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In previous works in this series, we focused on Hamiltonian renormalization of free field theories in all spacetime dimensions or interacting theories in spacetime dimensions lower than 4. In this paper, we address the Hamiltonian renormalization of the $U(1{)}^{3}$ model for Euclidian general relativity in four spacetime dimensions which is self-interacting. The Hamiltonian flow needs as an input a choice of ${\text{ }}^{*}$ algebra and corresponding representation thereof or state on it at each resolution scale. If one uses as input algebras and states in analogy to those used in the recent exact solutions of this model, then one finds that the flow finds as fixed point those exact solution theories.