Matrix–Eigenvalue Solutions and Complete Extensions for Three Advanced Fibonacci–Lucas Problems

We provide self-contained matrix–eigenvalue solutions of three advanced Fibonacci–Lucas problems. Each solution is then extended to a complete structural theory: the first alternating series is generalized to all nonzero integer companion orbits, the second convolution is embedded in a full one-parameter semigroup with explicit Gegenbauer representation, and the third reciprocal-square identity is lifted to arbitrary nondegenerate Lucas sequences with a rigidity theorem for its geometric weight. The three mechanisms are then unified into a complete rational coboundary theory for companion orbits: Laurent polynomial coboundaries are classified by orbit masses, rational coboundaries decompose into canonical endpoint bridge blocks, and the telescoping identity in the first problem is recovered as a universal two-step cluster bridge. All proofs are self-contained and use no external results.