= Solution
After separating an additive identity term and using dimensionless couplings, let $R$ be one blocking-and-rescaling step. A <renormalization-group fixed point> satisfies $R(u_*)=u_*$. Near it, linearization gives scaling coordinates $w_i$ with
$$
w_i'=b^{y_i}w_i+O(w^2).
$$
At a critical <renormalization-group fixed point>, no finite characteristic length survives. Since the <correlation length> measured in lattice units transforms as $\xi(Ru)=\xi(u)/b$, a nonzero scale-invariant critical length must be infinite. A sequence of systems approaching the same fixed point therefore has common long-distance behavior.
The <scaling hypothesis for critical phenomena> asserts that the singular <free-energy density> transforms homogeneously under these scaling coordinates. It follows from repeated blocking near the fixed point, once analytic backgrounds are removed and no <dangerously irrelevant coupling> supplies an additional singular dependence. The temperature and magnetic-field scaling directions set the principal departures from criticality; irrelevant directions label microscopic differences that disappear at long distances.
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