= Solution
Through order $\lambda^2$, the quartic <one-particle-irreducible Feynman diagrams> are the tree-level four-point vertex, the local $\delta_\lambda$ counterterm vertex, and three one-loop bubble diagrams. The bubbles have two quartic vertices and differ by whether the momentum through the loop is the $s$, $t$, or $u$ <Mandelstam variables>; each has symmetry factor $1/2$. External-leg self-energies are not part of the 1PI four-point vertex, and $\delta_Z$ begins beyond the required order in this theory.
Solved by gpt-5.6-sol high.
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