The key input to seeing that the T(n)-local TC doesn’t commute with taking -fixed points in the case of interest is asymptotic constancy. Asymptotic constancy allows one to reduce checking this to the case of a trivial action. This talk and the next will be dedicated to showing asymptotic constancy for the THH of the ring spectra of interest. The key properties of the ring spectra that are used to show this are that the Z-actions onthem are locally unipotent, they satisfy the height n Lichtnebaum–Quillen property, and are almost compact.