Coprime Level of Primes Is O (log log p): A Universal Hierarchy from 2, 3

We define the coprime level of a prime p: the number of recursive steps needed to express p as a smooth number plus a coprime residue, starting from the alphabet 2, 3. We prove a quadratic recurrence for the ±1 encoding, upgraded to a super-exponential recurrence R₊+₁ ≤ Cₖ · Rₖ^φ (P₊+₁) under the coprime encoding via negative association (Dubhashi-Ranjan). Combined with Cunningham immunity and a computational base (all 50, 847, 534 primes below 10⁹ have coprime level ≤ 3), this gives the universal bound L (p) = O (log log p) for all primes. The paper also establishes congruence obstructions, connects coprime residues to the Frobenius classification in cyclotomic fields, and shows that Connes' optimal set 2, 3, 5, 7, 11, 13 consists of the smallest Level 0+1 primes.