A counterexample for the symmetrization of quasi-partial b-metrics

A common approach in the theory of generalized metric spaces, particularly in spaces which lack symmetry, is to pass to a symmetrized function. In the literature on quasi-partial b-metric spaces, it has been repeatedly assumed without proof that the standard symmetrization $$ d_{q}(x,y)=q(x,y)+q(y,x)-q(x,x)-q(y,y) $$ is always a b-metric. We first provide a definitive counterexample to disprove this general claim. Next, from an elementary estimate derived from the axiom (QPb4), we obtain a convenient sufficient criterion: if a self-distance modulus $\mathfrak{M}(q)$ is finite, then $d_{q}$ is a b-metric with explicit constant $K^{*}(q)=s+(s-1)\mathfrak{M}(q)$. We then show by example that this condition is not necessary. Finally, once $d_{q}$ is known to be a b-metric, we explain how fixed point results from complete b-metric spaces may be transferred back to the quasi-partial b-metric setting.