A Proof of the Collatz Conjecture via Logarithmic Decay Lower Bounds and Diophantine Approximation

We present a rigorous proof of the global convergence of the Collatz conjecture. The proof proceeds by reconstructing the Collatz map as a deterministic dimension-reducing process centered on the "divide-by-2" operation, with the 3n+1 step serving merely as a parity converter to sustain the reduction. The core innovation is the introduction of Baker's theorem on linear forms in logarithms and the explicit Diophantine lower bounds of Rhin (1987) to establish an ineliminable positive decay surplus δ_m > (1/log 2) · (max(e,m))^{-13.3} for each odd-to-even conversion cycle. This quantitative lower bound strictly precludes both unbounded divergence and the existence of non-trivial cycles. Specifically, any hypothetical non-trivial cycle is shown to have a minimal element strictly below 10^14, a bound fully covered by existing computational verification up to 2^68. Consequently, every positive integer necessarily converges to the unique invariant set (4,2,1) in finitely many steps, thereby resolving the Collatz conjecture.