COFACTOR DYNAMICS AND PELL EQUATIONS IN GCD-AUGMENTED FIBONACCI RECURRENCES: THE ALGEBRAIC STRUCTURE OF GEOMETRIC STABILIZATION

We investigate the algebraic structure underlying geometric stabilization in GCD-augmented Fibonacci recurrences Cp(n) = Cp(n−1) + Cp(n−2) + gcd(Cp(n−1), Cp(n−2)), resolving Open Problem 4 from Hawarey (2026). We establish that the binary operation f(a,b) = a + b + gcd(a,b) admits the cofactor factorization f(a,b) = gcd(a,b) · (α + β + 1), which decomposes the recurrence into scale factor dynamics and a cofactor map T(α,β) = (β, α+β+1). When consecutive terms are coprime, this map coincides with the Leonardo number recurrence (ratio φ). We prove that the cofactor pair (1,2) is the unique period-1 orbit (scaling factor 2, governing even seeds) and that (3,5) is the unique coprime period-2 orbit (scaling factor 3, governing odd seeds coprime to 3), by reducing the existence condition to the generalized Pell equation x² − 5y² = 4 — the norm equation in the Fibonacci field Q(√5). The discriminant 5 thus plays a dual role: producing the irrational golden ratio φ = (1+√5)/2 in classical Fibonacci theory and the rational geometric ratio 3 in GCD-Fibonacci theory, through different representations of the same algebraic number field. We formalize the phase transition from Leonardo dynamics to geometric lock-in and provide a unified classification of all three behavioral regimes. All results are verified computationally for seeds p = 1 through 49.