Discreteness of sets generated by fixed-parameter extrapolation

Starting from $0$ and $1$, repeatedly apply the operation $(a,b)\mapsto(1-\lambda)a+\lambda b$, with a fixed complex parameter $\lambda$. We prove that the resulting set is discrete exactly when $\lambda$ is an algebraic integer whose other conjugates, apart from $\lambda$ and $\overline\lambda$, are real and lie in $(0,1)$. This gives the converse to the known sufficient condition for separated extrapolation sets. The proof adjusts integer polynomials within a residue class while preserving their values at $\lambda$. If there is an additional conjugate outside $(0,1)$, density of the corresponding field projections then produces nonzero generated points arbitrarily close to $0$.