The Global Divisor Rate and the Singular Boundary r=\log_2 3: Toward Structural Completeness in Collatz Dynamics

We reformulate the Collatz entropy framework by restructuring the critical constant $\log_2 3$ from a statistical limit (Tao) into a structural invariant derived from a weighted entropy functional ($S_\beta$). We then employ an analytic compensation framework to upgrade the ``almost global descent'' into a structurally guaranteed descent ($O(E^{-2})$), valid at every scale. Within this structure, the integers $2$ and $5$ emerge as dual convergence nodes (Gate--2 and Gate--5), which are shown to establish the global uniqueness of the $3n+1$ map within the generalized $Qn+1$ family.