= Solution
Let $I,L,V$ count internal lines, independent loops, and <interaction vertices> of a <connected Feynman diagram>. The graph identity $L=I-V+1$ and the two inverse powers from each <scalar propagator> give its <superficial degree of divergence>:
$$
\boxed{\omega=2L-2I=2(1-V).}
$$
The same formula applies to every connected subgraph. Graphs with $V\geq2$ have negative <superficial degree of divergence>, but this alone does not remove a possible <ultraviolet subdivergence>. A one-vertex subgraph with external legs in the <quartic scalar field theory> can diverge only by a <tadpole diagram>, whose subtraction was fixed in <1/c>. Subtract every such <tadpole diagram>; every remaining loop subgraph then has $V\geq2$ and negative <superficial degree of divergence>. Its <ultraviolet divergence> is absent. Equivalently, <normal ordering> the <quartic interaction> removes all self-contractions at a single <interaction vertex>. Thus \b[no new divergent field-dependent counterterm structure is needed], and in a <normal-ordering subtraction scheme in two dimensions> the one-loop mass subtraction exhausts these divergent coefficients.
The literal assertion that there are no further <counterterm> contributions in the <on-shell renormalization scheme> is stronger and generally false. A concrete counterexample is the two-loop <sunset diagram>. After analytic continuation below its three-particle threshold, the magnitude of its <self-energy> at $p^2=m^2$ contains the strictly positive finite integral
$$
\left|\Sigma_{\rm sunset}(m^2)\right|
=\frac{g_4^2}{6(4\pi)^2m^2}
\int_{x,y,z\geq0}dx\,dy\,dz\,
\frac{\delta(1-x-y-z)}{xy+xz+yz-xyz}>0
\quad(g_4\ne0).
$$
The integrable corners do not spoil convergence. Its nonzero value shifts the <pole mass>, requiring an additional \i[finite] <mass counterterm> if $m$ is to remain the exact <pole mass>; its derivative can likewise change the <pole residue>. These are compatible with the <power counting> result. If <vacuum diagrams> are retained rather than divided out in normalized <correlation functions>, a separate constant <vacuum energy> <counterterm> is also required: the one-vertex double-loop <vacuum diagram> is already divergent. The claim about the mass subtraction concerns field-dependent <counterterms> and normalized <correlation functions>.
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