[proofplan] We use the equivariant Darboux normal form at the toric fixed point to replace a neighbourhood of $p$ by the standard Hamiltonian $T^2$-model on $\mathbb C^2$, with the chosen Delzant coordinates matching the standard moment map. In that model, a ball of capacity $\varepsilon$ has moment image the open triangle $x_1+x_2<\varepsilon$ in the positive quadrant. The toric blow-up is the local symplectic cut which removes this ball and collapses the Hopf characteristic circles on its boundary, producing the new facet $x_1+x_2=\varepsilon$. Choosing $\varepsilon$ sufficiently small keeps the construction inside the fixed-point chart and away from all non-adjacent facets, so the rest of the polygon is unchanged. [/proofplan]

[step:Put the fixed point into the standard toric Darboux model]Let $A$ denote the chosen integral affine coordinate map from a neighbourhood of $v$ in $\mathfrak t^*$ to a neighbourhood of $0$ in $\mathbb R^2$, so that $A(v)=0$ and the two facets adjacent to $v$ are locally contained in $\{x_1=0\}$ and $\{x_2=0\}$. Since $p$ is a toric fixed point, the isotropy representation of $T^2$ on $T_pM$ has two primitive weights forming a $\mathbb Z$-basis of the weight lattice dual to the two primitive edge directions at the Delzant vertex $v$. By the equivariant Darboux normal form at a toric fixed point (citing a result not yet in the wiki: Equivariant Darboux normal form at a toric fixed point), there exist a $T^2$-invariant open neighbourhood $U\subset M$ of $p$, an open neighbourhood $W\subset\mathbb C^2$ of $0$, and a $T^2$-equivariant symplectomorphism \begin{align*} \Phi:W\to U \end{align*} with $\Phi(0)=p$ such that, after applying the integral affine coordinate map $A$, the moment map is \begin{align*} A(\mu(\Phi(z_1,z_2)))=(\pi |z_1|^2,\pi |z_2|^2) \end{align*} for every $(z_1,z_2)\in W$. Shrinking $W$ if necessary, choose $r>0$ such that \begin{align*} B_{\mathbb C^2}(0,r):=\{(z_1,z_2)\in\mathbb C^2: |z_1|^2+|z_2|^2<r^2\} \end{align*} is contained in $W$.[/step]

[guided]The first task is to make the words "corner at $v$" into the standard model where the computation is explicit. Let $A$ be the chosen integral affine coordinate map from a neighbourhood of $v$ in $\mathfrak t^*$ to a neighbourhood of $0$ in $\mathbb R^2$. Thus $A(v)=0$, and the two facets of the Delzant polygon through $v$ become, locally, the coordinate facets $\{x_1=0\}$ and $\{x_2=0\}$. Why is this coordinate choice compatible with the standard action on $\mathbb C^2$? At a toric fixed point, the isotropy representation of $T^2$ on $T_pM$ splits into two complex one-dimensional weight spaces. The Delzant condition says that the two primitive weights form a $\mathbb Z$-basis of the integral weight lattice. Therefore the integral affine coordinates determined by the two adjacent facets identify the local moment cone with the standard quadrant. We now use the equivariant Darboux normal form at a toric fixed point (citing a result not yet in the wiki: Equivariant Darboux normal form at a toric fixed point). It gives a $T^2$-invariant open neighbourhood $U\subset M$ of $p$, an open neighbourhood $W\subset\mathbb C^2$ of $0$, and a $T^2$-equivariant symplectomorphism \begin{align*} \Phi:W\to U \end{align*} satisfying $\Phi(0)=p$. The normalization of the theorem is exactly the normalization in which the pulled-back moment map is \begin{align*} A(\mu(\Phi(z_1,z_2)))=(\pi |z_1|^2,\pi |z_2|^2). \end{align*} This is the point where the integral affine coordinates matter: without choosing the Delzant coordinates determined by the primitive weights, the same formula could differ by an integral affine transformation. Finally, because $W$ is an open neighbourhood of $0$, there exists $r>0$ such that the Euclidean ball \begin{align*} B_{\mathbb C^2}(0,r):=\{(z_1,z_2)\in\mathbb C^2: |z_1|^2+|z_2|^2<r^2\} \end{align*} is contained in $W$. This radius will later impose one part of the smallness condition on $\varepsilon$.[/guided]

[step:Compute the moment image of the standard capacity ball] For $\varepsilon>0$, define the standard open ball of capacity $\varepsilon$ by \begin{align*} B_\varepsilon:=\{(z_1,z_2)\in\mathbb C^2:\pi(|z_1|^2+|z_2|^2)<\varepsilon\}. \end{align*} The standard moment map \begin{align*} \mu_{\mathbb C^2}:\mathbb C^2\to\mathbb R^2 \end{align*} is given by \begin{align*} \mu_{\mathbb C^2}(z_1,z_2)=(\pi |z_1|^2,\pi |z_2|^2). \end{align*} Therefore \begin{align*} \mu_{\mathbb C^2}(B_\varepsilon)=\{(x_1,x_2)\in\mathbb R^2:x_1\ge 0,\ x_2\ge 0,\ x_1+x_2<\varepsilon\}. \end{align*} Indeed, if $(z_1,z_2)\in B_\varepsilon$, then $x_i=\pi |z_i|^2\ge 0$ and $x_1+x_2<\varepsilon$. Conversely, if $x_1,x_2\ge 0$ and $x_1+x_2<\varepsilon$, choose $z_i\in\mathbb C$ with $\pi |z_i|^2=x_i$ for $i\in\{1,2\}$; then $(z_1,z_2)\in B_\varepsilon$ and $\mu_{\mathbb C^2}(z_1,z_2)=(x_1,x_2)$. [/step]

[step:Identify the local blow-up with the facet replacement]Assume $\varepsilon<\pi r^2$, so that $B_\varepsilon\subset W$. The toric symplectic blow-up of capacity $\varepsilon$ in the local model removes $B_\varepsilon$ and collapses the characteristic Hopf circles on $\partial B_\varepsilon$. Equivalently, it is the local symplectic cut model applied to the Hamiltonian circle action \begin{align*} e^{i\theta}\cdot(z_1,z_2)=(e^{i\theta}z_1,e^{i\theta}z_2) \end{align*} with Hamiltonian \begin{align*} H:\mathbb C^2\to\mathbb R,\qquad H(z_1,z_2)=\pi(|z_1|^2+|z_2|^2). \end{align*} By the [symplectic cut construction](/theorems/10074) [citetheorem:10074], the open region $H<\varepsilon$ is removed and the reduced boundary $H^{-1}(\varepsilon)/S^1$ becomes the exceptional divisor. Under $\mu_{\mathbb C^2}$, the level $H=\varepsilon$ is exactly \begin{align*} \{(x_1,x_2)\in\mathbb R^2:x_1\ge 0,\ x_2\ge 0,\ x_1+x_2=\varepsilon\}. \end{align*} Thus the local moment image after blow-up is the standard quadrant with the open triangle $x_1+x_2<\varepsilon$ removed and with the new boundary segment $x_1+x_2=\varepsilon$ included: \begin{align*} \{(x_1,x_2)\in\mathbb R^2:x_1\ge 0,\ x_2\ge 0,\ x_1+x_2\ge\varepsilon\}. \end{align*}[/step]

[guided]In the standard model, the blow-up is not merely the operation of deleting a ball; the boundary of the deleted ball is collapsed along a circle foliation. The relevant circle action is the diagonal action \begin{align*} e^{i\theta}\cdot(z_1,z_2)=(e^{i\theta}z_1,e^{i\theta}z_2). \end{align*} Its Hamiltonian, in the capacity normalization used in the theorem, is the map \begin{align*} H:\mathbb C^2\to\mathbb R,\qquad H(z_1,z_2)=\pi(|z_1|^2+|z_2|^2). \end{align*} The ball of capacity $\varepsilon$ is precisely the sublevel set $H<\varepsilon$, and its boundary is the level set $H=\varepsilon$. We now invoke the symplectic cut construction [citetheorem:10074]. Its hypotheses are satisfied in this local model: the diagonal $S^1$-action is Hamiltonian with moment map $H$, and on the sphere $H^{-1}(\varepsilon)$ the action is free because a point on this sphere is not $(0,0)$, so at least one coordinate is nonzero and a diagonal phase fixing the point must be the identity. The construction removes the open part $H<\varepsilon$ and replaces the boundary level by the quotient $H^{-1}(\varepsilon)/S^1$, which is the exceptional divisor of the blow-up. Now translate this operation into the moment image. Since \begin{align*} \mu_{\mathbb C^2}(z_1,z_2)=(\pi |z_1|^2,\pi |z_2|^2), \end{align*} the equation $H(z_1,z_2)=\varepsilon$ becomes \begin{align*} x_1+x_2=\varepsilon. \end{align*} The inequalities $x_1\ge 0$ and $x_2\ge 0$ remain because each $x_i$ is a squared norm multiplied by $\pi$. Therefore the collapsed boundary contributes the new closed segment \begin{align*} \{(x_1,x_2)\in\mathbb R^2:x_1\ge 0,\ x_2\ge 0,\ x_1+x_2=\varepsilon\}. \end{align*} The open triangle \begin{align*} \{(x_1,x_2)\in\mathbb R^2:x_1\ge 0,\ x_2\ge 0,\ x_1+x_2<\varepsilon\} \end{align*} is removed. Hence the post-blow-up local moment image is \begin{align*} \{(x_1,x_2)\in\mathbb R^2:x_1\ge 0,\ x_2\ge 0,\ x_1+x_2\ge\varepsilon\}. \end{align*} The distinction between the strict inequality for the removed ball and the non-strict inequality for the resulting polygon is exactly the boundary convention in the statement.[/guided]

[step:Choose the blow-up size so the cut affects only the chosen corner] Because $\Delta$ is a compact polygon and $v$ is a vertex, there is a neighbourhood $N\subset\mathbb R^2$ of $0$ in the chosen coordinates such that \begin{align*} A(\Delta)\cap N=N\cap\{x_1\ge 0,\ x_2\ge 0\}. \end{align*} Choose $\delta>0$ such that the closed triangle \begin{align*} T_\delta:=\{(x_1,x_2)\in\mathbb R^2:x_1\ge 0,\ x_2\ge 0,\ x_1+x_2\le \delta\} \end{align*} is contained in $N$. Also require $\delta<\pi r^2$, so that the corresponding capacity balls are contained in the Darboux chart. Set \begin{align*} \varepsilon_0:=\delta. \end{align*} Then for every $0<\varepsilon<\varepsilon_0$, the cut triangle lies inside the coordinate neighbourhood where the only facets of $\Delta$ present are the two facets adjacent to $v$, and the ball removed in the Darboux model lies inside $W$. [/step]

[step:Transport the local corner cut back to the toric surface] Let $0<\varepsilon<\varepsilon_0$. Through the equivariant symplectomorphism $\Phi$, the local blow-up of the standard model along $B_\varepsilon$ gives the toric symplectic blow-up of $(M,\omega,T^2,\mu)$ at $p$ of capacity $\varepsilon$. On $M\setminus U$, the construction is unchanged, so the moment image outside the chosen corner neighbourhood is the original polygon $\Delta$. Inside the coordinate neighbourhood, the preceding computation shows that the local moment image is obtained by imposing \begin{align*} x_1+x_2\ge \varepsilon. \end{align*} Therefore the blown-up moment polygon is \begin{align*} \Delta\cap\{x_1+x_2\ge\varepsilon\} \end{align*} in the chosen integral affine coordinates near $v$. The new facet has primitive inward conormal $(1,1)$ in these coordinates, so it is an integral affine facet. Its intersections with the adjacent coordinate facets occur at $(\varepsilon,0)$ and $(0,\varepsilon)$, hence the two adjacent facets are cut at integral-affine distance $\varepsilon$ from $v$. This proves the asserted corner-cut description of the toric blow-up. [/step]