Self-Contained Matrix–Sequence Solutions and Universal Extensions of Fibonacci Quarterly Problems B-1356 and B-1357

We solve two elementary Fibonacci–Lucas problems from The Fibonacci Quarterly using the companion matrix method. Both solutions are then extended to their natural general forms. The Padovan–Lucas alternating series is shown to be a specialization of a trace-index boundary telescope for any determinant-one matrix orbit. The exponential inequalities are shown to be special cases of a sharp tensor-square inequality for observable sequences on any real matrix state system, with an optimal universal constant. The paper is self-contained and proves all matrix identities, convergence estimates, and telescoping identities from first principles.