Proof. A path avoiding \(u,v\) and changing arcs has an edge whose endpoints lie in the two strict interiors. Its endpoints alternate with \(u,v\), which contradicts noncrossing. ◻

Proof. Suppose an edge \(uv\) of the path joined distinct gaps. Then each strict circular \(u\)–\(v\) arc contains a vertex of \(S\). A path inside \(S\) joining these two vertices would have to visit \(u\) or \(v\) by Lemma 5. Both endpoints lie outside \(S\), a contradiction. Thus each edge stays in one gap, and so does the whole path. ◻

Proof. If the walk avoids \(i\), take its first edge changing from one side of \(i\) to the other. The index \(i\) lies on one strict circular arc between its endpoints, and an index in the complementary closed arc lies on the other. They are therefore not cyclic neighbors. The final assertion is Lemma 5. ◻

Proof. Place the indices on a circle and draw each edge as a straight chord. The noncrossing condition makes this an outerplanar drawing. Add a new vertex in the complementary disk on the sphere and join it to every column vertex by a disjoint radial star. The resulting cone graph is planar. If \(G_I\) had a \(K_{2,3}\) minor, perform its deletions and contractions while retaining the new vertex and one edge from it to every contracted branch set. The resulting planar minor contains the cone over \(K_{2,3}\). That cone contains \(K_{3,3}\): add the cone vertex to the part of size two. This is impossible, since a simple bipartite planar graph on six vertices has at most eight edges, whereas \(K_{3,3}\) has nine. ◻

Proof. For an index \(i\) in the set, take a shortest path from \(o\) to \((i,H)\). If it visits a column \(j\) at height \(H'\), stop there and move vertically to \((j,H)\). The cost of this new ending is \(|H-H'|\), at most the vertical variation of the discarded suffix. Thus \(q_j\le q_i\le t\). The projected path therefore lies in the sublevel set and joins \(i\) to the source column. Taking these paths for all \(i\) proves connectivity. This argument also shows that every source-column representative belongs to the sublevel whenever the sublevel is nonempty. ◻

Proof. An entrance column has \(t\le b_i<t+a\) by (2). When \(t<30a\), all entrance midpoints have distance less than \(31a\) from \(o\). Any one is therefore within \(62a\) of every other, and within \(63a\) of every point on every entrance column. One portal suffices in this case.

Suppose \(t\ge30a\). Let \(L\) be the projection to columns of the closed ball \(B_{d_J}(o,t-12a)\). Shortest paths from \(o\) to the points of the ball stay in the ball, so \(L\) is connected. Every \(j\in L\) has \(b_j\le t-12a+a/2\), and hence \(L\) is disjoint from \(C\).

Assume there are three entrance midpoints \(x_1,x_2,x_3\) at pairwise distance greater than \(100a\). Starting from each \(x_k\), follow a shortest path toward \(o\) until its first visit to a column of \(L\). This portion has length at most \(13a\): by that length it reaches a point at radial distance \(t-12a\). Its projection joins \(C\) to \(L\). The three projected portions are vertex-disjoint. Indeed a common column would permit a vertical transfer of length at most \(a\), giving a path of length at most \(13a+a+13a<100a\) between the corresponding entrance points.

For each portion retain the segment after its last visit to \(C\) and before its first visit to \(L\). Its interior avoids both sets. It has an internal vertex, since its endpoints have radial values differing by at least \(12a-a/2>a\), while adjacent columns differ by at most \(a\). Contract the connected sets \(C\) and \(L\), and contract each of the three nonempty connector interiors to a separate vertex. This gives a \(K_{2,3}\) minor, contrary to Lemma 8.

A maximal set of entrance midpoints at mutual distances greater than \(100a\) thus has cardinality at most two. Maximality covers every entrance midpoint within \(100a\), and the extra vertical distance is at most \(a/2\). This proves the stated \(200a\) bound. Finally a representative of \(o\) has radial value at most \(a/2\). If \(t>a/2\), any nonempty high component is a proper subset of the connected graph and has an entrance. ◻

Proof. Each cell is the image of finitely many closed column intervals. Its column graph is connected, which makes that image path-connected. The same finite-union argument proves compactness of the cell and frontier.

Let \(q\) be the lower threshold of its layer. If \(q\le30a\), every point of the cell has distance less than \(D+31a\) from \(o\), so the diameter bound follows. Suppose \(q>30a\) and let \(C^+\) be the component of \(\{b_i\ge q\}\) containing the cell. It has at most two entrance portals by Lemma 10.

Take \(v\) in the cell and follow a shortest path to \(o\) until the first switch below threshold \(q\). Before that switch the projected path stays in \(C^+\): two adjacent high columns belong to the same high component. The last high column is an entrance, and the switch point has radial distance at least \(q-a/2\). Since \(d_J(o,v)<q+D+a/2\), the portion from \(v\) to that point has length less than \(D+a\). Some portal is within \(200a\) of that point. Thus the cell is contained in the union of two closed balls of radius \(D+210a\) around its portals.

A connected set contained in the union of two closed balls either lies in one of them or meets their intersection. To see the second alternative when neither ball contains the set, intersect both balls with the connected set; two nonempty disjoint closed subsets could not cover it. In that case the centers have distance at most \(2(D+210a)\). In either alternative the cell has diameter at most \(4(D+210a)<10D\). ◻

Proof. If a band path from \(v\) of length at most \(\rho\) visits column \(k\), join its visit point to \((k,H_*)\) and join \(v\) to \((i,H_*)\). The triangle inequality gives \[|b_k-b_i|\le\rho+a.\] Thus every visited column stays in \(i\)’s layer under (3); its projected path puts it in the same connected component of that layer. The same argument applies after any zero-cost switch chain at an endpoint, proving the assertion about all representatives.

Every frontier column has its radial value within \(a\) of one of the cell’s layer thresholds: its neighbor outside the cell cannot belong to the same layer. If \(z\) lies on a frontier column \(k\), then \[d_J(v,z)\ge |b_i-b_k|-a \ge \mathop{\mathrm{dist}}(b_i,T)-2a>\rho.\] The formula remains true for an empty frontier with distance \(+\infty\). The offset estimate is the length, at most \(2(\rho+2a)\), excluded in one period. Two representatives at height \(H\) have midpoint distance at most \(2|H-H_*|\le a\), since their points at height \(H\) are identical. This proves the representative assertion without a union bound.

For the numerical conclusion use \(0.02D-2a\ge0.018D>0.01D\). A path to a different cell must first visit a frontier column of the initial cell, so the same lower bound applies to its length. ◻

Proof. A connected component at a higher threshold is contained in a unique component at the preceding threshold, proving the parent assertion. For \(n\ge1\), repeat the root-geodesic argument in Lemma 12 at lower threshold \(n\delta\). A point of the node’s own layer has radial distance below \((n+1)\delta+a/2\). A child portal has radial distance in \([(n+1)\delta,(n+1)\delta+a)\). In either case its geodesic reaches an entrance of the parent node after length at most \(\delta+2a\), and an own portal is within a further \(200a\). The asserted upper bound follows.

An own portal has radial distance in \([n\delta,n\delta+a)\), whereas a child portal has radial distance at least \((n+1)\delta\). Their distance is therefore at least \(\delta-a\). At depth zero the upper bounds follow directly by taking \(o\), and a depth-one portal is at distance at least \(\delta\) from it. These estimates imply the stated bounds in every case. ◻

Proof. For each sign, the symmetric difference of the two half-lines has measure \(|\Delta h+\sigma\kappa\Delta P|\). Averaging the signs gives (5). The functions \(H\mapsto H\pm\kappa P(i,H)\) are increasing, and their average increment is the height increment.

For rebasing put \(\phi_\sigma(H)=H+\sigma\kappa\psi(H)\). These are increasing bijections of \(\mathbb R\). Define the new coordinate on the same label space by \(Z'(\omega)=\phi_{\sigma(\omega)}^{-1}(Z(\omega))\). Its pushforward measure is Lebesgue measure, since for \(a<b\), \[\mu\{a<Z'<b\} =\frac{\phi_+(b)-\phi_+(a)+\phi_-(b)-\phi_-(a)}2=b-a.\] Choose a sign coordinate unused anywhere earlier in this execution. It is independent of \(Z'\). Thus \((Z',\sigma')\) has the required joint pushforward, and, for every label, \[\{Z<\phi_\sigma(h)\}=\{Z'<h\}.\] This describes exactly the same sets by a chart with field zero. ◻

Proof. For a fixed label all active points of a column have the same component coordinate. On a vertical segment the vector changes only when activity changes, and its \(\ell_1\) cost is the original indicator cost. At an active switch the two columns belong to the same component; at an inactive switch both vectors are zero. Integration and any finite path give the upper bound. A membership difference always costs one, proving domination.

Every switch incident to an endpoint column of a non-neighbor link is at or below its tip. If the tip is inactive, nesting makes all such switches inactive. A path of column indices between the two strict sides must pass through an endpoint of the link, by noncrossing. No active path can do so. The active points therefore have different unit coordinates.

There are only finitely many possible components, which can be indexed by subsets of the columns. An active link can be tested at its maximum switching height, so all coordinates are measurable. Subtracting the vector assigned to one fixed reference point makes each assignment integrable: the integrated norm of the difference is bounded by the distance to that point. The same subtraction works for the unrefined indicator system. ◻

Proof. Since \(c_0<q/2\), the portal-neighborhood calculation above gives \(U_\nu(v)=L_\nu(v)=F_\pi(v)\) in the stated collar. The priority convention retains the parent atom even if \(T_\nu(v)=F_\pi(v)\).

For accuracy, the Lipschitz bound of \(p\) gives, at every point, \[ p(v)+\zeta\mathop{\mathrm{dist}}_{d_J}(v,K_\nu)+q \le T_\nu(v) \le p(v)+(2-\zeta)\mathop{\mathrm{dist}}_{d_J}(v,K_\nu)+2q. \tag{13}\] In the situations of (9), its upper bound is less than \(p(v)+3\delta\), by (6). This proves both accuracy assertions at the root.

Proceed by induction. At a nonroot node, the inductive hypothesis gives \(|F_\pi(k)-p(k)|\le5\delta\) for all its own portals. Therefore, writing \(r_\nu(v)=\mathop{\mathrm{dist}}_{d_J}(v,K_\nu)\), \[\begin{align*} A_\nu(v)&\ge p(v)-5\delta+\zeta r_\nu(v)-2q, \tag{14}\\ B_\nu(v)&\le p(v)+5\delta-\zeta r_\nu(v)+2q. \tag{15}\end{align*}\] At a child portal, (10) and \(a\le10^{-6}\zeta\delta\) imply \(\zeta r_\nu(v)\ge2q\). Hence \(U_\nu(v)\ge p(v)-5\delta\) and \(L_\nu(v)\le p(v)+5\delta\). Clamping a number in \([p(v),p(v)+3\delta]\) to an interval with these two properties gives a value in \([p(v)-5\delta,p(v)+5\delta]\).

On the own layer, the same argument without the favorable distance term gives \(U_\nu(v)\ge p(v)-5\delta-2q\) and \(L_\nu(v)\le p(v)+5\delta+2q\). Since \(2q\le\delta/10\), the clamped value has error at most \(5.1\delta<6\delta\). ◻

Proof. The closed ball is connected by geodesics through its center. Include all representatives of all its points in its column projection. For any two such representatives their radial values differ by at most \(2\rho+2a<\delta\), using the radial comparison with \(d_J(o,\cdot)\). Thus their layer depths take either a single value or two consecutive values \(n,n+1\). The projected graph is connected, and all its columns have \(b_i\ge n\delta\). They therefore belong to one superlevel node \(\nu\) at depth \(n\). In the single-depth case this proves the result.

Otherwise choose a represented point of layer depth \(n\) in the ball. For any represented point \(v\) of depth \(n+1\), concatenate its geodesic to the center with a geodesic to the chosen point. This is a band path of length at most \(2\rho\). Its last crossing from depth \(n\) to depth \(n+1\), when traversed toward \(v\), enters the child node containing the representative of \(v\). The entry column is an entrance column of that child. Hence \(v\) is within \(2\rho+200a<c_0\) of a portal of that child. The equality collar of Lemma 18 identifies its formula and its selected atom with those of \(\nu\) at \(v\). This proves the assertion for every representative. All the choices in this argument depend only on the band geometry and the ball. ◻

Proof. We first prove an obstruction to passing a parent atom through a distant node \(\mu\). Suppose \[ \zeta d_J(k,v)>13\delta\quad\text{for every }k\in K_\mu. \tag{16}\] Equations (13)–(15) give \[T_\mu(v)>p(v)+13\delta,\qquad A_\mu(v)>p(v)+7\delta,\qquad B_\mu(v)<p(v)-7\delta.\] If the parent value belonged to \([p(v)-6\delta,p(v)+6\delta]\), then \(L_\mu(v)=B_\mu(v)\) and \(U_\mu(v)=A_\mu(v)\), so \[F_\mu(v)=\min(A_\mu(v),T_\mu(v))>p(v)+7\delta.\] In particular, a value within \(6\delta\) of \(p(v)\) cannot be obtained at this node by selecting its parent atom. Notice that the lower bound on the tentative formula is required here as well as the two clamping bounds.

Let \(n\) be the depth of \(\nu\). If \(n\le M\), every ancestor is already in the asserted list. Otherwise consider its ancestor \(\mu\) of depth \(n-M\ge1\). For a point \(v\) of the ball, Lemma 19, radial comparison, and (11) give \[d_J(k,v)\ge M\delta-3a\qquad(k\in K_\mu).\] The definition of \(M\) implies (16). An atom originating strictly earlier than \(\mu\) could be selected by \(F_\nu(v)\) only by being inherited through every intervening node. All these inherited values would equal \(F_\nu(v)=Q(v)\), whose error is at most \(6\delta\). In particular \(\mu\) would select its parent atom with such a value, contradicting the preceding paragraph. This excludes all earlier origins. Each of the remaining \(M+1\) nodes contributes at most six atoms, proving the count \(N\). ◻

Proof. For two atoms originating at different depths, use the later atom’s own noise. Its intercept, whether \(p(k)\) or \(F_\pi(k)\), does not depend on that noise, because parent formulas use only earlier nodes. The earlier atom does not depend on it either. Thus, after conditioning on every other noise, the difference has the form \(c+\varepsilon\alpha\) with \(\varepsilon\in\{-1,1\}\) and \(\alpha\) uniform on \([q,2q]\). For two atoms at the same node their own noises are different and their parent intercepts depend on neither, so the same argument applies. The candidate list lies on one ancestor chain, so these are all cases. An independent comparison quantity can likewise be included in the conditioning.

For a continuous uniform noise an interval of length \(2t\) has probability at most \(2t/q\). For a grid with spacing \(\eta'\le\eta\), it contains at most \(2t/\eta'+1\) grid points; division by the total number \(q/\eta'+1\) gives at most \((2t+\eta')/q\). ◻

Completion of Theorem 17. Take the deterministic list furnished by Lemma 20. The union bound and Lemma 21, with \(t=4\rho\), show that the probability of a pair satisfying \(|A(x)-A'(x)|\le4\rho\) is at most \(N^2(8\rho+\eta)/q\). On the complementary event, two atoms in the list cannot be equal anywhere on the ball, since both are \(1\)-Lipschitz. The continuous function \(Q=F_\nu\) selects a value from this finite list at every point. Where the list values are distinct its selected name is locally constant: continuity and the positive minimum gap to the other finitely many values give a neighborhood with the same selection. A ball is connected by paths through its center, so the name is constant throughout the ball. This proves (8). The path version follows by applying the ball assertion to the paths, all of whose points lie in the ball. ◻

Proof. Every specified path lies in \(B_{d_{J_i}}(x,\rho)\) for each interpolation band \(J_i\). Use the deterministic bounded atom list for \(Q_i\) on that ball, with its chosen sign and shift. Adjoin the shifted \(g_j\) branches and, if present, \(f\). For pairs from one interpolation use Lemma 21. For pairs from different interpolations condition on all noises except the own noise of an atom in one interpolation; independence of the constructions makes the other atom independent of that noise. The same argument applies to a shifted \(g_j\) or the fixed branch \(f\). There is at most one branch without a fresh noise. Therefore every comparison of distinct names at \(x\) has the interval bound \((2u+\eta)/q_*\) at tolerance \(u\).

Take \(u=4L_*\rho\) and sum over at most \(S^2\) pairs. Off the event in (18), no two listed values agree anywhere on \(W\): their changes along each path from \(x\) are bounded by \(L_*\rho\). The expression is a continuous selection from the resulting finite list along each path. Its selected name is therefore constant on each path and, because all paths begin at \(x\), throughout \(W\). ◻

Proof. At a frontier point \(z\), the term with source \(z\) in (21) gives \(p_C(z)\le s(z)\). Thus \(F_C(z)\le s(z)-3E_A\). At either endpoint of the height band, (20) gives \[p_{h,C}\le s-1+10D+E_A<s.\] The replacement therefore equals the old field on every frontier and at every height endpoint, with a strict margin in the proposed branch. Different cell prescriptions agree at common quotient points. Within a cell, every constituent respects switches.

The old field and \(F_C\) have vertical Lipschitz constant at most \(1\); the height branch has constant \(3/(0.09a_1)\). The maximum and minimum in (24) have the maximum of these constants. The parameter choice makes its product with \(\kappa_A\) strictly less than \(1\). The vertical replacement and gluing rule for label sets consequently preserves nesting and the vertical upper bound.

An unchanged branch keeps its old ordinary chart. A winning atom of \(F_C\) has the form \(c\pm d_{J_1}(k,\cdot)\) and therefore gives an ordinary chart after converting back from the \(s\) orientation. A winning height branch is recorded by the height-only change of variables from the label-set construction, with a fresh constant chart. Identifiers can be chosen using the update data and the winning atom, with the inherited identifier retained wherever the old field wins.

Finally, the replacement never decreases \(s\), and \[F_C\le s_C+1-3E_A\le s+1+10D.\] Consequently \(0\le s'-s\le1+10D\). At a fixed point, a change \(s'-s\) in the common chart changes the label set by measure \(\kappa_A|s'-s|\), which proves the stated bound. ◻

Proof. Assume that the raise is at most \(2E_A\). The height interior requirement places \(w\) on the plateau of \(p_{h,C}\), where \[p_{h,C}(h(w))\ge s(w)+2-10D>s(w)+2E_A.\] It follows that \(F_C(w)\le s(w)+2E_A\), and (23) then gives \[ p_C(w)\le s(w)+7E_A. \tag{27}\] The constant branch \(s_C+1\) is strictly greater than the right-hand side of (27). The frontier is therefore nonempty, and there is a minimizing \(z\in\partial C\) with \[s(z)+(1-\zeta_A)d_{J_1}(w,z)\le s(w)+7E_A.\] Put \(d_0=d_{J_1}(w,z)\). The radial padding conclusion of Lemma 13 gives \(d_0\ge0.01D=10^{-5}\). The weak diameter estimate gives \(d_0\le10D<0.02\). Choose an intrinsic shortest path from \(w\) to \(z\). It remains in the old chart band by availability. Since \[\zeta_A d_0+7E_A \le (\epsilon/100)(10D)+7\epsilon/1000<\epsilon,\] this path satisfies (25).

For the last assertion, apply the old field’s Lipschitz bound to the remaining subpath, of length \(d_0-t\): \[s(\gamma(t))\le s(z)+(d_0-t) \le s(w)-t+\epsilon.\] ◻

Proof. The two endpoints lie in different width-\(D\) cells, since a cell has \(d\)-diameter at most \(10D<1\). Each of these cells is available: its radius-\(1\) neighborhood lies in the radius-\(5\) neighborhood of its endpoint, by the cell diameter bound and eligibility. Keep the coin at an endpoint with no descending path and suppress the coin of the other endpoint. This has probability \(1/4\). Lemma 26 raises the first comparison value by more than \(2E_A\) and leaves the second unchanged, so their new field difference has magnitude at least \(E_A\). Evaluate both new sets in their common old label variables. The common-chart symmetric-difference formula gives the asserted bound. ◻

Proof. Eligibility puts \(x,y,e\) in the same old ordinary chart. Write \(P=C_0+\tau s\) there and restrict labels to the sign \(\sigma=\tau\). On this half of the label space the threshold at \(v\) is \[T_v=h(v)+\kappa_A\tau C_0+\kappa_As(v).\] The other endpoint has \(s\)-value within \(E_A\) of \(s(w)\). Also all three heights lie in \(J_2\). Thus, for \(v=x,y\), \[T_v-T_e\ge\kappa_A(j-E_A)-a_2\ge\kappa_Aj/2,\] using \(E_A\le j/4\) and \(a_2\le\kappa_Aj/4\). The labels with \(\sigma=\tau\) and \(T_e\le Z<\min\{T_x,T_y\}\) have the required three membership conditions and mass at least \(\kappa_Aj/4\). Choices at the threshold endpoints have measure zero.

Lemma 16 separates the two active points into different components whenever the tip is inactive; each such label contributes \(2\) to the refined distance. After later changes, the subset of this label mass on which any of the three memberships changes has measure at most \(\Delta_x+\Delta_y+\Delta_e\). Apply the same component-refinement lemma to the surviving labels in the final nested system. This proves (29); no control of the changes in component names is required. ◻

Proof. Let \(B\) be such a ball, with arbitrary center \(z\). If it has no seeds there is nothing to prove. Otherwise fix \(w_*\in B\) among the seeds. Only a narrow band contained in a broad band can have seeds. Its broad band \(J_1\) is then unique, and the \(J_2\)-paths from \(z\) to the seeds show that all of them lie in one \(J_1\)-component.

All seeds of \(B\) have the same chart identifier in either system. Indeed \(d(z,w)\le4R<5\) for every such seed \(w\), so \(z\) lies in the common chart of its eligible witness pair. Since charts partition \(X\), those identifiers coincide. Write \(I\) for their common old system-A band and \(s=d_I(o,\cdot)\) for its distance field. All center-to-seed paths of length at most \(4R\), all descending witness paths, and the vertical intervals used below are contained in the relevant old chart band: they are at distance less than \(5\) from an endpoint of an eligible witness pair. For example, a point on a center-to-seed path has distance at most \(8R\) from \(w_*\), and a descending path has length at most \(0.02\). The additional vertical intervals have length at most \(a_1\). In particular \(J_1\subset I\) as height intervals: the full column interval \(J_1\) through any seed is within its radius-\(5\) neighborhood.

Set \(H=h(w_*)\) and form the profile \(q_i=s(i,H)\), with \(+\infty\) outside the old component as in Lemma 9. Put \[ t=s(w_*)-\tfrac12\,10^{-5},\qquad S=\{i:q_i\le t\}. \tag{31}\] For every seed \(w\in B\), the path through \(z\) gives \[|s(w)-s(w_*)|\le8R, \qquad |s(w')-s(w)|\le E_A.\] The terminal column \(i\) of either chosen descending path for \((w,w')\) therefore satisfies \[q_i\le s(w_*)+8R+E_A-10^{-5}+\epsilon+a_1<t,\] where we used \(8R+E_A+\epsilon+a_1<\tfrac12\,10^{-5}\). Thus \(S\) is nonempty, and it is connected in the active graph of \(I\) by Lemma 9.

Every column on any center-to-seed path of length at most \(4R\) lies outside \(S\). In fact, if such a path visits column \(i\) at point \(v\), then \[q_i\ge s(v)-a_1\ge s(w_*)-8R-a_1>t.\] All comparisons take place in \(I\), as verified above. The noncrossing gap property, Lemma 6, now places all these path words and their endpoint representatives in one linear circular gap of \(S\). Fix a representative of \(z\) to order this gap. Equivalent representatives also lie in the same gap, since a zero-length switch path satisfies the same avoidance argument.

Choose one representative seed from each group meeting \(B\). In the linear order of the gap, choose a seed of minimum index among those to the left of the center index, if there is one, and similarly a seed of maximum index on the right. Let \(P_-\) and \(P_+\) be shortest \(J_2\)-paths from \(z\) to these extreme seeds. When one side has no seed, use the constant path at \(z\) on that side. Each path has length at most \(4R\). For any chosen seed \(w\), at least one of these two path words either visits its representative column or makes a switch strictly over its index. This is the elementary skipping property of Lemma 7, applied between the center and the extreme index on that side.

If the path visits the column of \(w\), a vertical interval of length at most \(a_1\) connects \(w\) to a point of that path in \(J_1\). Consider instead a switch strictly over its index. Let \(a,b\) be the switching columns, ordered within the gap so that the representative of \(w\) is in \((a,b)\). The switch is not a circular-neighbor link, because its strict interval within the gap contains a column and the complementary arc contains the nonempty set \(S\). It is consequently a single switch at a tip \(e\), by the column-model hypothesis. This tip is on \(P_-\) or \(P_+\), has height in \(J_2\), and satisfies \[ d(w,e)\le8R<1. \tag{32}\] The interval \((a,b)\) is disjoint from \(S\).

We claim that the witness partner \(w'\) has representative strictly outside the closed interval \([a,b]\) in the circle. If its representative were in \((a,b)\), its descending path, whose terminal column belongs to \(S\), would have to visit column \(a\) or \(b\). This follows from noncrossing, or directly from Lemma 5. Join that visit vertically to \(e\), then follow the appropriate extreme path back to \(z\) and a shortest center-to-\(w\) path. The resulting path has length at most \[0.02+a_1+8R<1,\] contradicting \(d(w,w')\ge1\). If \(w'\) itself has representative \(a\) or \(b\), the vertical connection to \(e\) gives the same contradiction without the descending segment. This proves the claim.

The link at \(e\) thus separates the two witness representatives strictly. Because the witness pair was not excluded and (32) holds, its failure to satisfy the barrier-drop condition implies \[ s(e)\ge s(w)-j. \tag{33}\] The descending path from \(w\) must likewise hit column \(a\) or \(b\), since it begins in \((a,b)\) and ends in \(S\). Let \(v\) be its first such hit and let \(p\) be the arclength to \(v\). Lemma 26 and the vertical comparison with \(e\) give \[s(v)\le s(w)-p+\epsilon, \qquad s(v)\ge s(e)-a_1\ge s(w)-j-a_1.\] It follows that \(p\le j+\epsilon+a_1\). The descending prefix and the vertical interval from \(v\) to \(e\) both lie in \(J_1\), so \[ d_{J_1}(w,e)\le j+\epsilon+2a_1. \tag{34}\]

We have proved that every chosen seed is within \(\rho_{\mathrm{pack}}=j+\epsilon+2a_1\) in \(d_{J_1}\) of \(P_-\cup P_+\). Distinct chosen groups have distinct width-\(g\) cells: their bands, band component, and two old identifiers have already been shown to coincide. Their seed points are radially padded, so Lemma 13 gives pairwise \(d_{J_1}\)-separation at least \(b_0=0.01g\).

For each chosen seed, take one of the path points just constructed within \(\rho_{\mathrm{pack}}\), and assign it to one path containing it. Path points assigned to different seeds have \(d_{J_1}\)-distance at least \(b_0-2\rho_{\mathrm{pack}}\ge b_0/2\). Their arclength parameters on either path are therefore separated by at least \(b_0/2\). A path of length \(4R\) has at most \(1+8R/b_0\) such points. The two paths together have at most \[2+16R/b_0=2+1600R/g\] assigned points. This proves (30). ◻

Proof. Put \(D=d_I(k,w)\). If \(D=0\), the constant zero field suffices. If \(k,w\) have representatives on the same column, then \(D=|h(w)-h(k)|\) and the field \[L(v)=\min\{|h(v)-h(k)|,D\}\] suffices, with any two distinct indices as boundary columns: vertical travel attains this distance on that column, and every path has length at least its absolute height change.

Otherwise choose a representative column \(a\) of \(k\) and a different representative column \(b\) of \(w\). Write \[f_i(H)=\min\{d_I(k,(i,H)),D\},\qquad H\in I,\] where \(f_i=D\) on any component not containing \(k\). Every \(f_i\) is \(1\)-Lipschitz, and the values respect switches. Moreover, \[f_a(H)=\min\{|H-h(k)|,D\}=\min_i f_i(H). \tag{\ref{ord:support}.1}\]

For fixed \(H\) and \(t\le D\), every nonempty strict sublevel \[S_t(H)=\{i:f_i(H)<t\}\] is connected in the active column graph and contains \(a\). To see this, let \(i\in S_t(H)\) and take a path from \(k\) to \((i,H)\) of length less than \(t\). If the path visits column \(z\) at height \(T\), its remaining length is at least \(|T-H|\). Its prefix followed by vertical travel on \(z\) to height \(H\) therefore has length less than \(t\). Thus every visited column belongs to \(S_t(H)\), and projection of these paths to column indices connects \(S_t(H)\) to \(a\).

Consider a link with distinct endpoints \(p,q\) at height \(H\), and put \(c=f_p(H)=f_q(H)\). If neither endpoint is \(a\), let \([p,q]_a\) be the closed circular interval from \(p\) to \(q\) that does not contain \(a\). Then \[f_z(H)\ge c\qquad(z\in[p,q]_a). \tag{\ref{ord:support}.2}\] Indeed, a strict-sublevel path from any column \(z\) with \(f_z(H)<c\) to \(a\) avoids \(p,q\), whereas Lemma 5 forbids such a path when \(z\) is strictly inside \([p,q]_a\). If one endpoint is \(a\), both endpoint values equal the global minimum in [ord:root-minimum].

Cut the circular order at \(a\). For \(i\ne a\), let \([i,b]_a\) denote the closed interval between \(i\) and \(b\) in the resulting linear order, and define \[L(i,H)=\min_{z\in[i,b]_a} f_z(H),\qquad L(a,H)=f_a(H). \tag{\ref{ord:support}.3}\] These are finite minima of \(1\)-Lipschitz functions. Along either arc from \(a\) to \(b\) the intervals in [ord:projection] shrink, so their minima are nondecreasing; the initial root value is a lower bound for all of them by [ord:root-minimum].

We check switches explicitly. If a link meets \(a\), its other endpoint has the global minimum value, and so does its value of \(L\). Otherwise order its endpoints as \(p<q\) in the order cut at \(a\). By [ord:collision] the minimum of \(f\) on \([p,q]_a\) is their common value \(c\). If \(b\) lies between \(p\) and \(q\), both minima defining \(L(p,H),L(q,H)\) equal \(c\). If \(b\) lies outside that interval, the longer of the two minimizing intervals is the union of the shorter one and \([p,q]_a\). The minimum on the shorter interval is at most \(c\), because it contains a link endpoint. The two minima are again equal. Thus \(L\) respects every switch and is an ordered field on the quotient.

Finally, \(L(i,H)\le f_i(H)\le d_I(k,(i,H))\). At \(k\), its root value is zero. At \(w\), the minimizing interval is the singleton \(\{b\}\) and its value is \(D\). ◻

Proof. The image of \(H\mapsto(s(i,H),H)\) has Lipschitz constant at most \(4\) for the maximum norm, and distance to a fixed nonempty set is \(1\)-Lipschitz. Thus \(t_s\) is vertically \(4\)-Lipschitz on each open-arc column; it is identically zero on each boundary column. At a switch between columns on the same open arc, its values agree because both \(s\) and height agree. At a switch between opposite open arcs the common coordinate image belongs to \(Z_s\), so both values vanish. At a switch involving a boundary column, its image belongs to \(Z_s\) for the same reason. These checks also prove independence of representatives. Summing along any finite path of vertical motions and switches proves the \(d_I\) bound. ◻

Proof. Restrict and orient each path so that its endpoint values are \(u\) and \(v\). Fix \(H_*\in J\) and write \(q_i=s(i,H_*)\). The values \(q_i\) are nondecreasing in the linear order on this open arc, and \[|s(i,H)-q_i|\le4\eta\qquad(H\in J). \tag{\ref{ord:overlap}.1}\] Choose a point on \(\Gamma_x\) of value \(c=(u+v)/2\), with representative column \(i\). Then \(|q_i-c|\le4\eta\). The initial column of \(\Gamma_y\) has reference value at most \(u+4\eta<q_i\), and its terminal column has reference value at least \(v-4\eta>q_i\). Its column word therefore starts strictly before \(i\) and ends strictly after \(i\).

If \(\Gamma_y\) visits column \(i\), connect the two visits on that column vertically at cost at most \(\eta\), giving the claim. Otherwise Lemma 7 supplies a switch on \(\Gamma_y\) with endpoint columns \(p<i<q\). If its height is \(H_0\in J\), then \[|q_p-q_q|\le8\eta, \qquad q_p\le q_i\le q_q.\] In particular both endpoint reference values differ from \(q_i\) by at most \(8\eta\). The two endpoints of \(\Gamma_x\) have reference values respectively at most \(u+4\eta\) and at least \(v-4\eta\). Since \((v-u)/2>16\eta\), these lie strictly below \(q_p\) and strictly above \(q_q\), respectively. Thus the middle point on column \(i\) is strictly inside the interval bounded by the link \(pq\), whereas the endpoints of \(\Gamma_x\) are strictly outside it. Lemma 5 implies that \(\Gamma_x\) visits \(p\) or \(q\). Connect this visit vertically to height \(H_0\) and, if needed, use the zero-cost switch \(pq\). This joins a point of \(\Gamma_x\) to a point of \(\Gamma_y\) at cost at most \(2\eta\) (in fact \(\eta\) suffices when using the visited endpoint of that switch). The assumed connections to \(x,y\) complete the proof. ◻

Proof. Write \(s_0=s(x)\) and \(H_x=h(x)\). We first prove that \(Z_s\) does not meet the rectangle \[\mathcal Q= [s_0-0.2R,s_0+0.2R]\times [H_x-0.02R,H_x+0.02R]. \tag{\ref{ord:two-side}.1}\] Suppose otherwise that \((c,H_0)\in Z_s\cap\mathcal Q\). At the reference height \(H_0\), the endpoint columns of each low path have field value strictly below \(c\), and the endpoint columns of each high path have value strictly above \(c\). Indeed their terminal values are respectively below \(s_0-0.399R\) and above \(s_0+0.399R\), and moving their values to height \(H_0\) changes them by at most \[4(0.02R+\eta)\le0.080004R.\] These inequalities leave a strict margin from the interval \([s_0-0.2R,s_0+0.2R]\).

If \((c,H_0)\) is the image of a boundary column, these strict inequalities contradict that column’s extremality. Otherwise it is the image of a crosslink with one endpoint on each open arc. At height \(H_0\), monotonicity puts every column of value less than \(c\) on the root side of this crosslink and every column of value greater than \(c\) on its opposite side. The projected concatenation \((\gamma_z^-)^{-1}\gamma_z^+\) must therefore visit an endpoint of the crosslink by Lemma 5. Each such visit is reachable from \(z\) in length at most \(2R\), since it lies on one of the two original paths. Move vertically from the two visits to height \(H_0\) and use the crosslink if needed. This produces an \(x\)–\(y\) path of length at most \[4R+2(0.02R+\eta)<1,\] contradicting \(d(x,y)\ge1\). This proves [ord:empty-rectangle].

The maximum-norm ball of radius \(0.01R\) around either \((s(x),h(x))\) or \((s(y),h(y))\) is contained in \(\mathcal Q\). Hence \[|t_s(x)|,|t_s(y)|\ge0.01R. \tag{\ref{ord:two-side}.2}\] In particular both centers have representatives on open arcs, with a well-defined choice of arc.

Consider the value strip \[\mathcal S=\{v\in X_J: s_0-0.18R\le s(v) \le s_0+0.18R\}.\] No representative of a point of \(\mathcal S\) is on a boundary column, and no switch in \(\mathcal S\) joins opposite open arcs. Either event would give an image in \(Z_s\cap\mathcal Q\), because its height belongs to \(J\) and its value is within \(0.18R\) of \(s_0\). Thus any path contained in \(\mathcal S\) remains on a single open arc.

For \(z=x,y\), the concatenation \((\gamma_z^-)^{-1}\gamma_z^+\) starts below the strip and ends above it. The scalar field is continuous along this path, including at switches. Take its first visit to the upper strip boundary, and its last visit to the lower strip boundary preceding that visit. The intervening subpath \(\Gamma_z\) stays in \(\mathcal S\), traverses its full value interval, and lies on one open arc. Every point on it is reachable from \(z\) in length at most \(2R\). If \(\Gamma_x,\Gamma_y\) were on the same arc, Lemma 35, with common interval of length \(0.36R>32\eta\), would give \[d(x,y)\le4R+2\eta<1.\] Consequently their two arcs are opposite.

It remains to relate these full traversals to the centers. From either center \(z\), follow either one of its two original paths until its first exit from \(\mathcal S\). The resulting initial subpath \(\Pi_z\) lies on the center’s arc and its value range contains the interval between \(s(z)\) and the strip boundary it first reaches. This interval has length at least \(0.179R\). All its points are reachable from \(z\) in length at most \(2R\). If the two centers had the same arc, exactly one of the opposite full traversals \(\Gamma_x,\Gamma_y\) would lie on that arc. If it were \(\Gamma_x\), compare it with \(\Pi_y\); if it were \(\Gamma_y\), compare it with \(\Pi_x\). In either case the paths come from different centers and their value ranges overlap in an interval of length at least \(0.179R>32\eta\). Another application of Lemma 35 gives the same contradiction. Thus the centers have opposite arcs and, by [ord:positive-magnitude], \(t_s(x),t_s(y)\) have opposite signs.

Finally Lemma 34 and the assumed equality of band and unrestricted distances give \[|t_s(x)|+|t_s(y)| =|t_s(x)-t_s(y)| \le4d_I(x,y)=4d(x,y)\le8.\] This proves both upper bounds. ◻

Proof. First consider a point with \(d_F(v)=R\). The shallow branch satisfies \[T_F(v)\le p_c+0.05R-0.9R+E_B=p_c-0.849R.\] In an ordered chart a nearest admitted seed gives \(p_*(v)\ge p_c-0.65R\), leaving a margin of \(0.199R\). In an ordinary chart the same margin applies whenever that last inequality holds. Otherwise \(L\le p_*\) gives \[O_F(v)-p_*(v) \le0.05R+3(p_*(v)-p_c)+E_B<-1.899R.\] Thus the proposal has a uniform margin below the old field on every distance boundary. At the height ends the shield gives a margin as well: each domain point lies within \(R\) of a seed, which lies within \(10g\) in \(d\) of \(w_c\), and the old comparison has Lipschitz constant less than \(5\) along these eligible paths. Its difference from \(p_c\) is less than \(5(R+10g)<1/2\).

Every domain lies in its old chart, by seed eligibility. The comparison orientation and label variables are common on overlapping domains. Extend each proposal, after clamping against the old field, by the old field off its domain. The boundary margins and the clipping argument in Section 3 show that this extension glues vertically and respects switches. The finite maximum of these extensions has the same properties. The spatial branches have Lipschitz constant at most \(4\); the shield has constant \(3/(0.09a_2)\), which is less than \(1/\kappa_B\).

A winning atom of \(Q_F\) gives a constant plus or minus one point-source distance and hence a fresh ordinary chart on its narrow-band component. A winning \(O_F\) branch, after conversion back to the old orientation, becomes the ordered part of a new chart on the old band, with \(u=0\). It has slope at most \(4\). Additive constants preserve its order; if conversion negates the comparison field, reverse the two boundary columns as in Definition 31. This ordered part, its band, and its arc assignment remain fixed under the new identifier; subsequent increments change only its auxiliary field. A height branch is rebased by Lemma 15. Unchanged points retain their chart, and uses of the same inherited atom in the same interpolation keep one identifier.

Finally \(Q_F\) lies within \(E_B\) of \(0.9d_F\). Thus the shallow branch is at most \(p_c+0.051R\) and the old field is within \(5(R+10g)\) of \(p_c\) where the group can act. Taking a maximum over any number of groups does not add these bounds. Formula (5) gives a uniform symmetric-difference movement bound. ◻

Proof. At an admitted seed \(w\), \(|Q_F(w)|\le E_B\). Equations (35)–(36) imply that every applicable spatial branch exceeds \(p_*(w)+0.04R\): for the ordered branch use \(|L(w)-p_c|\le10g\). The height shield is on its interior plateau. The other pair endpoint is outside this group’s update domains, because the group’s seeds have unrestricted diameter at most \(10g\) and the pair distance is at least one.

At either endpoint all groups that could act have a seed within \(R\) in the narrow band. At most \(2M_{\mathrm{pack}}\) groups can therefore act in total at the two endpoints. Keeping the designated group and suppressing all other such groups has probability at least \(2^{-(2M_{\mathrm{pack}}+1)}\), a convenient lower bound. The desired ordered orientation, when needed, has probability \(1/2\).

Let \(B_x,B_y\) denote the old system-\(B\) sets and \(B'_x,B'_y\) the updated sets. On this event one endpoint is unchanged, while the movement of the other has measure at least \(0.04\kappa_BR\). If \(\mu(B_x\mathbin\triangle B_y)\ge0.01\kappa_BR\), the no-change mode already gives the required separation. Otherwise the triangle inequality gives \(\mu(B'_x\mathbin\triangle B'_y)>0.03\kappa_BR\), and in particular at least \(0.02\kappa_BR\). Each mode has probability \(1/4\), so both alternatives have uniform positive probability. ◻

Proof. The function \(t_s\) is \(4\)-Lipschitz in \(d_I\). The scalar function \(t\mapsto tH(|t|/r)\) has Lipschitz constant at most \(6\) and magnitude at most \(10r\). The distance factor has Lipschitz constant at most \(1/r\) for \(d\), hence for \(d_I\). Each increment has Lipschitz constant at most \(24+10=34\).

Its support is contained in \(0.005Rr<|t_s|<10r\). Since \(t_s\) is fixed for this identifier, the support intervals at distinct scheduled scales are disjoint when \(Q>2000/R\). In a comparison of two points at most two summands are nonzero at either endpoint. Subtracting their zero values at the other endpoint if necessary bounds the difference by \(68d_I(v,w)\). Thus \(\mathop{\mathrm{Lip}}(u)\le68<K\) at all times.

The distance factor makes the movement vanish continuously at the region boundary. Inside the region the new field has Lipschitz constant at most \(4+\lambda K<5<1/\kappa_B\). The vertical gluing argument applies, and the region partition does not change. The amplitude bound is \(10r\) before the factor \(\lambda\), proving the last assertion. ◻

Proof. Normalize the current scale to one. Ordinary charts admit every seed, so the old \(B\) chart is ordered. Failure of each admission test supplies a point on the corresponding \(d_{J_2}\)-sphere of radius \(R\) and an attaining path of length \(R\). Failure for \(P\) supplies a decrease of \(P\), and failure for \(-P\) supplies an increase. By (35), each change has magnitude greater than \(0.65R-60g\). The correction \(\lambda u\) changes by at most \(\lambda KR\) on either path. Thus the ordered part \(s\) has a positive and a negative change of magnitude greater than \[0.65R-60g-\lambda KR>0.4R\] along these paths from each endpoint.

A shortest \(x\)–\(y\) path has length at most \(2\) and stays in the eligible radius-\(5\) ball, hence in the old band \(I\). This proves \(d_I(x,y)=d(x,y)\) and bounds the change of \(u\) by \(2K\). The \(B\) cutoff therefore gives \[|s(x)-s(y)|\le10^{-4}R+2\lambda K<0.001R.\] The four excursion paths lie in the common band \(J_2\), whose width satisfies Lemma 36. Applying that lemma gives opposite signs of \(t_s\) and \(0.01R\le |t_s(x)|,|t_s(y)|\le8\).

These bounds put both endpoints on the plateau of the trapezoidal factor. Eligibility also puts their radius-\(5\) balls in the current region, so the distance factor in (41) equals one at both endpoints. For one of the two increment signs, \[|P_{\mathrm{new}}(x)-P_{\mathrm{new}}(y)| \ge \lambda|t_s(x)-t_s(y)|\ge0.02\lambda R.\] Formula (5) proves separation at least \(0.02\kappa_B\lambda R\). The favorable sign has probability at least \(1/2\). ◻

Proof. Lemma 24 bounds movement in the main clipping. Secondary movement is bounded by Lemma 37; increments are bounded by Lemma 39. These estimates apply even at points outside the finite vertex list. There are no simultaneous main updates at a point except at cell overlaps, where every update is unchanged. Secondary updates take a maximum, whose movement is bounded without summing over groups. Taking the largest of the bounds gives \(M\).

Work in scale units. If one of the cutoffs in (19) fails, no change preserves separation at least the minimum of \(10^{-4}a_2,\kappa_AE_A,\kappa_B10^{-4}R\). Otherwise the interior requirements hold with probability at least \(1/4\). Conditional on the bands and cells, a missing descending path is handled by Corollary 27: keep that endpoint’s cell and suppress the other, giving positive separation with probability at least \(1/4\) conditional on the main mode. If both paths exist, an excluded pair has the component certificate on a label set of measure at least \(\kappa_Aj/4\), which the no-change mode preserves.

For a surviving pair, admission at an endpoint is handled by Lemma 38, including its no-change alternative. If neither orientation admits either endpoint, apply Lemma 40. Each case uses one of four modes, at most one fair orientation, and at most \(2M_{\mathrm{pack}}+1\) prescribed group coins. It follows, for example, that \(p=2^{-(2M_{\mathrm{pack}}+10)}\) is a permissible lower bound after all interior and mode probabilities. Choose \(b>0\) below all the finitely many displayed gains and certificate masses. These choices depend only on the fixed parameters. ◻

Proof. Normalize \(r=1\) and write \(B=B_d(v,\rho)\). Every point of \(B\) has a shortest path to \(v\) contained in \(B\). Joining two such paths gives a path of length at most \(2\rho\) between any two points of \(B\). We will use these paths for all band-distance comparisons.

There are two ways an update could give different chart identifiers on \(B\): its domain could end inside the ball, or different formulas could win there. Strict margins at domain boundaries prevent the first kind of change. The comparison estimate of Corollary 22 controls the second.