A Vertical Zero-Free Region for Families of Dirichlet L-Functions

This paper establishes a uniform vertical zero-free region for families of Dirichlet L- functions by combining a forcing mechanism developed in earlier work with classical dispersion theorems. The proof combines three components: (1) a forcing mechanism from the explicit formula for hypothetical zeros, yielding energy concentration ≫X2δ/δ; (2) finite-rank projection removing low-prime common modes; (3) a mesoscopic ceiling ≪T1/2 via character sum dispersion controlled by BV/BDH. The contradiction between forcing and ceiling at δ= 5/16 establishes ℜ(ρ) ≤13/16 for zeros ρ in the family. The exponent 13/16 arises from the sharp unconditional range of the Bombieri-Vinogradov theorem. The restriction to moduli q ≤T2/5(log T)−B cannot be removed by the present method. Extension beyond ℜ(s) ≤13/16 would require either a stronger dispersion theorem or a forcing mechanism with super-quadratic amplification.