A metric and stationary measure do not determine a clock

Consider a two-vertex graph with distance one and uniform measure. For every $c>0$, the symmetric continuous-time generator

has that same metric space and stationary measure, but its nonzero eigenvalue is $2c$. Thus metric-measure information alone does not determine the time normalization or spectrum of a process. The random-walk and spectral companions require additional analytic inputs; they are not automatic consequences of a GHP record.