Near-Cramér Sato–Tate Equidistribution in Growing Arithmetic Progressions

## Overview Let \(E/ Q\) be a fixed non-CM elliptic curve, and let \(θ_p∈[0,π]\) denote its Frobenius angle at a prime of good reduction \(p\). This paper studies Sato–Tate equidistribution for primes constrained simultaneously by a literal moving hard interval \[X 1\), take \[h=√ X(log X)^A.\] If \[φ(q)=o\!((log X)A-1),\] then, under the corresponding finite twisted-symmetric-power GRH packet, \[I⊆[0,π]|μ^X,h;q,a(I)-μST(I)|E,A[{φ(q)}{(log X)A-1}(1+log{(log X)A-1}{φ(q)})]1/2.\] For fixed \(q\), this recovers the near-square-root discrepancy rate \[{√loglog X}{(log X)(A-1)/2}.\] ## Position in the literature Effective Sato–Tate theorems in arithmetic progressions, Sato–Tate results in Hoheisel-type short intervals, and GRH Cramér-scale prime localization in arithmetic progressions are all known separately. The contribution here is the combination of a literal moving hard prime window near the arithmetic-progression Cramér regime, a growing modulus, uniform Sato–Tate angular discrepancy, and finite twisted-symmetric-power analytic depth. No unconditional priority claim is made for this precise combination. ## Version Version v1.0.