Position-Space Renormalization and Half-Space Truncations in ${\phi }_4^4$

In this paper, we study half-space observables in the massive Euclidean ${\phi }_4^4$ theory. We prove that the renormalized correlators can be multiplied by half-space truncations without requiring any additional renormalization. More precisely, products with smooth approximations of the half-space indicator converge to well-defined distributional limits, uniformly in the ultraviolet cutoff. The proof relies on a position-space renormalization framework for the Wilson–Polchinski flow equation based on a hierarchy of power-counting spaces adapted to the singularity structure of the correlators. This yields uniform power-counting bounds and Besov–Hölder regularity estimates for the renormalized correlators. As a consequence, the correlators converge as the ultraviolet cutoff tends to infinity, and the convergence takes place in explicitly identified Besov spaces.