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 Rₖ₊₁ ≤ C_k · R_k^{φ(Pₖ₊₁)} under the coprime encoding via negative association (Dubhashi-Ranjan). Combined with Cunningham immunity and a computational base (all 50,847,534 primes below 10^9 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.