Relative Duality and Structural Reductions for the Symmetric Hit Problem in Four Variables

The symmetric hit problem asks whether the orbit sum of a monomial that is hit in $P(n)={\mathbf{F}}_2[x_1,\dots ,x_n]$ is hit in the invariant subalgebra $B(n)=P(n{)}^{{\mathrm{\Sigma }}_n}\cong H^{\ast }(BO(n);{\mathbf{F}}_2)$. The conjecture is known for $n\le 3$, while for $n\ge 4$ nontrivial stabilizers impose equivariant constraints. Let ${\bar{\iota }}_d:Q_d(B(n))\to Q_d(P(n))$ be induced by inclusion, and define ${\mathcal{C}}_d(n)=\ker ({\bar{\iota }}_d)$. We prove a natural isomorphism $${\mathcal{C}}_d(n{)}^{\ast }\cong \frac{K_d(\mathrm{D}\mathrm{S}(n))}{{\rho }_d(K_d(n))},$$ where $\mathrm{D}\mathrm{S}(n)$ is the dual symmetric algebra and ${\rho }_d$ is restriction from the ordinary divided-power Steenrod kernel. Thus a symmetric polynomial already hit in $P(n)$ can represent a relative obstruction only through a symmetric Steenrod-kernel functional modulo those extending from $P(n)$. Combining this duality with a stabilizer-compatible coset-parity formula, we reduce the four-variable conjecture to a uniform local descent hypothesis for monomials with repeated exponents. We also prove an unconditional family. For every $t\ge 0$, the ordered monomial $x^{{\lambda }_t}$ and the monomial symmetric function $m_{{\lambda }_t}$, where $${\lambda }_t=(10\cdot 2^t-1,10\cdot 2^t-1,2^{t+2}-1,2^{t+1}-1),$$ are hit in $P(4)$ and $B(4)$, respectively, although $\omega ({\lambda }_t){>}_l{\omega }_{\min }(\mathrm{\mid }{\lambda }_t\mathrm{\mid })$. The base identity $$m_{9,9,3,1}={\mathrm{S}\mathrm{q}}^8(m_{5,5,3,1})+{\mathrm{S}\mathrm{q}}^1(m_{10,6,4,1})$$ follows from an exact Lucas–Cartan enumeration, and the resulting zero cohit classes propagate under the ordinary and symmetric Kameko isomorphisms. Exact linear algebra gives ${\mathcal{C}}_8(4)={\mathcal{C}}_{12}(4)={\mathcal{C}}_{14}(4)={\mathcal{C}}_{22}(4)=0$. Finally, relative Kameko stability implies ${\mathcal{C}}_{26\cdot 2^t-4}(4)=0$ for every $t\ge 0$.