[proofplan] We use the standard bubble-tree compactness argument for bounded-energy harmonic maps from a closed surface into a compact real-analytic target. Small-energy regularity gives smooth convergence away from finitely many concentration points, and rescaling at each concentration scale extracts harmonic spheres after removable singularity is applied. The positive energy gap makes the extraction process finite. Finally, the assumed no-neck estimate removes the only possible missing energy, so the total limiting energy is exactly the energy of the weak limit plus the energies of the bubbles. [/proofplan] [step:Locate the finite concentration set and obtain smooth convergence away from it] Let $E_0 := \sup_k E(u_k;\Sigma) < \infty$. Choose a small-energy regularity threshold $\varepsilon_0 > 0$ for harmonic maps from two-dimensional domains into $N$. For each $k$, define the energy measure $\mu_k$ on $\Sigma$ by \begin{align*} \mu_k(A) := E(u_k;A) \end{align*} for each Borel set $A \subset \Sigma$. Since $\Sigma$ is compact and $\mu_k(\Sigma) \leq E_0$, the [compactness theorem](/theorems/2748) for finite Radon measures gives a subsequence, not relabelled, and a finite Radon measure $\mu$ on $\Sigma$ such that $\mu_k \overset{*}{\rightharpoonup} \mu$. Define the concentration set by the atoms of this limiting energy measure: \begin{align*} S := \{p \in \Sigma : \mu(\{p\}) \geq \varepsilon_0/2\}. \end{align*} If $p_1,\dots,p_m \in S$ are distinct, then finite additivity of $\mu$ on disjoint finite sets gives \begin{align*} m\frac{\varepsilon_0}{2} \leq \sum_{i=1}^m \mu(\{p_i\}) \leq \mu(\Sigma) \leq E_0. \end{align*} Thus $S$ is finite; write $S=\{p_1,\dots,p_J\}$. Let $K \subset \Sigma \setminus S$ be compact. For each $q \in K$, choose a geodesic ball $B_g(q,2r_q)$ whose closure is contained in $\Sigma \setminus S$ and whose boundary has $\mu$-measure zero. Since $q \notin S$ and $\mu$ has no atom of size $\varepsilon_0/2$ at $q$, shrinking $r_q$ if necessary gives $\mu(\overline{B_g(q,2r_q)}) < \varepsilon_0$. [Weak convergence](/page/Weak%20Convergence) of the finite measures and the zero-boundary condition imply \begin{align*} \lim_{k\to\infty}\mu_k(B_g(q,2r_q)) = \mu(B_g(q,2r_q)) < \varepsilon_0. \end{align*} Hence, for all sufficiently large $k$, $E(u_k;B_g(q,2r_q)) < \varepsilon_0$. A finite subcover of $K$ by the smaller balls $B_g(q,r_q)$ and the small-energy [regularity theorem](/theorems/2750) give uniform $C^m$ bounds on $K$ for every integer $m \geq 0$. The hypotheses of small-energy regularity are satisfied because the maps $u_k$ are smooth harmonic maps from two-dimensional coordinate disks and the local energies are below $\varepsilon_0$. By the Arzela-Ascoli theorem applied in these coordinate charts and a diagonal argument over a compact exhaustion of $\Sigma \setminus S$, a further subsequence converges smoothly on compact subsets of $\Sigma \setminus S$ to a smooth harmonic map \begin{align*} u_\infty: \Sigma\setminus S &\to N. \end{align*} Lower semicontinuity of the Dirichlet energy gives $E(u_\infty;\Sigma\setminus S) \leq E_0$. More precisely, for every open neighbourhood $V_j\subset\Sigma$ of $p_j$ with compact closure in the chosen coordinate chart, lower semicontinuity on compact subannuli and exhaustion imply \begin{align*} E(u_\infty;V_j\setminus\{p_j\}) \leq \liminf_{k\to\infty} E(u_k;V_j) \leq E_0. \end{align*} The removable singularity theorem for finite-energy harmonic maps from punctured disks applies near each $p_j$, because a conformal coordinate chart identifies a punctured neighbourhood of $p_j$ with a punctured disk and the limiting map has finite energy there. Therefore $u_\infty$ extends smoothly across every $p_j$, so $u_\infty:\Sigma\to N$ is harmonic. [guided] The point of introducing energy measures is to avoid the defect caused by a double $\liminf$: eventual small energy is needed in order to apply small-energy regularity. For each Borel set $A \subset \Sigma$, define \begin{align*} \mu_k(A) := E(u_k;A). \end{align*} These are finite Radon measures on the compact surface $\Sigma$, and their masses satisfy $\mu_k(\Sigma) \leq E_0$. The compactness theorem for finite Radon measures therefore gives a subsequence, still denoted by $u_k$, and a finite Radon measure $\mu$ such that $\mu_k \overset{*}{\rightharpoonup} \mu$. We define concentration by the atoms of the limiting measure: \begin{align*} S := \{p \in \Sigma : \mu(\{p\}) \geq \varepsilon_0/2\}. \end{align*} This definition has the right consequence: away from $S$, sufficiently small balls have limiting energy strictly below the regularity threshold, and weak convergence of measures turns that limiting statement into eventual small energy along the chosen subsequence. The set $S$ is finite because, for distinct $p_1,\dots,p_m \in S$, \begin{align*} m\frac{\varepsilon_0}{2} \leq \sum_{i=1}^m \mu(\{p_i\}) \leq \mu(\Sigma) \leq E_0. \end{align*} Thus \begin{align*} m \leq \frac{2E_0}{\varepsilon_0}, \end{align*} so there are only finitely many concentration points. Now fix a compact set $K \subset \Sigma \setminus S$. For each point $q \in K$, choose a geodesic ball $B_g(q,2r_q)$ with closure contained in $\Sigma \setminus S$ and with $\mu(\partial B_g(q,2r_q))=0$. Such radii exist because a finite Radon measure can charge at most countably many geodesic spheres centered at $q$. Since $q \notin S$, we may shrink $r_q$ so that \begin{align*} \mu(\overline{B_g(q,2r_q)}) < \varepsilon_0. \end{align*} The zero-boundary condition lets us pass from weak convergence of measures to convergence of the ball masses: \begin{align*} \lim_{k\to\infty}E(u_k;B_g(q,2r_q)) = \lim_{k\to\infty}\mu_k(B_g(q,2r_q)) = \mu(B_g(q,2r_q)) <\varepsilon_0. \end{align*} Therefore the local energy is below the small-energy threshold for all sufficiently large $k$. The small-energy regularity theorem applies on these coordinate balls because each $u_k$ is a smooth harmonic map, the domain dimension is two, the target $N$ is compact, and the relevant ball energy is below $\varepsilon_0$. It gives uniform $C^m$ estimates on smaller balls for every integer $m \geq 0$. Taking a finite subcover of $K$ by the smaller balls and applying the Arzela-Ascoli theorem yields smooth convergence on $K$ after passing to a subsequence. A diagonal argument over a compact exhaustion of $\Sigma \setminus S$ gives a smooth harmonic limit \begin{align*} u_\infty: \Sigma\setminus S &\to N. \end{align*} Finally, $E(u_\infty;\Sigma\setminus S) \leq E_0$ by lower semicontinuity. Near each $p_j$, a conformal coordinate chart turns the punctured neighbourhood into a punctured disk, so the removable singularity theorem for finite-energy harmonic maps from punctured disks applies. Hence $u_\infty$ extends smoothly and harmonically across every point of $S$. [/guided] [/step] [step:Extract a finite bubble tree at each concentration point] Fix $p_j\in S$. Work in a conformal coordinate chart \begin{align*} \varphi_j: U_j \to B(0,\rho_j) \subset \mathbb{R}^2 \end{align*} with $\varphi_j(p_j)=0$. The chart is chosen so that $U_j$ contains no concentration point other than $p_j$. Let $\tilde u_k:B(0,\rho_j)\to N$ be the coordinate representation $\tilde u_k(x) := u_k(\varphi_j^{-1}(x))$. We use the following scale-selection form of bubble-tree compactness for bounded-energy harmonic maps from planar disks into a compact target. Define the threshold \begin{align*} \eta:=\frac{1}{4}\min\{\varepsilon_0,\varepsilon_{\mathrm{neck}}\}. \end{align*} After passing to a subsequence, an iterative selection of maximal disks whose energy is at least $\eta$ produces a finite rooted tree $\mathcal{T}_j$ of concentration scales, together with ghost vertices for the base scale and for branching points that carry no nonconstant limiting sphere. The lemma asserts only the following facts: each non-ghost selected vertex produces, after filling its finite removable child punctures, a nonconstant finite-energy harmonic plane; vertices satisfy the parent-child and scale-separation relations stated below; each edge of the tree has an associated conformal annular neck; and every dyadic annulus in those edge necks has energy less than $\varepsilon_{\mathrm{neck}}$ for all sufficiently large $k$. The lemma is independent of the energy identity: it is proved by selecting a new scale whenever such a high-energy dyadic annulus exists, and finiteness follows because each non-ghost selected scale carries at least $\eta$ energy while the total energy is at most $E_0$. The hypotheses of this lemma are satisfied here: the maps $\tilde u_k$ are smooth harmonic maps on the fixed disk, the conformal coordinate preserves the two-dimensional Dirichlet energy, the target $N$ is compact, and the energies are bounded by $E_0$. Thus, after passing to a further subsequence, each vertex $a\in\mathcal{T}_j$ consists of centers $x_{k,a}\to 0$ and radii $r_{k,a}\downarrow 0$ and produces a nonconstant finite-energy harmonic map $v_a:\mathbb{R}^2\to N$ obtained as the smooth local limit away from finitely many child punctures of the rescaled maps \begin{align*} v_{k,a}(y) := \tilde u_k(x_{k,a}+r_{k,a}y). \end{align*} The scale-exhaustion clause just stated gives no energy identity and no no-neck conclusion. It only says that, after all selected vertices have been recorded, any remaining dyadic annulus with energy at least $\varepsilon_{\mathrm{neck}}$ would have triggered another selected scale. The vanishing of the energy in those remaining necks will be supplied separately by the assumed no-neck estimate. For each vertex $a\in\mathcal{T}_j$, define \begin{align*} \Omega_{k,a}:=\{y\in\mathbb{R}^2:x_{k,a}+r_{k,a}y\in B(0,\rho_j)\}. \end{align*} The rescaled map $v_{k,a}:\Omega_{k,a}\to N$ is given by \begin{align*} v_{k,a}(y) := \tilde u_k(x_{k,a}+r_{k,a}y). \end{align*} For every Borel set $A\subset\Omega_{k,a}$, conformal invariance of the two-dimensional Dirichlet energy gives \begin{align*} E(v_{k,a};A)=E(\tilde u_k;x_{k,a}+r_{k,a}A). \end{align*} The compactness theorem supplies a finite set $Y_a\subset\mathbb{R}^2$ of child punctures such that $v_{k,a}$ converges smoothly on compact subsets of $\mathbb{R}^2\setminus Y_a$ to a nonconstant finite-energy harmonic map $v_a:\mathbb{R}^2\setminus Y_a\to N$. The convergence conclusion follows from small-energy regularity on compact sets avoiding $Y_a$; its hypotheses are verified because every such compact set has local energy below the regularity threshold after all child scales have been removed. Lower semicontinuity gives \begin{align*} 0<E(v_a;\mathbb{R}^2)\leq E_0. \end{align*} For each $y\in Y_a$, the finite energy bound on small punctured disks around $y$ lets us apply the removable singularity theorem for finite-energy harmonic maps from punctured disks. Thus $v_a$ extends smoothly and harmonically across every finite child puncture in $Y_a$, and we henceforth regard $v_a:\mathbb{R}^2\to N$ as a finite-energy harmonic map. Stereographic compactification identifies $\mathbb{R}^2\cup\{\infty\}$ with $S^2$. The same removable singularity theorem applies at $\infty$, because $v_a$ is harmonic on $\mathbb{R}^2$ and has finite energy there. Thus $v_a$ extends to a nonconstant smooth harmonic sphere $\omega_a:S^2\to N$. We index this sphere as $\omega_{j\ell}$, where $j$ records the original concentration point and $\ell$ records the non-ghost vertex of the finite bubble tree over $p_j$. If secondary concentration points occur for $v_{k,a}$, the same construction is repeated at those points. The parent-child relation is defined by containment of scales: a child bubble $b$ of $a$ has centers and radii satisfying \begin{align*} \frac{r_{k,b}}{r_{k,a}} \to 0, \qquad \frac{x_{k,b}-x_{k,a}}{r_{k,a}}\to y_b\in\mathbb{R}^2, \end{align*} where $y_b$ is the child puncture coordinate in the rescaled parent plane. If two bubbles are not in a parent-child relation, their scales are separated by \begin{align*} \frac{r_{k,a}}{r_{k,b}}+\frac{r_{k,b}}{r_{k,a}}+\frac{|x_{k,a}-x_{k,b}|}{r_{k,a}+r_{k,b}} \to \infty, \end{align*} so their core regions are disjoint for large $k$. Because $N$ is compact and real-analytic, the Sacks-Uhlenbeck energy gap theorem for harmonic two-spheres gives a number $\varepsilon_* >0$, depending only on $N$, such that every nonconstant smooth harmonic sphere $\omega:S^2\to N$ satisfies $E(\omega;S^2)\geq \varepsilon_*$. Its hypotheses are satisfied because each recorded bubble is a smooth harmonic map from $S^2$ into the compact real-analytic target $N$. Since the total energy available is at most $E_0$, the number of nonconstant bubbles is bounded by \begin{align*} \left\lfloor \frac{E_0}{\varepsilon_*}\right\rfloor. \end{align*} Hence the bubble tree above each $p_j$ is finite. [guided] The rigorous way to organize bubble extraction is to use the scale-selection form of bubble-tree compactness, rather than to choose necks informally. Fix $p_j\in S$ and use the conformal chart $\varphi_j:U_j\to B(0,\rho_j)\subset\mathbb{R}^2$ with $\varphi_j(p_j)=0$. Let $\tilde u_k:B(0,\rho_j)\to N$ be defined by $\tilde u_k(x):=u_k(\varphi_j^{-1}(x))$. Define \begin{align*} \eta:=\frac{1}{4}\min\{\varepsilon_0,\varepsilon_{\mathrm{neck}}\}. \end{align*} The selection lemma repeatedly records a scale whenever a remaining dyadic annulus carries at least $\varepsilon_{\mathrm{neck}}$ energy. Each non-ghost recorded scale carries at least $\eta$ energy, so the number of non-ghost selections is bounded by \begin{align*} \frac{E_0}{\eta}. \end{align*} Ghost vertices are allowed at the base scale and at branching points; they carry no bubble energy but organize the complement into genuine annular edge necks. This step extracts finite scales, child-puncture limits, and exhausted neck annuli only; the energy identity remains to be proved below. The theorem returns a finite rooted tree $\mathcal{T}_j$. Each vertex $a\in\mathcal{T}_j$ has centers $x_{k,a}\to0$ and radii $r_{k,a}\downarrow0$. Define \begin{align*} \Omega_{k,a}:=\{y\in\mathbb{R}^2:x_{k,a}+r_{k,a}y\in B(0,\rho_j)\}. \end{align*} The rescaled map $v_{k,a}:\Omega_{k,a}\to N$ is \begin{align*} v_{k,a}(y):=\tilde u_k(x_{k,a}+r_{k,a}y). \end{align*} For every Borel set $A\subset\Omega_{k,a}$, conformal invariance gives \begin{align*} E(v_{k,a};A)=E(\tilde u_k;x_{k,a}+r_{k,a}A). \end{align*} The theorem also supplies a finite child-puncture set $Y_a\subset\mathbb{R}^2$ such that $v_{k,a}$ converges smoothly on compact subsets of $\mathbb{R}^2\setminus Y_a$ to a nonconstant finite-energy harmonic map $v_a:\mathbb{R}^2\setminus Y_a\to N$. This convergence is exactly the small-energy regularity conclusion after every smaller child scale has been removed. Because $v_a$ has finite energy, lower semicontinuity gives \begin{align*} 0<E(v_a;\mathbb{R}^2)\leq E_0. \end{align*} The finite-energy bound allows the removable singularity theorem to be applied at every point of $Y_a$, so $v_a$ extends to a harmonic map on all of $\mathbb{R}^2$. Stereographic compactification identifies $\mathbb{R}^2\cup\{\infty\}$ with $S^2$. The removable singularity theorem applies again at $\infty$: in a punctured coordinate disk around $\infty$, the map is harmonic and has finite energy. Hence $v_a$ extends to a smooth nonconstant harmonic sphere $\omega_a:S^2\to N$. The same compactness theorem gives the scale relations. If $b$ is a child of $a$, then \begin{align*} \frac{r_{k,b}}{r_{k,a}} \to 0, \end{align*} and \begin{align*} \frac{x_{k,b}-x_{k,a}}{r_{k,a}}\to y_b\in\mathbb{R}^2. \end{align*} If two vertices are not in a parent-child relation, then \begin{align*} \frac{r_{k,a}}{r_{k,b}}+\frac{r_{k,b}}{r_{k,a}}+\frac{|x_{k,a}-x_{k,b}|}{r_{k,a}+r_{k,b}} \to \infty. \end{align*} These relations are what make the later base, bubble, and neck regions disjoint. Finally, because $N$ is compact and real-analytic, the Sacks-Uhlenbeck energy gap supplies $\varepsilon_*>0$ such that every nonconstant harmonic sphere into $N$ has energy at least $\varepsilon_*$. Since the total available energy is at most $E_0$, the number of recorded bubbles is at most \begin{align*} \left\lfloor \frac{E_0}{\varepsilon_*}\right\rfloor. \end{align*} [/guided] [/step] [step:Define disjoint base, bubble, and neck regions] Choose $\delta>0$ so small that the geodesic balls $B_g(p_j,\delta)$ are pairwise disjoint and contained in the coordinate neighbourhoods used above. Define the base region by \begin{align*} \Sigma_\delta := \Sigma\setminus \bigcup_{j=1}^J B_g(p_j,\delta). \end{align*} For a bubble $a$ with center $x_{k,a}$ and radius $r_{k,a}$ in the original coordinate disk, choose a parameter $R>1$. Its bubble core is the coordinate disk \begin{align*} \mathcal{B}_{k,a}(R) := B(x_{k,a},Rr_{k,a}). \end{align*} If $b$ is a child of $a$, define the parent-scale puncture removed from the parent core by \begin{align*} \mathcal{P}_{k,b,\operatorname{par}}(R) := B\left(x_{k,b},\frac{r_{k,a}}{R}\right). \end{align*} The child bubble core itself is $\mathcal{B}_{k,b}(R)=B(x_{k,b},Rr_{k,b})$. Since $r_{k,b}/r_{k,a}\to0$, for fixed $R$ and all sufficiently large $k$ one has \begin{align*} \mathcal{B}_{k,b}(R)\subset \mathcal{P}_{k,b,\operatorname{par}}(R)\subset \mathcal{B}_{k,a}(R). \end{align*} After increasing $R$ and then taking $k$ large, the scale-separation relations from the previous step make same-generation truncated cores disjoint. Thus the truncated bubble region for $a$ is \begin{align*} \mathcal{C}_{k,a}(R) := \mathcal{B}_{k,a}(R)\setminus \bigcup_{b\text{ child of }a}\mathcal{P}_{k,b,\operatorname{par}}(R). \end{align*} The neck regions are the annular edge regions of the selected tree. For an edge from a genuine parent $a$ to a child $b$, the edge neck is $\mathcal{P}_{k,b,\operatorname{par}}(R)\setminus\mathcal{B}_{k,b}(R)$. Edges from the ghost root at the base scale to top-level children are also part of the scale-selection output; they are chosen as disjoint conformal annuli in the coordinate disk around $p_j$. Thus a region with several top-level bubbles is not treated as one multiply punctured annulus. Because the tree is finite, we choose $R$ outside a finite exceptional set and then take $k$ large so that all displayed inclusions, all same-generation disjointness relations, and all root-core containment relations inside the balls $B_g(p_j,\delta)$ hold simultaneously. By induction from the leaves toward the root, each genuine parent core is partitioned, up to boundary circles, into its truncated core, its child cores, and the parent-child neck annuli. The ghost-root edge annuli handle the region between the base scale and top-level children. Adding these edge annuli and the base region $\Sigma_\delta$ gives a disjoint cover of $\Sigma$ up to finitely many smooth curves, which have zero Dirichlet energy measure. On $\Sigma_\delta$, smooth convergence implies convergence of the energy densities uniformly, because $u_k\to u_\infty$ in $C^1(\Sigma_\delta)$ and the metrics are fixed. Hence \begin{align*} \lim_{k\to\infty} E(u_k;\Sigma_\delta)=E(u_\infty;\Sigma_\delta). \end{align*} Let $\mathcal{A}$ denote the finite set of all recorded bubble vertices. For a bubble $a\in\mathcal{A}$, let $\operatorname{Ch}(a)\subset\mathcal{A}$ denote its finite set of children, and let $y_b\in\mathbb{R}^2$ be the child puncture coordinate defined by \begin{align*} \frac{x_{k,b}-x_{k,a}}{r_{k,a}}\to y_b. \end{align*} For $R>1$, define the cap-puncture set in the parent bubble plane by \begin{align*} Q_a(R):=\{y\in\mathbb{R}^2:|y|>R\}\cup\bigcup_{b\in\operatorname{Ch}(a)}B(y_b,1/R), \end{align*} and let $\mathcal{Q}_a(R)\subset S^2$ be its image under stereographic compactification of $\mathbb{R}^2\cup\{\infty\}$. Indeed, under the parent rescaling $y=(x-x_{k,a})/r_{k,a}$, the outer boundary of $\mathcal{B}_{k,a}(R)$ becomes $|y|=R$, while $\mathcal{P}_{k,b,\operatorname{par}}(R)$ converges to the disk $B(y_b,1/R)$. For each fixed $R$, conformal invariance and smooth convergence of the rescaled maps on compact subsets of $\mathbb{R}^2\setminus Q_a(R)$ give \begin{align*} \lim_{k\to\infty}E(u_k;\mathcal{C}_{k,a}(R)) = E(\omega_a;S^2\setminus \mathcal{Q}_a(R)). \end{align*} Since each $\omega_a$ is smooth on $S^2$, absolute continuity of its energy measure gives \begin{align*} \lim_{R\to\infty}E(\omega_a;S^2\setminus \mathcal{Q}_a(R))=E(\omega_a;S^2). \end{align*} It remains to identify the annuli as necks covered by the theorem's hypothesis. Define $\mathcal{N}_{k,\delta}(R)$ to be the disjoint union of the annular edge necks supplied by the selected tree. For a genuine parent-child edge, this includes \begin{align*} \mathcal{P}_{k,b,\operatorname{par}}(R)\setminus\mathcal{B}_{k,b}(R). \end{align*} For a parent-child annulus, the ratio of outer to inner radii is \begin{align*} \frac{r_{k,a}/R}{Rr_{k,b}}=\frac{r_{k,a}}{R^2r_{k,b}}\to\infty \end{align*} as $k\to\infty$ for fixed $R$. The ghost-root edge annuli also degenerate in the ordered limit $k\to\infty$ followed by $\delta\downarrow0$, by the scale-separation and exhaustion clauses of the selection lemma. Thus each component is a degenerating conformal annulus in the ordered limits used below. We verify the dyadic small-energy condition using the scale-selection exhaustion clause. Let $\varepsilon_{\mathrm{neck}}>0$ be the threshold in the no-neck hypothesis. For each connected neck component $A_{k,\alpha}(\delta,R)\subset\mathcal{N}_{k,\delta}(R)$, every dyadic subannulus is disjoint from the base region and from all truncated bubble cores by construction. If such a dyadic subannulus carried energy at least $\varepsilon_{\mathrm{neck}}$ along a subsequence, then the scale-selection algorithm applied to that annulus would choose a center and radius inside the component: rescaling the annulus to a unit-size disk and using small-energy regularity away from the high-energy dyadic ring produces either a nonconstant bubble or a child puncture. The selected scale is separated from all recorded vertices because the annulus lies in the complement of their cores and parent-scale punctures. This contradicts the exhaustion clause of the finite tree $\mathcal{T}_j$. Hence every dyadic subannulus in every neck component has energy below $\varepsilon_{\mathrm{neck}}$ for all sufficiently large $k$. Each component $A_{k,\alpha}(\delta,R)$ is a degenerating conformal annulus in the ordered limit: first $k\to\infty$, then $R\to\infty$ for non-ghost parent-child edges, and finally $\delta\downarrow0$ for ghost-root edges. In the conformal coordinate around the corresponding concentration point, a parent-child neck is translated by $x_{k,b}$ and scaled to a round annulus centered at the origin; a ghost-root edge neck is already one of the conformal annuli supplied by the selection lemma in the coordinate chart around $p_j$. The two-dimensional Dirichlet energy is conformally invariant, so these translations, scalings, and conformal coordinate changes preserve the energy and the dyadic small-energy hypothesis. The no-neck estimate therefore applies to each component, and the number of components is finite. Thus \begin{align*} \lim_{\delta\downarrow0}\lim_{R\to\infty}\limsup_{k\to\infty} E(u_k;\mathcal{N}_{k,\delta}(R))=0. \end{align*} [guided] The purpose of the truncation is to separate three kinds of energy: base energy, bubble energy, and neck energy. Choose $\delta>0$ so that the geodesic balls $B_g(p_j,\delta)$ are pairwise disjoint and contained in the chosen coordinate neighbourhoods, and define \begin{align*} \Sigma_\delta := \Sigma\setminus \bigcup_{j=1}^J B_g(p_j,\delta). \end{align*} For a bubble vertex $a$, choose $R>1$ and define its core by \begin{align*} \mathcal{B}_{k,a}(R):=B(x_{k,a},Rr_{k,a}). \end{align*} If $b$ is a child of $a$, define the parent-scale puncture removed from $a$ by \begin{align*} \mathcal{P}_{k,b,\operatorname{par}}(R):=B\left(x_{k,b},\frac{r_{k,a}}{R}\right). \end{align*} The child core is $\mathcal{B}_{k,b}(R)=B(x_{k,b},Rr_{k,b})$, and $r_{k,b}/r_{k,a}\to0$ gives $\mathcal{B}_{k,b}(R)\subset\mathcal{P}_{k,b,\operatorname{par}}(R)\subset\mathcal{B}_{k,a}(R)$ for large $k$. The scale-separation relations imply that, after increasing $R$ and then taking $k$ large, same-generation truncated cores are disjoint. Hence the truncated bubble region is \begin{align*} \mathcal{C}_{k,a}(R):=\mathcal{B}_{k,a}(R)\setminus\bigcup_{b\text{ child of }a}\mathcal{P}_{k,b,\operatorname{par}}(R). \end{align*} The selected tree may have ghost vertices at the base scale or at branching points. These vertices are bookkeeping devices, not bubbles in the final sum. Their role is to replace multiply punctured complementary regions by finitely many genuine conformal annuli, one for each tree edge. For genuine parent-child edges the annulus is the parent-scale puncture minus the child core. For ghost-root edges over top-level bubbles, the scale-selection lemma supplies disjoint conformal annuli in the coordinate disk around $p_j$. Because the tree is finite, we choose $R$ and then $k$ so that all inclusions and disjointness relations hold at once. The base region, the truncated real-bubble regions, and these annular edge necks cover $\Sigma$ up to finitely many boundary curves, and those curves have zero Dirichlet energy measure. On $\Sigma_\delta$, the smooth convergence $u_k\to u_\infty$ in $C^1(\Sigma_\delta)$ gives \begin{align*} \lim_{k\to\infty}E(u_k;\Sigma_\delta)=E(u_\infty;\Sigma_\delta). \end{align*} Let $\mathcal{A}$ be the finite set of bubble vertices, let $\operatorname{Ch}(a)$ be the children of $a$, and let $y_b\in\mathbb{R}^2$ be defined by \begin{align*} \frac{x_{k,b}-x_{k,a}}{r_{k,a}}\to y_b. \end{align*} Under the parent rescaling, $\mathcal{P}_{k,b,\operatorname{par}}(R)$ becomes a disk converging to $B(y_b,1/R)$. Define \begin{align*} Q_a(R):=\{y\in\mathbb{R}^2:|y|>R\}\cup\bigcup_{b\in\operatorname{Ch}(a)}B(y_b,1/R), \end{align*} and let $\mathcal{Q}_a(R)\subset S^2$ be its stereographic image. On $\mathbb{R}^2\setminus Q_a(R)$ the rescaled maps converge smoothly to $\omega_a$, so conformal invariance gives \begin{align*} \lim_{k\to\infty}E(u_k;\mathcal{C}_{k,a}(R))=E(\omega_a;S^2\setminus\mathcal{Q}_a(R)). \end{align*} Since $\omega_a$ is smooth, its energy measure is absolutely continuous and therefore \begin{align*} \lim_{R\to\infty}E(\omega_a;S^2\setminus\mathcal{Q}_a(R))=E(\omega_a;S^2). \end{align*} The neck set $\mathcal{N}_{k,\delta}(R)$ is the finite union of these annular edge components, including the parent-child annuli $\mathcal{P}_{k,b,\operatorname{par}}(R)\setminus\mathcal{B}_{k,b}(R)$ and the ghost-root edge annuli over top-level bubbles. For a parent-child annulus, the ratio of outer to inner radii is \begin{align*} \frac{r_{k,a}/R}{Rr_{k,b}}=\frac{r_{k,a}}{R^2r_{k,b}}\to\infty, \end{align*} so it degenerates for fixed $R$ as $k\to\infty$. Ghost-root edge annuli degenerate in the ordered limit $k\to\infty$ and then $\delta\downarrow0$. The scale-selection exhaustion property says that if a dyadic subannulus in one of these components had energy at least $\varepsilon_{\mathrm{neck}}$ along a subsequence, then that subannulus would generate an additional recorded scale separated from every recorded bubble. This contradicts exhaustion of the finite tree. Thus every dyadic subannulus has energy below the no-neck threshold for large $k$. After translating, scaling, and using conformal coordinates, each component has the form required in the no-neck hypothesis, and conformal invariance preserves its energy. The assumed no-neck estimate applies to each neck component, and since there are finitely many components, \begin{align*} \lim_{\delta\downarrow0}\lim_{R\to\infty}\limsup_{k\to\infty}E(u_k;\mathcal{N}_{k,\delta}(R))=0. \end{align*} [/guided] [/step] [step:Pass through the ordered limits in the energy decomposition] Fix $\delta>0$ and $R>1$ chosen as above, and then take $k$ large enough that the finite tree's inclusions and disjointness relations all hold. The inductive parent-child decomposition covers each concentration ball by ghost-root edge annuli, truncated bubble regions, descendant edge annuli, and descendant cores, while $\Sigma_\delta$ covers the complement of the concentration balls. Boundary circles have zero two-dimensional measure, so they do not affect Dirichlet energy. Additivity of the Dirichlet energy over these disjoint measurable regions gives \begin{align*} E(u_k;\Sigma) = E(u_k;\Sigma_\delta) +\sum_a E(u_k;\mathcal{C}_{k,a}(R)) +E(u_k;\mathcal{N}_{k,\delta}(R)), \end{align*} where $a$ ranges over the finitely many extracted bubbles and $\mathcal{N}_{k,\delta}(R)$ denotes the union of all neck annuli left by the truncation parameter $R$. First let $k\to\infty$ with $\delta$ and $R$ fixed, but use limsup and liminf rather than assuming that the neck term has a limit. The base term converges by $C^1$ convergence on $\Sigma_\delta$. Each bubble term converges by conformal invariance and $C^1$ convergence of the rescaled maps on the fixed truncated sphere. Since all terms in the decomposition are nonnegative, additivity gives the upper bound \begin{align*} \limsup_{k\to\infty}E(u_k;\Sigma) \leq E(u_\infty;\Sigma_\delta) +\sum_a E(\omega_a;S^2\setminus \mathcal{Q}_a(R)) +\limsup_{k\to\infty}E(u_k;\mathcal{N}_{k,\delta}(R)). \end{align*} The corresponding lower bound follows by dropping the nonnegative neck term before taking liminf: \begin{align*} \liminf_{k\to\infty}E(u_k;\Sigma) \geq E(u_\infty;\Sigma_\delta) +\sum_a E(\omega_a;S^2\setminus \mathcal{Q}_a(R)). \end{align*} Next take the ordered neck limit. Absolute continuity of the smooth energy measures of the finitely many bubble maps $\omega_a:S^2\to N$ removes the cap and child-puncture errors as $R\to\infty$. The no-neck estimate gives \begin{align*} \lim_{\delta\downarrow0}\lim_{R\to\infty}\limsup_{k\to\infty}E(u_k;\mathcal{N}_{k,\delta}(R))=0. \end{align*} Consequently, after first letting $R\to\infty$ and then $\delta\downarrow0$ in the upper bound, we obtain \begin{align*} \limsup_{k\to\infty}E(u_k;\Sigma) \leq \lim_{\delta\downarrow0}E(u_\infty;\Sigma_\delta)+\sum_{j=1}^J\sum_\ell E(\omega_{j\ell};S^2). \end{align*} The lower bound gives, for every fixed $\delta>0$, \begin{align*} \liminf_{k\to\infty}E(u_k;\Sigma) \geq E(u_\infty;\Sigma_\delta)+\sum_{j=1}^J\sum_\ell E(\omega_{j\ell};S^2). \end{align*} Finally, since $u_\infty$ is smooth on the closed surface $\Sigma$, absolute continuity of its energy measure gives \begin{align*} \lim_{\delta\downarrow 0}E(u_\infty;\Sigma_\delta)=E(u_\infty;\Sigma). \end{align*} Letting $\delta\downarrow0$ in the lower bound and using the preceding upper bound proves \begin{align*} \lim_{k\to\infty}E(u_k;\Sigma) = E(u_\infty;\Sigma)+\sum_{j=1}^J\sum_\ell E(\omega_{j\ell};S^2). \end{align*} This is the asserted energy quantization identity, with finitely many concentration points and finitely many nonconstant harmonic spheres. [guided] For fixed $\delta>0$ and $R>1$, the regions $\Sigma_\delta$, the truncated bubble regions $\mathcal{C}_{k,a}(R)$, and the neck union $\mathcal{N}_{k,\delta}(R)$ of annular tree-edge components are disjoint and cover $\Sigma$ up to boundaries of zero energy measure. Additivity gives \begin{align*} E(u_k;\Sigma) = E(u_k;\Sigma_\delta) +\sum_a E(u_k;\mathcal{C}_{k,a}(R)) +E(u_k;\mathcal{N}_{k,\delta}(R)). \end{align*} The base term converges because $u_k\to u_\infty$ in $C^1(\Sigma_\delta)$: \begin{align*} \lim_{k\to\infty}E(u_k;\Sigma_\delta)=E(u_\infty;\Sigma_\delta). \end{align*} For each bubble vertex $a$, conformal invariance and smooth convergence on the truncated sphere give \begin{align*} \lim_{k\to\infty}E(u_k;\mathcal{C}_{k,a}(R))=E(\omega_a;S^2\setminus\mathcal{Q}_a(R)). \end{align*} We do not assume the neck term has a limit. Taking limsup and using nonnegativity yields \begin{align*} \limsup_{k\to\infty}E(u_k;\Sigma) \leq E(u_\infty;\Sigma_\delta) +\sum_aE(\omega_a;S^2\setminus\mathcal{Q}_a(R)) +\limsup_{k\to\infty}E(u_k;\mathcal{N}_{k,\delta}(R)). \end{align*} Dropping the nonnegative neck term before taking liminf gives \begin{align*} \liminf_{k\to\infty}E(u_k;\Sigma) \geq E(u_\infty;\Sigma_\delta) +\sum_aE(\omega_a;S^2\setminus\mathcal{Q}_a(R)). \end{align*} Now let $R\to\infty$. Absolute continuity of the smooth bubble energy measures gives \begin{align*} \lim_{R\to\infty}E(\omega_a;S^2\setminus\mathcal{Q}_a(R))=E(\omega_a;S^2), \end{align*} and the no-neck estimate gives \begin{align*} \lim_{\delta\downarrow0}\lim_{R\to\infty}\limsup_{k\to\infty}E(u_k;\mathcal{N}_{k,\delta}(R))=0. \end{align*} Finally, since $u_\infty$ is smooth on $\Sigma$, absolute continuity gives \begin{align*} \lim_{\delta\downarrow0}E(u_\infty;\Sigma_\delta)=E(u_\infty;\Sigma). \end{align*} Combining the upper and lower bounds proves \begin{align*} \lim_{k\to\infty}E(u_k;\Sigma) = E(u_\infty;\Sigma)+\sum_{j=1}^J\sum_\ell E(\omega_{j\ell};S^2). \end{align*} [/guided] [/step]