Proof. Base change and thin triangles give \[(y|z)_v=(y|z)_e+k-(v|y)_e-(v|z)_e+O(1), \qquad (v|y)_e=\min\{k,(x|y)_e\}+O(1).\] Thus \(a^{-(v|y)_e}\asymp D_y\). Exponentiating the first identity and using \((hy|hz)_e=(y|z)_v+O(1)\) proves (2). With \(z=z_0\), the factor \(D_{z_0}\) is bounded above and below, and \(d(y,z_0)\) is bounded above. When \(d(x,y)\le\varepsilon_*/2\), the latter is also bounded below. This proves (3). Finally, \((v^{-1}|v^{-1}z_0)_e=k-(v|z_0)_e+O(1)\), and \((v|z_0)_e\) is bounded in terms of \(\varepsilon_*\). ◻

Proof. Translate a bi-infinite geodesic so that it passes through \(v\). The ideal triangle with third vertex \(e\) puts \(v\) uniformly near one of its based sides; the nearby parameter differs from \(|v|\) by a bounded amount. A bounded change of basepoint changes products by a bounded amount, preserving (2). The two remaining assertions follow from (1) and the product estimate. ◻

Proof. Choose \(k\) with \(r_k\asymp t\). Rays ending in the given ball have their depth-\(k\) vertices in a uniformly bounded graph ball, by 2. At depth \(k+j\) there are at most \(CA^j\) possible continuations, where \(A>1\) depends on the valence. Endpoints whose rays share such a vertex are at visual distance at most \(C'r_{k+j}\). Choose the least \(j\ge0\) with \(C'r_{k+j}<s\). The desired bound follows with \(Q\ge\max\{2,\log_a A\}\). Coarse scales are absorbed by increasing the constant. ◻

Proof. For small \(s\), expand at \(x\) with \(R\asymp s\) and choose \(z_0\) a uniformly positive distance from \(x\); two fixed distinct points suffice for this choice. By (3), the two image points stay a uniform positive distance from \(p=hz_0\). Fix a homeomorphism \(\phi:Z\to\mathbb S^2\). Uniform continuity of \(\phi^{-1}\) places their spherical images a uniform distance from \(\phi(p)\). Join them outside a smaller round ball about \(\phi(p)\). Uniform continuity of \(\phi\) makes the pullback path stay a uniform \(d\)-distance from \(p\). The upper bound in (3) puts its preimage under \(h\) in \(B(x,Ls)\). Increase \(L\) for the remaining scales, using path connectedness of \(Z\).

For a continuum \(C\) in an open set \(V\), choose \(\varepsilon>0\) with its \(L\varepsilon\)-neighborhood in \(V\). Connectedness supplies a finite \(\varepsilon\)-chain in \(C\) between any two specified points: the set reachable by such chains is relatively clopen. Join consecutive points by the first assertion. ◻

Verification of the cited criterion. Corollary 3.5 of [8], derived from [6], applies to an arcwise connected, approximately self-similar metric sphere. Approximate self-similarity means that each open ball, with distances divided by its radius, is uniformly bi-Lipschitz to an open subset of the space. 1 supplies this property; intermediate and large radii change only the constant.

The criterion assumes bounded \(2\)-moduli of paths of diameter at least a sufficiently small \(d_0\) in successive graph approximations. For a dyadic scale \(2^{-m}\) choose \(n\) with \(2^{-m}/a<r_n\le2^{-m}\). Our cells form a fixed \(\kappa\)-approximation for any \(\kappa>2a\): the inner balls of radius \(2^{-m}/\kappa\) are disjoint, each is contained in its cell, and each cell is contained in the radius-\(\kappa2^{-m}\) ball about its center. The modulus is exactly \(M(n)\). Finitely many coarse scales have finite modulus. No Coxeter-group assumption occurs in this criterion. ◻

Proof. Let \(\psi:Z\to\mathbb S^2\) be the uniformizing homeomorphism and write \(\mathcal G=\{\psi g\psi^{-1}:g\in G\}\). The boundary cross-ratio estimate and the two fixed maps \(\psi,\psi^{-1}\) give one quasi-Möbius distortion function for every element of \(\mathcal G\). Fix \(f\in\mathcal G\) and \(x\in\mathbb S^2\), keep a fourth point away from \(x\), and let two points at equal spherical distance from \(x\) approach \(x\). In the cross-ratio inequality the factors involving the fourth point tend to one, both before and after applying \(f\). Thus the infinitesimal linear distortion has a bound independent of \(f\) and \(x\); the radius at which the estimate is accurate need not be uniform in \(f\). The metric characterization of planar quasiconformality, applied in conformal charts, makes each \(f\) quasiconformal. At its almost-everywhere nonsingular differentiability points, the same infinitesimal bound controls the ratio of singular values of \(Df\), and hence its analytic dilatation. This gives a common quasiconformal bound. For a reversing map apply the characterization after a conformal reflection; see [4] and the metric characterization in [17].

We recall explicitly the invariant-structure argument, including orientation reversal. This is the invariant-conformal-structure method of Sullivan, with the circumcenter construction detailed by Tukia [24]. Analytic quasiconformal maps have almost-everywhere defined nonsingular weak differentials, and they and their inverses preserve null sets [4]. Their differentials satisfy the almost-everywhere composition rule [2]. Since \(\mathcal G\) is countable, remove the \(\mathcal G\)-orbit of the union of all exceptional sets for these assertions and for all pairs of maps. Its complement \(X\) is invariant and conull. At \(x\in X\), let \(\mathcal C_x\) be the space of positive definite quadratic forms on \(T_x\mathbb S^2\) modulo positive scalars. This is a hyperbolic plane, and pullback by any invertible linear map, of either orientation, is an isometry between the corresponding fibers [4]. The orbit \(\{f^*q_{\rm round}(x):f\in\mathcal G\}\subset\mathcal C_x\) is uniformly bounded, by the common quasiconformal bound, and contains the round form. Its unique hyperbolic circumcenter defines a uniformly bounded measurable conformal structure \(q(x)\). Indeed, circumcenters of the increasing finite subsets from an enumeration are measurable and converge to that circumcenter: their radii increase, their centers stay in a fixed compact ball, and every subsequential limit is the unique center of the whole orbit. The composition rule permutes this orbit under pullback, so \(f^*q=q\) on \(X\) for every \(f\in\mathcal G\). Extend \(q\) arbitrarily across the null complement of \(X\).

The measurable Riemann mapping theorem [2] gives a quasiconformal homeomorphism \(F:\mathbb S^2\to\mathbb S^2\) with \(F^*q_{\rm round}=q\) almost everywhere. Invariance and the analytic composition rule show that every \(FfF^{-1}\) is conformal or anticonformal, according to the orientation of \(f\). Such sphere homeomorphisms are Möbius or anti-Möbius. Their unique extensions to hyperbolic isometries therefore define a homomorphism \(\rho:G\to\mathop{\mathrm{Isom}}(\mathbb H^3)\).

For completeness, properness and cocompactness follow directly from the action on ordered distinct triples \(\mathcal T(Z)\). On any compact set of triples all pairwise distances are bounded below. For \(|v|\to\infty\), at least two points of such a triple stay away from the expansion center for \(v^{-1}\); their image distance is \(O(r_{|v|})\). Thus \(v^{-1}K\cap K=\varnothing\) eventually for every compact \(K\subset\mathcal T(Z)\). Conversely, expand at one point of a closest pair with scale comparable to their distance. Formula (2) makes all three image distances uniformly positive: if the third point is nearby, all denominator factors have that scale; otherwise its distances to the pair cancel its denominator factor. Triples already uniformly separated need no expansion. Hence the action on triples is proper and cocompact.

These properties survive boundary conjugacy. Let \(I=\mathop{\mathrm{Isom}}(\mathbb H^3)\). The stabilizer \(H\) of a standard ordered boundary triple has two elements: the identity and reflection in its hyperbolic plane. Choose \(o\in\mathbb H^3\) fixed by \(H\), and let \(K_o\) be its compact stabilizer in \(I\). The equivariant map \[\mathcal T(\mathbb S^2)\cong I/H\longrightarrow I/K_o\cong\mathbb H^3\] is continuous, onto, and proper: isometries sending \(o\) into a compact set form a compact set. Properness and cocompactness therefore descend. The kernel fixes \(o\), so is finite. ◻

Proof. Cover \(Z\) by finitely many coordinate boxes \([-1,1]^2\), each lying in a chart containing \([-8,8]^2\) of diameter less than \(d_0\). A macroscopic path meeting an inner box must leave its larger box. Follow it to its first meeting with \(\partial[-2,2]^2\). If it exits through \(x=2\), its segment after the last preceding meeting with \(x=1\) crosses \([1,2]\times[-2,2]\) from left to right. The other three exit sides give analogous strips. Each strip extends by its height in both transverse directions inside \([-8,8]^2\).

Consequently there are finitely many crossing families \(\mathcal F_1,\ldots,\mathcal F_J\) such that every macroscopic path contains a member of one of them. Summing admissible weights gives \[M(N)\leq J\sum_{j=1}^J\mathop{\mathrm{Mod}}_2(\mathcal F_j,T_N) \leq J^2\max_j\mathop{\mathrm{Mod}}_2(\mathcal F_j,T_N).\] One strip realizes this bound along a subsequence. Affine coordinates give the asserted rectangles. ◻

Proof. For (i), suppose a lateral continuum and a top-to-bottom path are disjoint. Enlarge the rectangle, extending the continuum horizontally and the path vertically to the corresponding new edges. The extensions remain disjoint: a new horizontal and vertical segment can meet only at an original corner, already a common point of the original sets. Approximate the extended path, within its positive clearance from the continuum, by a polygonal path with interior in the enlarged rectangle. After subdivision at intersections it contains a simple polygonal arc between the top and bottom edges. The Jordan curve theorem, applied after adjoining a boundary route, shows that this arc separates the lateral edges. The connected continuum cannot avoid it.

For (ii), subdivide into finer square grids and mark the closed squares meeting the blocker \(K\). No edge-adjacent chain of unmarked squares joins top to bottom. Label \(A\) the unmarked squares reachable from the top through unmarked edge-adjacent squares, and append a top row of \(A\) squares. Label all remaining squares and an appended bottom row \(B\). Draw the interfaces between unlike labels. Interior vertices have even degree; each lateral boundary has an odd number of ends, since its labels begin with \(A\) and end with \(B\). There are no top or bottom ends. Some interface component therefore joins the lateral boundaries: otherwise every component contributes an even number of ends to the only lateral boundary it meets.

Along a path in this component, the adjacent \(B\) squares are original marked squares. An adjacent unmarked square would be reachable, while an interface with the appended bottom row would give an unmarked top-to-bottom chain. Successive \(B\) squares meet at least at a corner. Their union is a connected compact lateral crossing. A Hausdorff limit as the mesh tends to zero is connected, meets both lateral edges, and lies in \(K\). ◻

Proof. Let \(P\) minimize Euclidean norm in the convex hull of the finitely many cell-incidence vectors \(v\) of left-to-right crossings. Its coordinates are nonnegative and sum to at least one, so \(P\ne0\). Minimality gives \(\langle P,v\rangle\geq\|P\|^2\), making \(P/\|P\|^2\) admissible. Conversely any admissible \(\omega\) satisfies \(\langle\omega,P\rangle\geq1\). Thus the left-to-right modulus is \(\|P\|^{-2}\). By 6(i), each top-to-bottom incidence vector has inner product at least one with every \(v\), hence with \(P\). Set \(\rho=P\) and apply 5. ◻

Proof. We have \(F_N\geq U_N^0\) and \(H_N\leq F_N\); constant paths give the band values. Concatenation with nearly optimizing paths gives, for \(x,y\in K\), \[F_N(x)\leq\max\{\max_K U_N^0,F_N(y)\},\qquad H_N(y)\geq\min\{\inf_K F_N,H_N(x)\}.\] If \(F_N(x)>F_N(y)\), the first inequality places their values between \(\min_K U_N^0\) and \(\max_K U_N^0\). If \(H_N(x)>H_N(y)\), the second places them between \(\inf_K F_N\) and \(\sup_K F_N\). This proves (8). Short coordinate segments and continuity of \(U_N^0\) then prove continuity of both functions, including at the boundary.

Every point of \(\{H_N>t\}\) joins the bottom band within that set: choose a path with \(\inf F_N>t\) and use its tails. Likewise every point of \(\{F_N<t\}\) joins the top within \(\{F_N<t\}\). To connect a point \(x\) with \(H_N(x)<t\leq F_N(x)\) to the top, choose \(H_N(x)<s<t\) and follow any top-going path until its first point with \(F_N=s\). On this prefix \(F_N\geq s\), so concatenation with nearly maximizing paths from any prefix point \(z\) gives \[H_N(x)\geq\min\{s,H_N(z)\}.\] Hence \(H_N(z)\leq H_N(x)<t\) throughout the prefix. Its endpoint joins the top through \(F_N<t\), hence through \(H_N<t\). Thus both strict sides of the level join their respective bands, and then the outer top and bottom edges, without meeting the level.

The compact set \(\{H_N=t\}\) blocks all top-to-bottom paths by continuity and lies in \(\mathcal Q\). By 6(ii) it contains a lateral continuum \(C_{N,t}\); take \(d_1\) to be the distance between the lateral edges of \(\mathcal Q\). Complete any straddling path to a top-to-bottom path using the strict-side connections. The completed path meets \(C_{N,t}\) by 6(i), and the added pieces avoid the level, proving the last assertion. ◻

Proof. Join two points of \(\overline B(x,Jr_N)\) inside \(B(x,LJr_N)\) by 4. Vertices of triangles meeting this path lie in \(B(x,(LJ+1)r_N)\); any two such vertices join inside \(B(x,L(LJ+1)r_N)\subset U\) for large \(N\). The raw endpoint inequality bounds every difference of their values by the right side of (9), with \(C(J)=L(LJ+1)+3\). It therefore bounds the interpolant’s oscillation along the original path. Apply (8).

Squaring and summing costs a bounded factor by 3: both the number of nearby weights and the number of centers charging each weight are bounded. The same holds for \(br_N\)-separated centers, with constants depending on \(b>0\). A fixed \(g\) is bi-Lipschitz, so inverse images of target balls lie in source balls of radius \(C_gJr_N\), their centers are \(b_gr_N\)-separated, and all these source balls lie in a fixed compact subset of \(U\). The summation bound and (7) finish the proof. ◻

Proof. Choose \(R_*\) with \(4R_*<d_1\) and all closed \(5R_*\)-balls about \(K\) contained in \(U\). All following restrictions on \(m_0\) concern only \(r_N/R=a^{-(N-l)}\).

Expand at \(x,l\) by \(h\), and choose macroscopic admissible weights \(\omega\) at scale \(N-l\) with squared mass \(M(N-l)\). Put \[\mathcal A=\{q\in T_N:R/2\leq d(q,x)\leq3R\},\qquad \sigma_q=\sum_{\substack{\xi\in T_{N-l}\\ d(\xi,hq)\leq C_0r_{N-l}}}\omega_\xi, \quad q\in\mathcal A,\] where \(C_0>C_E+2\). Expansion on \(B(x,3R)\) separates the \(hq\) by a fixed multiple of \(r_{N-l}\). Packing bounds both overlap multiplicities, giving \[\sum_{q\in\mathcal A}\sigma_q^2\leq C M(N-l).\]

Let \(I\) be the open interval between the endpoint values. For \(t\in I\), the continuum \(C_{N,t}\) meets the given path and exits \(B(x,2R)\), since its diameter exceeds \(4R\). By 4, join these two continuum points in its \(r_N/4\)-neighborhood and retain a radial subpath \(\eta_t\) between radii \(R\) and \(2R\), contained in that closed annulus. Its diameter is at least \(R\), so \(h\eta_t\) is macroscopic by the choice of \(d_0\) in 2.

If a target cell \(B(\xi,2r_{N-l})\) meets \(h\eta_t\) at \(hy\), choose \(q\in T_N\) with \(d(q,y)\leq r_N\). For large \(m_0\), this center belongs to \(\mathcal A\) and \(d(hq,hy)\leq C_Er_N/R\), so \(\omega_\xi\) occurs in \(\sigma_q\). Some level point lies within \(r_N/4\) of \(y\), hence in \(\overline B(q,2r_N)\). Thus \(t\) lies in the closed range interval \(I_q\) of \(H_N\) on that ball. Accounting for all target cells gives \[1\leq\sum_{q\in\mathcal A}\sigma_q\mathbf1_{I_q}(t) \quad(t\in I), \qquad \Delta\leq C M(N-l)^{1/2} \left(\sum_{q\in\mathcal A}|I_q|^2\right)^{1/2}.\] The second inequality follows by integration and Cauchy–Schwarz. Apply (9) with \(J=2\) and its bounded-overlap summation. Taking \(C(2)r_N<R/4\) confines every charged center to \(R/4\leq d(q,x)\leq4R\), proving (10). ◻

Proof. Fix compact \(K\subset U\), and let \(x,y\in K\) with \(d(x,y)\leq r_k\). They join inside \(B(x,Lr_k)\). Choose \(b\) with \(a^b>2L\) and consider indices \(l\leq k-b\) above a fixed interior threshold, in an arithmetic progression of step \(b_1\) with \(a^{b_1}>16\). The annuli \(\{r_l/4\leq d(x,\cdot)\leq4r_l\}\) are disjoint. For \(N\) sufficiently large relative to \(k\), (10) and (6) give \[|H_N(x)-H_N(y)|^2\leq C\lambda^{-l} \mu_N\{r_l/4\leq d(x,\cdot)\leq4r_l\}.\] Summing over at least \(ck\) indices yields \(|H_N(x)-H_N(y)|\leq C_Kk^{-1/2}\). If \(\lambda>1\), the single choice \(l=k-b\) gives \(C_K\lambda^{-k/2}\) instead.

For each tolerance this controls a tail uniformly; the finitely many omitted functions are individually uniformly continuous. Thus the sequence is equicontinuous on \(K\). Arzelà–Ascoli on a compact exhaustion of \(U\), followed by weak compactness of the measures on \(Z\), gives the asserted subsequence. The two constant band values prove nonconstancy. The stronger estimate gives local Hölder exponent \(\log\lambda/(2\log a)\).

For (11), fix \(A>2L\) and choose \(As\leq R=r_l<aAs\). Any two points of \(\overline B(x,s)\) join in \(B(x,Ls)\subset B(x,R/2)\). The annular and record estimates give \[\bigl(\mathop{\mathrm{osc}}_{\overline B(x,s)}H_N\bigr)^2 \leq C\mu_N\bigl(\overline B(x,4aAs)\bigr)\] for large \(N\). Local uniform convergence passes the oscillation to the limit; the closed-set inequality for weak convergence bounds the right-hand limsup by \(C\mu(\overline B(x,4aAs))\). Take \(C_\mu=4aA\).

For each fixed \(t\), take a Hausdorff limit \(C_t\) of \(C_{N,t}\) in \(\mathcal Q\). It is connected and has diameter at least \(d_1\). Local uniform convergence gives \(u=t\) on \(C_t\cap U\). A fixed compact straddling path also straddles \(t\) for every sufficiently large \(H_N\), hence meets those \(C_{N,t}\) and their limit. The subsequence used here may depend on \(t\); no simultaneous selection is needed. ◻

Proof. Let \(I_q(f)\) be the closed range interval of \(f\) on the closed cell. For large \(n\) these cells lie in \(V\). The interval \(w(D_t)\) is covered by the range intervals of \(w\) on cells meeting \(D_t\), and each such cell has \(t\in I_q(f)\). Therefore \[b_0\leq\sum_{q:B(q,2r_n)\cap K\ne\varnothing} \mathop{\mathrm{osc}}_{\overline B(q,2r_n)}w\,\mathbf1_{I_q(f)}(t).\] Integrate over \(I\) and use \(|I\cap I_q(f)|\leq\mathop{\mathrm{osc}}_{\overline B(q,2r_n)}f\). Only finitely many interval indicators are integrated, so no measurable selection of \(D_t\) is required. Cauchy–Schwarz gives the final assertion. ◻

Proof. In a compact metric space \(X\), a point’s component is the intersection \(Q_x\) of its clopen neighborhoods. To verify connectedness of \(Q_x\), suppose it is separated into two compact pieces in disjoint open sets \(O_1,O_2\). Compactness of \(X\setminus(O_1\cup O_2)\) gives a finite intersection \(W\) of clopen neighborhoods of \(x\) contained in their union. Its intersection with the \(O_i\) containing \(x\) is clopen in \(X\) and misses the other piece of \(Q_x\), a contradiction. The same compactness argument shows that every open neighborhood of a component contains a clopen neighborhood of it.

Apply this to \(X=D\setminus B(p,r)\) and the component \(C\) of \(x\). If \(C\) missed the sphere, it would have a clopen neighborhood \(W\) in \(X\) missing the sphere. Compactness places \(W\) a positive distance from \(\overline B(p,r)\), so \(W\) is also clopen in \(D\). It is nonempty and misses \(D\cap B(p,r)\), contradicting connectedness. Thus \(C\) is the required subcontinuum. ◻

Proof. Write \(\delta(x)=\mathop{\mathrm{dist}}(x,Z\setminus O)\) and fix \(\eta>Q\). A finite cover of \(\overline O\) gives a uniform Hölder bound, with some exponent \(\alpha_0>0\), on a neighborhood of \(\overline O\). For all large \(k\) put \[A_k=\sup_{x\in O}\delta(x)^\eta \mathop{\mathrm{osc}}_{\overline B(x,r_k)}f.\] The balls lie in \(\Omega\), and \[ c a^{-Qk}\le A_k\le C a^{-\alpha_0 k}. \tag{14}\] For the lower bound, cover a compact variation path \(P\subset O\) by at most \(C r_k^{-Q}\) radius-\(r_k\) balls centered on \(P\), using 3. Their range intervals cover the interval \(f(P)\), so one ball has oscillation at least \(c r_k^Q\); also \(\min_P\delta>0\). The upper bound is the Hölder estimate.

Fix \(0<\alpha<\alpha_0\). Since \(A_k a^{\alpha k}\to0\), there are arbitrarily late indices attaining their maximum over the remaining tail. At these indices, \[ A_{k+j}\le a^{-\alpha j}A_k\qquad(j\ge0). \tag{15}\] Choose \(x_k\in O\) attaining at least half the supremum, and set \(\delta_k=\delta(x_k)\) and \(\sigma_k=A_k/\delta_k^\eta\). Boundedness of \(f\) and (14) give \(\delta_k^\eta\ge c r_k^Q\), hence \(\delta_k/r_k\ge c r_k^{Q/\eta-1}\to\infty\). Expand at \((x_k,k)\) and take a subsequence with \(p_k=h_kz_0\to b\).

For a compact \(K\subset Z\setminus\{b\}\), (3) gives \[ h_k^{-1}K\subset\overline B(x_k,C_Kr_k)\subset O \tag{16}\] eventually. The expansion formula gives \(d(h_k^{-1}z,h_k^{-1}w)\le C_Kr_kd(z,w)\) on \(K\). On every fixed multiple of this source ball, \(\delta\) is comparable to \(\delta_k\). Thus (15) implies \[ \frac{|f(y)-f(y')|}{\sigma_k}\le C a^{-\alpha j} \quad\text{if }d(y,y')\le r_{k+j},\quad j\ge0. \tag{17}\] For sufficiently small \(d(z,w)\), choosing \(j\ge0\) with \(a^{-j}\ge C_Kd(z,w)>a^{-j-1}\) proves a uniform local \(\alpha\)-Hölder estimate for \(F_k\) on \(K\).

For boundedness, 4 joins \(h_k^{-1}z\) to \(x_k\) inside a fixed multiple of \(B(x_k,r_k)\). Packing covers this path by a bounded number of radius-\(r_k\) balls centered on the path, each with oscillation at most \(C\sigma_k\). Their range intervals cover the path’s image, so \(|F_k(z)|\le C_K\). Compact exhaustion and Arzelà–Ascoli now give a locally uniform, locally Hölder limit. Finally, \(\mathop{\mathrm{osc}}_{h_k\overline B(x_k,r_k)}F_k\ge1/2\). The lower puncture bound places all these image balls in one compact set off \(b\), so local uniform convergence makes the limit nonconstant. ◻

Proof. For \(j\ge4\) put \(K_j=\{d(z,b)\ge1/j\}\) and \(L_j=\{d(z,b)\ge1/(2j)\}\). Choose the \(j\)th expansion so that \(h_j^{-1}L_j\subset U\) and its normalized \(u\)-transform differs from \(W\) by at most \(1/j\) on \(L_j\). Hold this expansion and its normalization fixed. Then choose \(N_j>N_{j-1}\) so large that \[\sup_{L_j}\frac{|H_{N_j}(h_j^{-1}z)-u(h_j^{-1}z)|}{\sigma_j} \le\frac1j, \qquad \mathcal E_{N_j}^{(j)}(f_j;K_j)\le\frac1j, \qquad (j+2)r_{N_j}<\frac1{2j}.\] The uniform approximation follows from 2. With \(h_j\) and \(\sigma_j\) fixed, 9 bounds the displayed energy by \(C(h_j,K_j,j)/(\sigma_j^2M(N_j))\), so the energy condition follows by increasing \(N_j\). The last condition keeps all the enlarged balls in \(L_j\). This proves local uniform convergence and vanishing energy at every fixed enlargement off \(b\).

For a fixed \(g\), choose a compact neighborhood \(K'\) of \(g^{-1}K\) avoiding \(b\). Pulling back each target ball and choosing a nearest source net point gives, by bi-Lipschitz distortion and packing, \[\mathcal E_n^{(J)}(F\circ g^{-1};K) \le C_g\mathcal E_n^{(J')}(F;K')\] for some fixed \(J'\ge2\) and all large \(n\). Each source center receives only boundedly many target centers. Eventually \(K'\subset K_j\) and \(J'\le j\), proving (18).

Uniform convergence makes the endpoint values of \(f_j\) straddle \(I\). The source level \(\tau_j(t)=u(x_j)+\sigma_jt\) therefore lies in \((0,1)\), and 8 supplies \(D_{j,t}=h_jC_{N_j,\tau_j(t)}\), meeting \(P\). Its source continuum has diameter at least \(d_1\), so contains a point at distance at least \(d_1/2\) from \(x_j\). By (3), \[\mathop{\mathrm{dist}}(b,D_{j,t})\le d(b,p_j)+Cr_{k_j}\longrightarrow0\] uniformly for \(t\in I\). ◻

Proof. Suppose \(x\in\mathcal N(W)\setminus\{c\}\) and \(V(x)\ne V(b)\). Choose \(\varepsilon>0\) and a coordinate disk about \(x\), compactly contained off \(b,c\), on which \(|V-V(b)|\ge4\varepsilon\). It contains a compact variation path \(P\) for \(W\). Choose a compact positive-length interval \(I\) strictly between its endpoint values, avoiding \(W(c)\). Continuity gives \(\epsilon_c>0\) such that \(\overline B(c,\epsilon_c)\) avoids \(b\) and its \(W\)-range has positive distance from \(I\). Consequently, for all large \(j\), the continua \(D_{j,t}\) from 14 avoid this closed ball.

Choose \(\rho>0\) so that \(\overline B(b,\rho)\) avoids \(c,P\) and \(|V-V(b)|\le\varepsilon\) there. For \(w_j=f_j\circ g^{-1}\), local uniform convergence gives \[|w_j-V(b)|\ge3\varepsilon\quad\text{on }P, \qquad |w_j-V(b)|\le2\varepsilon\quad\text{on }\overline B(b,\rho)\] for all large \(j\). Each \(D_{j,t}\) meets \(P\) and enters \(B(b,\rho)\). By 12, it has a subcontinuum joining a chosen point of \(P\) to the radius-\(\rho\) sphere and contained in \[K=\{z:d(z,b)\ge\rho,\ d(z,c)\ge\epsilon_c\}.\] This subcontinuum lies in the level \(f_j=t\) and has \(w_j\)-oscillation at least \(\varepsilon\). A fixed compact neighborhood of \(K\) avoiding both punctures eventually lies in \(h_jU\cap gh_jU\) and contains every closed radius-\(2r_{N_j}\) cell meeting \(K\). Hence 11 and Cauchy–Schwarz give \[\varepsilon|I|\le \bigl(\mathcal E_{N_j}^{(2)}(f_j;K)\bigr)^{1/2} \bigl(\mathcal E_{N_j}^{(2)}(w_j;K)\bigr)^{1/2}\longrightarrow0,\] a contradiction. All sets and \(g\) are fixed in this limit. Finally apply (19) at \(s^{-1}x\) with group element \(s^{-1}t\) to obtain (20). ◻

Proof. Let \(\xi\) be a based ray to \(b\). For \(q\in E\), \(|q|\le t\), thinness of the ideal triangle with sides \([e,q],\xi,q\xi\) puts \(q\xi(n)\) uniformly near \(\xi\) when \(n\) is sufficiently larger than \(t\). The nearby parameter lies in \([n-t-C,n+t+C]\). For one common integer \(n\), these vertices are distinct as \(q\) varies. Bounded valence therefore gives \[ \#\{q\in E:|q|\le t\}\le C'(t+1). \tag{21}\]

If another accumulation direction exists, take \(q_n\to x\ne b\) and \(q_n^{-1}\to y\) along a subsequence. Base change gives, uniformly on compact sets avoiding \(y\), \[(q_n|q_nz)_e=|q_n|-(q_n^{-1}|z)_e+O(1)\ge |q_n|-O(1).\] By 2, \(q_nz\to x\) there, and \(q_nb=b\) forces \(y=b\). Choose disjoint small neighborhoods of \(b,x\), with a point outside their union. For large \(n\), \(q=q_n\) sends the complement of the first into the second. Its positive iterates never return the outside point, so \(q\) has infinite order. It is therefore loxodromic, with endpoints \(b,x'\); see [9].

If some \(h\in E\) moved \(x'\), the cyclic subgroups generated by \(q\) and \(hqh^{-1}\) would intersect trivially: a nonzero common power would have the same pair of endpoints. Their \(m^2\) products \(q^i(hqh^{-1})^j\), \(1\le i,j\le m\), would then be distinct and have length \(O(m)\), contradicting (21). Thus every element of \(E\) fixes both endpoints. It maps a geodesic line between them to a line at uniformly bounded Hausdorff distance, so its orbit of \(e\) stays a fixed distance from that line. Its only possible accumulation directions are the two endpoints. ◻

Proof. Let \(E=\mathop{\mathrm{Stab}}(b)\) and let \(\Lambda_E\) be its at-most-two accumulation directions. Choose a variation region \(O\) for \(W\) whose closure avoids \(\{b\}\cup\Lambda_E\). Such a region exists: otherwise \(W\) would be locally constant on the connected sphere minus this finite set, and continuity would make it constant throughout its domain.

Apply 13 to \(W\) on \(Z\setminus\{b\}\), with this \(O\) and \(z_0=b\). Write the expansions as \(s_i=v_i^{-1}\), with centers \(y_i\in O\), depths \(\ell_i\to\infty\), and punctures \(p_i=s_i b\to b_*\). Positive affine normalizations of \(W_i=W\circ s_i^{-1}\) converge locally uniformly off \(b_*\) to a continuous nonconstant \(W_*\).

The punctures have a pairwise distinct subsequence. Indeed, if \(v_i^{-1}b=v^{-1}b\) infinitely often, then \(v_i v^{-1}\in E\). After a subsequence \(y_i\to y\in\overline O\). Since \(v_i\) lies at depth \(\ell_i\) on a ray to \(y_i\), 2 gives \(v_i\to y\) in boundary direction. Bounded right multiplication preserves this direction, implying \(y\in\Lambda_E\), a contradiction.

Choose two coordinate disks \(D_1,D_2\), with compact closures off \(b_*\), having disjoint \(W_*\)-range intervals and each containing a compact variation path. To do this, follow a variation path to its first hitting points of two distinct intermediate values. Just before either first hitting time the values differ, so the function is not locally constant at either point. Continuity supplies disks with disjoint ranges, and non-local constancy supplies a variation path in each.

For every sufficiently large \(i\), the puncture \(p_i\) is outside the closed disks. Uniform convergence of the normalized \(W_i\) preserves both the disjoint range intervals and the positive variation on the two fixed paths. Positive affine normalization preserves these properties, so \(\mathcal N(W_i)\) meets each disk. Fix two distinct such indices \(i,j\). By (20), \(W_j\) has the single value \(W_j(p_i)\) at every point of \(\mathcal N(W_i)\) in either disk. Taking one point in each contradicts the disjoint \(W_j\)-range intervals. ◻

Proof. Fix \(b\in[AR,B_0R]\). Each \(C_t\), \(t\in J\), meets the radius-\(b\) sphere, so the inner intervals of candidates with \(|d(x,y)-b|\leq r\) cover \(J\). By (24), two of them meet the respective quarters in (26) by at least \(\kappa\Delta\).

If both have \(t_y<1/8\), the lower-quarter outer interval ends below \(j_-+3\Delta/8\), and the upper-quarter interval starts above \(j_-+5\Delta/8\). Indeed each meets its quarter but overlaps \(J\) in length less than \(\Delta/8\); the assertion also holds for intervals extending beyond \(J\). Their distance exceeds \(\Delta/4\), making the node good. Otherwise one candidate has \(t_y\geq1/8\). Apply this at radial values spaced by \((H+3)r\): there are at least \(c_Ha^D\) such values, and the resulting centers are more than \(Hr\) apart. ◻

Proof. Join two candidates in a graph when their distance is at most \(Hr\). Packing bounds its degree, so graph coloring partitions \(Y\) into at most \(C_H\) separated classes, with \(C_H\) independent of \(D\). If the conclusion fails, then \[ \sum_{y\in Y}\min\{a^{-\gamma D}t_y,a^{\tau D}t_y^2\}<C_H. \tag{33}\] Call \(y\) large when \(t_y\geq a^{-(\tau+\gamma)D}\) and small otherwise. On large candidates the minimum is linear, so \[ \sum_{\text{large }y}t_y\leq C_Ha^{\gamma D}. \tag{34}\]

We will test levels at an intermediate radius \(s\), larger than the child radius \(r\) and smaller than the parent radius \(R\). This provides more radial tests than the large ranges can occupy. At the same time, it lets us detect the remaining levels with a modulus cost at the relative scale \(r/s\).

Remove the large ranges. Set \(D'=\lceil\beta D\rceil\) and \(s=a^{-D'}R\). For large \(D\), \(0<D'<D\) and \(s<R/4\). Choose \(s\)-spaced radial bins \(\mathcal B\) in \([(A+1)R,(B_0-1)R]\), so \(\#\mathcal B\asymp a^{D'}\). On \(\mathcal B\times J\) use counting measure times Lebesgue measure. Mark \((b,t)\) if a large candidate satisfies \[|d(x,y)-b|\leq5s,\qquad t\in I_y^{\mathrm{out}}.\] Each candidate meets at most 11 bins. By (25) and (34), the marked measure is at most \(CC_Ha^{\gamma D}\Delta\). Since \(\gamma<\beta\), the unmarked measure is at least \[ ca^{D'}\Delta. \tag{35}\]

Detect the remaining levels by modulus. Take a maximal \(s\)-separated set \(\mathcal Z_s\) in \(\mathcal A_x(R)\); its radius-\(s\) balls cover that annulus, and it has at most \(Ca^{QD'}\) points. Expand at each \(z\in\mathcal Z_s\) at depth \(k+D'\) by \(h_z\). Put \(q=r/s=a^{-(D-D')}\), and take macroscopic admissible weights \(\omega_\xi\), \(\xi\in T_{D-D'}\), of squared mass \(M(D-D')\). For fixed \(C_T>2+C_E\), define \[\sigma_y^z= \begin{cases} \displaystyle\sum_{\xi:\ d(\xi,h_zy)\leq C_Tq}\omega_\xi, &d(y,z)\leq3s,\\[3pt] 0,&d(y,z)>3s, \end{cases} \qquad \sigma_y=\sum_{z\in\mathcal Z_s}\sigma_y^z.\] Within \(\overline B(z,3s)\) the candidate images are separated by a fixed multiple of \(q\). Thus every sum has boundedly many terms and every target weight is charged boundedly many times. Thus \(\sum_y(\sigma_y^z)^2\leq CM(D-D')\) for each \(z\). Also each \(y\) lies within \(3s\) of boundedly many auxiliary centers. Packing and Cauchy–Schwarz give \[ \sum_y\sigma_y^2\leq Ca^{QD'}M(D-D'). \tag{36}\]

Fix an unmarked \((b,t)\). Choose \(w\in C_t\) on the radius-\(b\) sphere and \(z\in\mathcal Z_s\) with \(d(w,z)\leq s\). The diameter bound gives a point of \(C_t\) outside \(\overline B(w,2s)\). Join these points in the \(r/4\)-neighborhood of \(C_t\), by 4, and stop at the first radius-\(s\) hit about \(w\). The resulting path \(\pi\) has diameter at least \(s\) and lies in \(\overline B(w,s)\subset\overline B(z,2s)\). The scale-\(s\) expansion \(h_z\) therefore makes it macroscopic by the choice of \(d_0\) in 2.

The radial bin lies between \((A+1)R\) and \((B_0-1)R\), while \(s<R/4\). Thus every point of \(\pi\) lies between radii \((A+3/4)R\) and \((B_0-3/4)R\) about \(x\). In particular it lies in the candidate annulus and has a candidate center within distance \(r\). The same fixed source margin places every level point within \(r/4\) of \(\pi\) in \(K^+\subset U\). On these points the scalar-limit property gives \(u=t\), even though it does not assert this identity on parts of \(C_t\) outside \(U\).

For a target cell meeting \(h_z\pi\), choose a preimage hit \(v\in\pi\) and \(y\in Y\) with \(d(y,v)\leq r\). Then \(d(y,z)<3s\). If \(\xi\) is the target cell’s center, (2) gives \(d(\xi,h_zy)<(2+C_E)q<C_Tq\), so its weight occurs in \(\sigma_y^z\). A point of \(C_t\) is within \(r/4\) of \(v\), hence within \(5r/4<c_0r\) of \(y\). The preceding interior check gives \(t\in I_y^{\mathrm{out}}\). Since \(|d(x,y)-b|\leq s+r<5s\), unmarkedness makes \(y\) small. Admissibility therefore yields \[ \sum_{\substack{\text{small }y:\ |d(x,y)-b|\leq5s}} \sigma_y\mathbf1_{I_y^{\mathrm{out}}}(t)\geq1 \quad\text{on every unmarked pair}. \tag{37}\]

The unmarked levels now have an admissible detection weight supported on small candidates. We compare its squared mass with the surviving amount of level parameter.

Compare the energies. Integrating (37), using bounded bin multiplicity, gives \[ca^{D'}\Delta\leq C\Delta\sum_{\text{small }y}\sigma_y t_y.\] Only finitely many closed-interval indicators are integrated; no measurable selection of continua or paths is needed, and singleton ranges contribute zero. Cancel \(\Delta>0\). Cauchy–Schwarz and (36) imply \[ \sum_{\text{small }y}t_y^2 \geq\frac{ca^{-(Q-2)D'}}{M(D-D')}. \tag{38}\] Subexponential growth gives \(M(D-D')\leq a^{\tau D/4}\) for large \(D\): first bound \(M(j)\leq Ca^{\tau j/8}\) and then absorb \(C\) by increasing \(D\). Since \(D'\leq\beta D+1\) and \((Q-2)\beta<\tau/4\), \[\sum_{\text{small }y}a^{\tau D}t_y^2 \geq ca^{[3\tau/4-(Q-2)\beta]D} \geq ca^{\tau D/2}.\] On small candidates this is exactly the capacity, contradicting (33). Every threshold for \(D\) is uniform over nodes. ◻

Proof. By (42), (41), and Jensen’s inequality, \[\frac{E_0}{\Delta_0^2} \geq\sum_iP_i a^{D((b_*+\tau)m_i-\tau n)} \geq a^{D((b_*+\tau)\sum_iP_im_i-\tau n)}.\] Thus \[\frac1n\sum_iP_im_i\leq\frac{\tau}{b_*+\tau}+O(1/n) =\frac1{11}+O(1/n).\] Conservation of probability identifies the left side with the average bad mass over the first \(n\) generations. If good mass were eventually less than \(1/2\), that average would have lower limit at least \(1/2\), a contradiction. ◻

Proof. Centers assigned to the same depth-\(k\) vertex have mutual distance \(O(R)\). Their \(HR\)-separation and 3 bound their number. If several puncture balls share a point, their punctures have mutual products at least \(k-O_C(1)\). By (4) and 2, all corresponding inverse vertices lie in one bounded Cayley-graph ball. There are boundedly many such vertices, each with the bounded multiplicity just proved. ◻

Proof. Fix an ancestor \(A\) at generation \(n-m\), whose depth is \(k-l=k_0+D(n-m)\). We claim that only boundedly many descendants \(j\) have \(d(p_j,p_i)\le a^{-l}\). Split a geodesic word \[v_j^{-1}=w_ju'_j,\qquad |w_j|=l,\qquad |u'_j|=k-l.\] The puncture condition and (4) give \((p_j|p_i)_e\ge l-C\) and \((v_j^{-1}|p_j)_e\ge k-C\). Thus \(w_j\) lies a bounded distance from the depth-\(l\) point on a fixed ray to \(p_i\), leaving boundedly many possibilities.

Reversing and inverting the same word gives the geodesic \(v_j=(u'_j)^{-1}w_j^{-1}\). Its depth-\((k-l)\) prefix is therefore \((u'_j)^{-1}\). Since \[d(x_j,x_A)\le2(B_0+1)r_{k-l},\] thinness and 2 put this prefix a bounded distance from the depth-\((k-l)\) point on a fixed ray to \(x_A\). Hence \(u'_j\) also has boundedly many possibilities. Their products determine boundedly many vertices \(v_j\), and the first step of 20 bounds the nodes assigned to each. We count group elements here, so nonuniqueness of geodesic spellings causes no additional multiplicity. The source and inverse prefixes together account for the full word.

Each such descendant has \(P_j\le P_Aa^{-\gamma Dm}\) by (44). Sum first over these descendants and then over ancestors, whose probabilities sum to one. ◻

Proof. Write \(F_i=u\circ h_i^{-1}\) and \(F_j=u\circ h_j^{-1}\) on their open domains. If \(\mathop{\mathrm{osc}}_{E_i}F_j\ge\Delta_j/16\), endpoint \(j\) qualifies. Here we use \(\kappa\le1\). Otherwise, one pattern ball \(\mathcal B^s\) has its \(F_j\) range interval at distance at least \(\Delta_j/16\) from that on \(E_i\). Indeed, if neither did, the two pattern intervals would have distance less than \(3\Delta_j/16\), contradicting their separation by \(\Delta_j/4\).

Node \(i\) supplies a compact path \(\eta\subset\mathcal B^s\) with endpoint \(F_i\)-gap at least \(\kappa\Delta_i\). Suppose \(\mathop{\mathrm{osc}}_{E_j}F_i<\kappa\Delta_i/4\). Removing this latter range interval from the endpoint gap leaves a positive-length closed interval \(I\subset(0,1)\) strictly between the endpoint values and a positive distance from the range on \(E_j\). To obtain positive margins, choose a smaller closed interval inside one of the remaining open intervals. Its length need not be uniform over pairs.

Fix the pair and \(I\) before letting \(N\to\infty\). Put \(F_{i,N}=H_N\circ h_i^{-1}\) and \(F_{j,N}=H_N\circ h_j^{-1}\). The fixed range margins and local uniform convergence imply, for every \(t\in I\) and all sufficiently large \(N\), that \(h_iC_{N,t}\) hits \(\eta\) and avoids \(E_j\). It also reaches the inner \(i\)-puncture ball. For the first assertion, the endpoint values still strictly straddle all of \(I\), so 8 applies to the preimage path. For the second, the range on \(E_j\) remains disjoint from \(I\). By 12, it has a subcontinuum \(K_{N,t}\) containing a hit on \(\eta\), meeting the sphere \(d(z,p_i)=A_hR\), and avoiding \(B(p_i,A_hR)\). It remains in the level \(F_{i,N}=t\) and avoids \(E_j\). In particular, \[K_{N,t}\subset K_{ij}:= Z\setminus\bigl(B(p_i,A_hR)\cup B(p_j,A_hR)\bigr) \Subset h_iU\cap h_jU.\] Compact containment follows from (49) with half the radius. The sphere hit belongs to \(E_i\); no identification of that sphere with the boundary of a metric ball is needed.

Uniform convergence on \(\eta\cup E_i\) preserves the \(F_j\) range gap, giving \(\mathop{\mathrm{osc}}_{K_{N,t}}F_{j,N}\ge\Delta_j/32\) uniformly for \(t\in I\). The finite-cell comparison, 11, now gives \[\frac{\Delta_j}{32}|I|\le \sum_{\substack{q\in T_N\\B(q,2r_N)\cap K_{ij}\ne\varnothing}} \mathop{\mathrm{osc}}_{\overline B(q,2r_N)}F_{i,N}\, \mathop{\mathrm{osc}}_{\overline B(q,2r_N)}F_{j,N}.\] For large \(N\) these closed cells lie in the common domain. Both squared oscillation sums tend to zero by 9, so Cauchy–Schwarz contradicts the positive left side. Therefore \(\mathop{\mathrm{osc}}_{E_j}F_i\ge\kappa\Delta_i/4\), proving the alternative. The argument applies separately to every pair above the geometric threshold; no convergence rate uniform over pairs is required. ◻

Conclusion of the proof of 5. Assign each unordered far pair to a qualifying endpoint. At any point, (55) allows only \(Ca^{Ql}\) paying indices \(i\), and (54) allows boundedly many partners for each. Integrating this multiplicity bound gives \[ \sum_{\{i,j\}\text{ far, assigned to }i}\mu(\mathcal D_{ij}) \le Ca^{Ql}\mu(Z). \tag{56}\] This counts the enlarged balls directly and uses no invariance of \(\mu\). The assignment may be arbitrary when both endpoints qualify: all lower bounds hold for either endpoint, and the multiplicity estimate already counts every possible ordered contribution.

Conversely, every paying endpoint satisfies \[\Delta_i^2\ge\Delta_0^2a^{2\gamma Dn}P_i^2 \ge\Delta_0^2a^{2\gamma Dn}P_*^2.\] There are at least \(c\mathcal N^2\) far pairs, so (53) and (46) give total cost \[ c\mathcal N^2\kappa^2\Delta_0^2a^{2\gamma Dn}P_*^2 \ge c'k^{-2}a^{2\gamma Dn}. \tag{57}\] Since \(\kappa,\Delta_0>0\) are fixed and \(Ql=QD\lfloor\gamma n/Q\rfloor\le\gamma Dn\), comparison with (56) yields \[a^{\gamma Dn}\le Ck^2.\] This contradicts \(k=k_0+Dn\), excluding the unbounded subexponential case. ◻