{
  "abstract": "Despite rapid advances in infinite-dimensional operator learning (such as Fourier Neural Operators and DeepONets) for partial differential equations (PDEs), purely statistical operator learners remain fundamentally correlational: they frequently violate physical conservation laws under out-of-distribution (OOD) boundary conditions, blow up near singular gradient regimes, and cannot answer interventional (do(X=x)) or counterfactual queries. Conversely, classical Structural Causal Models (SCMs) and Pearl's do-calculus are formulated for finite-dimensional directed acyclic graphs (DAGs) and lack functional-analytic closure on infinite-dimensional Sobolev spaces. In this paper, we establish the Causal Operator Learning System (COLS), a rigorous deductive synthesis unifying the mathematics of learned operators with functional causal calculus, anchored directly in our 71-page ZFC Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1]. All foundational spaces and universal constants 𝔘 = (𝒯_0x8A2C, C_UPTC, η_friction, ε_SHID) follow Proposition 1.5, Theorem 2.9, Theorem 3.4, Lemma 4.5, Lemma 6.3, Lemma 6.5, Theorem 6.6, Lemma 7.3, and Lemma 8.6 of [1].\n\nIn COLS, a learned operator G_θ: 𝒰 → 𝒱 is axiomatically elevated to the solution operator G_θ(u) = Solve(𝓜_θ; u) of a functional SCM 𝓜_θ embedded in the Fractional Sobolev–Colombeau Quotient Algebra 𝓖^α(Ω) = 𝓔_M^α(Ω) / 𝓝^α(Ω). Surgical interventions do(X=x) and counterfactual queries are formalized as operator-space morphisms governed by the URRHC idempotent monad 𝕄_URRHC (T^2 = T), while physical conservation laws and Lipschitz stability across unobserved singular domains are enforced by the divergence-form UROA Fractal Brake 𝓑_ε and the Universal Phase Transition Capacity C_UPTC. We formulate the seven foundational axioms (COLS-1 through COLS-7) and prove six core theorems establishing interventional universal approximation, conservation-preserving OOD generalization, exact characterization of interventional equivalence, coupled error propagation inequalities, counterfactual operator identification, and stability–causality commutation. Complete Python numerical verification is provided in Appendix A.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Seonggil Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2q5e5nwrskbys.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23556758",
  "fulltext_url": null,
  "id": "2q5e5nwrskbys",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23556758",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23556758/files/The%20Causal%20Operator%20Learning%20System%20(COLS)_%20Unifying%20Infinite-Dimensional%20Neural%20Operators,%20Functional%20do-Calculus,%20and%20Conservation-Law%20Generalization%20within%20the%20ZFC%20Axiomatic%20Bridge.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2q5e5nwrskbys.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:20:43Z",
      "native_id": "23556758",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23556758"
    }
  ],
  "subjects": [],
  "title": "The Causal Operator Learning System (COLS): Unifying Infinite-Dimensional Neural Operators, Functional do-Calculus, and Conservation-Law Generalization within the ZFC Axiomatic Bridge",
  "title_text": "The Causal Operator Learning System (COLS): Unifying Infinite-Dimensional Neural Operators, Functional do-Calculus, and Conservation-Law Generalization within the ZFC Axiomatic Bridge",
  "type": "preprint",
  "updated_at": "2026-11-05T12:20:43Z",
  "url": "https://research.draconitic.com/paper/2q5e5nwrskbys",
  "venue": "Zenodo",
  "versions": []
}