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

We present a proof of the global convergence of the Collatz conjecture. By analyzing the relationship between the division-by-2 and 3n+1 operations in the Collatz map, we establish the decay structure of the iterative sequence. We prove that after 3n+1, an odd number necessarily becomes even, immediately triggering division by 2, ensuring the number of even steps (e) is no less than the odd steps (m). We mathematically rule out unbounded divergent trajectories by demonstrating an absolute algebraic contradiction: divergence requires the operational ratio to satisfy e ≤ m log₂ 3 ≈ 1.585m, whereas the natural density of 2-adic congruences forces this ratio to converge to e ≈ 2m for sufficiently long orbits. For closed cycles, we utilize Rhin's (1987) explicit Diophantine lower bounds for linear forms in logarithms to establish a positive decay surplus δ_m > (1/log 2) · (max(e,m))^{-13.3}. Combined with algebraic constraints on the minimal element of a cycle, this forces the minimal value of any hypothetical non-trivial cycle to be strictly less than 10^14. This theoretical upper bound is fully covered by existing computational verifications up to 2^68, confirming no non-trivial cycles exist. Therefore, every positive integer necessarily converges to the unique invariant set (4,2,1) in finitely many steps. The Collatz conjecture is thus proven.