= Solution
Within a finite blocking step, $g(u)$ is generally analytic near the critical couplings because only finitely many microscopic variables have been summed. If its regular contribution can be removed by the <analytic subtraction of an inhomogeneous renormalization recursion>, derivatives of that background are nonsingular. The critical powers are then determined by the homogeneous singular remainder and the relevant scaling dimensions, not by the normalization of the eliminated modes.
For example, in linear scaling coordinates a source monomial $g_{mn}t^mh^n$ is removed by choosing a background coefficient
$$
A_{mn}=\frac{g_{mn}}{1-b^{my_t+ny_h-D}},
$$
provided the denominator is nonzero. This explicitly demonstrates why the inhomogeneous term normally changes backgrounds and amplitudes rather than critical powers.
The qualification matters: at a <renormalization-group free-energy resonance>, $my_t+ny_h=D$, an analytic subtraction fails and logarithms such as $t^mh^n\log|t|$ can be needed. Repeated analytic blocking contributions are not automatically globally analytic at criticality. Even the field-independent $I_\pm$ in the supplied scaling form can affect a singular heat-capacity amplitude. Its omission is justified for leading exponents only when the remaining amplitudes are nonzero and the assumed pure power scaling is valid; it is not a license to discard the identity term from an exact <free energy>.
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