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The $4 n^2$-inequality for smooth points plays an important role in theproofs of birational (super)rigidity. The main aim of this paper is togeneralize such an inequality to terminal singular points of type $cA_1$, andobtain a $2 n^2$-inequality for $cA_1$ points. As applications, we provebirational (super)rigidity of sextic double solids, many other prime Fano3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$over $\mathbb{P}^1$ satisfying the $K^2$-condition, all of which have at mostterminal $cA_1$ singularities and terminal quotient singularities. These givefirst examples of birationally (super)rigid Fano 3-folds and del Pezzofibrations admitting a $cA_1$ point which is not an ordinary double point.

$2 n^2$-inequality for $cA_1$ points and applications to birational rigidity