Strict spectral supersaturation for cliques: extremal graphs and sharp thresholds

For every fixed $r\ge 3$ and all sufficiently large $n$, we determine the largest adjacency spectral radius of an $n$-vertex graph with fewer than $q{c}_r(n)$ copies of $K_{r+1}$, where $1\le q<n\mathrm{/}r$. Here $c_r(n)$ is the number of copies created by adding one edge to a largest part of the Turán graph. We also determine all extremal graphs. In most cases the extremal graph is obtained from an almost balanced complete multipartite graph by adding a star in one part. Two small values of $q$ require separate constructions, and an additional transition occurs when $r=3$ and $n\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$. Under the non-strict constraint, the unique extremal graph is obtained by adding a $q$-edge star to a largest part of the Turán graph. We determine the difference between the strict and non-strict values and prove that the sharp matching threshold is $\sqrt{2}(r-1)\mathrm{/}r$. The proof treats separately the ranges $q=o(n)$, $q\mathrm{/}m\to \gamma \in (0,1)$, and $q\mathrm{/}m\to 1$.