Conditional Proof of the Collatz Conjecture via Goldbach Partitions

The Collatz Conjecture asserts that the orbit of every positive integer under the map n → 3n + 1 (for odd n) and n → n/2 (for even n) eventually reaches the cycle (4, 2, 1).This paper presents a conditional proof of the conjecture by assuming the validity of the Goldbach Conjecture.We define two transformation functions, F1 (even) and F2 (odd), which map integers to the arithmetic mean of two prime numbers.We demonstrate that the Collatz dynamic is isomorphic to a sequence of "hops" between prime midpoints.Furthermore, we analyze the parity of these prime means to establish a probabilistic descent argument, showing that the trajectory is structurally bound to converge to unity.