Convergence and Dissipative Dynamics in the Family Qn+x

We use a logarithmic mean-drift framework to show that the singular anchors of the classical branch $3n+1$ occur precisely at the odd integers $1$ and $5$. These two nodes generate the only structurally relevant dyadic channels and correspond, in the general family $Qn+x$, to the distinguished $s\!-\!1\!-\!S$ configuration. Within this variational setting, the node $n=5$ emerges as the minimal nontrivial contraction anchor, establishing its structural uniqueness in the global descent geometry.