The Kodama Observable for Collatz Dynamics: Exact Decomposition, Residual Dynamics and Experimental Evidence up to 10^8

Summary This work introduces the Kodama Observable, a new mathematical quantity associated with Collatz trajectories (the 3x+1 problem). Main Contributions and Findings New Mathematical Framework: The Kodama Observable is formally defined for Collatz orbits. The paper derives exact identities that link this observable to the multiplicative structure of the orbits. Novel Dynamical Formulation: It introduces a residual formulation of the Collatz dynamics, offering a fresh perspective on the problem. Computational Evidence (up to 108108): Extensive computational experiments were performed for all integers up to 100 million (108108). The results suggest a bounded logarithmic conservation law for the observable. A strong empirical correspondence was found between the maxima of Collatz orbits and the minima of the Kodama Observable. Scope and Limitations: The work presents exact results and experimental observations. It does not claim a proof of the Collatz conjecture but offers new tools and strong empirical evidence for its study. Conclusion The research opens a new analytical pathway by introducing the Kodama Observable, which shows significant patterns and correlations with classical Collatz dynamics, supported by extensive computational verification up to 108108. This work introduces the Kodama Observable, a new quantity associated with Collatz trajectories. The paper derives exact identities linking the observable to the multiplicative structure of Collatz orbits and introduces a residual formulation of the dynamics. Computational experiments performed for all integers up to 10^8 suggest a bounded logarithmic conservation law and reveal a strong empirical correspondence between orbit maxima and minima of the Kodama observable. This work presents exact results and experimental observations. It does not claim a proof of the Collatz conjecture.