Fractional Geometry of Zeros for Terminal-Anchored Riemann–Liouville and Caputo Derivatives: A Fractional Gauss–Lucas Theory

We develop a terminal-explicit geometric theory for the algebraic zero sets associated with the Riemann–Liouville and Caputo fractional derivatives of a complex polynomial. Expanding the polynomial about an arbitrary complex terminal a reduces both operators to gamma-weighted polynomial transforms. The coordinate shift w=z−a is algebraically equivalent to a zero-terminal formulation; the terminal dependence re-enters through the shifted coefficients and through the pullback of the resulting geometry to the original z-plane. We derive exact barycenter identities, terminal-centered disk bounds, multiset convergence at the endpoint orders, first-order deformation formulas for simple roots, and solvable examples. The disk bounds provide a star-shaped localization framework rather than a convex-hull theorem. A regular-polygon family yields an exact homothety for the Riemann–Liouville roots, while numerical examples show curved generic trajectories and a nondegenerate Caputo evolution. The terminal therefore acts as a distinguished geometric anchor, with radial attraction occurring under additional algebraic symmetry.