= Solution
<Renormalization> compares the same long-distance physics at different observation scales. The starting object is a microscopic or cutoff <statistical Hamiltonian> $\mathcal H_g$ with every symmetry-allowed coupling $g$. Because a critical <correlation length> is much larger than a lattice spacing, repeatedly eliminating short-distance detail can reveal scale-invariant behavior.
In a real-space transformation choose a <normalized blocking kernel> $B(S|s)$ for retained variables. Define
$$
e^{-\mathcal H_{g'}(S)}=\sum_sB(S|s)e^{-\mathcal H_g(s)},\qquad\sum_SB(S|s)=1.
$$
Summing over $S$ proves exact preservation of $Z$ before truncation. The transformed <Hamiltonian> generally gains longer-range and many-body interactions and a field-independent constant. Observables must be blocked consistently. In momentum-shell language split $\phi=\phi_<+\phi_>$, integrate $\Lambda/b<|q|<\Lambda$, then use $x'=x/b$ and a field rescaling to restore cutoff and kinetic normalization. \b[Coarse-graining, rescaling and tracking generated interactions are distinct steps.]
Repeated transformations generate a flow $g'=R_b(g)$ with composition $R_bR_c=R_{bc}$ when the same exact coarse-graining family is used. Losing degrees of freedom generally makes this a semigroup, despite its conventional group name. A <renormalization-group fixed point> is invariant after rescaling. At a continuous transition the large-distance theory approaches a critical <renormalization-group fixed point>; finite <correlation length> instead eventually flows into an ordinary <thermodynamic phase> regime.
Linearize in scaling coordinates $u_i$:
$$
u'_i=b^{y_i}u_i.
$$
For increasing observation <length>, $y_i>0$ is relevant and must be tuned, $y_i<0$ is irrelevant, and $y_i=0$ is marginal and needs nonlinear analysis. Discrete multipliers $b^{y_i}$ must be compared with one, not with zero. The stable manifold forms the <critical surface> after relevant directions have been set to zero.
After subtracting regular additive contributions, ordinary <hyperscaling> gives
$$
f_s(t,h)=b^{-d}f_s(b^{y_t}t,b^{y_h}h).
$$
Choosing $b=|t|^{-1/y_t}$ or choosing the field as scale yields
$$
\boxed{\nu=1/y_t,\quad\Delta=y_h/y_t,\quad
2-\alpha=d/y_t,\quad\beta=(d-y_h)/y_t,\quad
\gamma=(2y_h-d)/y_t,\quad\delta=y_h/(d-y_h).}
$$
For a critical two-point function $G(r)\sim r^{-(d-2+\eta)}$, the <order parameter> <scaling dimension> is $x_m=(d-2+\eta)/2$, and its <conjugate field> has $y_h=d-x_m=(d+2-\eta)/2$. This recovers the Q1 identities while distinguishing the gap exponent $\Delta$ from the field's <scaling dimension> $y_h$.
As a concrete dimensional check, start from a scalar <gradient> action. Restoring its coefficient uses $\phi'(x')=b^{(d-2)/2}\phi_<(bx')$, so at the <Gaussian fixed point>
$$
r'=b^2r,\qquad u'=b^{4-d}u,\qquad h'=b^{(d+2)/2}h.
$$
Quartic interactions are relevant below four dimensions. They can produce an interacting <Wilson-Fisher fixed point>, whose exponents differ from naive <Gaussian> counting. Above four, the quartic coupling is irrelevant at the <Gaussian fixed point> but dangerous for ordered observables: $\phi_0^2=-r/u$ is singular when $u$ is dropped. Thus the simple two-variable <hyperscaling> formula is not universal above the <upper critical dimension>.
Different microscopic systems can flow to the same long-distance <renormalization-group fixed point>, explaining <universality> of exponents and normalized scaling forms. <Critical temperatures>, metric factors and regular <free energies> remain nonuniversal. Dimension, <order parameter> <symmetry>, interaction range and additional soft variables constrain the <universality class>. Truncations, perturbative expansions and numerics approximate the flow; they must not be confused with the exact elimination definition.<Statistical Field Theory by David Tong>[§3.1–§3.3: coarse-graining, scaling, relevance and dangerously irrelevant interactions]{ref}
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