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 $L(1/2, f \times g)$, where $g$ is a fixed self-dual dihedral cusp form associated with a nonquadratic class-group character of an imaginary quadratic field. $$(1-p^{-1})L_p(1/2, s_\theta \times g).$$ After heat regularization at the vanishing scale $$t_q = \frac{A}{\log q},$$ we prove strong Sobolev convergence of the regularized relative density $H_q$. More precisely, if $$K_q = \left\lfloor \kappa\frac{\log q}{\log p} \right\rfloor, \qquad 0 < \kappa < 1,$$ then, for $A$ sufficiently large, there exists $\eta > 0$ such that $$\Vert{}H_q - 1\Vert{}_{H^3_{\mathrm{cent}}(SU(2))} \ll (\log q)^{-1} + q^{-\eta}.$$ 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.