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

The motivic hit problem asks for a minimal set of generators of $H^{*,*}(BV_n;\mathbb{F}_2)$ as a module over the motivic Steenrod algebra. For the distinguished degrees $d=k+2d_1$ with $d_1=(n-1)(2^k-1)$, Kameko constructed a top layer spanned by the monotone translates of a monomial $z_k$ and showed that the Bockstein image there is contained in the span of pairwise sums of these translates. In this paper we work on the raw degree--$d$ component $N_n^{d,*}$, before quotienting by hit elements. We construct a local projection $\vartheta:N_n^{d,*}\to V$ onto Kameko's $M_1$--summand $V$, together with a parity functional $\varepsilon:V\to\mathbb{F}_2$, and prove that $$\vartheta\bigl(A_+^\sharp(N_n)\cap N_n^{d,*}\bigr)=\ker(\varepsilon).$$ Thus parity exactly describes the local top-layer image of hit elements. In particular, any element with odd local parity is non-hit, so every odd-parity linear combination of the translates of $z_k$ represents a nonzero class in the motivic hit quotient. We also show by a direct binary calculation that if $n=2^r+1$, $k=n-4$, and $r\ge 5$, then $\beta(d)>n$ for $d=(n-1)(2^{k+1}-2)+k$. Combined with Kameko's non-hit theorem, this yields a new infinite family of counterexamples to the motivic Peterson-type conjecture, distinct from Kameko's $k=n-3$ family; our parity criterion strengthens this by showing that every odd-parity linear combination is non-hit in these degrees. Finally, we prove that the local parity criterion and its consequences persist over any algebraically closed field of characteristic $0$.