Explicit cohit bases in the five-variable degrees $3\cdot2^s-3$

The Peterson hit problem, which asks for a minimal generating set of the polynomial algebra over the mod-$2$ Steenrod algebra, remains notoriously open for $k \\geq 5$ variables. A fundamental reduction strategy narrows the problem to generic degree families $n$ satisfying the condition $\\mu(n) < k$, where $\\mu(n)$ denotes the minimum number of summands of the form $2^u-1$ required to express $n$. In this paper, we investigate the five-variable case ($k=5$) for the degree family $n_s=3\\cdot2^s-3$, where $\\mu(n_1) = 1 < 5$ and $\\mu(n_s)=3 < 5$ for any $s > 1$. We construct explicit monomial representatives for these cohit spaces, with particular attention to the exceptional parameters $s=4,5,6$ where traditional inductive bounds fail. To overcome this, we introduce a global geometric restriction framework. By evaluating restrictions to all $155$ three-dimensional subspaces of $\\mathbb F_2^5$ and applying iterated Kameko maps, we exploit invariant idempotent structures to reduce the kernel independence problem to a single invertible $1085\\times1085$ matrix for all $s\\geq5$. This invariant matrix specifies independent kernel representatives and an ordered selection of admissible monomials, decoupling the proof from the increasing dimension of the algebra. Supported by exact Steenrod-image algebraic certificates, this framework rigorously establishes dimensions $1731$, $2511$, and $2791$ in degrees $45$, $93$, and $189$, respectively. The stable value $2790$ for $s\\geq7$ was previously stated by Nguyen Sum; our restriction-matrix approach provides an explicit coordinate realization of the representatives in this range. Furthermore, in the highly degenerate degree $45$ ($s=4$), where the global restriction has rank $1054$ on the $1090$-dimensional Kameko kernel, we systematically construct its $36$-dimensional defect kernel and the explicit quotient-coordinate functionals which detect it. The monomial representatives, source polynomials, and finite rank certificates are provided with the computational data. See links:https://doi.org/10.5281/zenodo.22763404 and https://doi.org/10.5281/zenodo.22767609