Occurrence of the periodic-type free differential phenomenon

Determine whether the periodic-type free differential phenomenon described in Section 6—where d_r(x)=ρ^r τ^n y with n>0 and a later d_s(z)=ρ^s y for some s>r causes d_s to be free even though d_s(τ^n z)=0—actually occurs in practical Ext computations for the ℝ-motivic or C2-equivariant ρ-Bockstein spectral sequences.

Background

The authors distinguish ‘free’ Bockstein differentials (with τ-free source and target) and discuss their τ^{2^n}-periodicity. They highlight a potential complication: if d_r(x)=ρ^r τ^n y and a later d_s(z)=ρ^s y occurs, the periodicity pattern may exhibit a subtle anomaly.

Whether such patterns truly occur affects the predictability of coperiodic/periodic families of differentials and influences how motivic information is transferred to the negative cone and to the C2-equivariant setting.