Fibonacci Quarterly Open Problems Collection

**2026f** **Title:** Self-Contained Matrix Solutions, Sharp Extensions, and Complete Limit Problems for Three Elementary Fibonacci Problems **Abstract:** We solve three elementary Fibonacci–Lucas problems using the companion matrix of the Fibonacci recurrence. The first reduces to two finite telescopes via Cassini's identity and a Fibonacci–Lucas product identity. The second constructs consecutive Fibonacci labels on a triangular grid by partitioning it into edge-stars; the construction works exactly when the side length is divisible by three. The third reduces to a sharp cyclic inequality via a denominator estimate and Cauchy's inequality. Each solution is then extended to a complete theory: all product tails are explicit, the grid construction is characterized by a bipartite reachability criterion, and the inequality is embedded in a universal variational theorem for homogeneous kernels. All proofs are self-contained and use no external results. **Keywords:** Fibonacci number, Lucas number, companion matrix, Cassini identity, triangular grid, edge-star construction, reachability criterion, homogeneous kernel, sharp inequality --- **2026g** **Title:** Matrix Orders, Exact Pisano-Period Counts, and a Complete Integral-Matrix Spectrum Theorem **Abstract:** We give a self-contained matrix solution of a Pisano-period counting problem. Using the Fibonacci companion matrix, the Pisano period is identified as the order of the matrix modulo n, which immediately proves finiteness of every exact-period set. The counting function is then obtained by Möbius inversion, and all possible periods are completely classified. The local structure is resolved by the Chinese remainder theorem and prime-power lifting. The entire argument is then extended to an arbitrary unimodular integer matrix: the content of A^s-I gives the universal modulus on which the order of A divides s; infinite-order matrices have finite exact-order sets governed by a content sieve, while finite-order matrices have a cofinite top exact-order set and finite lower sets. The Fibonacci companion matrix is recovered as the special case whose spectrum is explicitly computed. **Keywords:** Pisano period, Fibonacci number, Lucas number, companion matrix, matrix order, Möbius inversion, Chinese remainder theorem, prime-power lifting, integral matrix, spectrum theorem --- **2026h** **Title:** Self-Contained Matrix–Eigenvalue Solutions for Five Advanced Fibonacci Quarterly Problems H-951–H-955 **Abstract:** We solve five advanced Fibonacci–Lucas problems using a common matrix–eigenvalue method. The first problem reduces to a splitting identity that gives three telescoping series; the second is resolved by diagonalizing the companion matrices of second-order recurrences; the third is a genuine rational orbit telescope after Binet substitution; the fourth follows from a convex logarithmic inequality; and the fifth uses the differentiated cotangent generating function. Each mechanism is then extended to its natural limit: universal finite-boundary telescopes, a complete spectral trichotomy, a sharp entropy-cone inequality, a full zeta–Binet transform, and a complete rational q-coboundary classification. All definitions, convergence checks, and telescoping steps are included; no external result is invoked. **Keywords:** Fibonacci number, Lucas number, companion matrix, eigenvalue method, hyperbolic function, reciprocal square, Riemann zeta function, zeta–Binet transform, rational coboundary, telescoping sum