2-adic Twist Determinant Theorem for Weighted GCD Matrices: Introducing the Sakib Index and a Parity-Twisted Smith-Type Factorization

A classical theorem of H. J. S. Smith states that the determinant of the n × n GCD matrix (gcd(i, j))1≤i,j≤n equals Qn k=1 φ(k). Building on the standard meet/poset (Smith–Wilf–Lindstr¨om) determinant factorization framework for ma- trices of the form (f (gcd(i, j))), we introduce a 2-adically weighted family A(c) n = cv2(gcd(i,j)) gcd(i, j)
1≤i,j≤n, and derive a closed-form determinant factorization depending only on the 2-adic valuation counts up to n. The resulting correction factor (normalized by Qn k=1 φ(k)) is named the Sakib Index. A key parity-twisted specialization yields the explicit factor SI(n) = (−1)⌈n/2⌉ 3⌊(n+2)/4⌋, producing an unexpectedly simple 2-adic amplification of Smith’s determinant. Ten data-driven figures (OEIS-based and exact integer computations) and six concep- tual diagrams are provided.