The mystery of accuracy

Here are the Lucas polynomials taken from Wolfram. As x goes to 1, the approximation of Phi gets accurate. But the approzimation of Phi is the ratio of the two last Lucas numbers, and that accuracy grows as the maximum n gets larger. So x and n are related in adapted systems.

What is the limit? Well, obviously the finite bandwidth if the exchanges, or the limit on the power spectra of light.  Physicists call it relativity, the system cannot change to an increase in n without introducing endogenous errors and thus dropping n back down.  But how is that set? Or, what sets the limits of the vacuum to the fine structure?

I do not quite know, but I do know that my spreadsheet cannot find a better n, but then my spreadsheet is limits in the same way the proton is.