved this mystery of Nature with a concrete mathematical solution that can connect the classical atom, the scattering amplitude computations and the turbulence in hydrodynamics in one shot. This is indeed a joyous day for humanity.
Thanks,
Zulf