A Structural Characterization of the Hit Image in the Motivic Steenrod Algebra

Abstract

The motivic hit problem seeks a minimal set of generators for $H^{*,*}(BV_n; \mathbb{F}_{2})$ as a module over the mod $2$ motivic Steenrod algebra. Kameko demonstrated the failure of the motivic Peterson conjecture by constructing non-hit monomials $z_k$ in degree $d = k + 2d_1$. His analysis involves a distinguished summand in the quotient $N_n = M_n / (τ)$ spanned by monotone translates of $z_k$. In this paper, we isolate the local top-layer content of this summand before quotienting by hit elements. We construct a linear projection $\vartheta: N_n^{d,*} \longrightarrow V$ onto the $M_1$-summand $V$ and define a parity functional $\varepsilon: V \to \mathbb{F}_{2}$. We prove that the local image of hit elements is exactly the parity-zero hyperplane $\ker(\varepsilon)$, inducing a canonical exact sequence $0 \longrightarrow \vartheta(A^{\sharp}_{+}(N_n) \cap N_n^{d,*}) \longrightarrow V \xrightarrow{\ \varepsilon \ } \mathbb{F}_{2} \longrightarrow 0$. This provides a complete classification: an element is locally hit if and only if its $M_1$-component has even parity. Consequently, every odd-parity linear combination of the monotone translates of $z_k$ survives as a non-zero class in the motivic hit quotient. Furthermore, we provide a binary calculation verifying $β(d) > n$ for the infinite family $n = 2^r + 1, k = n - 4$ ($r \geq 5$), yielding new counterexamples distinct from Kameko's $k = n - 3$ family. These results are invariant under base change to any algebraically closed field of characteristic $0$.