Recurrence and a diagonal limiting schedule

This is the fixed-proxy obstruction in Hawk04-A/B/C CheungHawk04A,CheungHawk04B,CheungHawk04C.

If a continuum endpoint is approximated at increasing times, one must choose a diagonal regulator schedule satisfying, for example,

on the actual increasing windows $T_{\ell }$, together with the endpoint access comparison and decoupling estimates. For the sufficient selection in (eq:regulator-selection), occupation grows at least quadratically with $G{T}_{\ell }$, and the mesh must shrink accordingly. A finite-time theorem does not justify interchanging the regulator and late-time limits.