= Solution
There is an interpretation issue in this clause. A proper quartic vertex, defined as the fourth derivative of the <quantum effective action>, has no additional tree or reducible contribution in the bare cubic theory: the three boxes exhaust its one-loop diagrams. The strict local zero-momentum vertex of a <Wilsonian effective action> also has no high-mode exchange bridge, since the momentum carried by that bridge would be zero, outside the shell. Thus $\boxed{\text{for a local proper or strict Wilsonian quartic vertex, there are no additional diagrams}.}$
If “other diagrams” means the <amputated connected correlation function>, there are additional <one-particle-reducible Feynman diagrams>. Specify full external-propagator amputation and a vacuum with <tadpole subtraction>. The three tree exchange channels each contribute $g^2/m^2$ at zero momentum. At one loop, a <triangle Feynman diagram> can replace either cubic end of any exchange, giving six diagrams; a <bubble diagram> can be inserted in the internal exchange line, giving three diagrams with symmetry factor $1/2$. The proper lower-point vertices at zero momentum, from the same trace-log expansion, are
$$
\Gamma^{(3)}=g+g^3I_3,\qquad\Gamma^{(2)}=m^2-\frac{g^2}{2}I_2.
$$
Therefore the reducible part of the connected four-point amplitude is
$$
\boxed{\mathcal A_{4,\mathrm{exchange}}=\frac{3g^2}{m^2}+\frac{6g^4}{m^2}I_3+\frac{3g^4}{2m^4}I_2\quad\text{to one loop}.}
$$
Adding the three boxes gives $\mathcal A_4=3g^2/m^2+3g^4I_4+6g^4I_3/m^2+3g^4I_2/(2m^4)$ in this convention. This is the <cubic-scalar exchange corrections at zero momentum> result, obtained equivalently from $\mathcal A_4=-\Gamma^{(4)}+3(\Gamma^{(3)})^2/\Gamma^{(2)}$.
If the tadpole has not been subtracted, it is an additional one-loop subgraph that must not be forgotten. The proper one-point function is $t=gI_1/2$. Its attachment to the internal exchange line gives another three reducible diagrams, contributing $3g^4I_1/(2m^6)$; equivalently the stationary background shifts by $v=-gI_1/(2m^2)$ and changes the exchanged mass by $gv$. A linear <counterterm> enforcing $v=0$ cancels this contribution. External <self-energy> or <tadpole diagram> attachments are removed by the stated full-propagator amputation; using bare external amputation would retain them instead. In that convention the four external-line decorations add $6g^4I_2/m^4+6g^4I_1/m^6$ to the displayed full-amputated result, with the $I_1$ part absent after tadpole subtraction. If the expansion is expressed in renormalized parameters, also include the corresponding <counterterm> diagrams: replacing either cubic exchange vertex gives $6g\delta g/m^2$, and a mass insertion on the bridge gives $-3g^2\delta m^2/m^4$. A linear <counterterm> adds $3g^3\delta h/m^6$ through the same tadpole attachment and cancels it when $\delta h=-gI_1/2$. The kinetic <counterterm> has zero insertion at zero exchanged momentum. At one loop these exhaust the possibilities: the single cycle has length four, three, two or one, corresponding respectively to box, triangle, bubble or tadpole, with tree branches attached.
The <mass dimensions> are $[g]=0$, $[I_r]=6-2r$, $[m]=1$. Accordingly $g^2/m^2$, $g^4I_4$, $g^4I_3/m^2$, $g^4I_2/m^4$ and $g^4I_1/m^6$ all have dimension $-2$. These connected exchanges are generally nonlocal functions of external momentum before taking the zero-momentum value; they should not be silently reclassified as an independent local quartic coupling.
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