Gaussian Free-Shear Geodesics and Fourier Contact Thresholds

For the intrinsic Fourier length metric on Gaussian probability laws, we study a genuinely coupled two-dimensional family where correlation shear changes both the covariance and the mean. An exact mixed tangent formula reveals flat directions of the Fourier speed. Small nontrivial shears preserve the scalar geodesic length, and a Gaussian channel produces a finite global plateau whose threshold is characterized by a one-dimensional convex control problem. We give a complete absolutely continuous formulation of this reduction and an analytic asymptotic expansion of the threshold through its fifth-order coefficient. A scalar support-flow equation and an optimality-gap identity expose the frequency-contact mechanism. The result concerns intrinsic Gaussian paths, not a dependence measure comparing a joint law to a product of its marginals.