{
  "abstract": "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 \\(λ\\)-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 \\(λ\\)-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]}\\). The formalization has been accepted into the Palomar registry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.30782",
  "author_total": 1,
  "authors": [
    {
      "name": "Lars Warren Ericson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cgp8v3gfm46a1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.30782",
  "fulltext_url": null,
  "id": "cgp8v3gfm46a1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.30782",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1702|Logic, programming, and type systems"
    ]
  ],
  "openalex": "W7166882893",
  "pdf_url": "https://arxiv.org/pdf/2606.30782",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cgp8v3gfm46a1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2606.30782",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.30782"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A Lean 4 Formalization of Scott's Continuous Lattices (1972)",
  "title_text": "A Lean 4 Formalization of Scott's Continuous Lattices (1972)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/cgp8v3gfm46a1",
  "venue": "arXiv",
  "versions": []
}