Establish the asymptotic depth scaling of hierarchical transient gain

Determine the asymptotic dependence on hierarchy depth of the transient gain for the feed-forward epistemic hierarchy, including whether the observed approximate square-root growth persists asymptotically or instead saturates under specified perception-rate limits.

Background

The paper proves that feed-forward hierarchical epistemic operators can be non-normal and transiently reactive even when their spectral radius is arbitrarily small. It then reports numerical evidence that, over the accessible range of depths, the transient gain is well fit by a function of the form $1+c\sqrt{N}$.

The authors explicitly do not establish this scaling analytically. They also report that the gain saturates rather than grows in the instantaneous-perception limit, so the unresolved problem is to characterize the true asymptotic depth dependence and its dependence on the perception rate, distinguishing finite-range empirical behavior from a general law.