The Unique Increasing Composite and the Global Dynamics of a Prime Factor Sum Map

Let f (n) be the arithmetic function defined as the sum of the distinct prime factors of n, plus the total number of prime factors (counted with multiplicity). We prove that among all composite integers n ≥ 5, the only value satisfying f (n) ≥ n is n = 6. All other composite integers strictly decrease under iteration of f. For primes p ≥ 7, two iterations satisfyf (2) (p) ≤ p − 2, so primes cannot generate nontrivial cycles except through the special value 6. We show that the sequence inevitably decreases until it hits theboundary of 5, where the unique increasing property of 6 acts as a barrier against further descent. This definitively forces the sequence into the cycle 5, 6, 7, 8, excluding any other nontrivial periodic orbit for n ≥ 5