= Solution
Write $\beta_T=(k_BT)^{-1}$ for inverse temperature to distinguish it from the <order-parameter critical exponent>. A <normalized blocking kernel> $B_b(\sigma',\sigma)$ is a nonnegative conditional distribution of coarse spins given the fine spins, normalized by
$$
\sum_{\sigma'}B_b(\sigma',\sigma)=1.
$$
For deterministic blocking it can impose a majority rule or block average, with a <Dirac delta> replacing the sum constraint for continuous coarse variables. Each block occupies $b^D$ original cells, so the new lattice spacing and number of sites are
$$
\boxed{a'=ba,\qquad N'=b^{-D}N.}
$$
The physical volume $Na^D$ is unchanged. After $p$ blockings, $a_p=b^pa$ and $N_p=b^{-pD}N$.
Define the coarse <statistical Hamiltonian> by the exact Boltzmann-weight identity
$$
\sum_\sigma B_b(\sigma',\sigma)e^{-\beta_TH(u,\sigma)}
=e^{-\beta_T[H(Ru,\sigma')+N g(u)]}.
$$
The operator family must include all generated interactions for this <renormalization-group transformation> to be exact; restricting it to a few couplings is an approximation. The scalar $g(u)$ has energy units and is the contribution per original site that is independent of $\sigma'$. Restoring the original constant term $NC$ gives
$$
H(u',\sigma')+N'C'=H(Ru,\sigma')+N[C+g(u)],
$$
hence
$$
\boxed{u'=Ru,\qquad C'=b^D[C+g(u)].}
$$
This convention measures $g$ per original site; defining a constant per new site would move a factor $b^D$ into its definition.
The <identity-operator contribution to renormalization-group free energy> $C_p$ records the configuration-independent normalization generated by the eliminated spins. It includes their contribution to <free energy>, including <entropy> and measure factors, and need not be independent of temperature. It has no effect on normalized probabilities of the retained spins, but is indispensable for the total <partition function> and its temperature derivatives. Iterating its recursion gives
$$
C_p=b^{pD}\left[C_0+\sum_{j=0}^{p-1}b^{-jD}g(u_j)\right].
$$
Thus $N_pC_p$ is the original constant $NC_0$ plus the accumulated free-energy contributions of the eliminated degrees of freedom.
To obtain the <additive free-energy recursion under blocking>, use energy-valued <free energy> per current lattice site,
$$
F(u,C)=-\frac1{\beta_TN}\log Z(u,C,N)=C+f_0(u).
$$
The <blocking kernel> normalization implies
$$
Z(u,C,N)=Z(Ru,C',N/b^D),
\qquad F(u,C)=b^{-D}F(Ru,C').
$$
Substituting the recursion for $C'$ cancels $C$, giving
$$
f_0(u)=b^{-D}f_0(Ru)+g(u).
$$
A second step gives $f_0(u_0)=b^{-2D}f_0(u_2)+g(u_0)+b^{-D}g(u_1)$, and induction proves
$$
\boxed{f_0(u_0)=b^{-pD}f_0(u_p)+
\sum_{j=0}^{p-1}b^{-jD}g(u_j).}
$$
For dimensionless <free energy> $-\log Z/N$, multiply $F,f_0,g,C$ throughout by $\beta_T$.
The singular <free-energy density> must be distinguished from the exact per-site representative $f_0$. If $a(u)$ is a regular background and $f_s=f_0-a$, then
$$
f_s(u)=b^{-D}f_s(Ru)+g_s(u),\qquad
g_s(u)=g(u)-a(u)+b^{-D}a(Ru).
$$
Thus the same displayed inhomogeneous recursion holds with $f_s,g_s$ after subtraction. When a suitable analytic background solves $a(u)=g(u)+b^{-D}a(Ru)$, it leaves a homogeneous singular recursion. <Analytic subtraction of an inhomogeneous renormalization recursion> explains this separation. A <renormalization-group free-energy resonance> can obstruct analytic subtraction and produce a logarithm. The original function $g$ originates in the short-distance spins that were summed out, rather than in an extra external interaction.
Near a <renormalization-group fixed point> $u_*$, linearize the transformation:
$$
R(u_*+\delta u)=u_*+\mathsf A\delta u+O(|\delta u|^2).
$$
An <eigenvector> of $\mathsf A$ with <eigenvalue> $b^{y_i}$ defines a scaling coordinate $t_i$ with $t_i'=b^{y_i}t_i$ to leading order. A <relevant operator> has $y_i>0$: its coupling grows towards large distances, so it must be tuned to reach the fixed point. An <irrelevant operator> has $y_i<0$ and dies away; a <marginal operator> has $y_i=0$ and requires higher-order flow analysis. An operator of scaling dimension $x_i$ has $y_i=D-x_i$.
The coarse spins also require a multiplicative normalization. In continuum notation, write
$$
x'=x/b,\qquad \phi'(x')=b^{x_\phi}\phi_<(bx'),
\qquad x_\phi=\frac{D-2+\eta}{2},
$$
where $\phi_<$ is the retained field and $\eta$ is its <anomalous dimension>. This restores the normalization of the critical long-distance <connected correlation function>. Since a uniform <field conjugate to an order parameter> enters through $\int d^Dx\,H\phi$, it has eigenvalue exponent $y_h=D-x_\phi=(D+2-\eta)/2$.
Let $t$ be the thermal relevant coordinate with exponent $y_t>0$. After regular backgrounds have been removed, the leading singular <free-energy density> obeys
$$
f_s(t,H)=B^{-D}f_s(B^{y_t}t,B^{y_h}H)
$$
for a large total scale factor $B=b^p$. Choosing $B=|t|^{-1/y_t}$ gives the <scaling hypothesis for critical phenomena>
$$
f_s(t,H)=|t|^{D/y_t}
\Phi_\pm\!\left(H|t|^{-y_h/y_t}\right).
$$
Two temperature derivatives give the <heat-capacity critical exponent>; one conjugate-field derivative gives the <order-parameter critical exponent>; two such derivatives give the <magnetic-susceptibility critical exponent>. Scaling the <correlation length> gives $\xi(t)=B\xi(B^{y_t}t)$. Consequently,
$$
\alpha=2-\frac D{y_t},\qquad
\beta=\frac{D-y_h}{y_t},\qquad
\gamma=\frac{2y_h-D}{y_t},\qquad
\nu=\frac1{y_t}.
$$
At $t=0$, instead choose $B=|H|^{-1/y_h}$; differentiating gives $\delta=y_h/(D-y_h)$. These <scaling relations for critical exponents> assume that irrelevant variables can be set to their fixed-point values without singular consequences. A <dangerously irrelevant coupling> or logarithmic correction requires keeping the corresponding dependence.
For the <Gaussian field theory>, integrate out the <Fourier modes> in $\Lambda/b<|q|<\Lambda$ and retain $\phi_<$. The modes are independent at zero external field, so this <Gaussian thinning transformation> changes the normalization by a Gaussian determinant and leaves a quadratic <statistical Hamiltonian> for the retained modes. Restore the cutoff with
$$
x'=x/b,\qquad
\phi'(x')=b^{(D-2)/2}\phi_<(bx').
$$
In the integrated <statistical Hamiltonian>, the volume element contributes $b^D$, each <gradient> contributes $b^{-1}$, and each field contributes $b^{-(D-2)/2}$. Thus the coefficient $\kappa^{-1}$ of the <gradient energy> is unchanged, while
$$
(m^2)'=b^2m^2,\qquad h'=b^{(D+2)/2}h.
$$
In particular $y_t=2$, $y_h=(D+2)/2$ and $\eta=0$. The <Gaussian critical exponents> and the <Gaussian order-parameter scaling index> therefore give the requested indices
$$
\boxed{\alpha=\frac{4-D}{2},\qquad
\beta=\frac{D-2}{4}.}
$$
The sign convention $+h\phi$ corresponds to the physical conjugate field $H=-h$ and does not affect these powers.
There are physical qualifications to these Gaussian indices. The strictly quadratic <Gaussian field theory> is stable for positive $m^2$ and has no stable ordered phase for negative $m^2$; its purported spontaneous $\beta$ is therefore the formal <Gaussian order-parameter scaling index>. A stabilizing quartic term gives an actual ordered phase. Below four dimensions that term is relevant and changes the exponents; above four dimensions it is a <dangerously irrelevant coupling> and gives the actual mean-field $\beta=1/2$, rather than the Gaussian hyperscaling value.
The <heat capacity> power can also be checked directly. Apart from constants, the zero-field Gaussian <free-energy density> is
$$
f_0=\frac{k_BT}{2}\int_{|q|<\Lambda}
\frac{d^Dq}{(2\pi)^D}\log(\kappa^{-1}q^2+m^2).
$$
If $m^2\propto t$, its singular second temperature derivative is controlled by $\int d^Dq\,(q^2+\kappa m^2)^{-2}$. For $0<D<4$, rescaling $q=\sqrt{\kappa m^2}\,k$ gives a power $t^{D/2-2}$, as above. At $D=4$ there is a logarithm, and for $D>4$ an analytic background dominates the <heat capacity>. The negative formal Gaussian $\alpha$ then describes the subleading singular piece, with logarithmic resonances at appropriate even dimensions. This is the <Gaussian specific-heat infrared threshold>, and it prevents confusing singular indices with the entire observable.
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