The Nonexistence of Internal Fractional Models on $P_n$

We prove that the ordinary differentiation operator on the finite-dimensional polynomial space $P_n := \{p(x)\in \mathbb{C}[x] : \deg p \le n\}$ cannot serve as an internal model for classical fractional calculus. Here, by an \emph{internal model} we mean a family of linear endomorphisms acting on the same space $P_n$, 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 $P_n$: the former sends even the constant polynomial $1$ to a nonpolynomial function, while the latter either leaves $P_n$ or collapses to the zero operator at sufficiently high orders. Second, the differentiation operator $D_n:=\frac{d}{dx}\big|_{P_n}$ is a single nilpotent Jordan block. We show that such an operator admits no nontrivial $q$-th root for any integer $q\ge 2$. Consequently, no semigroup $(T_\alpha)_{\alpha\ge 0}\subset \mathrm{End}(P_n)$ with $T_1=D_n$ 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.