  Sum rules provide useful insights into transition strength functions and are
often expressed as expectation values of an operator. In this letter I
demonstrate that non-energy-weighted transition sum rules have strong secular
dependences on the energy of the initial state. Such non-trivial systematics
have consequences: the simplification suggested by the generalized Brink-Axel
hypothesis, for example, does not hold for most cases, though it weakly holds
in at least some cases for electric dipole transitions. Furthermore, I show the
systematics can be understood through spectral distribution theory, calculated
via traces of operators and of products of operators. Seen through this lens,
violation of the generalized Brink-Axel hypothesis is unsurprising: one
\textit{expects} sum rules to evolve with excitation energy. Furthermore, to
lowest order the slope of the secular evolution can be traced to a component of
the Hamiltonian being positive (repulsive) or negative (attractive).
