The renormalized Φ44 trajectory by perturbation theory in the running coupling (I). The discrete renormalization group

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We compute the renormalized trajectory of massless Φ 4 theory on four-dimensional Euclidean space-time by perturbation theory in a running coupling. We introduce an iterative scheme without reference to a bare action. The renormalized Φ 4 trajectory is described as an invariant curve in the center manifold of the trivial fixed point, tangent to the Φ 4 interaction. The expansion is proven to be finite to every order of perturbation theory. The proof includes a large momentum bound on the connected momentum space Green functions.