= Solution
Use two external slow-field legs and internal fast-mode propagators. Through the requested orders, the connected mass-correction topologies are:
* order $g_0$: one quartic vertex with one fast tadpole;
* order $g_0^2$: two quartic vertices joined either by three fast lines, or by two fast lines with a fast tadpole on the vertex carrying no external legs;
* order $\lambda_0$: one sextic vertex with two fast tadpole loops;
* order $\lambda_0g_0$: a sextic and a quartic vertex joined by two fast lines, with the remaining fast legs closed into tadpoles in the two possible external-leg placements; joined by four fast lines with two external legs on the sextic vertex; or joined by three fast lines with one sextic tadpole and one external leg on each vertex.
A nominal one-line bridge at order $\lambda_0g_0$ vanishes in a sharp momentum-shell scheme at small external momentum because that line would have to carry momentum outside the fast shell. These descriptions specify the same diagrams without depending on a particular drawing convention for vertices and external legs.
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