= Uniform-source vacuum functional and spacetime volume
{title2=$F=\mathcal V_d\,\mu^d f(\lambda,m^2/\mu^2,h/\mu^3)$}
A translation-invariant connected <generating functional> with a constant <external source> is extensive in spacetime volume $\mathcal V_d$. Its total logarithm is dimensionless, while its density has <mass dimension> $d$. Consequently its dimensionless representation at fixed physical volume must retain $\mu^d\mathcal V_d$. For a free massive scalar with source coupling $\mu^{-\epsilon/2}h\phi$, <completing the square> gives $F=\mathcal V_d\mu^{-\epsilon}h^2/(2m^2)$. This already disproves a literal rewriting of the full uniform-source logarithm using only $\lambda,m^2/\mu^2,h/\mu^3$ at fixed volume. In infinite volume the total functional is generally infinite; use the density, a source of compact support, or an explicit volume regulator. After writing the density as $\mu^d f$, its <renormalization group> equation has the engineering term $+df$. Ultraviolet subtraction alone does not establish infrared finiteness for a massless constant source.
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