The rank-five Peterson hit problem, the fifth Singer transfer, and a geometric generator in unoriented cobordism

The Peterson hit problem seeks a minimal set of generators for the polynomial algebra $P_s = \mathbb{F}_2[x_1,\dots,x_s]$ as an unstable module over the mod-2 Steenrod algebra $\mathcal{A}$. For rank five, general admissible bases fail, and the interplay between Kameko periodicity and modular invariants becomes computationally complex. In this paper, we study the rank-five cohit module in the generic family $N_d = 27\cdot 2^d - 5$. Exact sparse elimination in degree $49$ processes $292825$ monomials, yielding a hit rank of $289969$ and a cohit dimension of $2856$. We determine the exact weight summands and prove that the weight-$(3,3,2,2,1)$ summand is exactly the kernel of Kameko's operation, with dimension $1891$. These exact values systematically correct the corresponding rank-five kernel and dimension assertions in Nguyen Khac Tin's previous paper.
An exact invariant calculation shows that the general linear group invariants in degree $49$ form a one-dimensional space generated by a $283$-term polynomial, and we prove that the fifth Singer cohomological transfer is an isomorphism in this family. Geometrically, the Hilbert-Poincare series of the unoriented cobordism ring gives the dimension of the degree-$49$ cobordism group as $5692$. We prove that the Milnor hypersurface $H_{2,48} \subset \mathbb{R}P^2 \times \mathbb{R}P^{48}$ represents the unique nonzero indecomposable class by computing a tangential Stiefel-Whitney number, providing an explicit geometric generator. However, the evident map from $H_{2,48}$ to the classifying space $B(\mathbb{Z}/2)^5$ sends its fundamental class to a homology class with nonzero $Sq^2_*$. Consequently, this geometric generator cannot be identified with the functional dual of the algebraic invariant, establishing a precise boundary between the Steenrod-theoretic invariant line and the geometric cobordism generator.