= Solution
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. Thus
$$
\begin{aligned}
I_4&\longrightarrow\text{a finite limit},\\
I_3&=\frac1{64\pi^3}\log(\Lambda_0/\Lambda)+O(1),\\
I_2&=\frac{\Lambda_0^2}{128\pi^3}+O(\log\Lambda_0),\\
I_1&=\frac{\Lambda_0^4}{256\pi^3}+O(\Lambda_0^2).
\end{aligned}
$$
Consequently 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 $\phi^4$ coupling. More generally, cubic graph identities give the <superficial degree of divergence>
$$
D=6L-2I=6-2E,\qquad 3V=2I+E,\quad L=I-V+1.
$$
Vacuum, 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 $g$, $m^2\phi^2/2+g\phi^3/6$ 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.
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