Optimal error threshold for essential non-branching

Determine the optimal range of the entropy-error parameter \(\delta\) for which a metric measure space satisfying the almost barycenter curvature–dimension condition \(BCD_\delta(K,\infty)\) is necessarily essentially non-branching.

Background

The paper proves that if a complete separable geodesic metric measure space satisfies the additive-error condition $BCD_\delta(K,\infty)$ with $\delta<\frac12\log 2$, then the space is essentially non-branching. The threshold arises from the entropy gap $\log 2$ associated with an equal mixture of two mutually singular probability measures in the Rajala–Sturm branching argument.

The authors explicitly describe this bound as sufficient rather than optimal. Determining the largest error parameter—or the sharp dependence on $K$ and any other geometric data—for which essential non-branching still follows remains unresolved.