Fibonacci–Lucas Recurrence-Indexed Quaternionic Unitaries: Channel Convergence, Clifford Proximity, and Controlled Entanglement

We investigate the normalized operator geometry of the mixed Fibonacci–Lucas quaternion Qn = Fn + Lni + Fn+1j + Ln+1k. Standard quaternion representations associate this sequence with single-qubit unitary channels. We first establish a convergence theorem for quaternion sequences with two distinct real characteristic roots and a strictly dominant positive root. It gives an exact average gate infidelity to the limit and sharp projective-step asymptotics, including the degenerate constant-channel case. For the Fibonacci–Lucas assignment, orthogonal Binet coefficient vectors yield exact finite-index formulas with projective steps of order φ−2n and average gate infidelities of order φ−4n. Strict comparison of the five projective Clifford rotation classes proves that the limiting unitary has a unique nearest projective Clifford, the cyclic Pauli-axis permutation, with average gate fidelity 115+5/90≈0.884408. For the controlled family, product-input maximization gives concurrence 1−an2, while independent Haar averaging gives linear-entropy entangling power 21−an2/9. Both admit explicit algebraic limits. Reproducible matrix and high-precision scalar calculations verify these results. The analysis supplies exact convergence, finite-group, and entanglement benchmarks for a recurrence-indexed family without assuming Hamiltonian dynamics.