Unconditional Local Sato–Tate in Growing Arithmetic Progressions: Twist-Family Localization Beyond Endpoint Differencing

Let E be a fixed non-CM elliptic curve over the rational numbers Q. We prove an unconditional local Sato–Tate theorem for primes in simultaneously growing arithmetic progressions and moving short intervals. Fix exponents α, β, and γ satisfying 0 < α < 1/3, 0 ≤ β < 1/2 - α, 0 < γ < 1 - 2α - β. Uniformly for interval lengths h between X exp(-cₕ (log X)^γ) and $X/2$, progression moduli q between $2$ and exp(cq (log X)^β), and reduced residue classes a q, the Sato–Tate discrepancy is O((log X)-α). The proof localizes symmetric powers up to degree (log X)^α by combining a Dirichlet-twist-averaged log-free zero-density estimate with a zero-free region for twisted symmetric powers. A uniform short-interval prime number theorem in arithmetic progressions supplies the scalar normalization, including the contribution of a possible exceptional zero. Estimating the twist family before progression normalization avoids the full pointwise-character loss incurred by direct endpoint subtraction from the global estimate and yields the stated three-parameter range.