Joint Sato–Tate Localization near the Cramér Scale in Growing Arithmetic Progressions

Let E₁, E₂ / Q be fixed twist-inequivalent non-CM elliptic curves. This paper proves a conditional joint Sato–Tate theorem for primes constrained simultaneously to a literal moving interval $(X, X+h]$ and a reduced residue class a q, with the modulus $q = q(X)$ allowed to grow. The analytic input is a finite bidegree family of Dirichlet-twisted symmetric-power Rankin–Selberg L-functions L( s, Symᵐ π₁ × (Symⁿ π₂ ⊗ χ^*) ), 0 ≤ m, n ≤ M, under GRH. A two-scale plateau explicit formula, followed by Dirichlet-character recombination before hard-edge removal, yields a finite-depth joint rectangle discrepancy estimate of the form D⁽²⁾X,h;q,a E₁,E₂ 1/M + M² R(2),APM,q(X,h,Δ). At the balanced edge Δ φ(q)√ Xlog X, the natural Cramér-normalized variable is G = h/φ(q)√ Xlog X. Writing η = 1+log G/ G + h/Xlog X, and choosing the angular truncation depth at the scale M η-1/3, one obtains D⁽²⁾X,h;q,a E₁,E₂ η1/3. In particular, for h = √ X(log X)A, A > 1, and φ(q) = o((log X)A-1), the discrepancy satisfies D⁽²⁾X,h;q,a E₁,E₂,A [ φ(q)(log X)A-1 ( 1 + log (log X)A-1φ(q) ) ]1/3. The result combines near-Cramér hard-window localization, an individually growing arithmetic-progression modulus, and genuinely two-dimensional joint angular localization. The Rankin–Selberg conductor and zero-count bookkeeping remains linear in the bidegree dm,n = (m+1)(n+1), so no additional bidegree loss occurs beyond the standard two-dimensional M² truncation cost. This work is a distinct successor to two preceding hard-window Sato–Tate papers by the author. Quantitative Sato–Tate in Near-Square-Root Prime Intervals under Symmetric-Power GRH (Zenodo v3.0, DOI 10.5281/zenodo.22112048) provides the single-curve near-square-root finite-depth localization prototype. Near-Cramér Sato–Tate Equidistribution in Growing Arithmetic Progressions (Zenodo v1.0r1, DOI 10.5281/zenodo.22138400) introduces the growing-modulus arithmetic-progression realization and the recombination-before-hard-edge mechanism. The present paper carries this hard-window framework into the joint Sato–Tate setting through a growing bidegree Rankin–Selberg packet. It is a separate research article and is not a new version of either preceding paper.