Partial Fractional Integrals and Caputo Derivatives as a Commuting Shift Algebra on a Canonical Multi-Graded Space

Let d and let α=(α₁,,αd)∈(0,1)ᵈ. We prove that the algebraic direct sum G_αᵃˡᵍ := k₀ᵈCeₖ, eₖ(x) := ∏ⱼ₌₁ᵈ xⱼkⱼαⱼΓ(kⱼαⱼ+1),is the canonical multi-graded monomial space on which the partial Riemann--Liouville integrals Jⱼ:=₀ Ixⱼαⱼ for 1≤ j≤ d and the partial Caputo derivatives Cⱼ:=₀CDxⱼαⱼ for 1≤ j≤ d realize a commuting shift algebra. More precisely, for every k₀ᵈ and every j∈\1,,d\, Jⱼ eₖ = ek+eⱼ, and Cⱼ eₖ = cases 0, & kⱼ=0,\\ ek-eⱼ, & kⱼ≥ 1.cases Consequently, CⱼJⱼ=I, JⱼCⱼ=I-Πⱼ, and [Cⱼ,Jⱼ]=Πⱼ, where Πⱼ is the projection onto the coordinate-vacuum hyperplane \\,k₀ᵈ : kⱼ=0\,\. We further prove that the tuples (J₁,,Jd) and (C₁,,Cd) commute on G_αᵃˡᵍ, and that the model is unique up to a global scalar normalization among multi-indexed graded monomial lattices with one-dimensional homogeneous components. Finally, for every m₀ᵈ, the operator Jᵐ := J₁m₁⋯ Jdmd coincides with the partial Riemann--Liouville integral of order m α on all of G_αᵃˡᵍ, whereas Cᵐ := C₁m₁⋯ Cdmd coincides with the partial Caputo derivative of order m α on the natural tail subspace. Thus the multidimensional Caputo defect appears as a family of coordinate boundary layers rather than a single vacuum sector.