Analytic subtraction of an inhomogeneous renormalization recursion ($f(u)-a(u)=b^{-D}[f(R_{b}u)-a(R_{b}u)]$)

For the renormalization-group transformation $f(u)=b^{-D}f(R_{b}u)+g(u)$, a background solving $a(u)=b^{-D}a(R_{b}u)+g(u)$ leaves a homogeneous remainder. In linear scaling coordinates $t^{\prime }=b^{{\lambda }_t}t$ and $h^{\prime }=b^{{\lambda }_h}h$, an analytic source term $g_{mn}{t}^m{h}^n$ is removed by $a_{mn}=g_{mn}/[1-b^{m{\lambda }_t+n{\lambda }_h-D}]$ unless the denominator vanishes. Such a resonance may produce a logarithm and invalidate a pure homogeneous power law. The scaling hypothesis for critical phenomena concerns the singular remainder. A regular field-dependent background cannot generally be represented by a function of temperature alone; an arbitrary analytic identity term proportional to $h^2$ makes this explicit.