Robust protocols to reveal anyonic time-exchange phase

We consider hierarchical quantum Hall edge states with $N$ modes and a spatially local quantum point contact (QPC). In general, the field of an injected anyon does not directly acquire the universal statistical phase $\theta$. Short-range inter-edge interactions split the universal anyon charge and phase into $N$ fractionalized charges associated with nonuniversal phases $\pi\delta_m$. In contrast, their sum $\delta=\sum_{m=1}^N\delta_m$, which defines the local scaling dimension at the QPC, remains protected and is tied to the statistical angle through $\pi\delta=\theta$. If the injected anyon is of the same species as the one dominating backscattering at the QPC, time-domain braiding with phase $\theta$ is recovered either in the absence of inter-edge interactions with equal mode velocities, or by performing a spatially local anyon injection at the QPC. We then exploit a more robust \emph{local} anyonic time-exchange (ATE) link between anyons and quasiholes at the QPC, which is also a necessary ingredient for realizing such braiding. This allows us to propose minimal single-QPC protocols that do not rely on diluted anyon sources and that disentangle the role of $\theta$ as a genuine statistical phase from that as a scaling dimension. From the ATE link we derive two novel nonequilibrium fluctuation--dissipation relations (FDRs) that isolate $\theta$. They relate the DC backscattering noise either to an integral over the DC current or to the phase shift of the AC current with respect to an applied AC voltage (i.e., the phase of the admittance), accessible down to low frequencies. For thermalized edges, we show that in the quantum regime this admittance phase directly yields $\theta$ whenever $\delta>1/2$.