Traditional calculations in perturbative quantum chromodynamics (pQCD) are based on an order-by-order expansion in the strong coupling $��_s$. Observables that are calculable in this way are known as "safe". Recently, a class of unsafe observables was discovered that do not have a valid $��_s$ expansion but are nevertheless calculable in pQCD using all-orders resummation. These observables are called "Sudakov safe" since singularities at each $��_s$ order are regulated by an all-orders Sudakov form factor. In this letter, we give a concrete definition of Sudakov safety based on conditional probability distributions, and we study a one-parameter family of momentum sharing observables that interpolate between the safe and unsafe regimes. The boundary between these regimes is particularly interesting, as the resulting distribution can be understood as the ultraviolet fixed point of a generalized fragmentation function, yielding a leading behavior that is independent of $��_s$.

4+5 pages, 4 figures, 1 table. Version accepted for publication in PRD