Visualizing the Cosmic Nucleus in 3D
Mathematics

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Synchronized field or path integral
:

               
V(Φ) = ⎰ | ∏ cos(f0∙rsn + θtp) | dx


Where:
                V(Φ) = integral vacuum potential
                f
0 = nominal rate or frequency (=1)
                r
s = scaling ratio (producing scale-invariant wave set)
                Θ
tp = phase propagation (in parametric time)

In other words, each factor Φ
n = fntp) is a functional of the phase propagation, so that:

                V(Φ) = ⎰ | Φ0tp1tp2tp) etc.. | dx

So far it has numerically been shown that for rs=φ (Golden Ratio, 1.618..), n=0..2 and x/L0 >>1:

                
V(Φ) = ⎰ | ∏ Φntp) | dx ≅ A + B cos (2∙θtp)

With L
0 being the wavelength of f0. This means that the synchronized field produces an external (“2-theta”) oscillation with twice the frequency of the internal, nominal phase wave. In physics terms, this breaks the symmetry of the field.

The amplitude (top-top) of the 2-theta oscillation, B, is apprx. 4% of the ground level A. The latter in physics is hypothized to be the Vacuum Expectation Value (VEV).

Note: the standard use of the letter Phi for the field as well as for the Golden Ratio is mere coincidence. Either way the uppercase Φ refers to the field, and lowercase φ refers to the scaling ratio.



Animations:

The individual phase waves
Φ0, Φ1 and Φ2 are shown at the bottom, with the absolute value of the composite wave ∏Φn above it.The purpose of the model is to show the synchronized field’s “2-theta” wave A + B cos (2∙θtp) in a glance. For that purpose the ground level A is filtered out, and gain B is normalized, say from 0 to 1, so that the synchronized V(Φ)norm  ≅ cos (2∙θtp). In the animation this is the total white area under the curve of the composite wave. The model can be seen as a purely manual spherical projection of ∏Φn "seen from above”, whereby lighter colors indicate higher peak levels. The final rendering merges the composite wave with the integral potential through the multiplication ∏Φn∙V(Φ)norm, for each ∆x.
The sinusoïdal segment left from the centre traces the ideal “2-theta” wave, for easy verification and comparison.



Waterfall power spectrum of the "2-theta" wave generated out of the synchronized field, for axiomatic scaling ratio's ranging from apprx. 1.5 to 1.7. Exactly at Golden Ratio the spectrum peaks, proving the unique resonance point.


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