Proof. Put \(F=\widehat{\mathbb C}\setminus G\). To apply Moore’s decomposition theorem, we check that the decomposition relation is closed and that every element is nonseparating. Suppose first that \(x_n,y_n\) belong to the same decomposition element and converge to \(x,y\). If infinitely many of those elements are singletons, then \(x=y\). Otherwise, after discarding finitely many terms, let \(F_n\) be the component of \(F\) containing \(x_n,y_n\). The hyperspace of nonempty compact subsets of the sphere is compact, so a subsequence of \(F_n\) converges in Hausdorff distance to a nonempty compact set \(A\subset F\). This limit is connected and contains \(x,y\). Thus \(x,y\) lie in one component of \(F\), which proves closedness of the relation.

Now let \(Q\) be a component of \(F\). The connected set \(G\) lies in one component of \(\widehat{\mathbb C}\setminus Q\). Any other component \(V\) would be disjoint from \(G\), and its nonempty boundary would lie in \(Q\). The connected set \(Q\cup\overline V\subset F\) would then strictly contain \(Q\), a contradiction. Hence \(Q\) is nonseparating. Moore’s theorem now identifies \(X\) with a topological sphere; see [14]. The quotient is closed and has connected fibers. Since \(G\) is open and saturated, \(\pi|_G\) is a homeomorphism onto an open subset.

It remains to prove the inverse-image assertion and total disconnectedness of \(B\). Let \(A\subset X\) be a continuum. If \(\pi^{-1}(A)=P\cup Q\) were a separation into disjoint nonempty compact sets, every fiber over \(A\) would lie entirely in one of \(P,Q\). Consequently \(\pi(P)\) and \(\pi(Q)\) would be disjoint nonempty compact sets separating \(A\), a contradiction. A continuum contained in \(B\) therefore has connected inverse image in \(F\), which must lie in a single component of \(F\). The continuum is consequently a singleton. Finally, \(B\) is the compact image of \(F\). ◻

Proof. We use two facts about compact spaces. First, a component of a compact Hausdorff space is the intersection of its relatively clopen neighborhoods. Second, if \(C\) is a continuum and \(P\) is a proper closed subset of \(C\), every component of \(P\) meets the boundary of \(P\) relative to \(C\). Indeed, a component missing that compact boundary could be separated from it by a relatively clopen subset of \(P\). That subset would be both open and closed in \(C\), a contradiction. The second fact is the boundary-bumping principle.

For (i), suppose a nondegenerate continuum \(C\subset F\cup B\) contains \(x\notin F\). Choose \(y\in C\setminus\{x\}\) and a small closed metric neighborhood \(N\) of \(x\) missing both \(F\) and \(y\). The component of \(C\cap N\) containing \(x\) meets the boundary of \(N\) by boundary bumping. It is therefore a nondegenerate subcontinuum contained in \(B\), contradicting Lemma 2.

For the remaining parts, we first describe a common contour construction on the physical sphere. If \(A_0\) is a compact subset of a physical open set \(O\) away from infinity, a smooth cutoff supported in \(O\) and equal to one near \(A_0\) has a regular level between zero and one. This level is a compact smooth one-manifold, so it has finitely many components, each a smooth Jordan curve. The corresponding superlevel set contains \(A_0\) and has compact closure in \(O\). Taking its outer boundary curves fills its holes. If \(O\subset\pi^{-1}(V)\) for a Jordan domain \(V\subset X\setminus\{\infty\}\), each filled outer contour still lies in \(\pi^{-1}(V)\). Indeed, \(\pi^{-1}(X\setminus V)\) is connected by Lemma 2, contains infinity, and misses the contour; it must therefore lie on its unbounded side.

For (ii), take a Jordan neighborhood \(V\) of \(x\) whose closure is inside the prescribed neighborhood and misses infinity. By the first compact-space fact above, there is a compact relatively clopen set \(A_0\subset F\) with \(x\in A_0\subset V\). Choose a physical open neighborhood of \(\pi^{-1}(A_0)\) compactly contained in \[\pi^{-1}(V)\setminus\pi^{-1}(F\setminus A_0).\] Apply the contour construction there. Its boundary curves miss \(\pi^{-1}(F)\). The connected fiber \(\pi^{-1}(x)\) lies inside one of the resulting outer contours. Its filled disk projects to the required dust disk. Its closure is contained in \(V\) by the connected-complement observation above.

For (iii), the compact metric totally disconnected space \(B\) has a clopen base. Partition \(A\) into finitely many compact relatively clopen pieces \(A_1,\ldots,A_m\), each contained in a Jordan neighborhood \(V_i\) with closure in \(W\setminus(B\setminus A)\) and diameter less than \(\varepsilon\). The compact physical preimages of these pieces are pairwise disjoint. They therefore have pairwise disjoint open neighborhoods with disjoint compact closures, each lying in \(\pi^{-1}(V_i)\) and avoiding \(\pi^{-1}(B\setminus A_i)\). Make physical contours inside these neighborhoods. The boundary curves obtained for all pieces are mutually disjoint and miss the whole physical complement of \(G\). Their filled disks are either nested or have disjoint closures. Discard the nested ones and any disk missing \(A\). The remaining disks still cover \(A\), have disjoint closures, and each stays inside its corresponding \(V_i\).

For (iv), let \(Q\) be the union of the given closed disks. Its boundary is contained in the union of the given boundaries and hence misses \(F\). Use the regular-level construction to smooth a sufficiently small physical outer neighborhood of \(\pi^{-1}(Q)\). Take the smoothing collar around the boundary of \(\pi^{-1}(Q)\) disjoint from \(\pi^{-1}(F)\) and keep the neighborhood’s closure in \(\pi^{-1}(V)\). There are finitely many smooth boundary curves by the compact one-manifold argument above. Fill the outer contours and discard nested contours. The connected-complement observation keeps all filled disks in \(V\), and all new boundaries remain in \(G\setminus F\). ◻

Proof. We first prove the weighted estimate and finite length, then derive the topological conclusions from that length bound. Apply the coarea formula to the smooth circle-valued map \(u_i\pmod1\) on its masked domain; see [5]. Mask-boundary portions contribute zero after integration in the level parameter: each point of that smooth boundary occurs at just one level. Thus the domain term in Equation (2) equals its stated area integral. The dust term is bounded by Equation (1) and Tonelli’s theorem; only the countable support of \(m\) is involved. Cauchy–Schwarz bounds the right side when \(q\in L^2\). Taking \(q=1\) and \(m=0\) gives finite length for almost every level, since the spherical area is finite. Summing over the finite list proves (i).

For (ii), only compactness of \(H\) and the finite length in (i) are needed. Fix \(\varepsilon>0\) and choose a core \(K\) whose complementary dust disks have \(d_X\)-diameter less than \(\varepsilon\). Every continuum \(Q\subset H\) with \(\mathop{\mathrm{diam}}_X Q\ge\varepsilon\) meets \(K\): otherwise its connectedness places it inside one of those disjoint disks. Compactness of the physical set \(K\subset G\) and uniform continuity of \(\pi\) give \(r>0\) such that every closed spherical \(r\)-ball centered on \(K\) is contained in \(G\) and has image of \(d_X\)-diameter less than \(\varepsilon\). Decrease \(r\) so these balls are ordinary embedded spherical disks.

Choose \(p\in Q\cap K\). Since the closed spherical \(r\)-ball about \(p\) has quotient diameter less than \(\varepsilon\), \(Q\) exits this ball. By boundary bumping, the component of the intersection containing \(p\) meets the ball boundary. Spherical distance from \(p\), a \(1\)-Lipschitz function, maps this subcontinuum onto \([0,r]\). Therefore \[\mathcal H^1_{\mathrm{sph}}(Q\cap G)\ge r.\] For pairwise disjoint continua these measures add. Their number is at most \(\mathcal H^1_{\mathrm{sph}}(H\cap G)/r\), proving (ii).

The bound in (ii) now gives (iii). For each \(n\) there are only finitely many components of \(H\) with diameter at least \(1/n\), proving countability of the nondegenerate components. For local connectedness of every subcontinuum, let \(L\subset H\) be any subcontinuum, let \(p\in L\), and take a sufficiently small radius \(t>0\). Let \(A\) be the component through \(p\) of \(L\cap\overline B_X(p,t)\). If \(A\) were not a relative neighborhood of \(p\) in \(L\), there would be points \(p_n\to p\) in other components of this closed intersection. After passing to a subsequence those components are distinct: a finite union of components not containing \(p\) is compact and misses \(p\). If the closed intersection is all of \(L\), it has only one component, so this case is already excluded. Boundary bumping now shows that each of the distinct components meets \(\{x:d_X(x,p)=t\}\). For large \(n\) it consequently has diameter at least \(t/2\), in contradiction to (ii). Hence \(A\) is a connected neighborhood of \(p\). Arbitrarily small choices of \(t\) prove that \(L\) is locally connected.

Finally, a connected open subset \(V\) of a locally connected metric continuum is arcwise connected. One can see path connectedness directly by chaining connected relatively open sets of small diameter with closures in \(V\), and then recursively refining each link inside its predecessor. Connectedness guarantees finite chains between specified endpoints at every refinement. Choose the parameter subdivisions and image diameters to tend to zero. Completeness gives a continuous path between the endpoints whose image remains in \(V\). The elementary loop-erasure theorem for a path in a Hausdorff space then supplies a simple arc with the same endpoints inside its image. These standard continuum facts are also recorded in [21]. ◻

Proof. For \(\delta>0\), regularity of the finite Borel measure on the compact metric space \(L\) gives a relatively open set \(O\) with \[F\subset O\subset V,\qquad \nu(O)<\delta.\] Local connectedness makes each component \(O_k\) of \(O\) relatively open. There are at most countably many components, because disjoint nonempty open sets in a second-countable space are countable. Any two points of \(F\cap O_k\) can be joined by a simple arc in \(O_k\). Consequently \[\mathop{\mathrm{diam}}v(F\cap O_k)\le A\nu(O_k).\] Cover these real images by intervals, allowing arbitrarily small summable excess lengths. Countable subadditivity gives \[|v(F)|^*\le A\sum_k\nu(O_k)=A\nu(O)\le A\delta.\] Letting \(\delta\) tend to zero proves the first assertion. Apply it separately to the countably many local branches for the second. ◻

Proof. We first replace the coordinates outside a large core by shifted staircases. Compatibility will provide disks on whose collars at most one staircase varies. We can then fill those disks so that the extended differentials have rank at most one, and choose a metric from these differentials.

We regard \(\mathrm du_i\) as zero wherever its mask is absent; every use of this convention is away from mask boundaries. Fix \(\eta>0\), to be chosen in terms of \(\varepsilon\) at the end.

Step 1: random staircases. Choose a smooth core \(K\) containing \(K_0\), all mask boundaries, and the exterior \(\{|z|\ge R\}\) in its interior. Enlarge it until the original energies outside \(K\) are so small that \[ \sum_{i<j}\int_{G\setminus K} |\mathrm du_i\wedge\mathrm du_j| \le \sum_{i<j} \|\mathrm du_i\|_{2,G\setminus K} \|\mathrm du_j\|_{2,G\setminus K} <\frac{\eta}{8}. \tag{5}\] The mask intersections are implicit in these integrals. Take \(\chi\in C^\infty(G,[0,1])\) equal to \(0\) on a neighborhood of \(K\) and equal to \(1\) outside a larger compact set.

The staircases will vary only near a finite grid of levels modulo one. Let \(h=1/N\), where \(N\) is a positive integer. Independently for each \(i\), choose \(\theta_i\) uniformly in \([0,h)\). For \(0<\delta<h/4\) take a smooth nondecreasing staircase \(P_{i,\delta}\) with \[P_{i,\delta}(x+h)=P_{i,\delta}(x)+h, \qquad |P_{i,\delta}(x)-x|\le h,\] whose derivative is supported within distance \(\delta\) of \(\theta_i+h\mathbb Z\). It is obtained by translating one fixed staircase with an \(h\)-periodic derivative of integral \(h\) over a period. Therefore \[ \mathop{\mathrm{\mathbb E}}P_{i,\delta}'(x)=1 \quad\text{for every }x. \tag{6}\] Because \(1=Nh\), it also satisfies \(P_{i,\delta}(x+1)=P_{i,\delta}(x)+1\).

Use the cutoff to join each staircase to the original coordinate: \[ v_{i,\delta} =u_i+\chi\bigl(P_{i,\delta}(u_i)-u_i\bigr). \tag{7}\] The formula is interpreted in local branches and respects integer translations, so it is consistent for periodic coordinates. Set \[q_{i,\delta}=1-\chi+\chi P_{i,\delta}'(u_i), \qquad r_{i,\delta}=P_{i,\delta}(u_i)-u_i.\] Then \(q_{i,\delta}\ge0\), \(|r_{i,\delta}|\le h\), and \[\mathrm dv_{i,\delta} =q_{i,\delta}\,\mathrm du_i+r_{i,\delta}\,\mathrm d\chi.\] Equation (6) and independence for distinct labels give \(\mathop{\mathrm{\mathbb E}}q_{i,\delta}=1\) and \(\mathop{\mathrm{\mathbb E}}(q_{i,\delta}q_{j,\delta})=1\) for \(i\ne j\). Consequently the wedge sum \[W_\delta =\sum_{i<j}\int_{G\setminus K} |\mathrm dv_{i,\delta}\wedge\mathrm dv_{j,\delta}|\] satisfies \[ \mathop{\mathrm{\mathbb E}}W_\delta \le \sum_{i<j}\int_{G\setminus K}|\mathrm du_i\wedge\mathrm du_j| +(m-1)h\sum_i\int_{\mathop{\mathrm{supp}}\mathrm d\chi} |\mathrm d\chi|\,|\mathrm du_i|\,\mathrm da. \tag{8}\] The last integrals are finite because \(\mathop{\mathrm{supp}}\mathrm d\chi\) is compact in \(G\). With \(K\) and \(\chi\) fixed, choose \(h\) small enough that the right side is less than \(\eta/4\), uniformly in \(\delta\).

Step 2: choosing the grids and terminal indices. For the random shifts above, put \[F_i^{\rm grid} =\bigcup_{k=0}^{N-1}H_i(\theta_i+kh).\] Each grid is compact. In addition to \(F\), retain as terminal indices all flagged points lying in the grids of two distinct labels: \[S=F\cup \bigcup_{i<j}\bigl(\Sigma\cap F_i^{\rm grid} \cap F_j^{\rm grid}\bigr), \qquad J=|S|.\] At a nonsingular dust point, a band of length at most \(a_i(s)\) meets the translated lattice with probability at most \(\min(1,a_i(s)/h)\). The same bound holds at the singular point because there \(a_i(s)=1\). Thus, by independence, \[ h^2\mathop{\mathrm{\mathbb E}}J \le h^2|F|+ \sum_{i<j}\sum_{s\in\Sigma} \min(h,a_i(s))\min(h,a_j(s)) \longrightarrow0\qquad(h\downarrow0). \tag{9}\] Each summand tends to zero and is bounded by \(a_i(s)a_j(s)\), whose sum is finite by Cauchy–Schwarz. Dominated convergence proves the limit. The same estimate gives \(J<\infty\) almost surely for each \(h\).

Decrease \(h\), if necessary, so that \(h^2\mathop{\mathrm{\mathbb E}}J<\eta^2/4\) as well. Fix a deterministic sequence \(\delta_k\downarrow0\) with \(\delta_k<h/4\). Fatou’s Lemma and Equation (8) give \[\mathop{\mathrm{\mathbb E}}\bigl(\liminf_{k\to\infty}W_{\delta_k}\bigr)<\eta/4.\] Markov’s inequality shows that a set of positive probability satisfies both \[ \liminf_{k\to\infty}W_{\delta_k}<\eta, \qquad h\sqrt J<\eta. \tag{10}\]

We also require compatibility (P) for every choice of one lattice level for each of any collection of distinct labels. Each such tuple has an absolutely continuous distribution in its level variables, and there are only finitely many choices. Hence this additional requirement has probability one. Fix grids satisfying it and Equation (10).

The grids and their finite set \(S\) are now fixed. Choose terminal disks about the points of \(S\), with pairwise disjoint closures, in the region where \(\chi=1\). They contain no other point of \(S\), lie in the prescribed dust neighborhoods, and avoid all mask boundaries. For every nonsingular coordinate present at \(s\), its values in a sufficiently small neighborhood of \(s\), in an appropriate local lift, lie in its band enlarged by \(h\) at each end. We therefore choose \(T_s\) so that \[\mathop{\mathrm{osc}}_{\partial T_s}u_i\le a_i(s)+2h.\] The bound \(|P_{i,\delta}(x)-x|\le h\) gives, simultaneously for every \(\delta\), \[ \mathop{\mathrm{osc}}_{\partial T_s}v_{i,\delta}\le a_i(s)+4h. \tag{11}\] No real oscillation bound is required at a singular index. The terminal disks are now fixed, while the staircase width can still decrease.

Step 3: filling disks whose boundaries meet at most one grid. We claim that the remaining dust has a finite cover by disks, with pairwise disjoint closures and disjoint from the terminal disks, such that the boundary of each new disk meets at most one \(F_i^{\rm grid}\). All these disks can be placed where \(\chi=1\), inside the prescribed dust neighborhoods, and away from mask boundaries.

First observe that if \(s\in B\cap F_j^{\rm grid}\) is not a terminal index, then \(s\) is a singleton component of \[ B\cup\bigcup_{i<j}F_i^{\rm grid}. \tag{12}\] If \(s\) is flagged, it does not belong to any lower grid, by the choice of \(S\). If \(s\) is free, each coordinate defined at \(s\) has a unique trace. Thus, near \(s\), only one level from each of the grids occurring at \(s\) can be present; the other levels are closed and miss \(s\). If a nondegenerate component of the lower grid union contained \(s\), intersect that component with a sufficiently small closed neighborhood of \(s\). Boundary bumping makes the component of this intersection through \(s\) nondegenerate. It uses only the lower levels present at \(s\), contradicting the chosen instance of compatibility (P). Lemma 3(i) shows that adjoining \(B\) cannot create another nondegenerate continuum through \(s\). This proves the singleton assertion in both cases.

Process the labels in decreasing order. At the stage for \(j\), the remaining dust is a compact relatively clopen subset of \(B\), disjoint from the already chosen closed disks and from every higher grid. Use Lemma 3 to cover it by finitely many disjoint small preliminary disks whose closures avoid these obstacles. In each such disk \(V\), the remaining points of \(F_j^{\rm grid}\cap B\) form a compact set. Each is a singleton component of the closed set in Equation (12); hence Lemma 3 encloses it in a smaller disk with closure in \(V\) and boundary avoiding that closed set.

Take finitely many such disks covering the compact target. To merge them while preserving boundary avoidance, perturb their finitely many smooth boundaries into general position within the open complement of the obstacle, take the outer contours of their union, and smooth the corners there. Fill bounded complementary regions and discard nested contours. Every resulting disk stays in \(V\), because the complement of \(V\) is connected and contains infinity. The resulting closures are pairwise disjoint, their interiors still cover the target, and their boundaries avoid all lower grids. They also avoid the higher grids, since their closures lie in \(V\). Add these disks to the chosen collection and continue. After the last label, cover the dust still remaining by disks avoiding all the grids. This proves the claim. Call all disks chosen in this step filling disks.

Step 4: smooth fillings of rank at most one. The finite collection of terminal and filling disks covers \(B\) in its interiors. Its complement is therefore a compact subset of \(G\). On each filling boundary, every coordinate except possibly one avoids its grid. Choose connected collars, compact in \(G\), on which this avoidance still holds and \(\chi=1\). Every singular point lies in its assigned terminal disk, so the coordinates have single-valued lifts on each filling collar. When the collar avoids \(F_i^{\rm grid}\), a real lift has connected image in one interval complementary to the level lattice \(\theta_i+h\mathbb Z\). Compactness of the collar then makes its staircase constant for all sufficiently small \(\delta\). Call these constant coordinates inactive on the collar. There are only finitely many collars, so one width bound works for all of them.

By Equation (10) we may choose one of the \(\delta_k\) small enough for all these collar conditions and with \(W_{\delta_k}<\eta\). Fix this width and suppress it from the notation. Retain \(v_i\) on the complement of the selected disks. Across each filling disk extend the inactive coordinates constantly. Extend the one possibly varying coordinate from an inside collar by a smooth cutoff, making it constant on a smaller interior disk. These extensions agree with the retained coordinates on full collars and are therefore smooth. Their differentials have rank at most one at every point of a filling disk, so \[ \mathrm du_i^*\wedge\mathrm du_j^*=0 \quad\text{on every filling disk.} \tag{13}\] Here \(u_i^*\) denotes the resulting coordinate on the completed bordered surface \(Q\).

To check the periods, pull back by \(\pi\) the cyclic cover of the punctured mask. This cover is trivial on each filling disk. Perform the extension on one sheet and use its integer translates on the others. Equation (7) respects the same translations, so the completed coordinate has its original additive period. Under \(\pi\), a counterclockwise circuit about \(T_{p_i}\) winds once about \(p_i\), while a circuit about any other terminal disk in the mask has winding number zero. Thus the periods are exactly those asserted in the theorem.

Step 5: choosing a metric. The coordinates are now extended, with small wedge integrals outside \(K\) and zero wedge products in the filling disks. It remains to turn this control into small energy in one smooth metric. Let \(g_0\) be the spherical metric and, outside \(K\), form the smooth positive semidefinite tensor \[A=\sum_{i:\,M_i\text{ present}}\mathrm du_i^*\otimes\mathrm du_i^*.\] The sum at a point includes the masks containing that point. All mask boundaries lie inside \(K\), so this tensor is smooth where it is used. The differentials are single-valued even for periodic coordinates. For \(\tau>0\) put \(g_\tau=A+\tau g_0\). In a \(g_0\)-orthonormal frame, let \(\lambda_1,\lambda_2\ge0\) be the eigenvalues of \(A\). The sum of the energy densities of the coordinates in \(g_\tau\), expressed relative to \(\mathrm da_{g_0}\), is \[ \operatorname{tr}(g_\tau^{-1}A)\sqrt{\det g_\tau} =\frac{2\lambda_1\lambda_2+\tau(\lambda_1+\lambda_2)} {\sqrt{(\lambda_1+\tau)(\lambda_2+\tau)}} \longrightarrow 2\sqrt{\lambda_1\lambda_2}. \tag{14}\] In particular the limit is zero when \(A\) has rank at most one. The Cauchy–Binet identity gives \[\det A=\sum_{i<j}|\mathrm du_i^*\wedge\mathrm du_j^*|_{g_0}^2.\] The transferred differentials are bounded on the finitely many smooth compact pieces of the completed surface. The densities in Equation (14) therefore have a common integrable bound for \(0<\tau\le1\): each term \(\lambda_1\sqrt{(\lambda_2+\tau)/(\lambda_1+\tau)}\) is at most \(\sqrt{\lambda_1(\lambda_2+1)}\), and similarly for the other term. Dominated convergence is therefore applicable. By Equation (13) and the choice of the width, \[ \lim_{\tau\downarrow0} \sum_i\|\mathrm du_i^*\|_{2,(Q\setminus K)\cap M_i,g_\tau}^2 =2\int_{Q\setminus K}\sqrt{\det A}\,\mathrm da_{g_0} \le2W_\delta<2\eta. \tag{15}\]

Choose \(\tau\) so that the total energy outside \(K\) in \(g_\tau\) is less than \(3\eta\). With this regularization fixed, interpolate between \(g_0\) and \(g_\tau\) in a thin exterior collar of \(\partial K\), keeping \(g_0\) on a neighborhood of \(K\). The interpolated metrics are uniformly positive definite and their energy densities are bounded on the collar. We may therefore choose its thickness so that the interpolation adds less than \(\eta\) to the total energy. Extend the resulting metric smoothly across each terminal cap, first across its collar and then by interpolation with \(g_0\) inside the cap. This does not change the metric on \(Q\).

The tail energy is now less than \(4\eta\). Moreover, Equation (11) gives for each \(i\) \[\left(\sum_{s\ne p_i}[A_i(T_s)-a_i(s)]_+^2\right)^{1/2} \le4h\sqrt J<4\eta.\] Taking \(\eta<\min(\varepsilon^2/4,\varepsilon/4)\) establishes Equations (3) and (4). All alterations occurred outside a neighborhood of \(K\), completing the proof. ◻

Proof. The exterior \(\{|z|\ge R\}\) retains its original conformal structure. Write \[\frac{f_n(R\zeta)}R =\zeta+\sum_{j\ge1}c_{n,j}\zeta^{-j},\qquad |\zeta|>1.\] The exterior univalent area theorem gives \(\sum_{j\ge1}j|c_{n,j}|^2\le1\) [1]. Hence, for \(|\zeta|=r>1\), \[ \left|\frac{f_n(R\zeta)}R-\zeta\right| \le\left(\sum_{j\ge1}\frac{r^{-2j}}j\right)^{1/2} =\sqrt{-\log(1-r^{-2})}. \tag{18}\] For \(r\ge2\) the right side is at most \(\sqrt{\log(4/3)}<1\).

The image of \(\{|z|>2R\}\cup\{\infty\}\) is the unbounded side of the Jordan curve \(f_n(\{|z|=2R\})\). Injectivity therefore puts all other image points and all complementary disks on its bounded side, which lies in \(\{|w|<3R\}\) by Equation (18). For \(2R\le|z|\le4R\) that equation gives \(|f_n(z)|<5R\). This proves the uniform bound. The same argument applies without change to any normalized univalent map on \(G\).

On every relatively compact connected subdomain of \(G\setminus\{\infty\}\), the maps are eventually holomorphic, injective, and uniformly bounded. Diagonal normal-family compactness therefore gives a locally uniform holomorphic limit. The exterior estimate preserves its Laurent normalization, so the limit extends across infinity as \(f(z)=z+O(1/z)\) and is nonconstant. Hurwitz’s theorem for limits of injective holomorphic maps, applied on each of these subdomains, makes the limit injective on all of \(G\).

The centers and radii of the marked disks lie in a fixed compact parameter set. A further diagonal subsequence makes them converge for all countably many marks. Convergence of centers and nonnegative radii is precisely Hausdorff convergence of the disks, with radius zero giving a singleton. ◻

Proof. We first identify the components for an arbitrary normalized univalent \(f\), then treat terminal boundaries in a finite completion and pass their marked disks to the limit.

Let \(K_j\) be a core from the fixed exhaustion. It is the closed exterior of finitely many disjoint dust disks and contains infinity in its interior. Its image under \(f\) is a compact embedded bordered surface whose boundary curves are the images of the source boundary curves. Orientation fixes the image side of each boundary in a collar. The connected interior of the core, which contains infinity, must remain on that exterior side. Jordan separation therefore identifies \(f(K_j)\) with the closed exterior of the corresponding pairwise disjoint filled bounded Jordan disks.

Under refinement of the cores these image disks are nested in exactly the same way as their source dust disks. Strict nesting in the source gives \[\mathcal Q_f(C_{j+1}(b)) \subset\operatorname{int}\mathcal Q_f(C_j(b)).\] Their intersection is a nonempty compact continuum, strictly inside each earlier image disk. It misses \(f(G)\), because every point of \(G\) lies in the interior of some exhaustion core, while the intersection lies outside all image cores.

Conversely, a point outside \(f(G)\) lies inside exactly one image dust disk at every exhaustion level. Refinement makes the corresponding source disks nested. Their mesh tends to zero, so they determine a unique point \(b\in B\), and the target point belongs to \(K_b(f)\). A connected subset of the complement cannot meet two disjoint image disks at the same level. Thus the component containing a point of \(K_b(f)\) lies in every disk in Equation (19). The intersection is itself connected, so it is the whole component. Distinct dust points are separated at some exhaustion level. This proves the bijection.

For a finite completed surface, use cores bounded by successively smaller collars of its terminal boundaries. The same argument gives one end per terminal boundary; its nested image disks intersect in the corresponding round complementary disk. Fix such a disk \(D\) and a source collar curve \(C\). The disk \(D\) is compactly contained in the bounded side of the image of \(C\). Hence target points whose distance from \(D\) tends to zero eventually lie on that side, and their preimages lie between \(C\) and the terminal boundary. Taking arbitrarily small collars shows that the preimages approach this boundary as a set.

The transferred coordinate is smooth up to the terminal boundary, so its cluster values lie in its boundary range. At a nonsingular terminal index its period is zero, and a single-valued lift can be used on the whole collar. This gives the cluster oscillation bound without a boundary extension of the uniformizing map.

Finally fix a mark \(b\) and an exhaustion curve \(C_j(b)\). For all sufficiently large \(n\) this curve is retained on the core and the terminal disk indexed by \(b\) lies on its bounded source side. Jordan separation in that finite completed surface gives \[D_{n,b}\subset\operatorname{int}\mathcal Q_{f_n}(C_j(b)).\] Uniform convergence on the curve and stability of winding number put every Hausdorff limit point of these disks in \(\mathcal Q_f(C_j(b))\). Intersecting over \(j\) proves Equation (20). For strict trapping inside the image of a fixed collar, first use \(C_{j+1}(b)\) and then its compact containment in the bounded side of \(f(C_j(b))\). ◻

Proof. For each of the two incidence inequalities, its left side is a countably subadditive set functional of the residual image, continuous under increasing unions. Its value on \(\{s\}\) is bounded by the corresponding weight at \(s\). Consequently the inequalities for rational closed intervals extend first to arbitrary open time intervals by an increasing exhaustion. They then extend to arbitrary closed intervals by treating the two endpoints separately.

Restrict every old tag to the interior of the retained piece. Keep the restriction if the path is nonconstant there, and discard it otherwise. The residual image newly created by the discarded tags is countable: each discarded restriction has one constant image point, and there are only countably many tags. This proves the restriction claim, because countable image additions satisfy the incidence bounds by singleton estimates and have zero \(\lambda\) measure. Reversal and affine reparametrization are treated by pulling back the time intervals.

For the countable concatenation, keep these restricted tags in the piece interiors. The residual times fall into three groups: retained old residual times, times exposed by discarded constant tags, and junction or limit times. Fix a test interval \(I\) and a finite deletion set \(P\). For each open piece interval \(J\), let \(R_J\) be the pullback of the old residual times in \(J\cap\operatorname{int}I\). These times are still residual, because every new tag is a restricted old tag. Set \[A_J=(Z_{\mathrm{new}}\cap J\cap\operatorname{int}I)\setminus R_J.\] The image of \(A_J\) is countable: every such time belongs to a discarded constant-tag restriction. The remaining residual times \(T\) consist of junction or limit times, together with at most the two endpoints of \(I\), and hence are countable. The sets \(R_J\), \(A_J\), and \(T\) are pairwise disjoint as sets of times. The old bounds on open intervals apply to each \(R_J\). If \(A=T\cup\bigcup_J A_J\), its countable image satisfies, for either incidence functional \(\mathcal C\) and its corresponding weight \(w\), \[\mathcal C(\gamma(A)\setminus P) \le\sum_{s\in\gamma(A)\setminus P}w(s) \le\sum_{\substack{r\in A\\\gamma(r)\notin P}}w(\gamma(r)).\] Apply the union bound on image sets and sum the old bounds with this last one. All charged times are disjoint, so their sum is bounded by the required right side for the new path, even when their images overlap. The same decomposition proves the atomless condition. Every retained tag has its old tuple and hence still avoids the prescribed null set.

This proof explains the countability assumption on the extra times: countably many open piece intervals alone could leave a Cantor set of uncontrolled times. In the collar concatenations below the intervals accumulate only at explicitly specified dust endpoints, so the stated assumption holds. ◻

Proof. Tags and residual times. Start with \(C([0,1],X)\) and the countable product of the slot spaces. A slot is either inactive, or belongs to a countable disjoint union of spaces consisting of a type, a level tuple, and endpoints \(0\le a<b\le1\). These are standard Borel spaces. Give the inactive and active alternatives separate components. Image containment for a tag can be tested at rational times in \((a,b)\), using the closed level graphs. Nonconstancy can be tested by two such rational times with different images. Both requirements, as well as the tuple exclusion, are Borel. In this ambient coding the residual-time graph is closed relative to the code space times \([0,1]\), and in particular has compact time sections.

Incidence and visit counts. For Equation (23), introduce the sampled path \(\eta\) and two time variables \(r,u\). The incidence in question is equivalent to \[r\in Z_\Gamma\cap I,\qquad \gamma(r)=\eta(u),\qquad \gamma(r)\notin P,\] for some \((r,u)\in[0,1]^2\). Here \(P\) is finite. The last condition is a countable union of closed conditions \(\mathop{\mathrm{dist}}_X(\gamma(r),P)\ge1/n\); for \(P=\varnothing\) no condition is needed. Thus the witness sections are sigma-compact, and the Arsenin–Kunugui projection theorem for sigma-compact sections applies [10]. For Equation (24), use instead the closed incidence relation \(\gamma(r)\in H_i(v)\). The same argument applies. Integration against the fixed law \(\xi\) or Haar measure shows that the two left sides are Borel functions of the tagged code. The test \(s\in\gamma(Z_\Gamma)\) is likewise Borel in \((\Gamma,s)\); integration against \(\lambda\) gives the Borel atomless condition.

For a fixed point \(s\), at least \(k\) visits in a specified closed residual time set can be tested with \(k\) time variables. Distinctness of those times is the union, over \(n\), of the closed conditions \(|r_j-r_l|\ge1/n\) for \(j\ne l\). Projection again gives a Borel visit count \(N_s\in\{0,1,\ldots,\infty\}\). Since every counting \(\ell^2\) weight has countable positive support, \[\sum_{r\in\gamma^{-1}(B)}m(\gamma(r)) =\sum_{s:m(s)>0}m(s)N_s(\gamma)\] is Borel. The same argument treats the right sides of the two constraints. Omitting one or both endpoint times is handled by exhausting the resulting open or half-open parameter intervals.

Physical arclength. For a rational closed interval \(J\), the condition \(\gamma(J)\subset G\) is open in path space. On this set the physical restriction depends continuously on the path, locally uniformly in a compact subset of \(G\). Its length is the supremum of the partition sums, and rational partitions suffice by continuity. Hence the length is Borel. On finite-length restrictions, the integral of a bounded nonnegative continuous function \(f\) against arclength is the limit of sums \[\sum_j f(\gamma(t_j)) \mathop{\mathrm{length}}_{\mathrm{sph}}(\gamma|_{[t_j,t_{j+1}]}),\] over increasingly fine equal subdivisions. Uniform continuity of \(f\circ\gamma\) and finiteness of total length justify this limit. The full domain-part integral is the supremum of sums over finite disjoint collections of rational closed intervals contained in the open domain times. Such collections exhaust all those times, and arclength has no atoms. This proves the Borel length assertion. A countable determining family of continuous functions on \(G\) also shows that \(\gamma\mapsto\nu_\gamma\) is Borel into the space of finite Borel measures on \(G\). Consequently \(\int g\,\mathrm d\nu_\gamma\) is Borel for every fixed nonnegative Borel \(g\), by the usual monotone-class argument, including extended values.

The finite-length condition and all the constraints are therefore Borel, proving the assertion about the tagged space. Endpoint conditions are immediate. Avoidance of a fixed point on \([0,1)\) can be tested on the compact intervals \([0,1-1/n]\). Hit conditions for closed subsets of a compact collar have compact time sections; relatively open collar subsets are sigma-compact and give the same Borel conclusion. These observations prove the final assertion. ◻

Proof. We take limits of collar hit positions and fragment costs, then join actual fragments with summable extra cost. Shrinking disks will give continuity at the new endpoint times.

Choice of finite-cost collars. About \(p\) and \(q\) choose strictly nested dust disks \(V_j\) and \(W_j\) with closures shrinking to their respective points and with disjoint outer closures. Their boundary curves \(A_j=\partial V_j\) and \(B_j=\partial W_j\) can be chosen to satisfy \[\int_{A_j}g\,\mathrm ds<\infty, \qquad \int_{B_j}g\,\mathrm ds<\infty.\] Indeed each smooth boundary has a compact smooth annular collar in \(G\), and Fubini in its smooth parallel coordinates gives finite integrals for almost every parallel curve. Arbitrarily small perturbations preserve all the required nesting and separation. For part (ii), choose only the collars about \(p\), with the outer closed disk missing \(\overline F\). When joining collars disjoint from additional hit curves are available, choose them. For the nested gate curves used below, the intervening annuli provide these choices.

Cuts and limiting budgets. Consider part (i). After passage to a subsequence the two endpoints of \(\gamma_n\) lie in \(V_n\) and \(W_n\). For each fixed \(j\le n\), let \(\alpha_{n,j}\) be its first hit of \(A_j\) and let \(\beta_{n,j}\) be its last hit of \(B_j\). The cuts are ordered as \[\alpha_{n,n}\le\cdots\le\alpha_{n,1} <\beta_{n,1}\le\cdots\le\beta_{n,n}.\] The strict middle inequality follows from disjointness of the outer closed disks. The initial fragment between \(\alpha_{n,j+1}\) and \(\alpha_{n,j}\) lies in \(\overline V_j\), and the final fragment between \(\beta_{n,j}\) and \(\beta_{n,j+1}\) lies in \(\overline W_j\). Retain also the middle fragment between \(\alpha_{n,1}\) and \(\beta_{n,1}\).

Every cut point belongs to \(G\). Thus these fragments have disjoint interior time sets, no dust toll is duplicated at a cut, and arclength adds without endpoint atoms. A diagonal subsequence makes all the hit positions and all the fragment costs converge. Denote the limiting middle cost by \(c_*\) and the two families of limiting fragment costs by \(c_j^-\) and \(c_j^+\). Every finite partial sum is bounded by \(A\), so \[ c_*+\sum_{j\ge1}(c_j^-+c_j^+)\le A. \tag{26}\]

Joining actual fragments. On each fixed collar, two points sufficiently close to the same limiting hit position can be joined by a collar arc whose \(g\) integral is arbitrarily small. This is absolute continuity of the integral of \(g\) along that finite-integral smooth circle. Assign summable positive error allowances, with total less than \(\eta\), both to fragment costs and to all joining arcs. At each collar choose a neighborhood of its limiting hit position small enough for the assigned joining allowance. Select the middle fragment and every initial and final fragment from sufficiently far along its convergent subsequence so that its cost is at most its limit plus its allowance, and both endpoints are in the chosen collar neighborhoods. Adjacent selected fragments can then be joined with the assigned small costs.

Parametrize the resulting initial pieces on successive intervals accumulating at \(0\), the middle portion on a compact interior interval, and the final pieces on successive intervals accumulating at \(1\). The initial tails lie in shrinking \(\overline V_j\) and the final tails in shrinking \(\overline W_j\). Adding endpoint values \(p,q\) therefore gives a continuous path in \(X\). Equation (26) and the assigned errors bound its cost with the two endpoint tolls omitted by \(A+\eta\). Since \(g\ge c_0\), the sum of all domain-part lengths is finite. There are only countably many additional junction times. Lemma 14 makes the new path allowed.

For part (ii), use last hits of the collars about \(p\). Keep one actual prefix, beginning at its original point of \(F\), followed by the successive last-hit fragments. The same diagonal argument, budget inequality, and small collar arcs construct the required one-ended path. Finally, the construction uses only restrictions of original paths and arcs on the selected collars. This proves the asserted preservation of avoidance and hit restrictions. ◻

Proof. In the contradiction argument, shrinking new jumps collapse to dust junctions. Collar concatenation replaces the adjoining path pieces, and the new-jump tolls pay for the junction visits.

Fixing the jump pattern and its limiting costs. Suppose instead that covers of mesh tending to zero admit chains of cost less than \(\beta-\Delta\). All existing jump tolls and \(\rho\) are bounded below by a common positive number. The number of jumps in these chains is therefore uniformly bounded. After combining successive true paths and passing to a subsequence, fix the number of jumps, the pattern of old and new jumps, and the disk label at every old jump. The two endpoints of each new jump converge, after a further subsequence, to the same point \(p_i\in B\): its disk meets \(B\) and has diameter tending to zero. All true-piece costs and all new-jump costs may also be assumed convergent. Since \(\tau\) is fixed, the limiting cost of the new jump at \(p_i\) is at least \(m(p_i)\).

Replacing the blocks between old jumps. Break each chain into blocks separated by old jumps. The anchors of a block are \(E\), \(C\), or the open disk of the adjacent old jump. We will replace the block by an allowed true path between its two anchors, with cost at most the sum of its limiting costs plus arbitrarily small error. Different blocks need not choose the same domain point inside an old jump disk: that old jump can still join their chosen endpoints at its fixed toll.

Suppose first that a block has new-jump limits \(p_1,\ldots,p_d\). Collapse each run of equal consecutive limits to one junction and discard the intervening true pieces. This only decreases the available budget; one of the jump budgets in the run still pays at least the required single toll. Between successive distinct junctions apply Lemma 17(i) to the corresponding true pieces. Their endpoints lie in \(G\) and converge to those junctions. This produces an allowed path between them at its limiting true-piece cost plus an arbitrary small error, with the two junction tolls temporarily omitted.

We must also attach the replacement to its two outer anchors, with endpoints in the domain. At an outer anchor \(A\), if its adjacent junction \(p\) is outside \(\overline A\), use the one-ended version of Lemma 17, reversing paths if necessary. If \(p\in\overline A\), the anchor cannot be \(E\) or \(C\), since these are compact subsets of \(G\). It is therefore an old jump disk. Its boundary lies in \(G\), so \(p\in B\cap\overline A\) actually belongs to its interior. In this case discard the outer true piece and truncate the next constructed path at a domain point sufficiently close to \(p\) to lie in \(A\). Such points exist arbitrarily close to \(p\) in the collar construction. More generally, a nonconstant arc cannot have an initial interval contained in the totally disconnected set \(B\). The truncation is allowed by Lemma 14 and cannot increase the cost.

If a single remaining junction lies in both anchors, use instead a common domain point of their open intersection and the constant path. Such a point exists because \(G\) is dense in \(X\). If precisely one anchor contains the sole remaining junction, construct the one-ended path from the other anchor to that junction and truncate its end on a sufficiently deep domain collar inside the first anchor. The truncation adds no dust toll, and reversal gives the required order. If the block has no new junctions, simply retain an actual true path with cost sufficiently close to its limit.

Accounting for the dust junctions and reconnecting the blocks. Concatenate the retained constructed paths at each remaining dust junction using a single shared time, without adding a pause there. Exactly one previously omitted toll is then incurred at that time. It is covered by the limiting new-jump cost, which is at least \(m(p_i)\). Any additional internal visits to that point were already included in the true-piece costs. Infinite collar tails add only their specified endpoint times, and the construction uses countably many pieces in all. Its domain-part length is finite because \(g\ge c_0\). Thus Lemma 14 supplies the required allowed true path replacing the block.

There are finitely many blocks. Choose their errors to have total less than \(\Delta/2\) and reconnect them with the unchanged old jumps. The resulting old-level chain has cost at most \(\beta-\Delta+\Delta/2<\beta\), a contradiction. ◻

Proof of Theorem 13. Steps 1 and 2 reduce the endpoint hypothesis to a positive cost for reaching a fixed collar, with the relevant finitely many gates absorbed into a density on a penalty region. Steps 3–5 construct a separating function from chain costs, bound its topological bands, and smooth it with controlled energy while retaining those band bounds. Step 6 proves compatibility with the prior coordinates by applying the path-cost estimate to tagged arcs in their level unions.

Step 1: a finite collar already has positive cost. We claim that for some \(k\), every allowed path from \(E\) to \(C_k\) obeying the gates through index \(k\) has full cost at least \(1/2\). Otherwise choose violations with indices \(k_n\to\infty\) and cut each at its first hit of \(C_{k_n}\). The resulting paths do not hit \(b\) or any deeper prescribed collar. They consequently obey all the gates, including those with index greater than \(k_n\), since those have no hits. Their terminal points converge to \(b\). After taking a subsequence their costs converge to a number at most \(1/2\).

Choose strictly intermediate smooth collars between the prescribed ones, with finite \(g\) integrals. They avoid every prescribed hit curve. Apply the one-ended construction in Lemma 17 with error less than \(1/4\). It gives an allowed path from \(E\) to \(b\) with final toll omitted and cost less than \(1\). The retained original pieces avoid \(b\), the joining arcs lie in \(G\), and all gates are preserved. Thus its first visit to \(b\) is at the final endpoint, contrary to hypothesis.

Step 2: replace finitely many gates by a penalty density. In the ungated case set \(O=\varnothing\) and \(g_1=g\). With gates, form the compact set \[F=\bigcup_{j=1}^k\bigl(C_j\setminus f_0^{-1}(W_2)\bigr) \subset G.\] It is disjoint from \(E\), and its image misses \(W_1\). Compact separation gives a relatively compact open set \(O\subset G\) containing \(F\) whose closure misses \(E\) and whose image under \(f_0\) misses \(W_1\). Choose a bounded nonnegative density \(h\) vanishing outside \(O\) such that every path from \(E\) to \(F\) has \(h\)-length at least \(1\). For example, if \(F\ne\varnothing\), put \[\delta=\mathop{\mathrm{dist}}_{\mathrm{sph}}(F,\widehat{\mathbb C}\setminus O)>0, \qquad h=\delta^{-1}\mathbf 1_O.\] The last passage from outside \(O\) to \(F\) is entirely in \(G\) and has physical length at least \(\delta\). If \(F=\varnothing\), take \(h=0\) and \(O=\varnothing\). Set \(g_1=g+h\). This density is in \(L^2(G)\), remains bounded below by a positive constant, and agrees with \(g\) off \(O\).

An allowed path from \(E\) to \(C_k\) which fails a gate through index \(k\) reaches \(F\) and has \(g_1\)-cost at least \(1\). A path which obeys those gates has the \(1/2\) lower bound from Step 1. Cutting first at \(C_k\) if necessary shows that every allowed path from \(E\) to \(C_k\), without any gate restriction, now has full \((g_1,m)\)-cost at least \(1/2\).

Step 3: chain distance with progressively smaller jumps. Taking an infimum of path costs is the classical modulus-to-capacity construction; see [8]. Here shrinking jumps will control the bands at every dust point. Choose positive losses \(\Delta_j\) with \(\sum_j\Delta_j<1/8\). Starting with the true paths alone, add jump levels one at a time. At level \(j\) take \(\rho_j,\tau_j\downarrow0\) and a dust-disk cover of mesh tending to zero, with closures missing \(E\) and \(C_k\). Lemma 3 permits these covers, and Lemma 18 permits choosing each new mesh so small that the lower bound decreases by at most \(\Delta_j\). Thus all finite chains using any of the levels have cost at least \[\beta_*=\frac12-\sum_j\Delta_j>\frac14\] from \(E\) to \(C_k\). Each such chain uses only finitely many levels, so this assertion follows from the corresponding finite-stage bound.

For \(x\in G\) let \(D(x)\) be the infimum of the costs of these finite chains from \(E\) to \(x\), allowing the value \(\infty\), and define \[ u^0(x)=\min\{1,4D(x)\}. \tag{28}\] The empty chain shows \(u^0=0\) on \(E\). A jump disk cannot cross \(C_k\), because it is connected and its closure misses that curve. In a chain from \(E\) to a point of \(U_k\cap G\), some true-path piece must therefore cross \(C_k\). Truncating that chain at the crossing gives cost at least \(\beta_*\). Consequently \[ u^0=0\text{ on }E, \qquad u^0=1\text{ on }U_k\cap G. \tag{29}\] Reversal and concatenation give, for every allowed true path from \(x\in G\) to \(y\in G\), \[ |u^0(x)-u^0(y)|\le4\mathcal L_{g_1,m}(\gamma). \tag{30}\] This remains valid when one or both distances are infinite: a finite-cost connecting path makes finiteness of the two distances equivalent, and otherwise the inequality is vacuous or both truncated values are \(1\). Likewise any two domain points of a level-\(j\) jump disk \(V\) have value difference at most \(4c_j(V)\).

Step 4: topological bands. For each \(s\in B\), the level-\(j\) disk \(V_j(s)\) containing \(s\) contains all domain points sufficiently close to \(s\). Its oscillation bound therefore bounds the width of the topological band at \(s\) by \(4c_j(V_j(s))\). We claim \[\limsup_{j\to\infty}c_j(V_j(s))\le m(s).\] Indeed, for every \(\eta>0\) the set \(\{r\in B:m(r)\ge m(s)+\eta\}\) is finite and does not contain \(s\). As the mesh and \(\tau_j\) tend to zero, its points eventually lie outside the \(\tau_j\)-neighborhood of \(V_j(s)\). Also \(\rho_j\to0\). Equation (27) proves the claim. Hence every band has width at most \(4m(s)\). In particular \(u^0\) has a unique trace at every point of \[B\setminus\Sigma',\qquad \Sigma'=\Sigma\cup\{m>0\}.\] This is a countable enlargement of the flags.

Step 5: Sobolev estimates and band-preserving smoothing. The separating values and pointwise band bounds for \(u^0\) are now established. It remains to obtain smoothness and the energy estimate.

Measurability and weak derivatives. The finite-chain codes retain all true-path tags and are Borel by Lemma 15: matching endpoints, anchor membership, and membership in specified jump disks are Borel conditions, and their costs are Borel. Projection over such codes shows that \(\{x:D(x)<a\}\) is analytic for every \(a\). Thus \(u^0\) is measurable for completed area measure.

In a physical conformal chart, write \(\mathrm ds=\Lambda(z)|\mathrm dz|\). Ordinary axis segments in the chart are allowed and contain no dust tolls. Equation (30) gives \[|u^0(z_2)-u^0(z_1)| \le4\int_{[z_1,z_2]}\Lambda g_1\,|\mathrm dz|\] on every such segment. Fubini makes the line density integrable on almost every horizontal and vertical line in each compact chart rectangle. On each of these lines the displayed inequalities for all subsegments imply absolute continuity. Applying the usual slicing characterization of weak derivatives to an almost everywhere equal Borel representative gives \[|\partial_x u^0|,|\partial_y u^0|\le4\Lambda g_1, \qquad |\mathrm du^0|_{\mathrm{sph}}\le4\sqrt2\,g_1 \quad\text{almost everywhere}.\] Since \(0\le u^0\le1\) and spherical area is finite, this is a global \(W^{1,2}(G)\) function.

Smoothing with shrinking averaging neighborhoods. Choose a locally finite smooth partition of unity on relatively compact coordinate patches, and slightly larger patches in which to take positive mollified averages. The patches can be chosen so that every averaging sample contributing at a point \(x\) has \(X\)-distance from \(x\) tending to zero as \(x\) approaches \(B\). For example, first require their \(X\) diameters to be at most a fixed small multiple of their distance to \(B\), and then shrink the coordinate averaging radii further. On each patch approximate \(u^0\) by a positive convolution average. Choose its value and derivative errors in \(L^2\) so small that, including the derivatives of the partition functions, the sum of all errors is less than \(\varepsilon\). Local Sobolev approximation permits this on each of the countably many relatively compact patches. The resulting smooth function \(u\) satisfies \[0\le u\le1, \qquad \|\mathrm du-\mathrm du^0\|_{2,G}<\varepsilon.\] For precision, if \(\psi_j\) is the partition and \(v_j\) the local average, estimate \[\mathrm du-\mathrm du^0 =\sum_j\psi_j(\mathrm dv_j-\mathrm du^0) +\sum_j(v_j-u^0)\mathrm d\psi_j\] by the triangle inequality in \(L^2\), assigning summable tolerances to both terms.

Preservation of bands and constants. To verify the band assertion pointwise at every dust point, let \(\ell_s,r_s\) be the actual lower and upper topological limits of \(u^0\) at \(s\in B\). For every \(\eta>0\) all actual values of \(u^0\) in some punctured domain neighborhood of \(s\) lie in \([\ell_s-\eta,r_s+\eta]\). Changing to an almost everywhere equal Borel representative preserves these bounds almost everywhere in that neighborhood. If \(x\to s\) and a sample \(y\) contributes to the average at \(x\), the patch construction gives, for a fixed constant \(c\), \[d_X(y,s)\le d_X(x,s)+c\,\mathop{\mathrm{dist}}_X(x,B)\longrightarrow0,\] uniformly over the contributing patches and samples. Eventually all those averages, and then their positive partition sum, lie in the indicated interval. Letting \(\eta\downarrow0\) proves that all limiting values lie in the original band at \(s\), without an exceptional dust set. Thus the smoothed bands are contained in those of \(u^0\), and every trace at a new free point is unchanged. Equation (29) also permits exact preservation of the required constants. Choose \(l>k\) and arrange that every sampling patch meeting \(E_*\) lies in \(\operatorname{int}E\), while every sampling patch meeting \(C_l\) lies in \(U_k\cap G\). The two compact sets are disjoint and have the requisite open neighborhoods. Their averages are exactly \(0\) and \(1\), respectively.

The controlled size. Take \(a_u=4m\). Since \(g_1=g\) off \(O\), \[\|\mathrm du\|_{2,G\setminus O}+\|a_u\|_{\ell^2(B)} \le4\sqrt2\,\|g\|_2+4\|m\|_{\ell^2(B)}+\varepsilon \le8\bigl(\|g\|_2+\|m\|_{\ell^2(B)}\bigr)+\varepsilon.\] Total energy is finite as well, because \(g_1\in L^2(G)\).

Step 6: compatibility with every prior finite level union. Only (P) for the appended coordinate remains. We will show that its free trace values on each nondegenerate component of a generic prior finite level union form a null set.

Fix a type \(\boldsymbol\iota\) and choose a tuple outside \(N_{\boldsymbol\iota}\) and the null sets in Lemma 6 for the fixed weights \(g_1,m\). Put \(H=H_{\boldsymbol\iota}(\mathbf t)\). By Lemma 6 and its weighted coarea estimate, \(H\) has finite physical length in \(G\), its nondegenerate components are locally connected and countable in number, and \[ \nu_H =\mathbf 1_{H\cap G}g_1\,\mathrm d\mathcal H^1_{\mathrm{sph}} +\sum_{s\in H\cap B}m(s)\delta_s \tag{31}\] is a finite Borel measure. The integrability behind this last assertion is, explicitly, bounded by the finite sum of \[\int_{G\cap M_i}g_1|\mathrm du_i|\,\mathrm da +\sum_{s\in B}m(s)a_i(s)\] over the participating labels. Cauchy–Schwarz bounds each term. This also treats extended nonnegative values of \(g_1\), by truncation.

Let \(\alpha\subset H\) be a simple arc joining two new free points \(s,t\in B\setminus\Sigma'\). Its domain-part arclength is its physical image length and is finite. Give it the tag corresponding to the fixed permitted tuple. Subarcs between domain points are allowed, so Equation (30) bounds their value differences by four times their \(\nu_H\) measure. Domain points occur arbitrarily close to both ends of \(\alpha\), since \(B\) is totally disconnected. Passing to the free trace limits gives \[|u^0(s)-u^0(t)|\le4\nu_H(\alpha).\] The same trace inequality holds for \(u\), whose free traces were unchanged by smoothing. The measure \(\nu_H\) gives zero mass to the new free dust points. Apply Lemma 7 in each nondegenerate component of \(H\). The set of its free trace values has Lebesgue outer measure zero, hence zero image measure modulo \(1\). There are only countably many such components.

For the fixed generic prior tuple, almost every new level consequently contains no free point in a nondegenerate component of \(H\). The universal measurability of the level-continuum incidence from Section 2 permits slicing in the tuple variables. This is exactly (P) for the appended coordinate. Older instances of (P) persist because enlarging the flags only removes free points. The barrier and all its asserted bounds are proved. ◻

Proof. The Polish refinement theorem gives the topology in the statement; see [10]. We work in this topology. First we prove continuity under increasing unions and approximation by open neighborhoods, keeping the prescribed values of every density on reference-null sets. These two properties will give a compact subfamily of positive modulus. Separation on that compact subfamily will then produce the required probability law.

Increasing families. If \(A_j\uparrow A_\infty\), then \[ \mathop{\mathrm{Mod}}_\sigma(A_\infty)=\lim_j\mathop{\mathrm{Mod}}_\sigma(A_j). \tag{32}\] Only the case where the limit \(M\) is finite needs proof. Choose nonnegative Borel admissible metrics \(h_j\) for \(A_j\) whose squared norms tend to \(M\). Weak compactness in \(L^2(\sigma)\), followed by convex combinations of tails, gives a nonnegative limit \(h\) and finite convex combinations \(v_k\) of metrics with indices at least \(k\), such that \[\|h\|_2\le\sqrt M, \qquad \sum_k\|v_k-h\|_2<\infty.\] Choose a Borel representative of \(h\) finite everywhere. Keep the actual extended Borel representatives of the \(v_k\), and set \(D=\sum_k|v_k-h|\). This is an extended nonnegative Borel function with \(\|D\|_2<\infty\). For \(a\in A_\infty\), the metrics \(v_k\) are admissible at \(a\) for all sufficiently large \(k\). If \(\int D\,\mathrm d\nu_a=\infty\), then \(h+\varepsilon D\) is admissible there for every \(\varepsilon>0\). Otherwise \(\int|v_k-h|\,\mathrm d\nu_a\to0\), so either \(\int h\,\mathrm d\nu_a=\infty\) or convergence of the integrals gives \(\int h\,\mathrm d\nu_a\ge1\). Consequently \(h+\varepsilon D\) is admissible on the entire union. Letting \(\varepsilon\downarrow0\) proves Equation (32). The added density \(D\) accounts for every index at which convergence in \(L^2(\sigma)\) alone would not control integration against \(\nu_a\).

Open neighborhoods. For every subset \(F\subset A\) and every \(q>\mathop{\mathrm{Mod}}_\sigma(F)\), there is an open set \(V\supset F\) with \(\mathop{\mathrm{Mod}}_\sigma(V)<q\). To see this, first note that every nonnegative Borel \(h\in L^2(\sigma)\), including its prescribed extended values on null sets, has a nonnegative lower semicontinuous pointwise majorant with arbitrarily little increase of its \(L^2\) norm. Indeed, for \(e>0\) set \[h_e=e\sum_{j=0}^{\infty}\mathbf 1_{\{h>je\}}.\] At finite values, \(h\le h_e\le h+e\). By outer regularity, choose open sets \(U_j\supset\{h>je\}\) for which \(e\sum_j\sigma(U_j\setminus\{h>je\})^{1/2}\) is arbitrarily small. Then \(H=e\sum_j\mathbf 1_{U_j}\) is lower semicontinuous, majorizes \(h\) pointwise, and satisfies \[\|H\|_2\le\|h\|_2+e\sigma(Y)^{1/2} +e\sum_j\sigma(U_j\setminus\{h>je\})^{1/2}.\] Apply this construction to an admissible metric for \(F\) of squared norm less than \(q\), and scale its majorant slightly so that its integrals over \(F\) are strictly greater than \(1\), still with squared norm less than \(q\). The set \(V=\{a:\int H\,\mathrm d\nu_a>1\}\) is open by lower semicontinuity of integration and continuity of the measure map. It has the required modulus bound.

A compact subfamily. We now use the two modulus properties to retain positive modulus while restricting the codes to a compact set. Use a complete compatible metric on \(A\). At step \(j\), cover \(A\) by countably many closed balls of radius \(2^{-j}\). Starting with \(F_0=A\), Equation (32) permits selection of a finite union of these balls whose intersection \(F_j\) with \(F_{j-1}\) still has modulus greater than \(c\). The sets \(F_j\) are nested and closed. Every sequence \(a_j\in F_j\) has a Cauchy subsequence, by the finite ball covers at each fixed depth. Completeness shows that \(K=\bigcap_jF_j\) is nonempty and compact. The same argument shows that every open neighborhood of \(K\) contains all sufficiently late \(F_j\). If \(\mathop{\mathrm{Mod}}_\sigma(K)<c\), the open-neighborhood property would therefore contradict \(\mathop{\mathrm{Mod}}_\sigma(F_j)>c\). Hence \(\mathop{\mathrm{Mod}}_\sigma(K)\ge c\).

The compactness here comes from the nested finite covers and completeness. Individual closed balls in the refined topology need not be compact.

Separation and the barycenter. For every bounded continuous \(h\ge0\) with \(\|h\|_2\le1\), some \(a\in K\) satisfies \(\int h\,\mathrm d\nu_a\le C\). Otherwise \(h/C\) would be admissible on \(K\), contrary to \(C^{-2}<c\le\mathop{\mathrm{Mod}}_\sigma(K)\). Given finitely many such tests \(h_1,\ldots,h_n\), consider their integral vectors averaged against probability measures on \(K\). This is a compact convex subset of \(\mathbb R^n\). If it missed the orthant \(( -\infty,C]^n\), strict separation would give \(\alpha_j\ge0\), \(\sum_j\alpha_j=1\), such that \(\sum_j\alpha_j\int h_j\,\mathrm d\nu_a>C\) for every \(a\in K\). The test \(\sum_j\alpha_jh_j\) would contradict the preceding observation. Thus a probability measure meets all the finitely many inequalities.

Compactness of the probability measures on \(K\), and continuity of the integral tests there, now give one law \(P\) meeting every bounded continuous nonnegative test. Its barycenter \(\eta\) is finite: the continuous mass function \(a\mapsto\nu_a(Y)\) is bounded on \(K\). Scaling the tests, including arbitrarily large multiples of tests of zero \(L^2\) norm, gives \[\int h\,\mathrm d\eta\le C\|h\|_{L^2(\sigma)} \quad(h\ge0\text{ bounded and continuous}).\] Apply this also to \(|h|\). Density of bounded continuous functions in \(L^2(\sigma)\) and the Hilbert-space representation theorem give a nonnegative density \(q\) of norm at most \(C\). The finite measures \(\eta\) and \(q\sigma\) agree on bounded continuous functions and hence are equal. ◻

Proof. The reference space and path measures. Choose \(c_s>0\), \(s\in D\), with \(\sum_{s\in D}c_s^2\le1\), and put \[Y=G\sqcup D_{\mathrm{disc}}\sqcup\Pi, \qquad \sigma=\mathrm da\oplus\sum_{s\in D}c_s^2\delta_s\oplus\tau.\] Add the constraint \(\lambda(\gamma(Z_\Gamma))=0\) to the base path system. For its codes define \[ \nu_\Gamma=\ell_\Gamma\oplus \sum_{s\in D}c_sN^-_\Gamma(s)\delta_s\oplus\psi_\Gamma. \tag{37}\] Restrict to codes for which this measure is finite. This is a standard Borel family, and its map to \(\mathcal M_+(Y)\) is Borel by Lemma 15. In particular, the mean measures used below are ordinary Borel measure kernels.

A modulus bound independent of the prescribed tests. Let \(C_{\mathrm{bar}}\) denote the numerical constant in Theorem 13. The modulus of this family is at least \[ c_0=\left(\frac{\beta}{4C_{\mathrm{bar}}}\right)^2. \tag{38}\] Suppose, to the contrary, that it has an admissible extended Borel metric \(h\) with \(\|h\|_{L^2(\sigma)}<\beta/(4C_{\mathrm{bar}})\). Its domain and atom parts give \[g_0=h|_G, \qquad m_0(s)= \begin{cases}c_sh(s),&s\in D,\\0,&s\notin D, \end{cases} \qquad \|g_0\|_2^2+\|m_0\|_{\ell^2(B)}^2\le\|h\|_{L^2(\sigma)}^2.\] For a sufficiently small \(\delta>0\), set \(g=g_0+\delta\) and \(m=m_0+\delta\sum_{s\in D}c_s\mathbf 1_{\{s\}}\). Then \(g\) is bounded below by a positive constant and \[C_{\mathrm{bar}}\bigl(\|g\|_2+\|m\|_{\ell^2(B)}\bigr)<\beta/2.\] The small additions are permitted because \(\mathop{\mathrm{area}}(G)<\infty\) and \(\|(c_s)\|_{\ell^2(D)}\le1\).

In each type exclude the set on which the corresponding tuple part of \(h\) is infinite. It is Lebesgue null, because that component has a positive reference weight. We claim that every allowed path for this additional null-exclusion system has true \((g,m)\)-cost at least \(1\), omitting the final target toll. It suffices to consider one of finite true cost. The added tolls imply \(\sum_{s\in D}c_sN^-_\Gamma(s)<\infty\), so the non-tuple parts of Equation (37) are finite. All the tagged tuples have finite \(h\)-value. Reassign the countably many tags to successively larger unused slots so that the integral of the tuple part of \(h\) is less than any prescribed \(\varepsilon>0\). This leaves the path, tags, residual set, and every path-system constraint unchanged. The resulting code belongs to the family on which \(h\) is admissible. Since its non-tuple integrals are bounded by the true cost, \[1\le\int h\,\mathrm d\nu_\Gamma \le\mathcal L^-_{g,m}(\gamma)+\varepsilon.\] Here the superscript minus means that only the final target time is omitted. Let \(\varepsilon\downarrow0\). Paths of infinite cost satisfy the same lower bound automatically.

Theorem 13 applies to this path system, with its atomless constraint and its additional null tuple exclusions. Taking the error in that theorem sufficiently small produces a compatible barrier of budget less than \(\beta\), a contradiction. This proves Equation (38). Notice that its constant does not depend on \(D\), \(\lambda\), or the choices of \(c_s\).

Occupation, visit counts, and tuple distributions. Apply Lemma 20 with \(c=c_0/2\) and any fixed \(C>\sqrt{2/c_0}\). The resulting law has mean measure with density of \(L^2(\sigma)\) norm at most \(C\). On \(G\) and \(\Pi\), this gives Equations (34) and (36). At \(s\in D\), the mean mass is \(c_s\mathop{\mathrm{\mathbb E}}N^-_\Gamma(s)\), while the reference mass is \(c_s^2\). Consequently the contribution of the atom part to squared density norm is exactly \[\sum_{s\in D}\bigl(\mathop{\mathrm{\mathbb E}}N^-_\Gamma(s)\bigr)^2.\] Restoring the final visit adds one coordinate vector of norm \(1\), proving the count estimate in Equation (35).

The atomless test on the whole path image. It remains to verify the atomless-measure conclusion, which concerns the whole path image. Equation (36) implies absolute continuity of the tuple distribution in every fixed slot and type: that distribution, multiplied by \(2^{-k}\), is dominated by the mean weighted tuple measure. At a fixed free point \(s\), incidence of a tag of type \((i_1,\ldots,i_d)\) requires at least one of its levels to equal the unique trace of the corresponding coordinate at \(s\), modulo \(1\). This is a finite union of Lebesgue-null coordinate sections of the tuple torus. Thus a fixed free point is visited during an open tag interval with probability zero. Since \(\lambda(\Sigma)=0\), Fubini and the countable number of slots and types show that the expected \(\lambda\)-measure of all tagged dust contacts is zero. The residual contacts have \(\lambda\)-measure zero on every code in the family. Combining the two parts gives \(\int_Bm_\mu\,\mathrm d\lambda=0\). ◻

Proof. Fix a complete compatible metric on \(\mathcal M\). The defining relation and Tonelli’s theorem give the mixture identity \[ m_{\int z\,\mathrm dP(z)}(s)=\int m_z(s)\,\mathrm dP(z) \quad(s\in B) \tag{40}\] for every Borel probability measure \(P\) on \(\mathcal M\).

The small-set ideal. We first identify the subsets of candidate laws within which every prescribed countable and atomless test can still be met. Call \(A\subset\mathcal M\) small if some countable \(D\) and finite atomless \(\lambda\) admit no member of \(A\) satisfying Equation (39). Small sets form a sigma-ideal. Indeed, for a countable union take the union of the witnessing countable sets and a strictly positively weighted sum of the witnessing measures, with weights chosen to give finite total mass. A law satisfying the two tests for these combined data would satisfy the tests for each summand. By hypothesis \(\mathcal M\) is not small.

For a nonsmall closed set \(A\), remove every basis-open set \(V\) for which \(A\cap V\) is small. The remaining set \(A'\) is closed and nonsmall. Every neighborhood of each point of \(A'\) meets \(A\) nonsmally. Only countably many basis-open sets are removed, which is the reason this conclusion follows from the sigma-ideal property.

A finite split. Fix a closed nonsmall set \(A\), let \(A'\) be the set just obtained, and fix \(r>0\). An \(r\)-separated tuple in \(B\) has bounded cardinality, say at most \(N_r\). Consider the compact family \(T_r\) of measures \[\sum_{j=1}^n w_j\delta_{s_j}, \qquad 0\le n\le N_r, \qquad \mathop{\mathrm{dist}}(s_i,s_j)\ge r\ (i\ne j), \qquad w_j\ge0, \qquad \sum_jw_j^2\le1.\] The empty tuple gives the zero measure. Its probability barycenters form a compact convex set \(K_r\) of positive measures on \(B\). For every \(\zeta\in K_r\), the weights of its atomic part have \(\ell^2\) norm at most \(1\): on any fixed finite point set this follows from convexity of the Euclidean norm, and one then takes the supremum over finite sets. Write \(\zeta=\zeta_{\rm at}+\lambda\), where \(\lambda\) is atomless. Nonsmallness of \(A'\), tested against the countable atoms of \(\zeta\) and this \(\lambda\), gives a point \(z\in A'\) with \[\int m_z\,\mathrm d\zeta =\int m_z\,\mathrm d\zeta_{\rm at}\le C_0.\]

The following finite-mixture step uses the classical minimax strategy of the Kneser–Fan theorem as presented by Sion [20]. We give the finite-cover and separation proof here; the surrounding fusion construction is developed locally in this proof.

For fixed \(z\), the function \(\zeta\mapsto\int m_z\,\mathrm d\zeta\) is affine and upper semicontinuous on \(K_r\). Therefore the open sets \[\left\{\zeta\in K_r:\int m_z\,\mathrm d\zeta<C_0+\varepsilon/4\right\}, \qquad z\in A',\] cover \(K_r\). Choose a finite subcover with centers \(z_1,\ldots,z_k\). Their integral-vector image of \(K_r\) is convex and misses the open orthant in which every coordinate exceeds \(C_0+\varepsilon/4\). Separation from this open convex set gives coefficients \(\alpha_j\ge0\), \(\sum_j\alpha_j=1\), such that \[\sup_{\zeta\in K_r}\sum_j\alpha_j\int m_{z_j}\,\mathrm d\zeta \le C_0+\varepsilon/4.\] No closedness of the integral-vector image is needed here. The separator has nonnegative coefficients because the orthant is unbounded in every positive coordinate direction. Testing on \(T_r\) and using nonnegative Euclidean dual weights yields \[ \sup_{F\ r\text{-separated}} \left\|\sum_j\alpha_jm_{z_j}|_F\right\|_{\ell^2(F)} \le C_0+\varepsilon/4. \tag{41}\]

To extend this estimate to neighborhoods of the centers, keep the coefficients \(\alpha_j\) fixed and write its left side as \[\begin{split} \Phi_r(y_1,\ldots,y_k) &=\sup_{F\ r\text{-separated}} \left\|\sum_{j=1}^k\alpha_jm_{y_j}|_F\right\|_{\ell^2(F)}\\ &=\max_{\substack{0\le n\le N_r,\ s_1,\ldots,s_n\in B\\ \mathop{\mathrm{dist}}(s_i,s_l)\ge r\ (i\ne l)\\ w_i\ge0,\ \sum_{i=1}^n w_i^2\le1}} \sum_{i=1}^n\sum_{j=1}^k w_i\alpha_jm_{y_j}(s_i). \end{split}\] The case \(n=0\) contributes zero. The configuration and weight spaces in this maximum form a finite union of compact spaces. Joint upper semicontinuity of the hit functions therefore makes \(\Phi_r\) upper semicontinuous jointly in its centers. The margin in Equation (41) now gives each center a small closed neighborhood \(V_j\), of any prescribed sufficiently small diameter, such that \[ \left\|\sum_j\alpha_jm_{y_j}|_F\right\|_{\ell^2(F)}<C_0+\varepsilon \tag{42}\] for every simultaneous choice \(y_j\in V_j\) and every \(r\)-separated finite set \(F\). Take the neighborhoods to be closed balls contained in the requisite open neighborhoods. Each child \(A\cap V_j\) is closed and nonsmall: the ball \(V_j\) contains an open neighborhood of its center, and every such neighborhood meets \(A\) nonsmally. Discard children with zero coefficient.

Nested splits. Starting with \(A=\mathcal M\), perform the split at successive depths with \(r_n\downarrow0\). At depth \(n\), require every child to have diameter at most \(2^{-n}\). The result is a finitely branching tree of nonempty nested closed sets, with transition probabilities given by the coefficients \(\alpha_j\). Completeness gives a unique point of \(\mathcal M\) along every infinite branch. The branch-limit map is continuous: two branches with the same depth-\(n\) ancestor have limits in a set of diameter at most \(2^{-n}\). The transition probabilities give a Borel probability measure on branches and thus a Borel probability measure \(P\) on their limits. By mixture closure, \(\bar z=\int z\,\mathrm dP(z)\) belongs to \(\mathcal M\).

It remains to check every finite-set estimate for this one law \(\bar z\). Fix a depth \(n\) and an \(r_n\)-separated finite set \(F\). At a parent of that depth, let \(P_j\) be the distribution of the branch limit conditioned on taking its \(j\)th child. Its transition probability is \(\alpha_j\), and \(P_j\) is supported in that child’s closed set. These conditionings refer to tree nodes, so the closed child sets are allowed to overlap. Equation (42) holds for every simultaneous choice of points from the child sets. Integrating with respect to the product of their conditional distributions gives \[\begin{split} \left\|\sum_j\alpha_j\int m_y|_F\,\mathrm dP_j(y)\right\|_{\ell^2(F)} &\le \int\left\|\sum_j\alpha_jm_{y_j}|_F\right\|_{\ell^2(F)} \,\mathrm d(P_1\otimes\cdots\otimes P_k)(y_1,\ldots,y_k)\\ &\le C_0+\varepsilon. \end{split}\] The first inequality is convexity of the Euclidean norm. Its left side is the norm of the conditional mean hit vector at the parent. Average over the finitely many parents at this depth, and use Equation (40), to obtain \(\|m_{\bar z}|_F\|_2\le C_0+\varepsilon\). Every finite set of distinct points is \(r_n\)-separated for all sufficiently large \(n\). Taking the supremum over finite sets proves the lemma. ◻

Proof of Theorem 19. Fix the reference measure \(\tau\) on types and the constant \(C\) from Lemma 21. Let \(\mathcal M\) be the laws \(z\) on the base code space \(\mathscr A\) satisfying \[ \bar\ell_z=g_z\,\mathrm da,\quad \|g_z\|_2\le C; \qquad \|n_z|_{D_*}\|_{\ell^2(D_*)}\le C+1; \qquad \mathop{\mathrm{\mathbb E}}_z\psi_\Gamma=q_z\tau,\quad \|q_z\|_2\le C. \tag{43}\] We check the structural hypotheses of Lemma 22.

Borel structure and mixture closure. For a Borel kernel of finite measures on a Polish space with finite reference measure \(\sigma_0\), the condition that its mean have nonnegative \(L^2(\sigma_0)\) density of norm at most \(C\) is Borel. To verify this directly, choose a countable algebra generating the Borel sets, and test \[\int\!\left(\int h\,\mathrm d\nu_\Gamma\right)\mathrm dz(\Gamma) \le C\|h\|_{L^2(\sigma_0)}\] for all nonnegative rational simple functions on that algebra, including the constant function. These are countably many Borel inequalities. The constant test makes the mean measure finite. The inequalities define a bounded functional on the dense space of algebra-simple functions in \(L^2(\sigma_0)\); applying the estimate to absolute values treats signed functions. Its representing density gives the mean measure, first on the algebra and then on all Borel sets. Thus the tests are also sufficient.

Apply this observation to \(\ell_\Gamma\) and \(\psi_\Gamma\). The count functions \(N_\Gamma(s)\) are Borel by Lemma 15; finite partial sums of the squared mean counts test the middle condition of Equation (43). It follows that \(\mathcal M\) is a Borel subset of the standard Borel space of probabilities on \(\mathscr A\). The same integral inequalities and convexity of finite-dimensional Euclidean norms show that all three conditions are preserved by every Borel probability mixture of members of \(\mathcal M\). A mixture is still a law on the base code space, so it also preserves the endpoint conditions and path-system constraints.

Upper semicontinuity of hits. Forgetting tags gives a Borel map \[\mathcal M\longrightarrow \mathcal P\bigl(C([0,1],X)\bigr).\] Choose a Polish topology on \(\mathcal M\), with its same Borel sets, making this map continuous. The Borel probability mixtures just checked remain the same under this refinement. The relation \[\mathcal H_0= \{(\gamma,s)\in C([0,1],X)\times B:\gamma(t)=s \text{ for some }t\in[0,1]\}\] is closed: a sequence of witnessing times has a convergent subsequence. If \(z_j\to z\) in the chosen topology and \(s_j\to s\), the products of their untagged path laws with \(\delta_{s_j}\) converge weakly to the corresponding product for \(z,s\). The closed-set inequality therefore gives \[\limsup_jm_{z_j}(s_j)\le m_z(s).\] The hit functions have the joint upper semicontinuity required by Lemma 22. Their defining relation on tagged codes is Borel by the same closed hit test composed with the forgetful map.

The countable tests. The structural hypotheses are now established. The uniform extraction lemma supplies the remaining test hypothesis within this fixed candidate family. Given any countable \(D\subset B\) and finite atomless \(\lambda\), apply Lemma 21 to \(D\cup D_*\cup\{b\}\). The resulting law belongs to \(\mathcal M\), and \(m_z\le n_z\) gives \[\|m_z|_D\|_{\ell^2(D)}\le C+1, \qquad \int_Bm_z\,\mathrm d\lambda=0.\] All hypotheses of Lemma 22 hold with \(C_0=C+1\). Take \(\varepsilon=1\) there. Its law \(\mu\in\mathcal M\) satisfies \(\|m_\mu\|_{\ell^2(B)}\le C+2\), while membership in \(\mathcal M\) retains the occupation and mean-count estimates. Finally the last condition in Equation (43) implies absolute continuity in each slot and type, by the positive domination in Equation (33). This proves every conclusion with the common bound \(C+2\). ◻

Proof. Fix \(x\in G\). In a physical coordinate disk about \(x\), with coordinate \(z(x)=0\), choose \(r_0>0\) so that its closed \(r_0\)-disk lies in \(G\). Consider the nonnegative density, defined to be zero at \(x\) and outside the indicated punctured disk, \[\rho(z)=\frac{\mathbf 1_{\{0<|z|<r_0/e\}}} {|z|\log(r_0/|z|)}.\] The smooth spherical length factor is bounded above and below on this disk, and \[\int_0^{r_0/e}\frac{\mathrm dr}{r\log^2(r_0/r)}<\infty, \qquad \int_0^{r_0/e}\frac{\mathrm dr}{r\log(r_0/r)}=\infty.\] Consequently \(\rho\in L^2(G,\mathrm da)\), while every path that visits \(x\) and subsequently leaves this coordinate disk has infinite \(\rho\)-length on its domain times. To see the latter assertion without any regularity at the hitting time, stop between the circles of radii \(a\) and \(r_0/e\) and apply ordinary variation to the radial coordinate, then let \(a\downarrow0\). Every path under consideration ends in \(B\) and therefore must leave this disk after a visit to \(x\). On the other hand, \[\mathop{\mathrm{\mathbb E}}_\mu\int_{\gamma^{-1}(G)}\rho(\gamma)\,\mathrm ds =\int_G\rho g_\mu\,\mathrm da \le \|\rho\|_2\|g_\mu\|_2<\infty.\] This proves point avoidance.

For a path from outside \(\overline U\) to \(U\cap B\), the set of hitting times of \(\partial U\) is a nonempty compact set whose maximum is less than \(1\). The path lies strictly inside \(U\) after that maximum. An atom at a given last-hit location would give positive probability of visiting that fixed point of \(\partial U\subset G\), which has just been excluded. ◻

Proof. There are three cases to control: a residual contact of the later path, a residual contact of the earlier path with a later tag, and a contact between two tags.

Residual contacts of the later path. Let \(P_N\) be the finite deletions exhausting \(A_\xi\) in the residual intersection constraint. With \(Z_\gamma\) denoting the residual dust times of a tagged path, that constraint, independence, and Tonelli’s theorem give \[\begin{align*} &\mathop{\mathrm{\mathbb P}}\bigl( \gamma_\xi([0,1])\cap (\gamma_\mu(Z_{\gamma_\mu})\setminus P_N)\ne\varnothing \bigr)\\ &\hspace{15mm}\le \sum_{s\in A_\xi\setminus P_N}m_\xi(s)n_\mu(s) \le \|m_\xi\mathbf 1_{A_\xi\setminus P_N}\|_{\ell^2} \|n_\mu|_{A_\xi}\|_{\ell^2}. \end{align*}\] The last factor is finite by the assumed count bound, and the first tends to zero. Thus any meeting at a residual time of the later path occurs in \(A_\xi\), almost surely.

An earlier residual contact with a later tag. Next fix a prior coordinate \(i\), and delete finite sets exhausting \(\Sigma\) in its residual level constraint. Almost surely the weighted sum of visits of \(\gamma_\xi\) against \(a_i\) is finite, since its expectation is at most \[\|a_i\|_{\ell^2}\|n_\xi|_\Sigma\|_{\ell^2}.\] The deleted tails of this sum tend to zero. It follows directly from the residual level constraint that \[\bigl|\{t\in\mathbb T: H_i(t)\cap(\gamma_\xi(Z_{\gamma_\xi})\setminus\Sigma) \ne\varnothing\}\bigr|=0 \quad\text{almost surely}.\] This assertion holds simultaneously for all prior labels. Conditional on an earlier path with this property, a tag of the independent later path cannot meet its residual image off \(\Sigma\): each coordinate level in the tag has an absolutely continuous marginal distribution. There are only countably many tag slots and types, so all such meetings are excluded simultaneously.

Contacts between two tags. It remains to exclude a free common point lying on a tag of each path. For any fixed pair of slots and types, their two tuples are independent and absolutely continuous with respect to their level product measures. We can therefore assume simultaneously that all required instances of (P) hold and that levels belonging to the same label in the two tuples are distinct. At a free common point \(x\), choose the largest label \(j\) among the levels from either tag that actually contain \(x\). The unique trace at \(x\) implies that label \(j\) can occur at \(x\) in only one of the two tags. In the other tag, every level present at \(x\) has label less than \(j\).

Here the other tag supplies a nondegenerate continuum through \(x\) in those lower levels. Indeed its image is connected and nonconstant. Choose a compact parameter subinterval within the tag whose image contains both \(x\) and another point. Level sets in its finite union that miss \(x\) miss some neighborhood of \(x\), by closedness. Boundary bumping in a sufficiently small closed neighborhood now gives a nondegenerate subcontinuum through \(x\) using only the levels present there. This contradicts (P) for label \(j\).

All these assertions hold simultaneously for the countably many pairs of slots, types, and sublists. The incidence events are Borel or analytic by the path coding and compact level graphs; completed product measures therefore justify the conditioning and Fubini arguments. ◻

Proof. A common collar for uncountably many laws. Suppose the unavailable endpoints form an uncountable set. Choose distinct such endpoints \(b_\alpha\), \(\alpha<\omega_1\). Recursively apply Theorem 19 to obtain a law \(\mu_\alpha\) from \(E\) to \(b_\alpha\) at first arrival. At stage \(\alpha\), impose all the earlier laws in the residual intersection constraint. This is an admissible countable collection because \(\alpha\) is countable. The coordinate list and its flags remain fixed throughout this recursion.

Fix a number \(\delta>0\), to be made small below. For each endpoint \(p=b_\alpha\), shrinking its exhaustion disk gives \[ \|g_{\mu_p}\|_{2,U\cap G}<\delta, \qquad \|m_{\mu_p}|_{(U\cap B)\setminus\{p\}}\|_{\ell^2}<\delta \tag{44}\] for some exhaustion disk \(U\) containing \(p\). The first assertion is absolute continuity of the energy integral, since the disks shrink to a point outside \(G\). The second is the analogous tail property of a square-summable family after omitting its value at \(p\). There are only countably many disks in the exhaustion. Hence one common \(U\) satisfies Equation (44) for uncountably many of the chosen endpoints and their laws.

Alternating connections. For each of these laws, take the last-hit location on \(\partial U\). Its distribution is atomless by Lemma 23. Consequently there are four pairwise disjoint arcs \(I_1,I_2,I_3,I_4\) on \(\partial U\), listed in circular order, each having probability at least \(1/10\). The arcs can be chosen from a fixed countable family: use any fixed parametrization of the circle and endpoints in a countable dense set, choosing slightly smaller quantile arcs before this approximation. After another uncountable restriction the same ordered four arcs work for every law. Choose an earlier endpoint \(p\) and a later endpoint \(q\) from this subfamily.

Independently sample two paths to \(p\) and two paths to \(q\). With probability at least \(10^{-4}\) their last-hit locations fall, respectively, in \(I_1\) and \(I_3\) for the \(p\)-paths and \(I_2\) and \(I_4\) for the \(q\)-paths. The tails after these hits lie strictly inside \(U\), apart from their initial points. Concatenating the two tails through \(p\) gives a connection between its two boundary locations, and likewise for \(q\). Each connection contains a simple arc between its boundary endpoints. Jordan separation in the closed disk \(\overline U\) forces the two connections to intersect in \(U\); see Figure 2. Thus this event entails an intersection in \(U\) of one of the four independent cross-pairs consisting of a \(p\)-path and a \(q\)-path.

Dust intersections. We estimate the probability of such a cross-pair intersection. Lemma 24 removes all dust meetings outside a countable set. Points in that set with zero hit probability for either law can also be discarded. Independence and the union bound therefore give \[\begin{align*} \mathop{\mathrm{\mathbb P}}(\text{a common visit in }U\cap B) &\le \sum_{s\in U\cap B}m_{\mu_p}(s)m_{\mu_q}(s)\\ &\le m_{\mu_q}(p)+m_{\mu_p}(q) +\sum_{s\in(U\cap B)\setminus\{p,q\}} m_{\mu_p}(s)m_{\mu_q}(s)\\ &\le 2\delta+\delta^2. \tag{45}\end{align*}\] Here the target hit probabilities equal one, \(p\ne q\), and the last sum is estimated by Cauchy–Schwarz using Equation (44).

Domain intersections. For domain meetings, let \(\ell_\gamma\) denote a sample’s arclength occupation restricted to \(U\cap G\), and put \[I_a(\gamma,\widetilde\gamma) =\iint \frac{\mathbf 1_{\{\mathop{\mathrm{dist}}_{\rm sph}(x,y)<2a\}}}{a^2} \,\mathrm d\ell_\gamma(x)\,\mathrm d\ell_{\widetilde\gamma}(y).\] If the two paths meet at \(x\in U\cap G\), then for every sufficiently small \(a\) the physical ball of radius \(a\) about \(x\) is contained in \(U\cap G\). Each path has at least \(a/2\) of arclength in its ball of radius \(a/2\): after visiting \(x\) it must eventually leave the ball in order to reach its dust endpoint. Therefore \[\liminf_{a\downarrow0}I_a(\gamma,\widetilde\gamma)\ge\frac14\] on the domain-intersection event. The spherical kernel in this integral has uniformly bounded integrals in each variable, since spherical balls of radius \(2a\) have area at most a numerical constant times \(a^2\). Its operator norm on \(L^2(\mathrm da)\) is consequently bounded by a numerical constant \(C\). Independence yields \[\mathop{\mathrm{\mathbb E}}I_a \le C\|g_{\mu_p}\|_{2,U\cap G} \|g_{\mu_q}\|_{2,U\cap G} \le C\delta^2.\] Applying Fatou’s lemma along any sequence \(a\downarrow0\) gives \[ \mathop{\mathrm{\mathbb P}}(\text{a common visit in }U\cap G)\le4C\delta^2. \tag{46}\] The estimates concern the original paths, and hence also bound intersections of their last-hit tails.

The contradiction. Combining Equations [eq:dust-meeting-bound] and (46), and taking the union bound over the four cross-pairs, bounds the alternation event by \[4\bigl(2\delta+(1+4C)\delta^2\bigr).\] For sufficiently small \(\delta\) this is less than \(10^{-4}\), the contradiction. ◻

Proof. Discard the finitely many tangent lines and lines through point holes. For almost every remaining line the gradient integral in the bounded region is finite. The intersection of a chosen segment with \(D\) is a finite union of intervals. On each, ordinary variation bounds the change of \(v\) by the gradient line integral. Approach the endpoints of the complementary intervals from within \(D\). The difference across each such interval is at most \(A(T)\), by the cluster-interval hypothesis. Equivalently one can first use compact subintervals in \(D\) and then take endpoint limits; finite line variation supplies the limits whenever needed. Thus \[1\le \int_{D\cap\text{segment}}|\mathrm dv|\,|\mathrm dw| +\sum_{T:\,T\cap\text{segment}\ne\varnothing}A(T).\] Majorizing by the full-line gradient integral in the radius-\(L\) ball and by all disk hits makes this bound independent of any choice of segments. Integration in the transverse parameter and Cauchy–Schwarz give the first inequality in Equation (48). A round disk of radius \(r_T\) is met by a set of parallel lines of parameter length \(2r_T\). Finally the interiors of the disks are disjoint and lie in the radius-\(L\) ball, so \[\sum_T r_T^2\le L^2, \qquad \sum_T r_T A(T) \le L\Bigl(\sum_T A(T)^2\Bigr)^{1/2}.\] The proof uses the gradient only on the actual domain portions of the segments, which also proves the localized assertion. ◻

Proof. Fix \(b\notin S\). If \(K_b(f)\) had a projection of positive length, every selected barrier surrounding \(b\) would have to pay for that same length. Their arbitrarily small norms will contradict this requirement.

Suppose \(K_b(f)\) is nondegenerate. Some orthogonal projection of this compact continuum has positive length. Fix a compact interval \(J\) of positive length inside that projection. The target bound \(L\) from Section 3 can be increased once to contain all images of the outer zero circle \(E_*\) as well as all relevant inner curves and complementary disks. This choice is independent of the barrier: the inner curves lie in the bounded side of the fixed outer curve, whose image is uniformly bounded.

For each \(k\) choose a selected barrier \(u\) with \[\|\mathrm du\|_2+\|a_u\|_{\ell^2}<1/k\] whose inner dust disk contains \(b\). Its inner boundary curve \(C\) has value one, and \(E_*\) has value zero. The end-component description in Lemma 10 places \(K_b(f)\) strictly inside the bounded Jordan region of \(f(C)\), which in turn lies inside that of \(f(E_*)\). For this fixed test, local uniform convergence on the two curves and stability of winding imply the same inclusions of the fixed compact set \(K_b(f)\) for all sufficiently large finite maps \(f_n\).

Each line with parameter in \(J\) therefore meets the inner bounded region. Following it out toward infinity supplies a segment between \(f_n(C)\) and \(f_n(E_*)\), with test values one and zero. Both endpoints, and hence the segment, lie in the radius-\(L\) ball. Apply Lemma 26 to the transferred test on the finite target circle domain. Its energy norm has upper limit at most \(\|\mathrm du\|_2\), since it agrees with \(u\) on exhausting cores and its remaining energy tends to zero. Its boundary oscillations satisfy \[A_n(T)\le a_u(s_T)+e_n(T), \qquad e_n(T)\ge0, \qquad \sum_Te_n(T)^2\longrightarrow0.\] The terminal indices \(s_T\) are distinct. Consequently \[\limsup_n\Bigl(\sum_T A_n(T)^2\Bigr)^{1/2} \le\|a_u\|_{\ell^2(B)}.\] Taking the limit in Equation (48) gives \[|J|\le\sqrt{\pi}\,L\|\mathrm du\|_2+2L\|a_u\|_{\ell^2} \le \frac{(\sqrt{\pi}+2)L}{k}.\] The interval \(J\) is fixed and \(k\) is arbitrary, a contradiction. Further list extensions preserve both the old tests and their compatibility, so the argument applies to every stated extension. ◻

Proof. We first show that, at a generic independent level, free contacts inside a tag can occur only in singleton level components. A detour around such singleton contacts then gives a variation estimate for the prior coordinate along the tag. That estimate makes the free trace image null. Finally, coarea counts the remaining intersections.

Singleton contacts at an independent level. Fix a tag slot, its type, and a prior coordinate \(i\). For almost every pair consisting of the tag tuple and an independent level \(t\), every free contact of that tag with \(H_i(t)\) has a singleton component in \(H_i(t)\). To prove this, exclude equality between \(t\) and every level of label \(i\) in the tag, and impose the finitely many relevant instances of (P). If a level of some tag label \(j>i\) is present at a contact \(x\), (P) for \(j\) against \(i\) gives the assertion. If no such higher label is present, the tag yields a nondegenerate local continuum through \(x\) in labels lower than \(i\), exactly as in the proof of Lemma 24. This is forbidden by (P) for \(i\). Closedness permits levels absent at \(x\) to be removed locally. Absolute continuity of the tuple law in each slot and type, followed by Fubini, shows that for almost every sample this singleton assertion holds for almost every \(t\), simultaneously for all its tags.

A local variation bound. Near a free point in \(B\cap M_i\), choose an open topological disk \(V\) contained in the interior of the mask, away from the singularity if there is one. For a period coordinate choose a common lift on \(V\). Shrink \(V\) until all the defined values and bands of this branch lie in one real interval of length less than \(1/2\). Such a choice is possible because the point has a unique trace. A countable family of these disks covers the relevant free points.

Consider two free contact times \(r<s\) in one tag such that \(\gamma([r,s])\subset V\). Let their real trace values be \(\alpha\) and \(\beta\). We claim that \[ |\beta-\alpha|\le \int_{[r,s]\cap\gamma^{-1}(G\cap M_i)}|\mathrm du_i|(\gamma)\,\mathrm ds +\sum_{\substack{t\in[r,s]\colon\gamma(t)\in\Sigma\cap M_i}} a_i(\gamma(t)). \tag{50}\] Here the sum again counts parameter times. Assume \(\alpha<\beta\). For almost every real threshold \(v\in(\alpha,\beta)\) the preceding singleton assertion holds at its modulo-one level. We show that the segment must hit this threshold either at a domain point or in a flagged band.

Otherwise the compact set \[F_v=\gamma([r,s])\cap H_i(v\bmod1)\] consists only of free points that are singleton components of the closed level. They remain singleton components after adjoining \(B\), by the dust topology of Section 2. The small-disk construction of Lemma 3 covers \(F_v\) by finitely many open disks with closures in \(V\), missing the endpoints and with boundaries avoiding the entire closed level and \(B\). Process these disks successively on the current path segment, skipping any whose interior is no longer met. Replace the portion from first entry to last exit of a disk by an arc of its boundary. Each replacement removes the current level hits in that disk and adds none, so after the finite process all level hits have disappeared. Disks need not be disjoint for this argument.

The resulting path in \(V\) cannot exist. Off the level, the sign of the chosen branch relative to \(v\) is defined at domain points and at dust points by their bands, and is locally constant. At dust points this follows from the defining upper and lower limits of the band. The endpoint signs are opposite. The local range of length less than \(1/2\) ensures that no other integer translate of the level is involved. This proves the required domain-or-flag hit for almost every intervening threshold.

The set of values contributed by the domain times has length at most the integral in Equation (50): exhaust the countably many domain intervals and apply ordinary curve variation on their rectifiable physical portions. The flagged bands contribute at most the sum in that equation. These ranges cover almost every threshold between \(\alpha\) and \(\beta\), proving the claim. The case \(\beta<\alpha\) follows by reversing the endpoints.

Null trace image. For almost every sample, the right-hand costs above are the restrictions of a finite measure \(\kappa_i\) on path time: its domain part is \(|\mathrm du_i|\,\mathrm ds\) within the mask, and it has an atom of mass \(a_i(\gamma(t))\) at each flagged visit time. Indeed \[\mathop{\mathrm{\mathbb E}}_\mu\kappa_i([0,1]) \le \|g_\mu\|_2\|\mathrm du_i\|_{2,G\cap M_i} +\|n_\mu|_\Sigma\|_{\ell^2}\|a_i\|_{\ell^2}<\infty.\] The measure gives zero mass to free contact times. Fix a tag and one of the disks \(V\). By outer regularity, its free contact times in \(\gamma^{-1}(V)\) have an open cover, within that tag and \(\gamma^{-1}(V)\), of arbitrarily small \(\kappa_i\)-measure. The cover has countably many interval components. Within each component, Equation (50) bounds the diameter of the free trace image by the measure of that component. Summing proves that the free trace image in this tag and this disk has length zero. The disks and tags are countable.

On residual times, the same null-image conclusion follows directly from the residual level constraint by deleting the flags, as in the proof of Lemma 24. Thus all free contacts have null trace image. Singular points are flagged and play no role in this assertion.

Distinct intersection counts. The image of the domain times is countably rectifiable, and its physical length measure is dominated by the arclength occupation. The one-dimensional area formula applied to the smooth map \(u_i\) modulo one, in local physical charts and lifts, bounds the integrated number of distinct domain intersections by \[\int_{G\cap M_i}g_\mu|\mathrm du_i|\,\mathrm da.\] This also applies at mask boundaries by local smoothness; the occupation of each such fixed smooth boundary is zero almost surely because its area is zero. At a dust point, the parameter length of level incidence is at most \(a_i(s)\). The already proved null-image assertion removes free incidences under the joint law and level measure. On the countable flags, Tonelli gives the remaining bound \(\sum_s m_\mu(s)a_i(s)\), including a possible singular point with its budget one. Cauchy–Schwarz makes both bounds finite and proves Equation (49).

For a finite union, sum these bounds over its finitely many labels and apply Fubini to its level tuple. The null free-incidence assertion also passes to that union. Incidence and distinct finite-point counts are measurable in the completed sense supplied by the path coding, so these integrations are legitimate. ◻

Proof. Cut each path at its last hit of \(C\) and retain its terminal part, reparametrized on \([0,1]\). The anchor is outside \(\overline U\), so such a hit exists. Its last hitting time is strictly less than \(1\); after that time the path is strictly inside \(U\). The original path reaches \(b\) for the first time at its endpoint. Thus its terminal part has distinct endpoints and contains a simple arc with those endpoints. The gate condition passes to the last hitting point.

These operations can be performed measurably up to completion of the law. Last hitting times of a fixed closed set are Borel on the present path family. In the product of two path spaces, the conditions that the second path is injective, has prescribed endpoints, and has image contained in the first path’s image are Borel. For injectivity, use \[\min_{|s-t|\ge 1/n}d_X(\alpha(s),\alpha(t))>0 \qquad(n\in\mathbb N);\] image containment is closed for uniform convergence. The nonempty sections therefore admit a universally measurable selection by the Jankov–von Neumann theorem [10]. The pushforward of the completed probability measure is an ordinary Borel probability measure on path space.

The occupation measure of a selected simple arc is its domain image length measure. By the image-length observation in Section 4, this is dominated by the occupation of the original terminal path. There is no arclength atom at its initial point on \(C\). Consequently the unconditioned selected-arc law has occupation density at most \(\mathbf 1_{U\cap G}g_\mu\), and its hit probabilities are at most the original hit probabilities. The endpoint and interior conditions describe a Borel subset of path space. Inner regularity provides a compact subset of that set of probability at least \(1/2\). Conditioning on this subset proves Equation (51).

The indicators of \(U_j(b)\cap G\) decrease pointwise to zero. The square-summable function \(m_\mu\) has countable support, and the sets \((U_j(b)\cap B)\setminus\{b\}\) decrease to the empty set. Absolute continuity of the two squared integrals therefore proves Equation (52). Finally, all selected images are subsets of the original images. The level-intersection conclusions of Lemma 28 survive restriction, selection, and conditioning, with their expected counts increased by at most the conditioning factor. ◻

Proof. We first construct \(v\) and prove its energy estimate and boundary-measure identity. We then control its dust bands and free traces, and finally produce the smooth admissible coordinate \(u\).

Step 1: the lifted rays. Choose an orientation-preserving homeomorphism from \((\overline U,b)\) to the closed unit disk with its center. Write the cyclic cover of \(\overline U\setminus\{b\}\) as \[\mathcal H=\mathbb R\times[0,\infty),\] where the first coordinate is angle divided by \(2\pi\), the second is minus logarithmic radius, and the deck transformation is \((x,y)\mapsto(x+1,y)\). These are topological coordinates; all analytic estimates below will instead use physical conformal charts in \(G\).

For an arc \(\alpha\), choose the lift starting at \((a,0)\) with \(a\in[0,1)\). Its image \(\Gamma_\alpha\) is a proper simple ray with \(y\to\infty\) at its terminal end. Its integer translates are disjoint: an intersection of two translates would give a repeated projected point of the simple arc. In the one-point compactification of \(\mathcal H\), the translated ray \(\Gamma_\alpha+(k,0)\) is a crosscut from \((a+k,0)\) to the boundary point at infinity. Let \(R_k(\alpha)\) be its right-hand complementary component, the one incident to boundary points \((x,0)\) with \(x>a+k\). Jordan separation gives \[R_{k+1}(\alpha)\subset R_k(\alpha).\] Indeed the next ray lies on the right of the preceding ray because it is connected, is disjoint from that ray, and starts on its right. The right side of the next ray connects to boundary points right of both starts and is therefore contained in the preceding right side.

Step 2: the integer index and its local bounds. Compactness of the arc family has two consequences. The arcs approach \(b\) uniformly as the parameter tends to \(1\); and, for every \(\delta>0\), their portions with parameter in \([0,1-\delta]\) stay a uniform positive distance from \(b\). The first assertion follows from equicontinuity and the common terminal endpoint. The second follows from compactness and the fact that every member is injective and meets \(b\) only at parameter \(1\).

Fix \(0\le Y<\infty\). Uniform approach to \(b\) gives \(\delta>0\) such that every arc point with \(0\le y\le Y\) occurs before the common time \(1-\delta\). On this initial interval the arcs stay uniformly away from \(b\), so their projected circular positions are equicontinuous. Subdivide time into finitely many intervals on each of which all circular positions move by less than a semicircle. Starting with \(a\in[0,1)\), successive lifts then give a uniform bound for all lifted angles on this initial interval. Thus all unshifted rays in the strip \(0\le y\le Y\) lie in a common bounded horizontal interval.

For a compact subset of \(\mathcal H\), join its points vertically to the boundary. Sufficiently large positive or negative translates miss all those vertical segments. Their side at the boundary is fixed by the order of the starts, and it remains fixed along the segments. Hence, for every \(z\) off the translated rays, the set \(\{k\in\mathbb Z:z\in R_k(\alpha)\}\) is nonempty and bounded above. We may therefore define \[ J_\alpha(z)=\max\{k\in\mathbb Z:z\in R_k(\alpha)\}. \tag{56}\] The same argument bounds \(|J_\alpha|\) uniformly over the arc family on each compact subset of \(\mathcal H\), wherever it is defined. The definition also gives \[ J_\alpha(z+(1,0))=J_\alpha(z)+1, \qquad J_\alpha(x,0)=\lfloor x-a\rfloor \quad(x-a\notin\mathbb Z). \tag{57}\]

Step 3: measurability and an intersection inequality. The lifted arc on every initial compact time interval is a Borel function of the arc and its chosen starting angle, by path lifting. Off a ray, membership in its right component is Borel as well. One may test it by countable chains of rational open balls disjoint from the ray and joining the point to the appropriate boundary interval. Avoidance of a compact test path is Borel by the initial lifts and a countable exhaustion of \([0,1)\). Since a compact path disjoint from a closed proper ray has positive distance from it, these tests detect all complement connections. This proves measurability of the indices. On the coincidence set, where \(z\) lies on a translated ray, assign an arbitrary value whenever a random variable is required. We will take expectations only at points whose coincidence probability is zero.

Suppose a curve \(\delta\) in the cover projects injectively to \(\overline U\setminus\{b\}\) and its endpoints miss the sample rays. Every ray separating the endpoint indices must meet \(\delta\). Different such meetings project to different points because the projection of \(\delta\) is injective. Consequently \[ |J_\alpha(\delta(1))-J_\alpha(\delta(0))| \le \#\bigl(\alpha([0,1])\cap\operatorname{proj}\delta([0,1])\bigr). \tag{58}\] The right side counts distinct projected points; the inequality also holds when that count is infinite.

Step 4: averaging and continuity. The index is now defined, measurable, and locally uniformly bounded. We use these properties to average it. Lemma 23 applies to \(\eta\), since its arcs end at \(b\in B\) and have an \(L^2\) mean occupation density. Thus every fixed point of \(G\cap\overline U\), including a point of \(C\), is missed almost surely. For each lift \(z\) of such a point, define \[v(z)=\mathop{\mathrm{\mathbb E}}_\eta J_\alpha(z).\] Local uniform domination makes this expectation finite. For a convergent sequence in the lifted domain, discard the null coincidence events at its terms and its limit. For every remaining sample the index is locally constant at the limit. Dominated convergence proves continuity of \(v\), including at \(C\). The first identity in Equation (57) gives additive period one.

Step 5: the differential estimate. Work in a physical conformal rectangle compactly contained in \(G\cap U\), with a common lift. Here \(z=x+iy\) denotes a physical conformal coordinate, and the spherical metric is \(\lambda(z)|\mathrm dz|\); these \(x,y\) are unrelated to the topological angle and depth above. For a horizontal interval \(I\) and height \(y\) let \(N_\alpha(I,y)\) count the distinct intersections of the sample arc with \(I\times\{y\}\). Curve coarea, followed by the occupation bound, gives, for Borel sets \(A\) of heights, \[ \int_A\mathop{\mathrm{\mathbb E}}_\eta N_\alpha(I,y)\,\mathrm dy \le \int_{I\times A}\lambda(z)g_\eta(z)\,\mathrm dx\,\mathrm dy. \tag{59}\] To check the conformal factor, Euclidean image length is spherical image length divided by \(\lambda\), whereas spherical area is \(\lambda^2\mathrm dx\mathrm dy\). Projection coarea costs at most Euclidean image length. The same estimate holds after restricting counts to Borel portions of an interval.

Apply Equation (59) first to closed intervals with rational endpoints. Outside one null set of heights, the mean counting measure on every compact interior interval is bounded by \(\lambda g_\eta\mathrm dx\). This follows by the countable interval tests and then measure approximation. Averaging Equation (58) shows on those lines that \[|v(x_2,y)-v(x_1,y)| \le\int_{x_1}^{x_2}\lambda(x,y)g_\eta(x,y)\,\mathrm dx.\] Continuity permits arbitrary endpoints. Thus \(v\) is absolutely continuous on almost every horizontal line and \(|\partial_xv|\le\lambda g_\eta\) there. The vertical argument gives \(|\partial_yv|\le\lambda g_\eta\). The usual characterization by absolute continuity on lines yields local Sobolev regularity and \[|\mathrm dv|_{\mathrm{sph}}\le\sqrt2\,g_\eta \quad\text{almost everywhere}.\] An exhaustion by charts proves Equation (53).

Step 6: the boundary measure. The distribution \(\beta\) of the starting point has no atoms by point avoidance on \(C\). Between two ordered lifted boundary angles, the expected difference of the floors in Equation (57) counts the mean number of start translates in that interval. This proves Equation (54). In particular, for every continuous real function \(\varphi\) on \(C\), \[ \int_C\varphi\,\mathrm dv =\int\varphi(\alpha(0))\,\mathrm d\eta(\alpha). \tag{60}\] The left side is a Stieltjes integral. Only the orientation of the topological angular coordinate is used; no smoothness is required.

We have obtained a continuous period coordinate with the required energy bound and a positive boundary measure of total mass one. To make it admissible for transfer, we must still control its dust bands, verify compatibility, and smooth it.

Step 7: bands at the dust. Fix \(s\in B\cap U\setminus\{b\}\) and a lift of a neighborhood of \(s\). Consider two sequences of domain points tending to that lift. At all terms of both sequences the sample index is defined almost surely. For each sample, Step 2 shows that only finitely many translated rays meet a compact neighborhood of the chosen lift. Shrink to a connected neighborhood missing those rays that do not pass through the lift; their side memberships are then fixed. If the projected sample misses \(s\), the eventual index difference is therefore zero. Otherwise simplicity leaves just one ray through the lift. Only membership in its right-hand component can vary, so the index has at most two adjacent values. Hence the upper limit of the absolute index difference is at most one. The local uniform bound and the reverse Fatou inequality now give \[\limsup |v(z_n)-v(w_n)|\le m_\eta(s).\] Local boundedness makes the upper and lower bands finite, and choosing sequences approaching their extremes proves the asserted width bound. When \(m_\eta(s)=0\), the unique trace is \(\mathop{\mathrm{\mathbb E}}_\eta J_\alpha(s)\).

Step 8: compatibility. Put \(\Sigma'=\Sigma\cup\{s\in B:m_\eta(s)>0\}\), a countable enlargement of the flags. Fix a generic finite prior level union \(H\). By hypothesis the rule \[\nu(A)=\mathop{\mathrm{\mathbb E}}_\eta\#\bigl(A\cap H\cap\alpha([0,1])\bigr), \qquad A\subset H\text{ Borel},\] defines a finite measure. It gives zero mass to \(H\cap(B\setminus\Sigma')\), since almost every sample avoids this whole set. The counting integrals are measurable by compact incidence tests and finite distinct-point tests in path space; finite expected total count then defines the measure on all Borel subsets.

Let \(K\) be a nondegenerate component of \(H\) and work in a small disk \(V\) with closure in \(U\setminus\{b\}\) and with a common lift. For free points \(x,y\in K\cap V\) joined by a simple arc \(\delta\subset K\cap V\), their traces are expectations of the indices. Equation (58) gives \[ |v(x)-v(y)|\le\nu(\delta). \tag{61}\] Set \(F=K\cap V\cap(B\setminus\Sigma')\). We have \(\nu(F)=0\), and \(K\) is locally connected by Lemma 6. Equation (61) verifies the arc hypothesis of Lemma 7 for the continuum \(K\), the relatively open set \(K\cap V\), the measure \(\nu|_K\), and the free set \(F\). Hence its real trace image has Lebesgue outer measure zero. Countably many components \(K\) and lifted disks \(V\) suffice. The singular point \(b\) is flagged. Fubini in the level parameters therefore proves (P) for the new coordinate. All incidence statements may use completed measures, as in the earlier level arguments.

Step 9: smoothing. It remains to smooth \(v\) with an error that vanishes at every dust point, so that all nonsingular bands remain unchanged. Near the smooth curve \(C\), the continuous Sobolev branches extend across the boundary by reflection in smooth collar charts. Cover \(G\cap\overline U\) by a locally finite family of relatively compact simply lifted patches and take a smooth partition of unity. Smooth a branch on each patch; on other sheets use its integer translates. The local smoothing discrepancy is single-valued, so partition interpolation respects the additive period.

Choose a continuous positive tolerance \(e\) on the source patches with \[0<e(x)\le\tfrac12\min\{\varepsilon,\mathop{\mathrm{dist}}_X(x,B)\}, \qquad x\in G\cap\overline U.\] It extends positively across the compact collar of \(C\). On the \(j\)th patch choose the branch approximation error in values smaller than the positive infimum of \(e\) on the corresponding compact partition support. Simultaneously make its Sobolev error, including the derivatives of the partition function, less than \(\varepsilon 2^{-j}\). Local continuity and Sobolev approximation permit both requirements. Partition interpolation then gives \(|u-v|(x)\le e(x)\), while the differential errors are summable. The resulting \(u\) satisfies Equation (55) and is smooth up to \(C\). For every domain sequence tending in \(X\) to a nonsingular dust point, \(e(x)\) tends to zero; hence the upper and lower band limits are exactly unchanged in a common local lift. Its free traces are therefore unchanged. Since compatibility for a newly appended coordinate concerns only those traces against earlier level unions, (P) is preserved. ◻

Proof. Put \(\phi(w)=\mathop{\mathrm{dist}}(w,D_b)\). Its weak gradient has norm at most one. Apply Stokes’ theorem in the region inside \(F(C)\) after replacing each circular hole by a slightly larger concentric circle. There are only finitely many holes, so the offsets can be chosen disjoint and inside the domain. The exterior boundary integral is the wedge integral \(\int\mathrm d\phi\wedge\mathrm du^*\) plus the integrals around the offset circles, each with positive disk orientation.

On the distinguished offset of radius \(r_b+\rho\), \(\phi\) is the constant \(\rho\). The period of \(u^*\) there is one, so its contribution is \(\rho\) and tends to zero.

At every other hole the period is zero, so there is a single-valued branch. Subtracting a constant from that branch and integrating by parts bounds the boundary integral by \[2\pi(r_T+\rho)\bigl(A_u(T)+o(1)\bigr).\] Here \(|\mathrm d\phi|\le|\mathrm dw|\) along the Euclidean offset circle, and the oscillation on small offsets tends to at most \(A_u(T)\) by the collar correspondence in Lemma 10. This argument also covers a hole of radius zero. The remaining region has Euclidean area at most \(\pi L^2\), so Cauchy–Schwarz bounds the wedge integral by \(\sqrt\pi L\|\mathrm du^*\|_2\). Letting the offsets tend to zero proves Equation (62).

Disjointness of the disk interiors gives \(\sum_T r_T^2\le L^2\). The terminal indices are distinct, and the index \(b\) occurs only at the distinguished hole. The positive-excess estimate of Theorem 8 therefore gives \[\left(\sum_{T:\,s_T\ne b}A_u(T)^2\right)^{1/2} \le \|a_u|_{B\setminus\{b\}}\|_{\ell^2}+o(1).\] Conformal invariance of Dirichlet energy and the energy estimate of that theorem also give \(\|\mathrm du^*\|_2\le\|\mathrm du\|_{2,G\cap U}+o(1)\). A second application of Cauchy–Schwarz proves Equation (63). ◻

Proof. We first fix geometry that will serve both alternatives. In the barrier case it gives a family of line segments with a fixed positive transverse width. In the law case it keeps the arc starts a fixed positive distance from \(D_0\).

Fixed geometry near the protrusion. Write \(K=K_b(f_0)\). There is a boundary point \(x\) of \(K\) outside \(D_0\): for example, maximize distance to the center of \(D_0\) on \(K\). Choose a bounded open ball \(W_2\) about \(x\) whose closure misses \(D_0\), and a smaller closed ball \(W_1\) about \(x\) contained in \(W_2\). Boundary bumping, applied in a still smaller closed ball, gives a nondegenerate subcontinuum \(Q\subset K\cap\operatorname{int}W_1\). The ball can be chosen small enough that \(K\) also has points outside it, because \(D_0\subset K\) is separated from \(x\).

Every boundary point of a complementary component is approached from the domain. Indeed, if a ball about such a point missed the domain, the connected ball would lie in that same complementary component, contradicting that the point was a boundary point. Thus choose a small closed ball \(A\subset f_0(G)\cap\operatorname{int}W_1\) and nested smaller balls strictly inside \(A\). Their preimages give an anchor \(E\) and a compact testing set \(E_*\subset\operatorname{int}E\). Start the collar sequence deep enough that its closed disks miss \(E\).

There is a direction and a closed interval \(J\) of positive transverse width such that every line with transverse parameter in \(J\) joins a point strictly inside \(f_0(E_*)\) to a point of \(Q\), by a segment in \(\operatorname{int}W_1\). To see this, first direct a line from the center of the smallest anchor ball to a point of \(Q\). That point projects strictly inside the ball’s transverse projection. Perturb the direction, if necessary, so that the transverse projection of \(Q\) is nondegenerate. The projection is an interval, and it overlaps the interior of the anchor-ball projection in positive length. Take \(J\) compactly inside that overlap. Choose the starts on the middle chords of the anchor ball; they then have a uniform interior buffer. Convexity of \(W_1\) puts the segments inside it.

These sets and \(J\) are fixed before the test’s collar and penalty region are chosen. For every prescribed collar \(C_l\), the compact continuum \(Q\) lies strictly inside the bounded side of \(f_0(C_l)\), whereas the buffered starts lie outside. After a collar is fixed these inclusions persist under sufficiently small changes of the map, by winding number on \(C_l\) and on the boundary of \(E_*\). In particular the buffered starting points remain images of points in \(E_*\).

The barrier case. Use the endpoint problem with these anchors and with gates prescribed by \((f_0,W_1,W_2)\). Let \(c_L\) be a constant large enough for the line-payment estimate in the common target size bound. Fix \(\tau>0\) with \(c_L\tau<|J|/2\). If there is a compatible barrier \(u\) with \[\|\mathrm du\|_{2,G\setminus O}+\|a_u\|_{\ell^2}<\tau,\] append it. It equals zero on \(E_*\) and one on some fixed inner collar \(C_l\). In sufficiently close finite-map data the segments just constructed start in the zero region and reach the side inside \(F(C_l)\); stop each segment at its first hit of that curve. The compact set \(\overline O\) has image disjoint from \(W_1\) under \(f_0\), and this separation persists under convergence on \(\overline O\). Thus all gradients paid along these segments lie off the image of \(O\).

Lemma 26, with the transfer error tending to zero, gives \[|J|\le c_L\bigl(\|\mathrm du\|_{2,G\setminus O} +\|a_u\|_{\ell^2}\bigr)+o(1),\] a contradiction to \(c_L\tau<|J|/2\). To justify the restricted energy estimate, retain \(\overline O\) in the unchanged core and use the original energy off \(O\) there, together with the transfer error on the rest. All required separations involve fixed compact sets after \(u\), \(C_l\), and \(O\) have been chosen. This excludes a relative neighborhood of \(f_0\), and hence a relative neighborhood of \((f_0,D_0)\).

The law case. If no such barrier is available, Theorem 19 provides an endpoint law \(\mu\) obeying all prescribed gates. We will make the Stokes upper bound small while keeping the boundary integral bounded below. Put \[d_0=\mathop{\mathrm{dist}}(\overline W_2,D_0)>0.\] Apply Lemma 29 at a sufficiently deep collar \(C=\partial U\), and then Lemma 30. The resulting smooth period coordinate \(u\) can be chosen so that \[ \sqrt\pi L\,\|\mathrm du\|_{2,G\cap U} +2\pi L\,\|a_u|_{B\setminus\{b\}}\|_{\ell^2} <d_0/4. \tag{64}\] Indeed the energy norm and the nonsingular budget norm tend to zero with the depth. The singular budget is one and is absent from the Stokes upper bound. The gate puts all arc starts in \(f_0^{-1}(\overline W_2)\), so the unsmoothed coordinate satisfies \[ \int_C\mathop{\mathrm{dist}}(f_0(\cdot),D_0)\,\mathrm dv =\int\mathop{\mathrm{dist}}(f_0(\alpha(0)),D_0)\,\mathrm d\eta(\alpha) \ge d_0. \tag{65}\]

We next preserve this lower bound while smoothing. On the smooth compact curve \(C\), the function \(\psi=\mathop{\mathrm{dist}}(f_0(\cdot),D_0)\) is Lipschitz. Since \(u-v\) is single-valued on \(C\), integration by parts for a continuous function of bounded variation gives \[\left|\int_C\psi\,\mathrm d(u-v)\right| =\left|\int_C(u-v)\,\mathrm d\psi\right| \le \sup_C|u-v|\,\operatorname{Var}_C(\psi).\] After the collar is fixed, choose the smoothing accuracy so fine that this error is less than \(d_0/4\), while maintaining Equation (64). Thus \[\int_C\mathop{\mathrm{dist}}(f_0(\cdot),D_0)\,\mathrm du>3d_0/4.\] For fixed smooth \(u|_C\), this integral varies continuously with the map uniformly on \(C\) and with the disk in Hausdorff distance. Consequently it remains greater than \(d_0/2\) on a relative neighborhood \(\mathcal V\) of \((f_0,D_0)\).

Append \(u\) and retain \(b\) as a mark. In a finite transfer, the transferred coordinate agrees with \(u\) on \(C\) for all sufficiently late stages. If the limiting pair belonged to \(\mathcal V\), the outer integral in Lemma 31 would therefore have liminf greater than \(d_0/2\). Equations (63) and (64) give a limsup less than \(d_0/4\). This contradiction excludes \(\mathcal V\).

Persistence. In either case the contradiction uses one fixed appended coordinate, fixed compact source sets, its fixed energy and band bounds, and the vanishing errors for its eventual transfers. Enlarging the coordinate list or the flag set does not change these data. Later transfers still satisfy all the estimates once this coordinate and the required mark are included. The exclusion therefore persists under all further compatible countable extensions. ◻

Proof. First selection with the seed retained. Start the selection in Section 6 with \(\mathscr L_0\) and \(\Sigma_0\). Every application of Theorem 25 permits precisely this arbitrary current countable list and permits a countable flag enlargement. At a successor stage append the selected barrier after the existing well-order. At a countable limit stage take the order-preserving union of all earlier lists and the union of the flags. These unions are countable. Each compatibility requirement concerns one new label and finitely many lower labels, so it already holds at a previous stage; increasing the flags can only remove free points. Thus compatibility persists, while all previously chosen functions and their bounds remain fixed.

The countable-basis argument in Section 6 makes this process stop at a countable ordinal. It produces a countable set \(S_1\) as in Equation (47), with \(K_b(f)\) a singleton for every \(b\notin S_1\) in every later compatible transfer limit. Put \(S=S_0\cup S_1\). This enlargement is countable and contains the seed singularities. The newly selected barriers are real and introduce no singularities. In particular, Theorem 27 applies outside \(S\).

Second selection at every mandatory mark. Consider the second-countable disjoint union \[\mathcal Q=\coprod_{b\in S}\mathcal Q_b.\] Starting with an empty excluded open subset of \(\mathcal Q\), select an uncovered bad pair whenever one exists. Apply Lemma 32, append its coordinate, and add its excluded relative neighborhood to the open set. At countable limit stages take the order-preserving unions of coordinate lists and flags. All newly introduced period singularities belong to \(S\).

This selection also stops at a countable stage. Each strict increase of the excluded open set contains a basic open set not contained in the preceding one. Once chosen, that basic set is contained in all later excluded open sets, so it cannot be chosen again. An \(\omega_1\)-long selection would contradict the countability of the basis. At termination every bad pair is covered, and the accumulated list and flags are countable. The exclusions persist by Lemma 32.

Finite transfers and preservation of the original tests. Enumerate the underlying countable set of final labels and take increasing finite subsets \(I_n\) exhausting it, each ordered by the inherited well-order. A countable well-order need not have finite initial segments exhausting it; none are needed here. Similarly choose increasing finite subsets exhausting \(S\) and include in the \(n\)th prescribed terminal indices every singularity of \(I_n\). Apply Theorem 8 to \(I_n\), with error \(\varepsilon_n\), prescribed dust-cover mesh tending to zero, and input core containing \(H_n\), the preceding core, and all additionally specified compact source sets. One may include compact neighborhoods of these sets, so the resulting cores are nested and contain \(H_n\) in their interiors. Include the fixed exterior region and all current mask boundaries as required by that theorem.

Finite circle-domain uniformization gives \(f_n\) and \(\Omega_n\). Exact core agreement, the total tail bound, and the individual positive-excess bounds are exactly the conclusions of Theorem 8. They apply to every fixed seed coordinate for all sufficiently large \(n\). No selected flag enlargement changes a coordinate’s original band budget. The terminal disks shrink in the quotient metric by the prescribed cover meshes. Normalized compactness and the uniform radius bound follow from Lemma 9.

Identification of the limiting components. Pass to a subsequence on which the maps and all marked disks converge. Lemma 10 gives \(D_b\subset K_b(f)\) for \(b\in S\). If one inclusion were strict, \((f,D_b)\) would be a bad pair in a previously excluded neighborhood. The fixed coordinate responsible for that neighborhood occurs in every sufficiently late finite sublist. Its exact core values and vanishing transfer errors contradict its exclusion estimate. Thus \(D_b=K_b(f)\) for every \(b\in S\). All components outside \(S\) are points by Theorem 27, and these are all components by Lemma 10. Hence \(f(G)\) is a circle domain.

Every argument just used depends on fixed retained coordinates and marks and survives further compatible extensions. A new marked end outside \(S\) already has a singleton component; its distinguished disk is contained in that singleton and therefore equals it. This also proves the stated persistence for additional singularities and marks.

Finally, retain a fixed contour and its collar in all sufficiently late cores. Orientation and Jordan separation give its bounded-side correspondence in each finite surface. For the bound on its image, choose \(r_C>2R\) so large that the source contour is inside \(|z|<r_C\). The exterior estimate in Equation (18) bounds \(f_n(\{|z|=r_C\})\) uniformly; the contour’s image and its bounded side lie inside that outer image curve. This gives a bound depending only on \(R\) and the fixed contour, independent of the number of tests. Contours inside \(|z|\le4R\) have the common bound \(5R\). ◻

Proof of Theorem 1. For a proper sphere domain make an initial Möbius change of source coordinate so that infinity belongs to the domain. Apply Theorem 33 with the empty seed list and no prescribed marks. It supplies a conformal circle-domain image. The whole-sphere case is immediate. ◻

Proof. If \(B=\varnothing\), take \(Q_n=K_n=G=\widehat{\mathbb C}\), \(f_n\) the identity, and \(v_{i,n}=u_i\); all tail and oscillation sums vanish. We may therefore assume that \(B\) is nonempty.

Each test is bounded by compactness of \(X\), has full mask, and has singleton band \(\{u_i(b)\}\) at every \(b\in B\). It is therefore a coordinate with budget \(a_i=0\), and the initial flag set may be empty. We verify compatibility in any chosen ordering of the finite list.

For distinct labels \(i,j\), Equation (68) implies that the image of \(B\) under \((u_i,u_j)\) modulo one has zero Haar measure on \(\mathbb T^2\): decompose \(\mathbb R^2\) into countably many integer translates of a unit square. For a label \(j\) and a finite tuple of lower labels \(i_1,\ldots,i_d\), a common dust point between \(H_j(t)\) and any \(H_{i_k}(t_k)\) requires \((t,t_k)\) to lie in this null pair image. Each such condition is a null cylinder in \(\mathbb T^{d+1}\), by Fubini. Their finite union is null, including when lower labels are repeated. Almost every tuple therefore has no such common dust point at all, which is stronger than compatibility (P).

Apply Theorem 33 with these seeds, choosing every finite sublist to contain all \(m\) seeds from the outset. Put \(v_{i,n}=u_{i,n}\circ f_n^{-1}\). Exact core agreement gives Equation (69), and conformal invariance of energy gives Equation (70). Since all seed budgets vanish, Equation (67) gives Equation (71). The cluster-interval assertion is Lemma 10. All remaining conclusions are those of the theorem. ◻