Sieve Dimension Zero and a Structural Reduction of Artin's Conjecture for a = 10

We establish that the sieve for Artin's primitive root conjecture has dimension κ = 0, eliminating the parity barrier and reducing the analytic requirements to their minimum. The sieve lower bound yields S (A, z) ≥ 0. 236·π (x) unconditionally. The exceptional Siegel zero is controlled unconditionally via Goldfeld–Gross–Zagier. The surplus is decomposed into a controlled range (closed by effective Chebotarev and the linear disjointness of the splitting fields Kq) and a residual range, which we reduce to a single mean-value hypothesis on primitive prime divisors of cyclotomic polynomials Φₖ (10). This hypothesis is a weakening of a conjecture of Erdős studied by Murty (Mathematika, 2012) and is verified computationally for k ≤ 10⁴ (mean ω* (k) = 0. 67). Third paper in the trilogy C10–C11–C12 on the multiplicative order of 10 modulo primes.