  Traditional calculations in perturbative quantum chromodynamics (pQCD) are
based on an order-by-order expansion in the strong coupling $\alpha_s$.
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 $\alpha_s$
expansion but are nevertheless calculable in pQCD using all-orders resummation.
These observables are called "Sudakov safe" since singularities at each
$\alpha_s$ 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
$\alpha_s$.
