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At fixed $m>0$ and $\Lambda >0$, the radial integrands in the scalar shell integral in six dimensions behave as $q^{5-2r}$ in the ultraviolet. ThusConsequently the proper box quartic vertex is ultraviolet finite. The reducible triangle insertion contains the logarithmic cubic-vertex divergence. The internal bubble contains the mass divergence; its nonzero-momentum expansion additionally has a logarithmic kinetic-term divergence. An unsubtracted tadpole attachment has the quartic one-point divergence and is canceled by tadpole subtraction, or incorporated consistently in the chosen stationary background.

These divergent lower-point subgraphs require counterterms, not a new independent divergent ${\varphi }^4$ coupling. More generally, cubic graph identities give the superficial degree of divergenceVacuum, one-, two-, and three-point functions need vacuum-energy, linear, mass, wave-function renormalization and cubic-coupling counterterms. Proper higher-point functions have negative superficial degree and are finite after subtraction of divergent subgraphs. Hence the perturbative renormalizability of cubic scalar theory in six dimensions permits a continuum perturbative expansion with finitely many renormalized parameters; induced irrelevant higher-field and derivative interactions are finite predictions at finite scale, not an infinite list of ultraviolet parameters.

This perturbative statement does not establish a nonperturbative real Euclidean measure. For any nonzero real $g$, $m^2{\varphi }^2/2+g{\varphi }^3/6$ tends to $-\infty$ in one field direction, so even its constant-field integral is divergent. Thus $\boxed{\text{perturbatively renormalizable, with no convergent positive real Euclidean cubic measure}.}$ Stability or a specified analytic continuation would be additional input beyond these loop calculations.