An entropy bridge from weighted to spectral Turán theorems

We establish an entropy bridge, and then use it to prove that for any color-critical graph $F$ with chromatic number $χ(F)=r+1\ge 3$, there exists a constant $λ_0=λ_0(F)$ such that if $G$ is an $F$-free graph with $λ(G)\ge λ_0$, then for every $\ell\ge 1$ with $(r,\ell )\neq (2,2)$, \[ λ^\ell (G) \l