On $\mathbb{A}$-generators of the cohomology $H^{\ast }(V^{\oplus 5})=\mathbb{Z}\mathrm{/}2[u_1,\dots ,u_5]$ and the cohomological transfer of rank 5

Computing the cohomology of the 2-primary Steenrod algebra $\mathbb{A}$ is a central problem in algebraic topology, as it forms the $E_2$-term of the Adams spectral sequence converging to the stable homotopy groups of spheres. The Singer cohomological transfer, ${\phi }_n$, is a key homomorphism for characterizing this cohomology. Singer conjectured that ${\phi }_n$ is always a monomorphism. The Singer transfer is closely linked to the Peterson hit problem, which seeks a minimal generating set for the $\mathbb{A}$-module $H^{\ast }(V^{\oplus n})=\mathbb{Z}\mathrm{/}2[u_1,\dots ,u_n]$, also unsolved for $n\ge 5$. In this paper, we study the hit problem for $H^{\ast }(V^{\oplus 5})$ and verify Singer’s conjecture for the case $n=5$ in the general degree $d=2^{t+5}+2^{t+2}+2^{t+1}-5$ for any non-negative integer $t$. We demonstrate that the Singer cohomological transfer is an isomorphism for $n=5$ in degree $d$. This provides a positive answer to Singer’s conjecture in these specific cases. The appendix provides our new algorithm implemented on the computer algebra system OSCAR, through which all principal results of this paper have been completely verified.