Self-Contained Matrix–Eigenvalue Solutions and Universal Extensions of Fibonacci Quarterly Advanced Problems H-946–H-949

**Abstract:** We give self-contained matrix solutions to four advanced problems from *The Fibonacci Quarterly* and then formulate their natural universal extensions. The companion matrix organizes Fibonacci numbers, Lucas numbers, and their one-parameter generalizations. The limits, Diophantine equations, arctangent telescoping, and binomial identities are solved by eigenvalue cancellation, seed-filter equalization, arctangent coboundary classification, and Cayley–Hamilton expansions. The deeper extensions provide a complete Lipschitz-tail criterion, multi-channel filter rigidity, non-abelian group cohomology, matrix Lie product convergence, and Ho–Kalman Hankel realization, connecting the problems to control theory, Lie groups, and signal processing. **Keywords:** Fibonacci number; Lucas number; companion matrix; matrix sequence; seed-filter system; tensor observable; spectral expansion; eigenvalue cancellation; Lipschitz-tail criterion; arctangent coboundary; Lie-group telescope; non-abelian cocycle; control realization; Ho–Kalman realization; balanced Gramian; Cayley–Hamilton theorem; Krylov quotient; Hankel rank