The Nonexistence of Internal Fractional Models on Pₙ

We prove that the ordinary differentiation operator on the finite-dimensional polynomial space Pₙ: = \p (x) Cx: p n\ cannot serve as an internal model for classical fractional calculus. Here, by an internal model we mean a family of linear endomorphisms acting on the same space Pₙ, indexed by nonnegative orders, satisfying the semigroup law and extending the first derivative at order 1. Two independent obstructions are established. First, the classical Riemann--Liouville and Caputo fractional derivatives do not act internally on Pₙ: the former sends even the constant polynomial 1 to a nonpolynomial function, while the latter either leaves Pₙ or collapses to the zero operator at sufficiently high orders. Second, the differentiation operator Dₙ: =ddx|₏䂸 is a single nilpotent Jordan block. We show that such an operator admits no nontrivial q-th root for any integer q 2. Consequently, no semigroup (T_) ₀ End (Pₙ) with T₁=Dₙ can exist. The negative conclusion is therefore structural: the failure lies not in the operator-theoretic idea of fractional powers itself, but in the choice of a finite-dimensional ordinary polynomial state space.