Algebraic redshift in the $C_2$-equivariant Adams spectral sequence

We study $v_n$-periodic phenomena in $C_2$-equivariant stable homotopy through the lens of the $C_2$-equivariant Adams spectral sequence at the prime 2. In particular, we construct/detect certain classes related to powers of the $v_n$ generators of ${\pi }_{\ast }(BP)$ in the cohomology of certain finitely generated subalgebras $A^{C_2}(m)$ of the $C_2$-equivariant Steenrod algebra. We define the notion of classes in ${\text{Ext}}_{A^{C_2}}({\underset{‾}{H}}^{\star },{\underset{‾}{H}}^{\star })$ being $v_n$-periodic or $v_n$-torsion and exhibit a chromatic filtration by showing that $v_n$-torsion classes are also $v_k$-torsion for $0\le k<n\mathrm{.}$ We also promote the Lin-Davis-Mahowald-Adams splitting of Ext of the suitable version of ``$R{P}_{-\mathrm{\infty }}^{\mathrm{\infty }}$" to the $C_2$-equivariant setting and use this to define appropriate algebraic versions of Mahowald’s root invariant. We establish that whenever a class corresponding to a power of $v_n$ is nonzero in ${\text{Ext}}_{A^{C_2}(m)}({\underset{‾}{H}}^{\star },{\underset{‾}{H}}^{\star }),$ then the same power of $v_{n-1}$ is also nonzero in ${\text{Ext}}_{A^{C_2}(m-1)}({\underset{‾}{H}}^{\star },{\underset{‾}{H}}^{\star }),$ and its algebraic Mahowald invariant $M_m^{C_2-alg}(v_{n-1}^{2^f})\subset {\text{Ext}}_{A^{C_2}(m)}({\underset{‾}{H}}^{\star },{\underset{‾}{H}}^{\star })$ contains class(es) corresponding to $v_n^{2^f}\mathrm{.}$ Real motivic versions of these results hold as well.