{S\over O}=1 ; justification: energy and mass are conserved, lossless system
{S\over O}={A\over{1+A\beta}}
1={R_\infty\over R_H}-{m_e\over m_p} & (using m_e\over m_p instead of {{\pi r_pR_H}\over \alpha^2})
R_\infty={m_ee^4\over 8{\epsilon_0}^2h^3c}
(will fill in math steps later, refer to The Universe is a Feedback System)
A_v={S\over O}={A\over{1+A\beta}}
A_v=1
1+A\beta=A
1=A-A\beta
A={m_ee^4\over 8{\epsilon_0}^2h^3cR_H}
\beta={{8{\epsilon_0}^2h^3cR_H}\over{m_pe^4}}
A, open loop forward gain:
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| This is approximately equal to 1 due to artificial tuning of constants |
\beta, reverse gain:
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* Universe of a Hydrogen atom
Moar later.
These values of A and \beta are reminders that this equation and this problem is rooted in the physical single hydrogen atom - a dynamic between a proton and electron. This mathematical tool of transforming the equation to an analogous "system", one can pull out key features (with good math tools/models). The dynamic between the proton and electron captured in a simple diagram. The hydrogen atom, 1H, has a full analytical solution (full Wave Equation or combined Schrödinger wave equation for both proton and electron or however it's done).
