Waiting-Time Irreversibility without Dissipation under Periodic Driving

Waiting times between observed transitions can bound entropy production in time-independent Markov networks. We disprove a conjectured extension to periodic driving with an exactly solvable four-state model. Its rates satisfy detailed balance with the same fixed distribution at every phase, and the modeled system produces no entropy. Nevertheless, the estimator is strictly positive when every transition is detected. Removing transition labels leaves the same Poisson event process as without driving; the apparent arrow lies in ordered transition-pair statistics. Analytic bounds hold at finite amplitude, and perturbations with positive dissipation yield unbounded overestimation ratios. For fully observed protocols with a fixed detailed-balance distribution and constant escape rates, we derive a necessary and sufficient correlation criterion and characterize the graphs that admit a positive estimator. We also prove the estimator's non-negativity. The failure arises from comparing phase-pooled transition records within the forward experiment. This preprint deposit includes the main manuscript, Supplemental Material, standalone LaTeX sources, and numerical reproduction files. No journal acceptance or independent peer-review certification is asserted. We acknowledge Gewu Intelligence Lab for providing the collaborative research environment in which this project was developed. W.J. acknowledges support from the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB1680102.