Matrix Solutions, Extensions, and Final Problems for Three Advanced Fibonacci–Lucas Identities

We provide self-contained matrix solutions of three advanced Fibonacci–Lucas problems using the companion matrix of the Fibonacci recurrence. The first is a bilateral arctangent identity evaluated via branch correction and alternating cancellation; the second is an odd-power polynomial identity whose coefficients are forced by Cassini's relation; the third is a reciprocal series evaluated by diagonalizing the companion matrix. Each solution is then extended to a complete finite theory: a finite-shift correlation formula with even-shift cancellation, a quadratic-product lower-envelope theorem with the full Lucas boundary classification, and a rational additive q-coboundary criterion. A final sequence-space closure upgrades these mechanisms to ℓ¹ shift kernels, locally uniform nonnegative infinite series, and absolutely summable one-sided first differences. All proofs are elementary and self-contained.