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

For an integer k 3, let (n) denote the n-th k-gonal number. We organize the consecutive gnomon intervals ( (n), (n+1) ] of all regular polygons into a doubly indexed spatial matrix to analyze the distribution of almost-primes (P₂). 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 0. 98 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 10^65. Through topological inheritance, this explicitly secures a P₂ 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.