A Modular Roof Structure for Collatz Dynamics II: A Closed Modular System for Roof Descent

This work complements and extends the results of the author's previous manuscript, where a modular structure was introduced to analyze the dynamics of the Collatz function through a modular transformation \(S(n)\) and a roof function \(T(n)\). In this continuation, we show that the dynamics of roofs are governed by a closed modular system consisting only of the classes \(4,16,22,34 {36}\). After the first cycle, odd numbers congruent to \(3 {6}\) disappear from the orbit, eliminating the classes \(10\) and \(28 {36}\). A complete classification according to \(n {6}\), \(n {9}\), and \(n {4}\) shows that stable roofs arise exclusively from odd numbers \(5 {6}\), and that all roofs satisfy \(T(n)≡ 16{36}\). This closed modular system explains why every chain of roofs inevitably descends to the minimal roof \(16\), reducing the global Collatz dynamics to a discrete process defined solely on roofs. The modular elevation operator introduced in the appendix shows that every unstable roof can be promoted to a stable one without altering the orbit, providing a theoretical explanation for the modular rigidity observed in the previous work.