Establish a covering-number lower bound for online probe generators

Establish a matching lower bound on the covering number N(ε) of a concrete online probe generator, in order to determine whether probe rotation provides asymptotic protection against structural attackers beyond the generator’s effective covering complexity.

Background

The paper proves that rotation prevents an attacker from defeating every freshly sampled probe through simple memorization, but argues that this guarantee does not extend automatically to structural attacks. If exploitable probe behaviors have an ε-cover of size N(ε), an attacker may generalize from representative probes, making the sample complexity of defeating rotation Θ(N(ε)) rather than Θ(N), where N is the fresh-probe pool size.

The unresolved issue is whether a concrete online probe generator can be shown to have a sufficiently large covering number, or whether low-complexity generators admit small covers that substantially weaken the protection supplied by rotation. The paper identifies distributional drift and surface diversity as design mechanisms intended to enlarge this covering number, but does not establish a matching lower bound.

References

This is a design argument rather than a guarantee; establishing a matching lower bound on N (ε) for a concrete online generator is left to future work.

— Harness-agnostic detection and immunization of reward hacking in self-evolving language models  (2609.04665 - Yang et al., 4 Sep 2026) in Appendix D.3, “Covering-number argument for structural attacks”