[proofplan] Fix an interior support point $x_0 \in \operatorname{spt}T \setminus \operatorname{spt}\partial T$. The argument first passes to tangent cones of the area-minimizing integral current at $x_0$, then uses the classification of two-dimensional codimension-one area-minimizing cones in $\mathbb R^3$ to show that every tangent cone is a multiplicity-$q$ plane. This flat tangent cone triggers the multiplicity version of Allard interior regularity, giving a $C^{1,\alpha}$ graphical surface near $x_0$. Finally, the weak minimal surface equation and elliptic bootstrapping upgrade the graph to smoothness, and constancy of integer multiplicity on each connected sheet follows from the constancy theorem for integral currents. [/proofplan]

[step:Choose an interior ball on which the current has no boundary] Let $x_0 \in \operatorname{spt}T \setminus \operatorname{spt}\partial T$. For a nonempty set $A\subset\mathbb R^3$, let $\operatorname{dist}(x_0,A):=\inf\{|x_0-a|:a\in A\}$ denote the Euclidean distance from $x_0$ to $A$. Since $U\subset\mathbb R^3$ is open and $x_0\in U$, define \begin{align*} d_U:=\operatorname{dist}(x_0,\mathbb R^3\setminus U)>0. \end{align*} Since $x_0\notin\operatorname{spt}\partial T$ and $\operatorname{spt}\partial T$ is relatively closed in $U$, there exists $d_\partial>0$ such that \begin{align*} B(x_0,d_\partial)\cap U\cap\operatorname{spt}\partial T=\varnothing. \end{align*} Set \begin{align*} r_0:=\frac{1}{2}\min\{d_U,d_\partial\}>0. \end{align*} Then $B(x_0,r_0) \subset U$ and $B(x_0,r_0)\cap\operatorname{spt}\partial T=\varnothing$, so $\partial(T\llcorner B(x_0,r))=0$ in $B(x_0,r)$ for every $0<r<r_0$. The restriction $T\llcorner B(x_0,r)$ is still area minimizing among integral $2$-currents in $B(x_0,r)$ with the same boundary, because any compactly supported competitor in $B(x_0,r)$ is also an admissible interior competitor for $T$ in $U$. [/step]

[step:Identify every tangent cone as a multiplicity plane]For $0<r<r_0$, define the dilation map $\eta_{x_0,r}:\mathbb R^3\to\mathbb R^3$ by \begin{align*} \eta_{x_0,r}(y)=\frac{y-x_0}{r}. \end{align*} For each compact set $K\subset\mathbb R^3$ and all sufficiently small $r>0$, the set $x_0+rK$ is contained in $B(x_0,r_0)$; on such $K$, define the local blow-up by restricting $(\eta_{x_0,r})_{\#}T$ to $K$. Equivalently, for each fixed $R>0$ and all $r<r_0/R$, the blow-up in $B(0,R)$ is \begin{align*} (\eta_{x_0,r})_{\#}(T\llcorner B(x_0,Rr)). \end{align*} By the existence and integrality of tangent cones for area-minimizing integral currents, every sequence $r_j \downarrow 0$ has a subsequence, still denoted $r_j$, for which these local blow-ups converge as integral currents on compact subsets of $\mathbb R^3$ to an area-minimizing integral cone $C$ of dimension $2$ in $\mathbb R^3$. By the [classification of two-dimensional codimension-one area-minimizing cones in $\mathbb R^3$](/theorems/???), there exist an oriented $2$-plane $P \subset \mathbb R^3$ and an integer $q\in\mathbb N$ such that \begin{align*} C=q[[P]]. \end{align*} Here $[[P]]$ denotes the integral current of integration over $P$ with its chosen orientation. Since $x_0\in\operatorname{spt}T$, the density at $x_0$ is positive, so $q\geq 1$.[/step]

[guided]The purpose of the blow-up is to replace the possibly complicated current near $x_0$ by a scale-invariant limiting object. For each $0<r<r_0$, the map $\eta_{x_0,r}:\mathbb R^3\to\mathbb R^3$ defined by \begin{align*} \eta_{x_0,r}(y)=\frac{y-x_0}{r} \end{align*} translates $x_0$ to the origin and magnifies distances by the factor $r^{-1}$. The blow-up is understood locally: for each compact set $K\subset\mathbb R^3$ and all sufficiently small $r>0$, we restrict $(\eta_{x_0,r})_{\#}T$ to $K$; equivalently, on a fixed ball $B(0,R)$ we use $(\eta_{x_0,r})_{\#}(T\llcorner B(x_0,Rr))$ once $r<r_0/R$. We now verify the hypotheses of the existence and integrality theorem for tangent cones of area-minimizing integral currents. The current $T$ is an area-minimizing integral $2$-current by hypothesis. The previous step gives an interior ball $B(x_0,r_0)\subset U$ such that $\partial(T\llcorner B(x_0,r))=0$ in $B(x_0,r)$ for every $0<r<r_0$, and the same step also shows that these restrictions remain area minimizing for compactly supported competitors inside the ball. Thus the tangent-cone theorem applies at $x_0$: the blow-ups have subsequential limits on compact subsets, and each limit is an area-minimizing integral cone $C$ of dimension $2$ in $\mathbb R^3$. The special dimension and codimension now matter. The [classification of two-dimensional codimension-one area-minimizing cones in $\mathbb R^3$](/theorems/???) says that such a cone cannot have a genuine singular cone structure: it must be an integer multiple of an oriented plane. Thus there are an oriented $2$-plane $P\subset\mathbb R^3$ and an integer $q\in\mathbb N$ for which \begin{align*} C=q[[P]]. \end{align*} The notation $[[P]]$ means the integral current associated to integration over the oriented plane $P$. The integer $q$ is nonzero because $x_0\in\operatorname{spt}T$; if $q=0$, the tangent cone would have zero mass in every ball, contradicting positive density at a support point of an integral current.[/guided]

[step:Apply flat tangent cone regularity with multiplicity]Choose one tangent cone $C=q[[P]]$ obtained in the previous step; no uniqueness of tangent cones is being used, only the existence of one blow-up sequence converging to a flat multiplicity-$q$ plane. We apply the flat-tangent-cone [regularity theorem](/theorems/2750) for codimension-one area-minimizing integral currents. Its hypotheses are: the current is integral and area minimizing in an open ball, has no boundary in that ball, and has at the base point a tangent cone equal to $q[[P]]$ for an oriented plane $P$ and an integer $q\geq 1$. These hypotheses hold at $x_0$: integrality and area minimality are part of the theorem statement, the first step gives a ball $B(x_0,r_0)$ on which the current has no boundary and remains area minimizing for compactly supported competitors, and the previous step gives the tangent cone $q[[P]]$. The regularity theorem includes the varifold compactness and excess-improvement conclusion needed here: it rules out cancellation in the blow-up limit by using area minimality and monotonicity, gives convergence of the associated mass measures along the flat blow-up sequence, and produces a scale at which the density, mass ratio, height excess, tilt excess relative to $P$, stationarity, zero mean curvature, and no-boundary hypotheses in the multiplicity-$q$ Allard theorem hold in the theorem's quantitative form. Applying its conclusion, there exist $0<r_1<r_0$, an [open set](/page/Open%20Set) $V\subset P$, a number $\alpha\in(0,1)$, and a map \begin{align*} f: V &\to P^\perp \end{align*} of class $C^{1,\alpha}$ such that \begin{align*} \Sigma_1:=\{z+f(z):z\in V\} \end{align*} is an embedded $C^{1,\alpha}$ surface, relatively closed in $B(x_0,r_1)$ and without boundary in $B(x_0,r_1)$. The integral-current conclusion of the same theorem gives the oriented current identity \begin{align*} T\llcorner B(x_0,r_1)=q[[\Sigma_1]], \end{align*} where $[[\Sigma_1]]$ denotes integration over $\Sigma_1$ with the orientation induced from $P$ through the graph parametrization.[/step]

[guided]This is the delicate point in the proof: current convergence alone is not enough to pass from a flat blow-up to small excess, because integral currents can converge with cancellation. We therefore invoke the flat-tangent-cone regularity theorem for codimension-one area-minimizing integral currents, whose proof combines monotonicity, lower semicontinuity of mass, the constancy of density for the limiting flat cone, and Allard-type excess improvement. We verify the theorem's hypotheses one by one. The current $T$ is an integral $2$-current and is area minimizing in $U$ by the theorem statement. The first step produced a radius $r_0>0$ such that $B(x_0,r_0)\subset U$, $\partial(T\llcorner B(x_0,r))=0$ in $B(x_0,r)$ for every $0<r<r_0$, and the restrictions remain area minimizing for compactly supported competitors. The tangent-cone step produced an oriented plane $P\subset\mathbb R^3$ and an integer $q\geq 1$ such that one tangent cone at $x_0$ is \begin{align*} C=q[[P]]. \end{align*} Thus the input data match the theorem exactly: an integral area minimizer, no interior boundary, and a multiplicity-$q$ flat tangent cone. The theorem's quantitative content is stronger than ordinary current convergence. It asserts that area minimality prevents mass cancellation in the blow-up sequence and yields convergence of the associated mass measures to the mass measure of $q[[P]]$. Consequently, after passing to a sufficiently small physical scale, the mass ratio is within the required tolerance of $q$, the height excess over $P$ is small, the tilt excess relative to $P$ is small, the generalized mean curvature is zero because the current is area minimizing, and there is no boundary in the ball. These are precisely the density, mass-ratio, height/excess, stationarity, mean-curvature, and no-boundary hypotheses used in the multiplicity-$q$ Allard regularity conclusion. The conclusion is not merely that the support is graphical as a varifold. The integral-current version identifies the current itself. Hence there exist $0<r_1<r_0$, an open set $V\subset P$, a number $\alpha\in(0,1)$, and a $C^{1,\alpha}$ map \begin{align*} f: V &\to P^\perp \end{align*} such that \begin{align*} \Sigma_1:=\{z+f(z):z\in V\} \end{align*} is an embedded $C^{1,\alpha}$ surface, relatively closed in $B(x_0,r_1)$ and without boundary in $B(x_0,r_1)$, and the current is exactly \begin{align*} T\llcorner B(x_0,r_1)=q[[\Sigma_1]]. \end{align*} The orientation on $\Sigma_1$ is the one transported from the oriented plane $P$ by the graph parametrization $z\mapsto z+f(z)$, so the notation $[[\Sigma_1]]$ is an integral current, not just a set-theoretic support.[/guided]

[step:Upgrade the graphical regularity to a smooth minimal surface] The graph $\Sigma_1$ is stationary for area in $B(x_0,r_1)$ because $T\llcorner B(x_0,r_1)$ is area minimizing there. In coordinates on $P$, the function $f:V\to P^\perp$ is scalar-valued after choosing a unit vector in the one-dimensional space $P^\perp$, and stationarity gives the weak minimal surface equation \begin{align*} \operatorname{div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}}\right)=0 \end{align*} on $V$. Since $f\in C^{1,\alpha}(V)$, the coefficient matrix obtained by linearizing this uniformly elliptic quasilinear equation is Hölder continuous on compact subsets of $V$. By [elliptic smoothness for graphical solutions of the minimal surface equation](/theorems/???), $f\in C^\infty(V)$. Hence $\Sigma_1$ is a smooth embedded minimal surface in $B(x_0,r_1)$. [/step]

[step:Conclude regularity and local constancy of multiplicity] Set $r:=r_1$ and $\Sigma:=\Sigma_1$. The preceding steps give \begin{align*} T\llcorner B(x_0,r)=q[[\Sigma]], \end{align*} where $q\in\mathbb N$ and $\Sigma$ is a smooth embedded minimal surface that is relatively closed in $B(x_0,r)$ and has no boundary in $B(x_0,r)$. This is precisely regularity of the interior point $x_0$. On any connected regular sheet $\Sigma'\subset \operatorname{spt}T\setminus\operatorname{spt}\partial T$, the multiplicity function is integer-valued and locally constant by the [constancy theorem for integral currents on connected oriented manifolds](/theorems/???), applied to the boundaryless current obtained by restricting $T$ to coordinate patches of $\Sigma'$. Therefore away from $\operatorname{spt}\partial T$, the support of $T$ is a smooth minimal surface with locally constant integer multiplicity on each connected regular sheet. [/step]