PRH | Aux | 4.3.6 • Introduction to Lyapunov and a Toy Collatz–Like Certificate

This note introduces a minimal, self-contained illustration of the Lyapunov method through a discrete “toy–Collatz” system that mirrors the structure of the Collatz descent proof. We first recall the Lyapunov principle—constructing a potential function (V) that decreases along every trajectory—and show how a simple logistic recurrence admits such a function explicitly. We then build a two–residue integer map whose potential
$$
V(n)=\log n+\phi(n\bmod 2^k)
$$
and finite residue inequalities replicate the Collatz certificate’s logic: each step may raise $\log n$ but lowers (V) through a residue–dependent correction. The parameters $\rho$ and $\delta$ serve distinct roles—$\rho$ as a blur or uncertainty budget, and $\delta>0$ as a strict safety margin—together guaranteeing a per-step drop of at least $\delta+\rho-\varepsilon(n)$. This finite, periodic construction exemplifies how a bounded Lyapunov potential on residues can enforce global descent in an infinite process.