Vanishing-Scale Sobolev Equidistribution for Rankin–Selberg-Weighted Satake Measures

We study the local Satake parameters at a fixed split prime p in a prime-level family of holomorphic newforms, equipped with Petersson harmonic weights further tilted by the central Rankin–Selberg values L(1/2, f × g), where g is a fixed self-dual dihedral cusp form associated with a nonquadratic class-group character of an imaginary quadratic field. (1-p⁻¹)Lₚ(1/2, s_θ × g). After heat regularization at the vanishing scale tq = A/log q, we prove strong Sobolev convergence of the regularized relative density Hq. More precisely, if Kq = κlog q/log p , 0 < κ < 1, then, for A sufficiently large, there exists η > 0 such that Hq - 1_H³cent(SU(2)) (log q)⁻¹ + q-η. Consequently, the relative entropy, relative Fisher information, and a growing finite-mode discrepancy all decay quantitatively to zero. The proof isolates a general analytic mechanism: quantitative Fourier or character control on an expanding representation window, together with vanishing-scale spectral smoothing, yields strong Sobolev equidistribution and nonlinear energy-functional control. This paper provides the forward model case in a broader program relating arithmetic cancellation in automorphic families to energy functionals of their local spectral measures.