Convergence and Dissipative Dynamics in the Family Qn+x

We prove that for a general affine system $T_{Q,x}$ to admit a Collatz-type deterministic descent loop, its orbits must intercept the singular boundary $Qn+x=2^t$ at nodes satisfying the variational threshold $t>\log_2 Q$. By evaluating the underlying modular resonance $2^t \equiv x \pmod Q$, we establish that only the classical $(3,1)$ branch minimizes the expansive baseline to yield a global dissipative anchor (the Gate-5 channel). Consequently, the $3n+1$ map is structurally unique among all affine branches in algebraically and geometrically realizing global terminal descent.