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We formulate a renormalized running coupling expansion for the$\beta$--function and the potential of the renormalized $\phi^4$--trajectory onfour dimensional Euclidean space-time. Renormalization invariance is used as afirst principle. No reference is made to bare quantities. The expansion isproved to be finite to all orders of perturbation theory. The proof includes alarge momentum bound on the connected free propagator amputated vertices.Comment: 14 pages LaTeX2e, typos and references correcte

Running Coupling Expansion for the Renormalized Φ 4 4 Trajectory from Renormalization Invariance