I am only an amateur math lover. I am not familiar with mathematical jargons nor am I trained in professional expositions. So the presentation might looks sloppy.
I did not work out the analytical form of the inverse "$h$" function.
Due to the lack of professional math training, I did not try to build a comprehensive number theory to justify why the "$h$" function should be elementary.
I am not a professional programmer, either. Many of the functions I wrote is simply to validate whether my math thinking is tenable, and I was not able to solve some numerical problems. For instance, calculating an "$h$" function with a large power parameter requires some techniques of arbitrary precision arithmetic. I did not have such expertise, and instead, I only use the Matlab's built-in function "vpa". Similarly, the inverse "$h$" function is written with a primitive numerical algorithm. Because of these restrictions, most of the programs are only valid when the arguments are limited within a certain range.