Fractional Shift Algebra on a Canonical $\alpha$-Graded Space

Let $0 < \alpha < 1$, and define $e_n(x):=x^{n\alpha}/\Gamma(n\alpha+1)$ for $n\ge 0$. We prove that the algebraic direct sum $\mathcal{G}_{\alpha}^{\mathrm{alg}}:=\bigoplus_{n=0}^{\infty}\mathbb{C}e_n$ is the distinguished $\alpha$-graded monomial space on which the order-$\alpha$ Riemann--Liouville integral $J_\alpha:={}_0 I_x^\alpha$ and the order-$\alpha$ Caputo derivative $C_\alpha:={}_0^{\mathrm{C}}D_x^\alpha$ act as a unilateral shift pair, namely $J_\alpha e_n=e_{n+1}$ for $n\ge 0$, $C_\alpha e_0=0$, and $C_\alpha e_n=e_{n-1}$ for $n\ge 1$. It follows that $C_\alpha J_\alpha=I$, $J_\alpha C_\alpha=I-\Pi_0$, and $[C_\alpha,J_\alpha]=\Pi_0$, where $\Pi_0$ denotes the projection onto the vacuum component. We further establish a uniqueness theorem: among graded monomial chains with one-dimensional homogeneous components, $\mathcal{G}_{\alpha}^{\mathrm{alg}}$ is, up to multiplication of the entire basis by a single nonzero scalar, the unique chain on which $J_\alpha$ and $C_\alpha$ act as forward and backward shifts with vacuum annihilation. Finally, for every $m\in\mathbb{N}$, we show that $J_\alpha^m={}_0 I_x^{m\alpha}$ on all of $\mathcal{G}_{\alpha}^{\mathrm{alg}}$, whereas $C_\alpha^m={}_0^{\mathrm{C}}D_x^{m\alpha}$ holds on the tail subspace $\mathcal{G}_{\alpha}^{(\ge m)}:=\bigoplus_{n=m}^{\infty}\mathbb{C}e_n$. Hence the failure of the full semigroup property for Caputo derivatives is localized precisely in a finite-dimensional low-grade defect sector.