Prove the asymmetric recovery criterion under dynamic execution priorities

Prove or refute the Asymmetric Recovery Criterion for a cooperator interacting with a defector when execution probabilities are updated after every operation rather than fixed at their initial values; specifically, determine whether the condition E_max > \tilde{E}^* guarantees the stated positive expected energy differential at replication completion.

Background

The Asymmetric Recovery Criterion addresses a cooperator entering an encounter at maximum energy and a low-energy defector whose stealing can generate net energy for itself. The conjecture derives a threshold \tilde{E}^* under a frozen-priority approximation, in which each program’s execution probability remains equal to its initial energy share.

The paper acknowledges that actual execution probabilities change after every operation. Repeated stealing can increase a defector’s execution priority, while costly computation can reduce a cooperator’s priority, so the conjectured threshold is not established for the full dynamic process.