Exclusion of Linear Orbit Growth in Discrete Deterministic Dynamical Systems A Unified JNLDZ–JRNK+–JLR Framework

Abstract This work establishes a structural exclusion result for discrete deterministic dynamical systems.Under a unified framework composed of the JNLDZ Law (area separation), the JRNK+ Rule (explicit structured growth), and the JLR Formula (deterministic trajectory with re-entry), it is shown that linear growth of orbit length or positive density coverage of the interval [1,M][1, M][1,M] is impossible. The argument relies on a separation-of-areas principle, bounding the cumulative sum along an orbit relative to its maximal excursion. Any hypothetical family of orbits exhibiting asymptotically linear length would necessarily violate this structural constraint. As a consequence, an entire class of extreme behaviors in discrete dynamical systems is excluded. The result applies to deterministic systems satisfying the stated hypotheses and provides a structural contribution to the analysis of Collatz-type dynamics and related iterative processes. This manuscript is a preprint and has not undergone peer review.