Determine asymptotic behavior beyond the finite computational horizon

Determine whether the long-lived carpet-like regimes and other dynamical behaviors identified for time-dependent modular Laplacian schedules persist asymptotically or exhibit aperiodicity beyond the finite computational horizons studied.

Background

The study investigates periodic modular schedules of the form [2,k,2^s]^∞ and reports long-lived, densely occupied carpet-like regimes over finite observation windows. It also identifies schedule-dependent densification, collapse suppression, and synchronization of trajectories generated by different inserted moduli.

The paper explicitly limits its claims to finite computational horizons and does not establish whether these observed behaviors continue indefinitely, eventually collapse, become periodic, or become aperiodic. Establishing the asymptotic behavior would clarify whether the observed carpet-like dynamics represent transient finite-time effects or genuinely persistent regimes.