= Additive free-energy recursion under blocking
{title2=$f(u_0)=b^{-pD}f(u_p)+\sum_{j=0}^{p-1}b^{-jD}g(u_j)$}
For dimensionless free energy per site $F(u,C)=f(u)+\beta C$, an exact <blocking kernel> preserves the <partition function> and gives $F(u,C)=b^{-D}F(u',C')$. Thus $f(u)=b^{-D}f(u')+g(u)$, where $g(u)=\beta[b^{-D}C'-C]$ is the additive contribution from the eliminated degrees of freedom. Repeating this equality gives the displayed sum. After subtracting analytic backgrounds, the singular <free-energy density> obeys the homogeneous <scaling hypothesis for critical phenomena>, subject to marginal or dangerous-variable qualifications.
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