The Collatz conjecture states any number, N0, after successive computations will always yield one, initiating the recursive sequence of 1→4→2→1→4 because 1 equals itself via (3n+1) 2 for n = 1. Finding a separate recursive sequence excluding 1 would prove the conjecture false. Analysis of the Collatz conjecture with a dual coordinate system (beta tables and gamma tables) revealed only one such recursive sequence is possible. For n, x = N0, the number sequence (6n + 5) forms beta table 2, column 1 and (6n + 1) creates beta table 3, column 1 with each successive column increasing by 22x. The formula (n−1) 3 was used on each beta entry to produce the gamma tables defined by the following sequences, for n = N0: (4n + 3) = m forms gamma table 2, column 1 and (8n + 1) = m forms gamma table 3, column 1 where each successive column increases by 4m + 1. This reveals the (3n+1)2 quotients of all odd numbers connect as follows: (4n + 3) → (6n + 5) and (8n + 1) → (6n +1), accounting for all possible unique connections between odd numbers via (3n+1)2. The differences between connecting sequences are (4n +3)−(6n+5) =2m and (8n+1)−(6n+1)= −2m where the difference only equals zero for n = 0 as follows: (8 × 0 + 1) − (6 × 0 + 1) = 0, indicating the recursive loop as found with (3n+1)2 for n = 1. Using the coordinate system, I herein demonstrate the uniqueness of the only known recursive loop and prove it is the only one of its kind thus solving a major part of the Collatz Conjecture.