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

The motivic hit problem seeks a minimal set of generators for $H^{\ast ,\ast }(B{V}_n;{\mathbb{F}}_2)$ 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 $N_n=M_n\mathrm{/}(\tau )$ 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 $\vartheta :N_n^{d,\ast }⟶V$ onto the $M_1$-summand $V$ and define a parity functional $\epsilon :V\to {\mathbb{F}}_2$. We prove that the local image of hit elements is exactly the parity-zero hyperplane $\ker (\epsilon )$, inducing a canonical exact sequence $0⟶\vartheta (A_{+}^{\mathrm{♯}}(N_n)\cap N_n^{d,\ast })⟶V\stackrel{\epsilon }{\to }{\mathbb{F}}_2⟶0$. 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 $\beta (d)>n$ for the infinite family $n=2^r+1,k=n-4$ ($r\ge 5$), 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 $0$.