= A superficially convergent graph can have a divergent subgraph
In massive four-dimensional <phi-fourth theory>, a six-point triangle is <one-particle irreducible> and has superficial degree $-2$. Insert a <tadpole diagram> self-energy into one of its internal lines. The resulting graph is still one-particle irreducible and still has degree $-2$, but its tadpole loop diverges. Recursive local <counterterms> remove the subgraph divergence before overall power counting is applied. Convergence requires attention to all ultraviolet subgraphs, not merely to simultaneous scaling of all loop momenta.
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