Escape from Geometric Lock-in in GCD-Augmented Fibonacci Recurrences

We study the recurrence C(n) = C(n−1) + C(n−2) + gcd(C(n−1), C(n−2)) through its states, the coprime pairs obtained by dividing consecutive terms by their gcd. A step is an amplification when this gcd grows, and once the states reach a cycle the sequence grows exactly geometrically. The author’s earlier work left open whether every seed reaches a cycle. We show that it does not: the amplifications can stop, and this escape can be certified by a finite computation. Between amplifications the shifted states evolve by the Fibonacci matrix and their norm N alternates in sign, so a further amplification can only come from a prime dividing N² − 1, and finitely many orbit computations modulo these primes decide, once N² − 1 is factored, whether it ever comes. The seed (11, 5) is never amplified: C(n) = 6L_n−1 − 1 for all n, with L_n the Lucas numbers. The seed (1, 135) joins this orbit after three amplifications, and infinitely many seeds escape. Every escaping orbit has N divisible by 30, and its growth ratio tends to the golden ratio. Among the 24,463 coprime seeds with entries at most 200, at least 956 escape (907 certified within 1,200 steps and 49 by an extended check), 77% of them on the orbit of (11, 5), while 2,320 remain undecided.