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

Let d and let = (₁, , d) (0, 1) ᵈ. We prove that the algebraic direct sum G^alg: = ₊䃐㵧Ce₊, e₊ (x): = ₉=₁ᵈ xⱼ^kⱼⱼ (kⱼⱼ+1), is the canonical multi-graded monomial space on which the partial Riemann--Liouville integrals Jⱼ: =₀ Iₗ䲛^ⱼ for 1 j d and the partial Caputo derivatives Cⱼ: =₀^CDₗ䲛^ⱼ for 1 j d realize a commuting shift algebra. More precisely, for every k₀ᵈ and every j\1, , d\, Jⱼ e₊ = e₊+₄䲛, and Cⱼ e₊ = cases 0, & kⱼ=0, \\ e₊-₄䲛, & kⱼ 1. cases Consequently, CⱼJⱼ=I, JⱼCⱼ=I-ⱼ, and Cⱼ, Jⱼ=ⱼ, where ⱼ 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^alg, 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 J^m: = J₁^m₁ Jd^md coincides with the partial Riemann--Liouville integral of order m on all of G^alg, whereas C^m: = C₁^m₁ Cd^md coincides with the partial Caputo derivative of order m on the natural tail subspace. Thus the multidimensional Caputo defect appears as a family of coordinate boundary layers rather than a single vacuum sector.