Numerical Universality and Cyclic Attractors in a Two-Branch Piecewise Arithmetic Mapping

This paper is the story of a personal math experiment. I wanted to see what happens to numbers if we change the rules of standard math puzzles. I decided to add a rule: f(n) = n/2 + 1. I wrote a Python script on Google Colab to test what happens. My computer calculated the first 100,000,000 numbers. I thought some numbers would fly away to infinity, but they didn't. Absolutely every single number got trapped into exactly 7 repeating loops. I am presenting these results to the world as an open conjecture.