Curvature-Controlled Infrared Regularization of Crystalline Membranes

We show that negative Gaussian curvature regularizes the infrared sector of crystalline membranes. In a covariant formulation of embedded elasticity, the Green strain contains a symmetry-required linear coupling between the normal phonon field and extrinsic curvature. Integrating out the in-plane phonons converts this coupling into a finite quadratic contribution to the inverse flexural response. Anomalous roughening is thereby replaced by curvature-controlled saturation, and the mechanism survives on minimal hyperbolic patches. Hyperbolic geometry preempts anomalous elasticity before the flat infrared regime is reached, implying the absence of a crumpling phase at the harmonic level. The same Gaussian-order coupling admits sound propagation in the infrared.