Jarzynski equality for counterwork under reversed memory-filtered driving
We introduce a counterwork functional generated by a sign-inverting memory-filtered effective protocol. Given an imposed protocol $λ(t)$, the effective protocol $Λ(t)$ is obtained by applying an active protocol-memory kernel to $\dotλ(t)$, rather than by invoking a passive bath response. The counterwork is then the ordinary Hamiltonian work associated with $H(Γ,Λ(t))$, so that Jarzynski's equality applies directly to it. When $Λ(t)$ reverses the endpoints of $λ(t)$, the corresponding free-energy difference satisfies $ΔF_C=-ΔF_W$, and the exponential average of the counterwork is the reciprocal of that of the original work. We derive the kernel normalization realizing this reversed displacement and show, by Jensen's inequality, that the product of the exponentials of the average work and counterwork is bounded by unity, implying $\langle C\rangle+\langle W\rangle\geq0$. Thus negative average counterwork is possible only when compensated by the average work of the original operation. We further discuss the counteroperation under incomplete thermodynamic information, showing that the robust strategy is to enforce endpoint reversal while minimizing dissipated counterwork.