5x5: Polynomial Convergence of 5 Constants - Combined Newton-Raphson Iteration Technique - Standard Approach

The Algorithm of post 9/5/2022 new-algorithm-idea-multi-variate CONVERGES when holding the proton mass and electron mass (thus the $m_p\over m_e$ mass ratio) constant and iterating on these 5 constants:

$$e$$

$$\epsilon0$$

$$h$$

$$c$$

$$R_H$$

The initial values and tolerances need to be corrected and re-run the algorithm. Here is a table of the constants NIST/CODATA and what I used in the Newton_Raphson Iteration NRI method.

Image 1. Constants: NIST/CODATA & NRI

NRI method showing electron charge error getting smaller and smaller
(i.e., more negative on the log Y-scale)

https://phxmarker.blogspot.com/2024/10/showing-99-iterations-on-plots-e-e0-h-c.html 

This link is to the post that shows the color points on the graph with the inputs shown above in Image 1.

The algorithm is described in this post, only modification was to hold the proton and electron mass constant: https://phxmarker.blogspot.com/2022/09/new-algorithm-idea-multi-variate.html

A re-run is needed with the corrected inputs - I didn't use all the digits from NIST/CODATA.

The interesting thing is it appears to converge to a stable solution.  I need to investigate if it is UNIQUE by using different starting values and verfiy if it always converges to these same values.

Another interesting thing is a MAJOR NOTE: One does not need the proton radius solution to perform this NRI method of determining the constants. Simply don't drop the term, the reduced mass approximation (effective masses) when determining the constants. 

There may be some reason the $m_e\over m_p$ term was dropped from the polynomial - as I understand it, that assumption allows one to proceed with an analytic solution to Schrodinger's wave equations for solid-state theory. And that assumption does not have to interfere with determining the coefficients like anyone would do for their boundary value problems.

When you zoom in on the solutions, the last series of iterations, it curves back, very similar to the Lambert W function that is used to solve the iterative Widlar current source circuit.

Zoom in on last iterations of electron charge

https://en.wikipedia.org/wiki/Lambert_W_function

The dream lives! The dream of solving for the constants. 

I'm looking into applying the constants to analyzing the "Island of Stability" - i.e., 

™Resonance & Harmony™ and MetaMaterials†™:

https://en.wikipedia.org/wiki/Island_of_stability 

https://en.wikipedia.org/wiki/Island_of_stability

The Surfer, OM-IV

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