  In a previous work, we established perturbative renormalizability to all
orders of the massive $\phi^4_4$-theory on a half-space also called the
semi-infinite massive $\phi^4_4$-theory. Five counter-terms which are functions
depending on the position in the space, were needed to make the theory finite.
The aim of the present paper is to prove that these counter-terms are position
independent (i.e. constants) for a particular choice of renormalization
conditions. We investigate this problem by decomposing the correlation
functions into a bulk part, which is defined as the $\phi^4_4$ theory on the
full space $\mathbb{R}^4$ with an interaction supported on the half-space, plus
a remainder which we call "the surface part". We analyse this surface part and
establish perturbatively that the $\phi^4_4$ theory in
$\mathbb{R}^+\times\mathbb{R}^3$ is made finite by adding the bulk
counter-terms and two additional counter-terms to the bare interaction in the
case of Robin and Neumann boundary conditions. These surface counter-terms are
position independent and are proportional to $\int_S \phi^2$ and $\int_S
\phi\partial_n\phi$. For Dirichlet boundary conditions, we prove that no
surface counter-terms are needed and the bulk counter-terms are sufficient to
renormalize the connected amputated (Dirichlet) Schwinger functions. A key
technical novelty as compared to our previous work is a proof that the power
counting of the surface part of the correlation functions is better by one
power than their bulk counterparts.
