On the Global Convergence of the 3n+1 Problem: A Hybrid Telescoping Sieve, Diophantine Metric Gap, and Automated SMT Refutation of Non-Trivial Cycles

The Collatz (3n+1) conjecture asserts that the iterative discrete piecewise map T(n) = n/2 if n is even and T(n) = (3n+1)/2 if n is odd universally converges to the global attractor x_final = 1 for all positive integers. In this monograph, we establish an exhaustive algebraic and automated constraint framework that refutes the existence of non-trivial cycles and infinite divergence. We prove that all closed periodic orbits of k odd steps and m even divisions correspond exactly to integer solutions of the Telescoping Fixed-Point Identity x0 = C / (2^m - 3^k). Through a multi-stage parity and residue sieve, candidate cycle generators are restricted to the prime wheel {6k ± 1}. We derive the unified metric congruence (C - D)(C + D) ≡ 0 (mod 24) where D = 2^m - 3^k, and formulate the Semiprime Entanglement property C = p * q. Finally, we show that non-trivial candidates x0 ≥ 5 require an analytical power gap 2^m / 3^k < 1.100, which creates an irreducible contradiction across the continued fraction spectrum of log2(3), confirmed as universally UNSAT by automated formal SMT verification in Z3.