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 $\pi_*(BP)$ 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 $\text{Ext}_{A^{C_2}}(\underline{H}^\star, \underline{H}^\star)$ 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 $0\le k < n.$ We also promote the Lin-Davis-Mahowald-Adams splitting of Ext of the suitable version of ``$R P_{-\infty}^\infty$" 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)}(\underline{H}^\star, \underline{H}^\star),$ then the same power of $v_{n-1}$ is also nonzero in $ \text{Ext}_{A^{C_2}(m-1)}(\underline{H}^\star, \underline{H}^\star),$ and its algebraic Mahowald invariant $M_m^{C_2-alg}(v_{n-1}^{2^f}) \subset \text{Ext}_{A^{C_2}(m)}(\underline{H}^\star, \underline{H}^\star)$ contains class(es) corresponding to $v_n^{2^f}.$ Real motivic versions of these results hold as well.