Searching for a Decreasing Function in the Collatz Conjecture
This paper investigates whether a Lyapunov-style function can be found that decreases along Collatz trajectories, and concludes that no such function is established here: the work is a structured research programme, not a proof. Starting from the failed candidate F(n) = n, the logarithmic potential F(n) = log₂ n reveals an average downward drift once the process is compressed to odd integers; residue classes modulo powers of 2 then explain how many divisions by 2 follow each odd step, and a residue-dependent correction on classes modulo a power of 2 turns part of the problem into an optimisation problem. The aim is to show how an elementary-looking problem becomes a serious investigation spanning dynamical systems, probability, number theory, optimisation and computation.