The Coprime Level of Every Prime Is Effectively Bounded: A Universal Hierarchy from {2, 3}

We define the coprime level L(p) of a prime: the number of recursive steps needed to express p as a smooth number plus a coprime residue, starting from the base prime set {2,3}. We prove L(p) = O(log* p), with effective bounds: • Unconditionally, L(p) ≤ 4 for all primes p ≤ 10^(3.67 × 10^11).• Assuming p_4* > P_12 (Conjecture C3), L(p) ≤ 4 for all primes p ≤ 10^(10^13). The proof rests on a coprime recurrence Rₖ₊₁ ≤ C_k · R_k^φ(Pₖ₊₁) and the enlarged-modulus construction, with Cunningham chain rigidity neutralizing the principal Erdős-Selfridge obstacle. The level is nevertheless unbounded (sup_p L(p) = ∞), but each new level first appears at a scale requiring an exponential tower one step taller. Direct computation through 10^12 finds L(p) ≤ 3 for all 37,607,912,016 primes in that range. This is v2 of the prior deposit "Coprime Level of Primes Is O(log log p)". Main change: bound sharpened from O(log log p) to O(log* p) with explicit two-tier effective bounds; Cunningham obstruction resolved via matching-side rigidity.