= Identity-operator contribution to renormalization-group free energy
{title2=$f(u)=b^{-D}f(R_bu)+g(u)$}
If an energy per site $C$ multiplies the identity operator, blocking sends $C$ to $b^DC+c(u)$. 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.
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