The Prime Jump System: Nine Elementary Theorems and the Universal Termination Conjecture

This paper presents the “Prime Jump System” (PJS), a discrete additive dynamical system over N that couples a linear Fibonacci recurrence with a primality-stopping condition. We prove nine fundamental theorems governing the predictability, strict monotonicity, and parity of the orbits. We demonstrate that Mersenne primes act as perfect absorbing fixed points within the system. Finally, we formulate the Universal Termination Conjecture and provide heuristics regarding the density of attractors, supported by constructive Chinese Remainder Theorem (CRT) sieves for arbitrarily long composite sequences.