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 $\mathbb{Q}$. We prove an unconditional local Sato–Tate theorem for primes in simultaneously growing arithmetic progressions and moving short intervals. Fix exponents $\alpha$, $\beta$, and $\gamma$ satisfying $$0 < \alpha < \frac{1}{3}, \qquad 0 \le \beta < \frac{1}{2} - \alpha, \qquad 0 < \gamma < 1 - 2\alpha - \beta.$$ Uniformly for interval lengths $h$ between $X \exp(-c_h (\log X)^\gamma)$ and $X/2$, progression moduli $q$ between $2$ and $\exp(c_q (\log X)^\beta)$, and reduced residue classes $a \pmod q$, the Sato–Tate discrepancy is $O((\log X)^{-\alpha})$. The proof localizes symmetric powers up to degree $(\log X)^\alpha$ 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.