Harmonics of S4 Universe and physics constants

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Mathematics of spherical harmonics is classical, but in the theory of spherical harmonics most well-known are harmonics of the 2-sphere. The harmonics of the 4-dimensional unit sphere are less well-known but a number of authors have found derivations of these. The level k eigenvalues of the Laplace-Beltrami operator in this case are:

(k+4)(k+3)/12

It follows that for the scaled 4-dimensional sphere of radius R, the eigenvalues of Laplacian are:

(k+4)(k+3)/( 12 R^2)

The zero-th eigenvalue of the Laplacian corresponds to the energy quantum, thus

h = R^{-2},

now h = 6.2606833x10^{-34} Joule x second, and thus R = 3.9965857 x 10^16 meters. This is the RADIUS OF THE UNIVERSE.

The constant curvature of the Universe is thus also equal to Planck's constant. S4 is Einstein and thus any 3-submanifold is also, giving Einstein's field equations validity for any 3-submanifold including possibly ones that are time-dependent. Note that the 'quantum hypothesis' is unnecessary, as classical mechanics on S4-universe of radius R will lead to quantization of all smooth functions, including the energy spectra of atoms and molecules.

THE FOLLOWING IS SPECULATIVE AND JUST TRIAL-AND-ERROR, NOT RESULTS

In S4 Universe, an embedded 4-ball of radius D conjecturally should have harmonics of a 4-sphere of radius R/D. The equitorial radius of Earth is 6,378,100, thus R/D = 6.2661 x 10^9, and the corresponding harmonics are the harmonics of the 4-sphere scaled by 2.54 x 10^{-20}. Let us call this spectrum the "Earth energy spectrum".

If we go down by 67 octaves, we will have the spectrum that unit 4-sphere harmonics scaled by 2.54 x 1.4757 = 3.7484, for which the first few values are:

3.7484
6.2473
9.3710
13.1194
17.4925
22.4904
28.1130
34.3603
41.2324
48.7292
56.8507

The "planetary octaves" in Hertz (1/second) are -- these are the harmonics of the 3-dimensonal BALL of radius of the Earth.

7.8
13.7
19.6
25.5
31.4
37.3
43.2

The spectra are not exactly the same. But as a quick look, this is promising.