Almost-Primes P2 in the Gnomons of Regular Polygons: Explicit Bounding via a Transposed Matrix Structure

For an integer $k \ge 3$, let $\Pk(n)$ denote the $n$-th $k$-gonal number. We organize the consecutive gnomon intervals $\bigl(\Pk(n), \Pk(n+1)\bigr]$ of all regular polygons into a doubly indexed spatial matrix to analyze the distribution of almost-primes ($P_2$). While the vertical traversal of this matrix yields intervals of length $O(X^{1/2})$---traditionally rendering explicit sieve thresholds ineffective---we demonstrate that the required analytical threshold for existence strictly monotonically decreases as one descends the columns. By executing a strictly optimized explicit weighted linear sieve along the first row (width $O(X^{2/3})$) and replacing asymptotic error approximations with an exact finite Buchstab descent, we first establish a direct unconditional bound of $n \ge 0.98 \times 10^{130}$ for the classical Legendre interval. To circumvent this astronomical requirement, we evaluate the sieve on our spatial matrix to establish a rigorously optimized absolute bound of $j \approx 10^{65}$. Through topological inheritance, this explicitly secures a $P_2$ element in continuous discrete intervals of size $O(X^{1/2})$ for specific polynomials, drastically compressing the theoretical integer spectrum and bypassing the degenerate analytical bounds.