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.