 Solution

The single-field inflation squeezed-limit consistency relation for the comoving curvature perturbation isEquivalently, $B_{\mathcal{R}}/P_{\mathcal{R}}(q)=-(3+k{\partial }_k)P_{\mathcal{R}}(k)$. The scalar shift in parts a--c is an internal symmetry and leaves the hard fields unchanged, so its right-hand side vanishes. A long adiabatic curvature mode instead acts as a spatial dilation of the hard coordinates, and its response is the scale dependence measured by the scalar spectral index.