Missing Descendants in the Carrollian Conformal Family

In the Carrollian conformal algebra, the relation $[P^0,K^0]=0$ implies that $K^0$ annihilates all operators generated from a primary by temporal translation $P^0$. The standard conformal family defined by translation descendants does not contain the independent descendant chain that $K^0$ lowers successively to the primary. We construct the complete Carrollian conformal representation by including this chain together with all of its translation descendants. We derived the corresponding local operators, orbit structures, and checked their Casimirs in 2D and 3D. For each orbit configuration, the global two-point Ward identities fix the kinematic factors and selection rules for both magnetic non-contact branches and electric contact branches. Unlike ordinary conformal symmetry, Carrollian conformal symmetry generally determines the correlators only up to arbitrary functions of Carrollian invariants rather than constants. The complete representation contains sectors with $\mathcal{C}_2=m^2=\kappa\rho-\beta^2>0$ for 2D and $\mathcal{C}_2=m^2=\kappa\rho-\vec{\beta}^{\,2}>0$ for 3D, which may describe massive states in flat holography.