By restricting k to rational numbers, the Schrödinger wave equation Ψ = Ae^{-i/ℏ(Et+mrdr/dt)} = Ae^{-i π k} can be converted into a polynomial with the base π. Ultimately, this leads to the action S for each object:
S_Object =(2π)⁴ E_t + (2π)³ E_r i^t + (2π)² E_φ i^t-1 + (2π)E_θ i^t-2
t in Z If 2 objects and an observer have a common center of gravity, the energies can be related and calculated using a single polynomial. The integer quantum numbers E_t, E_r, E_φ and E_θ ensure cohesion and lead to the four fundamental natural forces. Our worldview, with 3 isotropic dimensions x, y and z and rotations with 2π must be distinguished from this. The polynomials are transformed by simple operators (addition) for parity, time and charge. The 3 spatial dimensions result from regularly recurring parity operators. Numerous calculations are given for the orbits in the solar system and for the masses of the elementary particles, e.g.:
m_neutron / m_e =(2π)⁴ +(2π)³+(2π)²-(2π)¹-(2π)⁰-(2π)^-1+2(2π)^-2+2(2π)^-4-2(2π)^-6 +6(2π)^-8=1838.6836611
The charge operator for all particles is:
C = - π + 2π^-1 - π^-3 + 2π^-5 - π^-7 + π^-9 - π^-12
Together with the neutron mass, the result for the proton is:
m_proton=m_neutron + C m_e= 1836.15267363 m_e
The probabilities for the correct representation of the neutron and proton mass have been calculated and are greater than 0.99997. The muon and tauon masses can be calculated in the same way.
Fine structure constant:
1/α = π⁴+π³+π²-1- π^-1 + π^-2-π^-3 + π^-7 - π^-9- 2π^-10-2π^-11-2π^-12
For an observer and two objects, from the torque and angular momentum alone, a common constant of h, G, and c can be derived, giving a ratio of meters and seconds:
h G c⁵ s⁸ /m¹⁰ Math.sqrt( π⁴- π²- π^-1- π^-3 ) = 1.00000
How fast the surface of a body moves relative to its radius is determined purely by the smallest possible ratio of its rational coordinates.
2π c(α) meter orbital period = Math.sqrt(diameter)
2π c meter day = (Earth's diameter)²