The paper extends the existing leading (LO), next-to-leading (NLO), and next-to-next-to-leading order (NNLO) parton distribution functions (PDFs) to approximate next-to-next-to-next-to-leading order (aN³LO). The authors construct an approximation to the N³LO splitting functions, incorporating all available partial information from fixed-order computations and small and large \(x\) resummation. They estimate the uncertainty on this approximation by varying the set of basis functions used to construct it. Known N³LO corrections to deep-inelastic scattering structure functions are included, and the FONLL general-mass scheme is extended to \(\mathcal{O}(\alpha_s^3)\) accuracy. A set of aN³LO PDFs is determined, accounting for both the uncertainty on splitting functions due to incomplete knowledge of N³LO terms and the uncertainty related to missing higher corrections (MHOU), estimated through a theory covariance matrix formalism. The perturbative stability of the resulting PDFs is assessed, and the impact of MHOUs on them is studied. The authors compare their results to those from the MSHT group and examine the phenomenological impact of aN³LO corrections on parton luminosities at the LHC, as well as on the Higgs and Drell-Yan total production cross-sections. They find that the aN³LO NNPDF4.0 PDFs are consistent within uncertainties with their NNLO counterparts, improve the description of global datasets, and enhance the perturbative convergence of Higgs and Drell-Yan cross-sections. Additionally, they observe that MHOUs on PDFs decrease substantially with the increase of perturbative order.The paper extends the existing leading (LO), next-to-leading (NLO), and next-to-next-to-leading order (NNLO) parton distribution functions (PDFs) to approximate next-to-next-to-next-to-leading order (aN³LO). The authors construct an approximation to the N³LO splitting functions, incorporating all available partial information from fixed-order computations and small and large \(x\) resummation. They estimate the uncertainty on this approximation by varying the set of basis functions used to construct it. Known N³LO corrections to deep-inelastic scattering structure functions are included, and the FONLL general-mass scheme is extended to \(\mathcal{O}(\alpha_s^3)\) accuracy. A set of aN³LO PDFs is determined, accounting for both the uncertainty on splitting functions due to incomplete knowledge of N³LO terms and the uncertainty related to missing higher corrections (MHOU), estimated through a theory covariance matrix formalism. The perturbative stability of the resulting PDFs is assessed, and the impact of MHOUs on them is studied. The authors compare their results to those from the MSHT group and examine the phenomenological impact of aN³LO corrections on parton luminosities at the LHC, as well as on the Higgs and Drell-Yan total production cross-sections. They find that the aN³LO NNPDF4.0 PDFs are consistent within uncertainties with their NNLO counterparts, improve the description of global datasets, and enhance the perturbative convergence of Higgs and Drell-Yan cross-sections. Additionally, they observe that MHOUs on PDFs decrease substantially with the increase of perturbative order.