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

Critical Metrics and Covering Number