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

Let $d\in\mathbb{N}$ and let ${\alpha}=(\alpha_1,\dots,\alpha_d)\in(0,1)^d$. We prove that the algebraic direct sum $$ \mathcal{G}_{{\alpha}}^{\mathrm{alg}} := \bigoplus_{\mathbf{k}\in\mathbb{N}_0^d}\mathbb{C}e_{\mathbf{k}}, \quad e_{\mathbf{k}}(x) := \prod_{j=1}^d \frac{x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)},$$is the canonical multi-graded monomial space on which the partial Riemann--Liouville integrals $J_j:={}_0 I_{x_j}^{\alpha_j} \text{ for } 1\le j\le d$ and the partial Caputo derivatives $C_j:={}_0^{\mathrm{C}}D_{x_j}^{\alpha_j} \text{ for } 1\le j\le d$ realize a commuting shift algebra. More precisely, for every $\mathbf{k}\in\mathbb{N}_0^d$ and every $j\in\{1,\dots,d\}$, $J_j e_{\mathbf{k}} = e_{\mathbf{k}+\mathbf{e}_j},$ and $$C_j e_{\mathbf{k}} = \begin{cases} 0, & k_j=0,\\ e_{\mathbf{k}-\mathbf{e}_j}, & k_j\ge 1.\end{cases}$$ Consequently, $C_jJ_j=I,$ $J_jC_j=I-\Pi_j,$ and $[C_j,J_j]=\Pi_j,$ where $\Pi_j$ is the projection onto the coordinate-vacuum hyperplane $\{\,\mathbf{k}\in\mathbb{N}_0^d : k_j=0\,\}$. We further prove that the tuples $(J_1,\dots,J_d)$ and $(C_1,\dots,C_d)$ commute on $\mathcal{G}_{{\alpha}}^{\mathrm{alg}}$, 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 $\mathbf{m}\in\mathbb{N}_0^d$, the operator $J^{\mathbf{m}} := J_1^{m_1}\cdots J_d^{m_d}$ coincides with the partial Riemann--Liouville integral of order $\mathbf{m}\odot {\alpha}$ on all of $\mathcal{G}_{{\alpha}}^{\mathrm{alg}}$, whereas $C^{\mathbf{m}} := C_1^{m_1}\cdots C_d^{m_d}$ coincides with the partial Caputo derivative of order $\mathbf{m}\odot {\alpha}$ on the natural tail subspace. Thus the multidimensional Caputo defect appears as a family of coordinate boundary layers rather than a single vacuum sector.