  In both quantum computing and black hole physics, it is natural to regard
some deformations, infinitesimal unitaries, as \emph{easy} and others as
\emph{hard}. This has lead to a renewed examination of right-invariant metrics
on $\operatorname{SU}(2^N)$. It has been hypothesized that there is a critical
such metric -- in the sense of phase transitions -- and a conjectural form
suggested. In this note we explore a restriction that the ring structure on
cohomology places on the global geometry of a critical metric.
