Earliest swapping for every chain length: Kamin et al. (2023), Section IV D 1, Eq. (41)

The all-length earliest-swapping conjecture in Section IV D 1, Eq. (41), of Kamin, Shchukin, Schmidt and van Loock (2023) states that repeatedly merging an adjacent pair with smallest maximum ready time minimizes total dephasing. We give a self-contained potential proof for arbitrary nonnegative real ready times and every tie choice. The optimum is the largest ready time plus the sum of adjacent maxima minus the sum of ready times. An exact nonnegative slack formula characterizes all optimal histories. The pathwise inequality also answers the exponential-dephasing question in Section IV D 5, for the same fixed-readiness swapping model: earliest swapping simultaneously minimizes expected dephasing and maximizes every expected exponential attenuation factor.