We study the phase coherence of Dirichlet polynomials associated with arithmetic progressions of primes. For a k-term prime progression (p, p+d, ..., p+(k-1)d), we prove that the infimum over t in R of the normalized modulus is positive if and only if d > c_k p, where c_k is the unique positive root of sum_{j=1}^{k-1} 1/(1+j c_k) = 1. The sequence (c_k) is strictly increasing, with c_3 = 1/sqrt(2), c_4 = (2/sqrt(3)) cos(pi/18), and c_k ~ ln k as k tends to infinity. Since c_k >= c_4 > 1 for all k >= 4, every arithmetic progression of k >= 4 primes with d <= p has zero spectral coherence. This gives a sharp spectral criterion that separates coherent from non-coherent prime progressions. Numerical verification on more than 1.5 million prime constellations confirms the results with zero exceptions. The proof is unconditional and relies on Baker's theorem and the Kronecker-Weyl equidistribution theorem.