A Second-Order-of-Accuracy Derivation of the Newton Gravitation Constant From The GEM Unification Theory

In the GEM theory of unification,\footnote{Brandenburg, J. E., (1995), Astrophysics and Space Science, 227, p133.} a combination of the Kaluza-Klein and Sakharov unification theories, the appearance of the Kaluza-Klein 5$^{th}$ dimension allows the separate appearance of electrons and protons from the vacuum. This occurs by having the masses of the electron and proton merge smoothly as the radius of spacetime curvature approaches the Planck length, and also separate smoothly as a new 5$^{th}$ dimension is inflated to its ``compact'' size r$_{o}$= e$^{2}$/m$_{o}$c$^{2}$, in esu, where m$_{o}$ = (m$_{p}$m$_{e})^{1/2}$ where e, and m$_{p}$ and m$_{e}$, are the electron charge, proton and electron masses respectively. The 5$^{th}$ dimension appearance inflates from the Planck Length: r$_{P}$=(G$\eta $/c$^{3})^{1/2}$ An improved model features the relationship between $\sigma $ = (m$_{p}$/m$_{e})^{1/2}$ and the radius of curvature r$_{c}$ where the 5$^{th}$ dimension inflates from r$_{c}$ = r$_{P}\rightarrow$ r$_{o}$ and $\sigma $ =1$\rightarrow$42.8503 is now rationalized near the Planck length to be ln(r$_{c}$/r$_{P})=\sigma $/(1+0.8473/$\sigma ^{3})$ so that $\sigma $=1 inside the event horizon 2r$_{P}$. and the formula gives a more accurate expression for the value of $\sigma $ after inflation. The formula after inflation is inverted to find a new 2$^{nd}$ order expression for G: G=e$^{2}$/(m$_{p}$m$_{e}) \quad \alpha $ exp( -2($\sigma $ - 0.847319574/$\sigma^{2}${\ldots}.) = 6.67419 x10$^{-8}$ dyne-cm$^{2}$gm$^{-2}$, where $\alpha $ is the fine structure constant, and is within experimental accuracy of the measured value.