A Lean 4 Formalization of Scott's \emph{Continuous Lattices} (1972)

We present a complete machine-checked formalization of Dana Scott's landmark 1972 paper \emph{Continuous Lattices} \textbf{[Sco72]}, carried out in Lean 4 against mathlib and including the March 1972 Milner correction in \textbf{[Sco72]} (pp.~135--136). Scott's paper develops a model for $\lambda$-calculus from a topological starting point. He defines \emph{injective} $T_0$-spaces -- those with a strong extension property for continuous maps -- and shows that they are exactly the \emph{continuous lattices}: complete lattices whose Scott topology is determined by the order via the way-below relation ($\ll$). On this foundation he studies projections, retractions, products, function spaces, and inverse limits. The capstone (Theorem 4.4) constructs an inverse limit $D_\infty$ of function-space approximants and proves $D_\infty \cong [D_\infty \to D_\infty]$, yielding a purely mathematical model for Church's untyped $\lambda$-calculus. Our development formalizes \textbf{43 numbered results} from Scott's Sections 1--4 (Propositions, Corollaries, Lemmas, and Theorems), each as a sorry-free Lean theorem, together with supporting infrastructure (step functions, the $\Uparrow a$ basis of Scott opens, Milner's coarser-than-Scott hypothesis, the function-space tower, and the $i_\infty$/$j_\infty$ pair). The formalization is \textbf{classical} (uses \texttt{Classical.choice} transitively) and follows Scott's proof dependency order. Where the Lean proof required choices not visible in the original -- or where dead ends were encountered -- we record detailed notes in Section 5. All proofs check with the standard footprint $\texttt{[propext, Classical.choice, Quot.sound]}$.