Finite quartic graph and superficial scaling conventions — Examples

In Euclidean momentum dimension $D=4$, take the connected graph with two quartic vertices joined by two internal edges and with two external half-edges at each vertex. Its counts are $V=2$, $I=2$, $E=4$, and $L=1$. Its superficial degree is therefore zero, so it has logarithmic superficial degree under the stated power-counting convention.

Separately, on punctured Euclidean ${\mathbb{R}}^4$ let $C_0(x)=1\mathrm{/}(4{\pi }^2\mathrm{\mid }x{\mathrm{\mid }}^2)$ and $u=C_0^2=1\mathrm{/}(16{\pi }^4\mathrm{\mid }x{\mathrm{\mid }}^4)$. For any $\chi \in \mathcal{D}({\mathbb{R}}^4)$ equal to one near $0$, and every $\mu >0$, define $T_{\mu }(\phi )={\int }_{{\mathbb{R}}^4\setminus \{0\}}[\phi (x)-\chi (\mu x)\phi (0)]\mathrm{/}(16{\pi }^4\mathrm{\mid }x{\mathrm{\mid }}^4)d^{4}x$ for $\phi \in \mathcal{D}({\mathbb{R}}^4)$. Conditional on the current QFT-IR04 proposition Exact four-dimensional Euclidean logarithmic scale anomaly with its explicit Axiom-of-Choice hypothesis, this family has the exact logarithmic scale-change identity stated below and no same-degree homogeneous extension. This is a separate Euclidean distribution-extension calculation; it is not identified with the integrated amplitude of the graph. The graph count alone does not establish convergence or divergence of an actual loop integral.