Weighted Fibonacci Identities Derived from the Matrix $S_4^n(F_s, F_{s+1})$

This study presents a matrix-algebraic framework for deriving weighted Fibonacci identities from the $4 \times 4$ matrix $S_4^n(F_s, F_{s+1})$. By matching the closed-form entries of this matrix with the corresponding binomial expansions obtained from basis decomposition and structural convolution kernels, a new class of combinatorial identities is derived. These identities relate weighted binomial sums to Fibonacci subsequences of the form $F_{(s+1)n+k}$ for $k \in \{-1,0,1\}$. The method exploits the sparsity and commutativity of the basis matrices to produce parity-dependent convolution operators in a persymmetric setting. The resulting framework provides a systematic way to generate weighted Fibonacci identities from structured matrix relations.