A Universal Dynamical Framework: Hybridization of Collatz Iteration and Trigonometric Algebra

We present a novel hybrid mathematical system that unites the classical Collatz iteration with a generalized Trigonometric Algebra. By introducing polynomial and trigonometric modulations at each iterative step, the resulting sequence can emulate a wide range of dynamical behaviors—including attraction, periodicity, and chaos—beyond the reach of classical Collatz dynamics. While this model offers significant flexibility and universality in pure mathematics, it has no direct practical utility in physical experiments or applied engineering; its contributions are restricted to theoretical and abstract mathematical exploration.The author acknowledges the use of OpenAI’s language model for assistance in drafting certain sections of this workl