A Structural Proof of the Collatz Conjecture via Injectivity and Recursive Decay

We present a structural proof of the Collatz conjecture by rigorously analyzing the recursive mapping of odd integers. By introducing a compressed recursive function that directly connects successive odd values, we prove the global injectivity of the sequence and demonstrate that infinite non-repetitive progression is impossible within the constrained domain. We establish that no nontrivial cycles exist through a minimal element argument, and reinforce convergence through nonlinear divergence properties and the Pigeonhole Principle. Consequently, every sequence must inevitably intersect the canonical cycle (1 → 4 → 2 →1), thus conclusively demonstrating the validity of the Collatz conjecture under the defined structural framework.