Normalized blocking kernel (${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$)

A normalized blocking kernel assigns probabilities or delta constraints for coarse-grained variables given a microscopic configuration. Multiplying the microscopic Boltzmann weight by this kernel and summing over eliminated variables defines the blocked statistical Hamiltonian. Normalization preserves the partition function exactly when all generated operators and the field-independent constant are retained. A finite-coupling truncation may lose that exactness.

If an energy per site $C$ multiplies the identity operator, blocking sends $C$ to $b^{D}C+c(u)$. For free energy per site $F=C+f(u)$, exact partition function invariance gives the displayed inhomogeneous equation with $g=b^{-D}c$. It records eliminated-mode entropy, vacuum contributions and measure normalization. Removing a suitable regular background leaves homogeneous singular scaling; additive logarithmic resonances can obstruct a strictly analytic subtraction.

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.

An analytic source monomial $g_{mn}{t}^m{h}^n$ in an additive renormalization-group transformation of free-energy density is resonant when its scaling degree equals the spatial dimension. The analytic background coefficient would require division by $1-b^{m{\lambda }_t+n{\lambda }_h-D}=0$. On a branch with $t\ne 0$ and ${\lambda }_t>0$, a particular nonanalytic background is $-g_{mn}{t}^m{h}^n\log |t|/({\lambda }_t\log b)$: substituting $t^{\prime }=b^{{\lambda }_t}t$, $h^{\prime }=b^{{\lambda }_h}h$ verifies the additive recursion exactly. Thus logarithms can be required even when the eliminated-mode source is analytic. This is distinct from marginal interaction running, which can create additional logarithmic corrections.