Fractional Shift Algebra on a Canonical -Graded Space

Let 0 < < 1, and define eₙ (x): =x^n/ (n+1) for n 0. We prove that the algebraic direct sum G_^alg: =₍=₀^Ceₙ is the distinguished -graded monomial space on which the order- Riemann--Liouville integral J_: =₀ Iₓ^ and the order- Caputo derivative C_: =₀^CDₓ^ act as a unilateral shift pair, namely J_ eₙ=e₍+₁ for n 0, C_ e₀=0, and C_ eₙ=e₍-₁ for n 1. It follows that C_ J_=I, J_ C_=I-₀, and C_, J_=₀, where ₀ denotes the projection onto the vacuum component. We further establish a uniqueness theorem: among graded monomial chains with one-dimensional homogeneous components, G_^alg is, up to multiplication of the entire basis by a single nonzero scalar, the unique chain on which J_ and C_ act as forward and backward shifts with vacuum annihilation. Finally, for every m, we show that J_ᵐ=₀ Iₓ^m on all of G_^alg, whereas C_ᵐ=₀^CDₓ^m holds on the tail subspace G_^ (m): =₍=₌^Ceₙ. Hence the failure of the full semigroup property for Caputo derivatives is localized precisely in a finite-dimensional low-grade defect sector.