= Model of a regularity structure
{title2=$(\Pi,\Gamma)$}
A <model of a regularity structure> realizes the abstract symbols of a <regularity structure> $(A,T,G)$ as concrete <distributions>, consistently across expansion points. It comprises continuous linear realization maps $\Pi_x:T\to\mathcal S'(\mathbb R^d)$ and maps $\Gamma_{xy}\in G$ satisfying
$$
\Gamma_{xx}=I,\qquad\Gamma_{xy}\Gamma_{yz}=\Gamma_{xz},\qquad\Pi_x\Gamma_{xy}=\Pi_y.
$$
Thus $\Gamma_{xy}$ transports an expansion from $y$ to $x$ without changing the represented <distribution>.<A theory of regularity structures by Martin Hairer>[Definition 2.17, printed pp. 25–26, algebraic identities]{ref}
Analytic bounds connect each symbol's assigned order $\alpha$ to its actual small-scale size. For ordinary isotropic scaling, put $\varphi_x^\lambda(z)=\lambda^{-d}\varphi((z-x)/\lambda)$. For $\tau\in T_\alpha$ one requires
$$
|\langle\Pi_x\tau,\varphi_x^\lambda\rangle|\leq C\|\tau\|_\alpha\lambda^\alpha,\qquad
\|\Gamma_{xy}\tau\|_\beta\leq C\|\tau\|_\alpha|x-y|^{\alpha-\beta}\quad(\beta<\alpha).
$$
These bounds are uniform on each <compact set> of base points, below each fixed upper order, for $0<\lambda\leq1$ and smooth <test functions> supported in the unit ball with $C^r$ norm at most one; choose an integer $r>\max(0,-\inf A)$. Other scaling choices change the rescaled <test functions> and distance. Negative $\alpha$ allows singular <distributions>, whose small-scale averages can grow as $\lambda$ shrinks.<A theory of regularity structures by Martin Hairer>[Definition 2.17, equation 2.15; Remark 2.24, interpretation for negative orders]{ref}
In the one-dimensional <polynomial model of a regularity structure>, $(\Pi_xX^n)(z)=(z-x)^n$ and $\Gamma_{xy}X=X+x-y$. For $x=1,y=2$,
$$
\Pi_1\Gamma_{12}X^2=\Pi_1(X-1)^2=((z-1)-1)^2=(z-2)^2=\Pi_2X^2.
$$
A change of variable $z=x+\lambda w$ in the <test function> pairing gives exactly $\lambda^n\int w^n\varphi(w)\,dw$, explaining the analytic bound for the <monomial> $X^n$. The realization maps supply information absent from the abstract triple alone.
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