Setting Up The Problem - Multi-Dimensional Roots of Proton to Electron Mass Ratio Equation(s)

(DRAFT  - a few minor correction and comments are required.  11/11/17 MR)
This post is concerning "Rest Mass Physics", 0K, the physics of rest mass of proton and electron from first principles and fundamental constants.  The dynamics is the hard problem. 11/11/17 MR

from: https://www.slideshare.net/NorthWestMax/linked-in-slides-hydrogen-atom

You can do it the easy way, or the hard way, or both.

Is it possible, mathematically and in Nature, from a "Golden Ratio" of physical constants, to determine those physical constants to any desired precision via numerical methods?  That's what this is about, so if it's already known about the stability of solutions or convergence issues, we'll find out as this investigation proceeds.

Example Approach to Multi-Dimensional Polynomial Root Finding:

$$F\left(x_0,x_1,\cdots,x_n\right)={e^4m_e\over{8h^3c\epsilon_0^2R_{\infty}}}-{m_e\over m_p}=1$$

$x_0=e=$ elementary charge

$x_1=m_e=$ eletron mass

$x_2=h=$ Planck's Constant

$x_3=c=$ Speed of Light

$x_4=\epsilon_0=$ permittivity of free space

$x_5=R_{\infty}=$ Rydberg constant

$x_6=m_p=$ proton mass

$$F\left(x_0,x_1,\cdots,x_6\right)={x_0^4x_1\over8x_2^3x_3x_4^2x_5}-{x_1\over x_6}=1$$

$err=F\left(x_0,x_1,\cdots,x_n\right)-1$

Use precise starting values or seeds values for the constants.

Iterate until $err\rightarrow0$.

Requires extra precision calculations (>64bit?)

Google Calculator check of F(x) <-- click to see calculation of identity
(check#2) <-- something is amiss??? Bad starting point for coeficient values?

So, this is a basic statement of the problem and one potential numerical method solution.

Update
A more precise defination of the Rydberg constant is:
$$R_H\equiv{m_em_p\over{m_e+m_p}}{e^4\over8c\epsilon_0^2h^3}$$
$m_pr_p={2h\over\pi c}={4\hbar\over c}$  <~~~ use this for proton mass-radius product!
Need to double check equations again...  ;-)
(it's an ongoing project)
Correction may be:

$$F\left(x_0,x_1,\cdots,x_n\right)={e^4m_e\over{8h^3c\epsilon_0^2R_{\infty}}}=1+{m_e\over m_p}$$

$$F\left(x_0,x_1,\cdots,x_n\right)={x_0^4x_1\over8x_2^3x_3x_4^2x_5}=1$$

(the dream since 1981 or was it 1977???)

The Surfer, OM-IV

 (this post needs a little re-working... 11/11/17MR)

Ω