The Causal Operator Learning System (COLS): Unifying Infinite-Dimensional Neural Operators, Functional do-Calculus, and Conservation-Law Generalization within the ZFC Axiomatic Bridge

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]. In 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.