Matrix Solutions, Strengthenings, and Extensions of Three Elementary Fibonacci–Lucas Problems

We solve three elementary Fibonacci–Lucas problems using the companion matrix of the Fibonacci recurrence. Each solution is strengthened beyond the original statement: the infinite product telescopes to a whole seeded family, the rigidity problem reduces to the equality case of Cauchy’s inequality, and both inequalities are sharpened with exact difference formulas. The paper then closes the three natural terminal extensions: rational multiplicative coboundaries are fully classified by a finite divisor equation; Lorentz-cone sum inequalities are proved in full generality, invariant companion cones are classified, and single-orbit containment is characterized by an open-semicircle criterion; and every nonconstant integer trace datum with positive and monotone even traces satisfies the exact pointwise lower envelope min{L₂ₙ, 2ⁿ⁺¹}. No external results are invoked; all definitions and identities are proved from the matrix model.