Hi Branko, I get Eq.(17) = 0.000000574908201 using the CODATA values in your paper. This not a tricky calculation, so I wonder how you get 10^{-14} which is 7 orders of magnitude smaller. Maybe you are reporting the squared error? Considering that G or any physical constant is not that well determined, it seems unreasonable to claim such accuracy. Having said that, your hypothesis is supported to the extent allowed by the CODATA values.
I feel uneasy with the idea that physical constants are a consequence of mathematics, or that mathematics is a consequence of physics. It takes the mathematical universe concept too far in my opinion, but the possibility is interesting and should not be dismissed because of intuitive considerations. Your essay certainly fuels the imagination. - cw