A Higher-Order Clique Density Theorem

Reiher's clique density theorem determines the sharp lower envelope for the density of $K_r$ at fixed edge density. We prove a higher-order version in which the prescribed quantity is itself a clique density. For every $3\le s<r$, we determine the minimum possible $K_r$-density among graphons with prescribed $K_s$-density. For $s\ge3$ the constraint is genuinely nonlinear and leaves the edge density undetermined; nevertheless, on the positive range the sharp lower boundary is the classical multipartite edge-to-clique profile, reparametrised by $K_s$-density. We also prove stability on the positive branches of this profile: at every interior point, near extremality forces cut-distance closeness to the corresponding extremal family at the induced edge density.