Real-analytic finitely forcible kernels

Lovász and Szegedy asked whether a nonconstant continuous, or even smooth, finitely forcible kernel exists. For every $0<λ\leq1/128$, we construct an explicit nonconstant real-analytic kernel $W_λ$ taking values in $(1/4,7/8)$. A single finite family of simple graphs, independent of $λ$, forces every $W_λ$ among all bounded symmetric real-valued kernels on $[0,1]^2$.