Exactly Three Single-Amplification Cycles in GCD-Augmented Fibonacci Recurrences

The integer recurrence C(n) = C(n−1) + C(n−2) + gcd(C(n−1), C(n−2)) reduces, after division by the gcd of consecutive terms, to a map on coprime cofactor pairs. A step of this map is an amplification when the gcd grows. We determine all cycles of the map with exactly one amplification per period. There are exactly three: the fixed point (1, 2), the 2-cycle (3, 5) → (5, 9), and a new 4-cycle (29, 47) → (47, 77) → (77, 125) → (125, 203). Their cycle multipliers are 2, 3 and 7. A sequence whose cofactor pairs reach one of these cycles therefore eventually grows by exactly 2 per step, 3 per two steps or 7 per four steps. The proof shifts the cofactors by one, which makes the dynamics between amplifications linear in the Fibonacci matrix. The unique solution of the resulting linear system has an explicit Fibonacci norm, which forces the cycle multiplier to be a Lucas number or one more than a Lucas number; gcd identities for Fibonacci and Lucas numbers then exclude every period except 1, 2 and 4. As consequences, we give an algebraic proof that the period-2 orbit is unique, and we correct a claim in the author’s earlier work that no other periodic orbits exist. We also show that the seeds (1, 9), (1, 15) and (1, 27), described as non-geometric in the author’s earlier work, grow by exactly 7 every four steps. For cycles with any number of amplifications, the same linearization gives a cycle equation and shows that the cycle multiplier exceeds φ^q, where q is the period and φ the golden ratio. In a census of 8,771 coprime seeds, the new cycle attracts 35.9%.