Let $U \subset \mathbb{R}^n$ be open, let $V$ be a stationary integral $m$-varifold in $U$, and let $x_0 \in \operatorname{spt}\|V\|$. For $r>0$ with $B(x_0,r)\subset U$, define the density ratio
Then there exists a sequence $r_j \downarrow 0$ such that the rescaled varifolds
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\begin{align*}
V_j := (\eta_{x_0,r_j})_{\#}V,
\qquad
\eta_{x_0,r_j}: U \to \mathbb{R}^n,
\qquad
x \mapsto \frac{x-x_0}{r_j},
\end{align*}
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converge as integral varifolds on compact subsets of $\mathbb{R}^n$ to an integral varifold $C$. Every such tangent varifold $C$ is stationary in $\mathbb{R}^n$ and conical about the origin. Moreover, for every $R>0$,