Occurrence of a fourth ‘mixed’ type of Bockstein differential involving τ-torsion sources

Ascertain whether the hypothesized fourth type of Bockstein differential—where x and y are τ-free with d_r(x)=ρ^r τ^n y and there exists a later differential d_s(z)=ρ^s y from a τ-torsion class z—actually occurs in practical Ext computations for the ℝ-motivic or C2-equivariant ρ-Bockstein spectral sequences.

Background

Beyond the three established classes of Bockstein differentials (γ→γ, Q→Q, and Q→γ), the authors point out a theoretically possible fourth pattern where a later differential arises from a τ-torsion source to a τ-free target previously hit by another differential.

Confirming or ruling out this phenomenon would sharpen the taxonomy of differentials and clarify whether additional unexpected interactions between τ-torsion and τ-free classes must be accounted for in computations.