Proof. Almost surely only finitely many Voronoi cells meet \(\overline D\), and each cell clipped to \(\overline D\) is closed and convex. Every connected component of a finite union of such clipped cells is path connected, by concatenating segments through a chain of intersecting cells. Thus, almost surely in the geometry and for every coloring, our connected-set crossing event agrees with the continuous closed-cell crossing event in (Vanneuville 2025, Definition 1.1).

After an orientation-preserving isometry, write \(\overline D=[-w,w]\times[-h,h]\), with the chosen sides vertical. Scaling by \(\varepsilon^{-1}\) sends the full-plane intensity-\(\varepsilon^{-2}\) Poisson process to unit intensity and this rectangle to \([-\lambda R,\lambda R]\times[-R,R]\), where \(\lambda=w/h\) and \(R=h/\varepsilon\). The Voronoi cells and their conditional fair coloring commute with this similarity. Therefore (Vanneuville 2025, Theorem 1.3) gives an absolute \(\alpha>0\) and a constant \(C_\lambda<\infty\) such that, for \(0<\varepsilon\le h\), \[\mathop{\mathrm{Var}}_{\mathrm{geom},\varepsilon}(Q_\varepsilon) \le C_\lambda(\varepsilon/h)^\alpha.\] Let \(m_\varepsilon=\mathbb P(\mathcal C_\varepsilon(D;a,b,c,d))\). Conditioning on the geometry gives \(\mathbb E_{\mathrm{geom},\varepsilon}Q_\varepsilon=m_\varepsilon\), while Theorem 1 gives \(m_\varepsilon\to F(x_D)\). Hence \[\mathbb E_{\mathrm{geom},\varepsilon}|Q_\varepsilon-F(x_D)|^2 =\mathop{\mathrm{Var}}_{\mathrm{geom},\varepsilon}(Q_\varepsilon)+|m_\varepsilon-F(x_D)|^2 \le C_\lambda(\varepsilon/h)^\alpha+|m_\varepsilon-F(x_D)|^2 \longrightarrow0.\] Markov’s inequality gives the assertion in probability. ◻

Proof. For any even set \(D\) of vertices of a connected finite graph, the subgraphs with boundary \(D\), when nonempty, form an affine translate of the cycle space. Toggling a path between two prescribed defects gives such a translation. In particular, the even and two-defect ensembles have the same cardinality; subdividing an edge does not change that cardinality.

In the present two-defect ensemble, ordinary vertices have occupied degree zero or two, while the degree-two marked vertices have occupied degree one. Thus there is one simple defect path and a collection of disjoint circuits. Fix a possible circuit \(L\). When it is occupied, every third edge at a vertex of \(L\) is unoccupied. A defect path must consequently lie in \(\Gamma\setminus L\). The circuit can occur in the defect ensemble only if \(m,n\) lie in the same component of this complement. If they do, fix a path between them in that component. Toggling it bijects even subgraphs containing \(L\) with two-defect subgraphs containing \(L\). Hence \[\mathbb P_{m,n}(L\text{ occupied}) =\mathbb P_{\varnothing}(L\text{ occupied}) \mathbf 1_{\{m,n\text{ in one component of }\Gamma\setminus L\}}.\] Summing over \(L\) proves the assertion. If the two marks lie on one edge, its open interior connects them whenever that edge is not in \(L\), even when both endpoints of the edge belong to \(L\). ◻

Proof. For the off-diagonal identity, start with uniform oddness at the two dual vertices \(t,u\). Toggle the short segment from \(t\) to \(m_j(t)\) and that from \(u\) to \(m_k(u)\). This moves the defects to the two marked points and preserves uniformity. By 4, it suffices to take the \(\sum_{j,k}\omega^{j-k}\) Fourier sum of the resulting circuit counts.

Delete the interiors of the two small stars. For a fixed initial configuration, what remains consists of circuits and paths pairing its occupied star ports. Each star has either one or three such ports. The outside circuits contribute a constant. If a star has one occupied port, the modified defect path uses that port regardless of which short segment was toggled, so the circuit count is independent of that star’s index. These terms vanish in the Fourier sum.

It remains to consider three occupied ports at each star. There are two possibilities. With one through path, the two other ports at each star are paired by an outside cap. If \(r_t,r_u\) are the through ports, the additional circuit count after moving the defects is \[\mathbf 1_{\{j=r_t\}}+\mathbf 1_{\{k=r_u\}}.\] It is a sum of one-index terms and again has zero double Fourier sum. With three through paths, let \(\pi\) be their matching of ports. An additional circuit is formed exactly when \(k=\pi(j)\): the other two through paths then join into one circuit. Disjoint paths between the two boundary circles of the punctured sphere reverse their positive cyclic orders, so \(\pi(j)=\ell-j\pmod 3\) for some \(\ell\). Their contribution is \[\sum_{j=0}^2\omega^{j-\pi(j)} =\omega^{-\ell}\sum_{j=0}^2\omega^{2j}=0.\] This establishes the cancellation for each initial configuration, including when \(t,u\) are adjacent.

For the norm, a circuit avoiding \(t\) places the three nearby spoke points in one component, whose Fourier sum is zero. If \(t\) is occupied, exactly one circuit passes through it. The two used spokes belong to that circuit, while the missing spoke belongs to exactly one complementary component. Thus \(Z_t\) has precisely one nonzero coordinate, of modulus one, in this case. The three distinct primal vertices of \(t\) are not all the same color with probability \(3/4\), proving the second identity. Equivalently, any diagonal kernel in this calculation can be evaluated using two distinct points of the same spoke, which have identical \(H\)-vectors; coincident defects are never needed. ◻

Proof. The triangles with centers in \(C\) are connected through shared sides. A boundary side separates such a triangle from a triangle visited by \(L\); the latter’s missing spoke leads into \(C\). This identifies the boundary sides with the tags.

At a primal vertex, the sectors belonging to \(C\) form one cyclic interval, unless they fill its entire neighborhood. Otherwise choose two separated runs of \(C\)-sectors. Each intervening interval contains an \(L\)-sector: an immediately adjacent untouched triangle would still connect to \(C\). Extend two selected \(C\)-sectors and two intervening \(L\)-sectors to alternating points of a sufficiently small circle about the primal vertex, using short segments inside the sectors. Outside that circle they are joined in the two disjoint connected sets \(C\) and \(L\). Two disjoint connected sets in the complementary disk cannot join alternating boundary contacts, a contradiction.

It follows that the union of the closed \(C\)-triangles is a connected subsurface of the sphere with disjoint simple boundary circuits. Each boundary circuit faces a different complementary disk. The connected circuit \(L\) avoids this subsurface and faces it across every boundary side, so there can be only one such disk and one boundary circuit. This is \(P\). The missing side at an occupied triangle has equal-colored endpoints; hence all vertices of \(P\) have the same color. Following either side of a disagreement circuit gives a connected walk of constant color: within a visited triangle the adjacent site either stays the same or changes across an edge joining two sites of that color. The vertices facing \(P\) therefore lie in one opposing cluster.

Any directed edge of \(P\) identifies its occupied right triangle and thus \(L\); its left triangle identifies \(C\). Finally, a component with no dual vertex must be one open edge with both endpoints on \(L\). Those endpoints give its two tags, whose side vectors are opposite. This proves (7). Their separate \(U\)- and \(Q\)-contributions need not vanish. ◻

Proof. An alleged bypass may be made simple. It is a crosscut of the right-hand disk and separates its two nonempty rim arcs. Choose an edge on each arc. Its facing triangle lies on that arc’s side of the crosscut, since a primal graph path cannot cross the interior of a face. Its opposing vertex does not lie on the bypass. Thus the two facing vertices are in different parts of the cut disk. Their opposing cluster can cross neither the monochromatic rim nor the monochromatic bypass, contradicting 6.

For the converse, let \(uv\) be an edge of the given circuit and \(w\) the third vertex of its right triangle. If \(w\) had the circuit color and lay strictly on the right, \(u,w,v\) would be a bypass. If \(w\) belonged to the circuit, one of \(uw,wv\) would be a right-side chord, unless the right disk were that single triangular face. The latter possibility has no opposite-colored site and is excluded by hypothesis. Every facing vertex is therefore of the opposite color.

Choose the occupied dual circuit \(L\) through one facing triangle. It cannot cross the monochromatic circuit, so it stays on its right. The triangles on the left are untouched and form part of one component \(C\) off \(L\). Its rim \(P^*\) includes the chosen edge and lies in the closed right side of the original circuit, since its left subsurface contains the original left disk. If the two circuits differed, a segment of \(P^*\) between successive contacts with the original circuit would be a right bypass. Such a segment has distinct endpoints by simplicity: follow \(P^*\) from the common edge to its first departure and then to its next contact. This contradicts the hypothesis, proving equality. ◻

Proof. Take a maximal biconnected block of the black graph containing the given black circuit. The face of this block toward the white circuit has a simple boundary. That boundary lies between the two given circuits: the block contains the black one and cannot cross the white one. A black bypass in this face would add an ear between two distinct vertices of the block and enlarge the block, contrary to maximality. The white circuit supplies an opposite-colored site in the face. Apply 7. In the full plane the same argument uses the finite black cluster containing the given circuit. ◻

Proof. Work first at spacing one. Anchor each rim at the average of its vertex positions, denoted \(b(P)\). This choice is translation equivariant. For \(s>0\), retain only those rims all of whose tagged barycenters are within distance \(s\) of their anchor, and define the stationary random measure \[\mu_s=\sum_{P\text{ retained}}|Q_P(1)|^2\delta_{b(P)}.\] For anchors in a square \(W_R\) of side \(R\), the full sum of every retained rim equals its sum against \(\mathbf 1_{W_R^{+s}}\), where \(W_R^{+s}\) is the \(s\)-neighborhood of the square. Thus (5) and stationarity give \[\mathbb E\mu_s(W_R) \le \frac34\mathbb E\sum_{z_t\in W_R^{+s}}|q_t|^2 =c_q\mathop{\mathrm{Area}}(W_R^{+s}).\] The finite constant \(c_q\) exists by 12. Dividing by \(R^2\) and letting \(R\) grow shows that the intensity of \(\mu_s\) is at most \(c_q\), uniformly in \(s\). Every rim has finitely many tags, so monotone convergence gives a finite intensity for \[\mu=\sum_P|Q_P(1)|^2\delta_{b(P)}.\] Let \(\theta(M)\) be the intensity restricted to rims of diameter at least \(M\). Since each rim is finite and \(\mu\) has finite mean on a unit square, \(\theta(M)\to0\) as \(M\to\infty\).

Rescaling a spacing-one configuration by \(\varepsilon\) multiplies each \(Q\) sum by \(\varepsilon\) and anchor area by \(\varepsilon^2\). Consequently, the expected squared mass of spacing-\(\varepsilon\) rims of diameter at least \(a\), with anchors in \(B\), is \(\mathop{\mathrm{Area}}(B)\theta(a/\varepsilon)\). A rim contained in the box has its anchor in the box, so this bounds the expression in (8). Finally, \(V_P(1)\) is the tangent sum around a closed polygon and is exactly zero. ◻

Proof. Reimer’s fair-bit inequality (Reimer 2000) states that \(|A\mathbin\square B|\le |A\cap\overline B|\), where \(\overline B=\{\mathbf1-b:b\in B\}\) denotes bit complementation, not set complementation. Apply it on slices of pairs of configurations. For each pair \((x,y)\in\{0,1\}^n\times\{0,1\}^n\), fix the set \(K\) on which they agree and their common values there. On the other coordinates the two strings are complementary. If \(x\in A\mathbin\square B\), its disjoint certificates, restricted to the coordinates outside \(K\), certify the two restricted events on that slice. Reimer’s inequality bounds the number of such pairs in the slice by the number with \(x\in A\) and \(y\in B\). Summing over the slices gives \(2^n|A\mathbin\square B|\le |A||B|\), which is (11). A fixed bounded region meets only finitely many Voronoi cells almost surely, so the conditional percolation tests indeed use a finite product. ◻

Proof. If an empty disk of radius at least \(\rho\) contains a point of a region \(K\), it contains an empty subdisk of radius comparable to \(\rho\) with center in an \(O(\rho)\)-collar of \(K\). A grid of mesh a small multiple of \(\rho\) reduces this to finitely many empty-disk tests. The Poisson void probability of each test is at most \(\exp(-c\rho^2/\varepsilon^2)\), and the number of tests for a bounded region is at most a constant times \((1+\mathop{\mathrm{diam}}(K)/\rho)^2\).

A triangle meeting \(K\) has an empty circumdisk containing its meeting point. The same observation applies to a cell meeting \(K\) with a remote site. Excluding the above empty subdisks makes all relevant circumdisks small and places them inside the retained collar. Both the positive empty-circle tests defining those triangles and the exclusion of larger ones are then local; they remain valid after any change outside that collar. This proves (i), including shrinking collar radii much larger than \(\varepsilon\sqrt{\log(1/\varepsilon)}\), and yields the exponential bounds in (ii).

On the event that all incident circumradii are at most \(R\), the relevant vertices lie in a box of side \(O(R)\). The incident triangle count is bounded by a polynomial in the number of Poisson sites there, and every diameter weight is bounded by a polynomial in \(R\). Splitting into successive values of \(R\), the exponential tail and the Poisson count moments give all the asserted moments. Tile \(S\) by \(O((s/\varepsilon+1)^2)\) squares of side \(\varepsilon\). Scaling the unit-square estimate and using the triangle inequality in \(L^p\) proves (12); no independence between adjacent squares is needed. Stationarity and scaling give (13).

For a barycenter mask, the spatial ergodic theorem for the Poisson process, applied at spacing one to expanding squares, proves the degree-two limit after rescaling. The higher moment bounds imply convergence in every finite moment norm. Triangles meeting the square but with barycenter outside it are contained in a thin boundary neighborhood on a good mesh event. First fix the width of that neighborhood, use the same moment bound there, and then let the width tend to zero; the bad mesh event is removed by higher moments and its vanishing probability. This proves the statement for intersecting triangles and for the other indicated weights.

Finally, in a thin collar of a circle, follow successive triangles around the circle and use their primal edges to form a walk with nonzero winding. On a sufficiently fine mesh the walk stays in the collar and follows the circle in order. Decomposing this walk into simple cycles gives a simple cycle with nonzero winding. Choosing the collar within a prescribed margin gives (iv), with simultaneous locality for any fixed finite list. ◻

Proof. Attach four exterior vertices, successively colored black, white, black, white, and fan each onto its side; close the exterior to a sphere. The four color switches give four boundary ends of disagreement strands. Their noncrossing pairing forces a connection between the two black fans or between the two white fans. Following the adjacent sites of a strand gives the corresponding primal connection inside the disk. For the last assertion, color exactly the vertices of \(X\) black and every other vertex white. A white crossing of the blocked pair is impossible, so the other crossing uses vertices of \(X\) only. ◻

Proof. Fan the two end sides to two uncolored exterior terminals. Crossings become simple terminal-to-terminal paths, internally disjoint and ordered from left to right. Conversely, first-hit and last-hit trimming recovers proper crossings in the same order.

Fix a first color. If there is a path of that color, add an artificial terminal-to-terminal edge on the exterior right. The union of this edge with all simple paths of the chosen color is biconnected: it is a union of simple cycles sharing that edge. Its left outer boundary is therefore a simple path of the chosen color. Call it the leftmost path. It has no same-colored bypass strictly on its left between distinct vertices, including the terminals. Conversely this property characterizes it. Indeed a portion of any other such path lying strictly to its left, between successive contacts, would be a bypass. Uniqueness also follows at the first divergence of two candidates, using the order at the start terminal or the order from their common incoming edge.

The event that a specified path is leftmost is determined by its own spins and the spins on its left. All vertices strictly to its right retain independent fair colors. The leftmost path can replace the first path of any ordered disjoint family: it lies on or to the left of that path, while all the others lie strictly to its right. Delete its internal vertices and apply induction in the graph on its right. For a fixed first color this shows that the remaining color sequence does not affect the probability. A global color interchange changes the first color as well. The proof never uses the presence of every vertex of the original rectangle, so also applies to a subgraph. ◻

Proof. We may take \(S/s\) large. Choose locally determined primal boundary cycles strictly inside the annulus and trim the four witnesses to disjoint proper paths between them. Every step is made on a good mesh event; its complement has vanishing probability at these fixed scales. Select an adjacent pair of the four paths of opposite colors. In their gap there is a spanning disagreement strand \(\gamma\): no strand can leave through a monochromatic side path, and the odd number of switches at each annular end forces a pairing from one end to the other. Stop \(\gamma\) at its boundary edges. The sites of its visited triangles are disjoint from the two remaining paths, also at the ends because all four paths are proper.

Sum over each fixed geometric candidate for \(\gamma\) and each consistent spin pattern on its visited sites. Its occurrence is determined by those spins. Conditional on them, delete those sites and slit the annulus along the strand. The remaining sites are free fair bits in a planar rectangle, and the two avoiding paths have a linear order. By 14, the probability of the union of their four possible color sequences is at most four times the probability of any specified sequence. Choose the sequence compatible with alternating four arms. The two missing arms are supplied by the opposite-colored banks of \(\gamma\). The sites beside each bank form a connected arm, are disjoint from the two avoiding paths, and lie on their respective extreme sides in the slit: they connect to the strand through its visited triangles without crossing either avoiding path. The resulting four arms thus have the required cyclic order.

Let \(N\) count proper spanning wall strands in the working annulus, each stopped at its first boundary edges and counted once. These are the spanning components after the walls are cut at all contacts with the two boundary cycles. Let \(G\) be the geometry event on which localization holds and a fixed buffered subannulus has quenched white-circuit probability at least \(\delta>0\). A fixed finite rectangle chain supplies this circuit bound by 10. In particular, \(\mathbb P(G^c)=o(1)\) for each fixed \(s,S\), and \(\delta\) is independent of \(S/s\). The preceding sum over strands gives \[ \mathbb P(\text{mixed four arms}) \le o(1)+C\mathbb E\bigl[\mathbf 1_G N\mathbf 1_{A_4^\varepsilon(r,R)}\bigr], \tag{15}\] where \(r\asymp s\), \(R\asymp S\) are deterministic buffered square radii. To make this final transfer, first stop all the cell paths at their first hit of the square outer boundary and then trim their starts at their last hit of the inner boundary. Planar order and disjoint used site sets are preserved. The summation is over all fixed strands; no selection of a globally first strand is used when claiming that the remaining bits are free.

On \(G\), the variable \(N\) has conditional moments of every fixed order, uniformly in the annulus ratio. Indeed its disjoint spanning strands have adjacent black banks. A gap between consecutive strands can account for at most the two black banks bordering it; black paths in different gaps cannot meet or cross a strand. This also separates their endpoint sites: a stopped strand ends inside a boundary edge between opposite-colored sites, and each black boundary site belongs to only one gap. Thus \(N\ge2k\) yields at least \(k\) black crossings with disjoint site witnesses, including their endpoints. A black crossing of the working annulus is incompatible with the white circuit in the fixed subannulus, so its conditional probability is at most \(1-\delta\). BK gives \[\mathbb P^{\eta}(N\ge2k)\le(1-\delta)^k\quad\text{on }G, \qquad \mathbb E[\mathbf 1_GN^q]\le C_{q,\delta}\quad(q<\infty).\] Keeping the indicator of \(G\) in (15), Hölder’s inequality and (10) bound its second term by \[C_q\mathbb P(A_4^\varepsilon(r,R))^{1-1/q} \le C_q(s/S)^{(1+c_4)(1-1/q)}.\] Choose \(q\) large enough that the displayed exponent is greater than one, and reduce \(c\) accordingly. Taking the microscopic limit proves the claim. No estimate of \(N\) on bad geometry is required. ◻

Proof. Suppose first that the rim \(P\) is black. For a large enough ratio \(S/s\), take four nested primal cycles near radii \(2s,5s,S/5,S/2\). They are locally determined and pairwise disjoint on a good mesh event. The two visits and the excursions in both rim directions give four vertex-disjoint black journeys between the cycle near \(2s\) and the cycle near \(S/2\). Trim them properly and denote them by \(P_1,\ldots,P_4\) in their cyclic order. The margins at both ends ensure that the vertices between the middle cycles are interior vertices of the original rim journeys, with both incident rim edges still present. Retain these four complete journeys throughout the argument.

Each oriented journey has its right side in one of the two adjacent annular gaps. Since a gap has only two sides, at least two distinct gaps have a right-facing rim side. Fix one such gap. The vertices of the inner middle cycle block every connection between its annular ends. Apply the indicator-coloring assertion of 13, coloring exactly those vertices black. It gives a transverse crossing between the two journey sides, using only vertices of that cycle. Trim this to a proper crosscut. Next work in the subdisk between this crosscut and the outer end. The outer middle cycle still separates the two ends, because the first crosscut lies strictly inside it. The same indicator argument gives a second proper transverse crosscut using its vertices. This successive construction orders the crosscuts even when the original journeys make extra visits to either middle cycle.

Consider the quadrilateral between these crosscuts. There is no actual black crossing between its two journey sides. Otherwise take a simple such crossing and start just after its last contact with the right-facing side. Both incident rim edges at that departure vertex belong to the original journey; the new edge therefore enters the open right sector of the whole rim. Follow the crossing only until its first subsequent contact with the whole rim, including any other rim portions in the quadrilateral. Such a contact exists because the other selected side is part of the rim. Simplicity makes the two contact vertices distinct. Until this contact the path stays in the right disk and has no rim vertices in its interior, so it is a bypass forbidden by 7. This argument does not require the quadrilateral itself to be contained in the smaller annulus between the middle cycles. Hex in the actual coloring now gives a white crossing between the two transverse cuts. Trim it properly between the middle cycles.

Doing this in two distinct original gaps gives two disjoint white arms: the boundaries shared by original gaps are black journey vertices, which the white paths cannot use. These white arms divide the middle annulus into two interval quadrilaterals. Arbitrarily trim the original black journeys to obtain four disjoint black arms in this annulus. It remains to ensure that both white intervals contain a black arm; arbitrary trims alone need not ensure this if a journey revisits a middle boundary.

For this purpose let \(X\) be the set of vertices of all four original journeys, restricted to one of the white intervals. Every path between its two white sides must meet \(X\). Otherwise it would join two distinct original gaps while avoiding all four paths that separate those gaps in the larger working annulus. Apply 13 with the artificial black set exactly \(X\). It gives a crossing between the middle cycles using sites of \(X\) and hence genuinely black sites. The same argument works in the other white interval. If the four arbitrary black trims already occupy both intervals, retain them. If all lie in one, retain three and add the newly obtained black crossing in the other. The white arms separate the intervals, so the four chosen black paths are still vertex-disjoint. The new crossing need not be a subpath of a single original journey; membership of its sites in \(X\) is enough.

We now have four black arms and two white arms, with at least one black arm in each interval between the whites. Choose one black arm in each interval to obtain alternating four, add one of the two remaining black arms as the extra arm for \(A_5\), and reserve the last black arm for \(A_1\). These witnesses use disjoint site sets. Transfer all six paths to deterministic buffered square boundaries, stopping at first outer hits and last inner hits to preserve their order. The event implies \(A_5^\varepsilon(r,R)\mathbin\square A_1^\varepsilon(r,R)\) for \(r\asymp s\), \(R\asymp S\).

Conditional on \(\eta\), 11 bounds its probability by \(p_{5,\varepsilon}^\eta(r,R)p_{1,\varepsilon}^\eta(r,R)\). Averaging and applying Cauchy–Schwarz and (9) gives \[\mathbb E[p_{5,\varepsilon}^\eta(r,R)p_{1,\varepsilon}^\eta(r,R)] \le C\alpha_5^\varepsilon(r,R)\alpha_1^\varepsilon(r,R) \le C(s/S)^{2+c_1}.\] For a white rim interchange the colors, at a cost of a factor two. Remove the mesh exception and take the limit superior. The global rim was used only for no-bypass; the resulting arm event is local, so the same bound covers existence of any such rim in a bulk completion. ◻

Proof. Vertex Menger gives a set of at most two black vertices meeting every black crossing. Boundary vertices are allowed in this cut. Temporarily recolor these vertices white; there is now no black crossing of the annulus. Attach a black cap vertex to each boundary cycle, fanning it onto that cycle. The two cap vertices are not black-connected.

An occupied dual circuit separates the caps. To see this directly, sites lying in the same component off all disagreement walls are joined in their color: a curve in that component, made transverse to dual edges, crosses only unoccupied dual edges and hence only equal-colored primal adjacencies. Since the caps are not connected, at least one wall separates them. The neighboring white walk along that wall uses original annular sites and is deformable to the wall while avoiding the caps. It therefore has nonzero annular winding. Decompose the walk into simple cycles and retain one with nonzero winding; it is an essential white circuit. On restoring the original colors, only the cut vertices can become black, so the circuit has at most two black vertices. Applying the construction in a working annulus with the stated margins gives the last assertion. ◻

Proof. Write \(A=L_s^+\cap L_t^-\) and \(\sigma=(\sigma_s,\sigma_t)\). The endpoint exterior restrictions are independent, so \(\nu(A\mid\sigma)\ge\beta^2\). For each color, continue through its open layer by a finite connected network of buffered crossing corridors. At each end include a transverse bar in the nominal slab \(5<|u|<6\), crossing the entire corresponding window with margin; join these bars through the layer. All continuation queries lie strictly outside the state disks. A bar intersects every successful side path, since that path traverses the slab through the window. Retention and buffers allow passage between cell and primal crossings without changing this separation argument.

Here is the corridor construction in the form that will also be used for local preparations. Approximate a compact trace in an open corridor by finitely many horizontal and vertical polygonal legs on a sufficiently fine fixed square grid. Put thin crossing rectangles along the legs, with both horizontal and vertical crossings in successive overlap squares, and add the same overlaps at branch points. Crossing paths therefore join into the prescribed network. Slightly lengthening and shrinking the rectangles leaves room to convert them into retained primal paths. Along a charted simple trace the steps may be chosen with arbitrarily small longitudinal extent, so concatenation in order gives any prescribed positive bound on longitudinal backtracking.

Let \(E\) assert the retained-mesh and quenched rectangle lower bounds in these finitely many continuation footprints. Fixed collars keep its point queries outside \(O_s\cup O_t\), independently of the states. Proposition 10 and Lemma 12 give \[\nu(E^c\mid\sigma)\le\delta_\varepsilon\longrightarrow0\] uniformly over the usable states. No good-geometry requirement deep inside an arbitrary fixed state is needed: its successful side paths already use retained edges.

Condition now on all point geometry and on \(\sigma\). On sites queried by black constraints order the bits toward black, and on the disjoint set queried by white constraints order them toward white. Both \(A\) and the continuation success \(K\) are increasing in this one product order; constraints of the same color may overlap. The two sets of colors have positive separation by the layer and chart-tolerance margins. On \(E\), conditional positive association and the crossing input give \[\nu(A\cap K\mid\eta,\sigma) \ge \nu(A\mid\eta,\sigma)\nu(K\mid\eta,\sigma) \ge q_e\nu(A\mid\eta,\sigma),\] where \(q_e>0\) is fixed. Since \(A\cap K\cap E\subset S_e\), integration before subtraction yields \[\nu(S_e\mid\sigma) \ge q_e\nu(A\cap E\mid\sigma) \ge q_e(\beta^2-\delta_\varepsilon).\] This proves (18). It does not infer any quenched black side propensity from an annealed one. If the two white side propensities are themselves bounded below after fixing geometry, the white-only version of the same product argument gives \(q_e^W\beta^2\) without this annealed integration. For fixed normalized templates, finitely many buffered rectangle chains give common positive constants by scaling; rigid motions preserve the laws. ◻

Proof. The root components join negative to positive side vertices within their central windows. Concatenate these paths with the successful sector paths. For both colors one obtains a monochromatic closed walk of longitudinal index one, respectively in the disjoint layers \(-.75<v<.75\) and \(-2.3<v<-.9\). This index follows by lifting the longitude: root passages cross the central slabs, and sector passages go from one positive slab to the next negative one without passing the end section cuts. Arbitrary local backtracking is harmless. Fine mesh keeps the edge segments in the stated layers. Each walk contains an essential simple circuit. Lemma 8 supplies a black rim between them. The farther subbands on either side lie in the two exterior sides of this sandwich, so the chosen rim is confined to \(-2.35<v<.8\).

This rim visits both the inside and outside of either gate: it crosses section \(0\) deep inside and also reaches the other plugs. A black rim can meet a gate with at most two black sites only at those sites. By simplicity it must meet exactly two, once each, and cross at both. A tangency would not change side, leaving too few contacts. Following a gate edge between the two contacts is also impossible: the complementary rim arc would then have to visit both sides without another contact. Thus the incoming and outgoing edges are strictly in opposite gate fans.

All visits to the central section belong to the single inner gate arc. That arc lies in the end chart, since the gate and its bounded inside lie in \(O\) and all other bands avoid its obstacle. The full central slab is inside the first gate, and compact positive and negative chart portions meeting the gates are separated. The total positive longitudinal index one therefore forces negative entry and positive exit. This uses ordered passage, or equivalently lifted longitude, without requiring a transversal intersection with section \(0\). Outside its inner first-gate arc the rim cannot cross that section. Consequently the outside arc from \(s\) to the next \(t\) stays in their intervening sector. Since each inner arc contains all visits to its own cut and these chart slabs are disjoint, the inner arcs have the asserted cyclic order. ◻

Proof. Order rays in the outgoing fan counterclockwise between its two boundary rays. For any incoming ray in the opposite open fan, the counterclockwise sector from that ray to an outgoing ray contains the same initial portion of the outgoing fan. Hence, if the two outgoing rays differ, one enters the other’s open right sector, independently of the differing incoming rays. Suppose, for instance, that \(\gamma'\) leaves to the right of \(P\).

Follow \(\gamma'\) to its first subsequent contact with the whole rim \(P\), not merely with \(\gamma\). Such a contact exists no later than \(w\). It is not \(v\), because \(\gamma'\) is simple. As \(\gamma'\) is a path in \(P\)’s graph, planarity makes the first contact a vertex contact; a shared edge is first reached at its endpoint. Before this contact the path is connected and disjoint from \(P\), so it stays in the open right component into which it departed. It is therefore a monochromatic right bypass between distinct vertices of \(P\), contradicting Lemma 7. An earlier return on the complementary portion of \(P\) gives exactly the same contradiction. The other case uses \(P'\) in its own graph.

Thus the first edges agree. At any later first divergence the incoming edge is common, and cyclic order from that incoming ray again puts one outgoing edge into the other’s open right sector. The same whole-rim first-return argument rules this out. The paths agree to \(w\). ◻

Proof. The second gate is in its prescribed outer ball, compactly inside \(O\); its bounded inside is in the same ball. This gives one deterministic envelope, also containing the gate neighborhoods. For an outside path use the closed inset footprint \(-2.35\le v\le.8\) in the whole sector. It has positive lateral margins. The chart–connector seams are interior product-band seams. At the two longitudinal ends the full section and a central slab are compactly inside the endpoint states, which shield the cuts. Hence this footprint is compactly contained in \(\operatorname{int}D_e\). Its collar is fixed before selecting a rim; there is no requirement that the rim follow a particular witness of \(S_e\).

If a triangle is incident to a vertex in such an envelope and its circumradius is at most \(\rho_{\varepsilon,h}\), its whole circumdisk is within \(2\rho_{\varepsilon,h}\) of that envelope. For small \(\varepsilon\) the disk lies in the retained region, so its vertices and empty-circumdisk condition are unchanged. Mesh control in both configurations excludes additional large incident triangles on either side. Thus all incident triangles, not only the edges of a preselected path, agree. The required mesh probabilities follow from ordinary control and the fixed density bounds for the experiments. ◻

Proof. For each type, fix a short unperturbed reference completion, for example using a sufficiently separated copy. Define \(m_s^\varepsilon(\sigma_s)\) as the conditional mean of its cutoff mass given \(\sigma_s\). Couple any other experiment to this completion sharing only that state and sampling the remaining data conditionally independently. Inner second-gate consistency identifies the cutoff mass on their common mesh-good event. Their \(L^2\) bounds show that their difference is \(o_{L^1}(1)\); indeed \(\mathbb E[|Y|\mathbf 1_E]\le\|Y\|_2\mathbb P(E)^{1/2}\) on a bad event \(E\). Using two copies also in the reference completion and conditional Jensen shows that its mass is \(o_{L^1}(1)\) from its own conditional mean. Consequently \(\widehat m_s=m_s^\varepsilon+o_{L^1}(1)\) in every completion.

Next fix one two-plug completion of \(e\) and condition its sector mass on \(\sigma_s,\sigma_t\) and the exterior sector datum \(\xi_e\). Call the result \(j_e^\varepsilon(\sigma_s,\sigma_t,\xi_e)\). These retained data have the same marginal in every completion, by (19). Coupling and outside first-gate consistency, again with a second copy for the defining completion, give \(\widehat j_e=j_e^\varepsilon+o_{L^1}(1)\) in every experiment containing \(e\).

In a fixed cycle, conditional on its complete state vector, the data \(\xi_e\) are independent, each with the conditioned law determined by its two endpoint states. The closed estimate (8) and the cycle density show that \[\sum_e(m_s^\varepsilon+j_e^\varepsilon)=o_{L^1}(1).\] For \(Q\) the selected rim is contained in a fixed box and has a fixed positive minimum diameter, since it visits distinct plugs; for \(V\) its complete tangent sum is zero. Now resample one \(\xi_e\) independently given the states, leaving all other sectors fixed. Subtracting the two small cycle sums gives \(j_e^\varepsilon-(j_e^\varepsilon)'=o_{L^1}(1)\). Conditional Jensen therefore gives \[j_e^\varepsilon- \mathbb E[j_e^\varepsilon\mid\sigma_s,\sigma_t]=o_{L^1}(1).\] The conditional law used in this last expectation depends only on those two states. Set \[i_e^\varepsilon=m_s^\varepsilon+ \mathbb E[j_e^\varepsilon\mid\sigma_s,\sigma_t].\] This proves both statements in (21). The defining short completions and Jensen give the stated \(L^2\) bounds. All replacements use only bounded second moments and high-probability path equality; uniform integrability of their squares is unnecessary. The choices are measurable because the locally finite graphs have countably many finite paths and walls, and different choices agree through the same path comparison. ◻

Proof. Apply (21) to the two cycles and subtract the common \(i_c^\varepsilon\). Adding independent unused states does not change an \(L^1\) error. A complementary connector always exists: two disjoint closed Jordan disks joined by the attached rectangular band form a Jordan disk, with the unused ports on its boundary. In its exterior, the Jordan–Schoenflies theorem gives another handle band joining those ports. The transverse parametrizations match because departure and arrival sections have opposite induced boundary orders. The remaining deformation assertion follows from the collar construction and finite chaining below. ◻

Proof. Extend a short piece of each given end collar and taper its transverse width to be very small at the far tip. The full port is usable initially: the compact middle arc is separated from it, while the given product collar handles its immediate neighborhood. Straighten a slightly extended middle arc by an orientation-preserving plane arc chart, with its left side above a horizontal segment.

From the upper and lower corners of each narrow collar tip, follow constant small signed transverse coordinates in that end collar for a short additional distance. In the straightened coordinates these side paths lie on the corresponding sides of the midline and progress along it with arbitrarily small error. Stop at their first hits of prescribed vertical lines, chosen in order at the two ends. Join the upper stopped paths, and separately the lower ones, by height graphs strictly on their side of the middle segment and as close to it as desired. The compact middle interval is separated from the initial collar pieces and disks, so these joins fit in the specified neighborhood and avoid all the specified compact sets. First-hit stopping keeps the side paths simple.

Each joined side, the appropriate half-tip segments and the centerline bound a Jordan rectangle. The two rectangles lie on opposite sides in the straightened coordinates and may be taken thin there. The initial disk-with-collar interiors abut their tips on the other side, so they cannot cross either boundary curve. Fill the rectangles by Jordan–Schoenflies, prescribing the centerline and boundary parametrizations, and glue them to the end collars. This gives the required band with the full original port attachments. ◻

Proof. Consider the mixed experiment with successes \(S_{e_g}\) and \(S_{c_0}\), with the nominal states and roots. Their exterior queries are still separated, and Lemma 23 applies. All allowed tubes lie in slightly widened, separated layers of the nominal annular band, with the same longitudinal order. This follows by the local tolerance where anything moved; elsewhere the system is unchanged. Thus the mixed experiment has the same confined-rim and gate conclusions, and we may use deterministic nominal masks for its masses.

Compare its \(e\) contribution with the pure \(e_g,c_g\) experiment by sharing the states and the exterior part of the nominal \(e_0\) sector. Every possible pulled-forward inset \(e_g\) path outside the states, together with all queries of \(S_{e_g}\), is buffered in that shared region. The queries of both \(S_{c_g}\) and \(S_{c_0}\) avoid it. Consequently the shared marginal in both experiments is the ordinary one conditioned on \(S_{e_g}\) at the two sampled states: the other success factor integrates to one given the states and affects only other exterior data. This also follows directly from the product density (19), irrespective of how unused points are assigned to the sampling regions.

The common collar and the two gate comparisons identify the first-gate path outside the cutoff balls and the starting cutoff contribution. Thus the extracted \(e\) value in the mixed experiment is \(i_{e_g}^\varepsilon\) up to \(o_{L^1}(1)\). The same argument identifies its other contribution with \(i_{c_0}^\varepsilon\). Closing the mixed rim therefore gives \(i_{e_g}^\varepsilon+i_{c_0}^\varepsilon=o_{L^1}(1)\), whereas the nominal system gives \(i_{e_0}^\varepsilon+i_{c_0}^\varepsilon=o_{L^1}(1)\). Subtraction proves the claim. ◻

Proof. Join identical unrotated reference copies on a horizontal line, using their fixed stubs and a straight gap. Their center separation is \(\ell h\), where \(\ell>D\) for a fixed sufficiently large \(D\). Write \(A_\varepsilon^X(\ell)=\mathbb E_\mu i_e^\varepsilon\). A remote complementary connector through the unused stubs closes both this transport of length \((\ell+r)h\) and the two transports obtained by inserting a middle reference copy. Lemma 29 gives \[A_\varepsilon^X(\ell+r)=A_\varepsilon^X(\ell)+A_\varepsilon^X(r)+o(1), \qquad \ell,r>D.\] Translation and rotation of the entire experiment, including the root ordering, preserve its law and rotate both fields as vectors. If a conditional-average or gate tie-breaking convention was chosen separately at the translated placement, its transported version is an alternative defining completion. Gate consistency with common gates in the shared states and Proposition 28 compare the two choices to \(o_{L^1}(1)\). Thus covariance to this accuracy is sufficient. Color exchange preserves the law as well.

The ratios \(A_\varepsilon^X(\ell)/h\) are bounded uniformly for \(D+1\le\ell\le D+4\), in the limit superior: these transports have bounded normalized size and uniformly buffered short completions. For each fixed larger length there is likewise a finite bound, not necessarily uniform in that length.

Only countably many lengths are required: the standard routes below are fixed by grid terminal pairs, finitely many types and integer twists. Include the positive integers and take the rational span of these lengths, a countable additive subgroup \(\mathcal L\subset\mathbb R\). Extract limits of \(A_\varepsilon^X(\ell)/h\) for all \(\ell\in\mathcal L\) with \(\ell>D\). Their function \(A\) is additive whenever both arguments exceed \(D\). Extend it to \(x\in\mathcal L\) by \[\widetilde A(x)=A(u+x)-A(u),\qquad u,u+x>D.\] The value is independent of a sufficiently large \(u\): comparing two choices \(u,v\) uses \(A(u+x)+A(v)=A(u+v+x)=A(v+x)+A(u)\). Choosing all offsets large similarly proves additivity on \(\mathcal L\). For small \(x\), take a fixed rational \(u\) in the interior of \([D+1,D+4]\); the interval bound gives a common bound on \(\widetilde A(x)\). Additivity then gives continuity at zero: if \(|x|<\delta/N\), a bound for \(\widetilde A(Nx)\) gives the same bound divided by \(N\) for \(\widetilde A(x)\). Since \(\mathcal L\) contains the rationals, continuity implies \(\widetilde A(x)=a_h^X x\). The interval bound also bounds \(|a_h^X|\) independently of \(h\). Covariance yields (22) in every direction. Countable diagonal extraction makes these choices simultaneously as asserted. ◻

Proof. We give the mask argument also for curvilinear end charts. In an unperturbed straight experiment choose a deterministic polygonal Jordan quadrilateral \(R_e\) approximating the sector from section \(0\) to section \(0\) with transverse interval \([-2.7,1.5]\). Its boundary is close in the ordered parametrization. The transverse end cuts remain in the first gate balls, and its top and bottom sides lie beyond the possible rim layer \([-2.35,.8]\) on its left and right, respectively. Permit a small longitudinal enlargement only inside the cutoff balls. By compact margins, outside slightly smaller cutoff balls the mask contains exactly the relevant sector side of every confined rim, with room, and excludes the complementary side.

Here the required polygonal approximation follows directly from Jordan simplicity. Trace a polygonal closed walk uniformly close to the boundary in order, split any crossings in general position and retain a simple subcycle with positive winding about a fixed inner point. Dropped parameter gaps have endpoints tending together, and therefore are short by injectivity of the limiting Jordan parametrization. They cannot be almost a whole turn: the retained cycle would then be local and have zero winding about the inner point. This preserves the ordered approximation and the four chosen sides. Fix one such quadrilateral in a sufficiently long template. For larger \(\ell\), stretch only its straight middle region horizontally, keeping both end regions rigid. The margins persist for each fixed \(\ell\), and the total length of the two end cuts is \(O(h)\) uniformly in \(\ell\).

On good geometry, polygonal intersection with the left side of \(P\) gives \[\int_{P\cap R_e}\mathrm dz=\ell h+O(h).\] Indeed this integral is minus the integral over the portion of \(\partial R_e\) on that left side. The entire top side contributes with reversed orientation, the bottom side does not contribute, and the end cuts cost \(O(h)\). Their endpoint locations differ from the plug centers by \(O(h)\). A fixed cut has probability zero of an accidental exact overlap with a random primal edge or vertex degeneracy.

We next compare this geometric integral with the barycenter mask \(V_P(\mathbf 1_{R_e})\). On fine mesh a discrepancy can involve only rim edges whose tagged triangles meet a cut segment. Put \(s=\varepsilon^\alpha\) with \(\alpha<1\) sufficiently close to one and cover these finitely many segments by \(O_e(s^{-1})\) boxes of side \(s\). In any one enlarged box the \(L^2\) norm of the total candidate edge length is \(O(s^2/\varepsilon)\), by the scaled local triangle moments. The presence of a candidate rim edge implies a monochromatic arm from scale \(s\) to a fixed positive distance for this pattern, since the rim also visits a different plug. Its ordinary probability is \(O_e(s^c)\), apart from a negligible mesh event. Cauchy–Schwarz and summation thus bound the expected discrepancy by \[C_e s^{-1}\frac{s^2}{\varepsilon}s^{c/2} =C_e\varepsilon^{\alpha(1+c/2)-1}\longrightarrow0\] when \(\alpha>1/(1+c/2)\). Bad mesh events are discarded using the same local moments. The bounded density of this fixed experiment transfers the assertion to its law. Alternatively, convergence in probability and the Bessel \(L^2\) bound suffice for the corresponding mean limit.

Outside the two cutoff balls, \(R_e\) gives exactly \(\widehat j_e\). Its difference from \(\widehat m_s+\widehat j_e\) consists of deterministic masks inside the endpoint cutoff balls. Couple at either endpoint state to a fixed short completion. The inner second-gate paths and their triangles agree, so these correction means are \(O(h)+o(1)\) by the short second-moment estimate. Their constant is independent of \(\ell\): the end masks were fixed before the middle strip was stretched. By Proposition 28 and (22), \[|a_h^V\ell-\ell|\le C\] for arbitrarily large positive integers \(\ell\). This proves (23). ◻

Proof. The smoothing and full-port collars were established after Lemma 30. Identify each hole boundary with the circle of directions at its center. Fix a radial identification of the plane with the two punctures blown up and the exterior of the circular holes: in disjoint neighborhoods of the centers, send radius zero to the hole radius by an increasing radial coordinate, agreeing with the identity outside those neighborhoods. Apply its inverse to each radial-ended proper arc and fill in the punctures. The result is a smooth embedded compact interval, radial near its ends, and the fixed identification recovers exactly the original proper arc after blowing up. Any two such smooth intervals, with ordered endpoints matched, are carried one to the other by an orientation-preserving plane diffeomorphism. To obtain this smooth extension, first map smooth strip-and-cap neighborhoods of the two intervals by an orientation-preserving diffeomorphism, preserving their ordered axes. Choose smooth disk boundaries inside these neighborhoods. The prescribed Jordan–Schoenflies theorem extends this map across their complementary surfaces, retaining its smooth values on collars of those boundaries. Hatcher’s smoothing theorem for surface homeomorphisms, in its form relative to a boundary neighborhood (Hatcher 2025, Theorem B and the following refinement, p. 2), replaces the exterior homeomorphism by a diffeomorphism without changing these collars. Gluing with the retained neighborhood map gives the required plane diffeomorphism and preserves the two intervals and their endpoints exactly.

An orientation-preserving plane diffeomorphism has a locally smooth deformation to the identity. After subtracting its value at a base point, the maps \(x\mapsto F(tx)/t\) tend on compact sets to its positive differential as \(t\downarrow0\); restore the translation and join the positive linear map to the identity. During the resulting deformation of the interval, postcompose by the unique direct similarity restoring the two current ordered endpoint positions to the two fixed punctures. This correction is the identity at both ends of the deformation. Thus the punctures stay fixed throughout.

Blowing up their directions gives a deformation of proper arcs whose ends move on the two hole circles. Lift both moving endpoint angles to real-valued continuous functions. Compensate each angle in a disjoint boundary collar, tapering the compensating rotation to zero away from that boundary. The deformation now fixes the port points. At its end the two compensations are integral full twists, since the original and final port points agree. These are the only possible differences between the fixed-end deformation classes. The compensated arcs are transverse to the hole circles. Compactness of the parameter interval provides uniformly small product collars in which all these transverse germs can be straightened to radial germs, continuously in the parameter. Take the straightening correction to be the identity at both terminal times, where the germs are already radial. The resulting deformation therefore has the full fixed-end collar form required for the common-closure comparisons.

They are detected separately by net arguments. For the starting puncture, lift the argument along the deforming arc. The difference of its final and initial net argument changes is minus the net turn of the starting direction loop: the direction at the other puncture has zero net argument change about the starting puncture. For the arriving puncture the analogous identity has the opposite sign. Hence equal net argument changes at both centers make both compensations trivial at the end, giving an endpoint-fixed deformation of the kind used in Lemma 29. Conversely integer full turns at the ends realize the corresponding changes; the explicit disjoint representatives are constructed below.

Choose the signs of the two integers so they record the additional argument turns in the orientation from \(s\) to \(t\). Their absolute sum is bounded by the displayed net changes divided by \(2\pi\), plus the bounded turns of the reference route. Normalizing the Jordan disks does not change winding differences between two arcs with the same ends: these differences are integer winding numbers of their joined closed curves about the respective centers. For allowed perturbed charts, pull the route back first to the nominal disks. The arc stays outside the fixed central balls, and the small supported change has only a bounded effect on the endpoint argument choices. This gives the same bound for \(k\) up to a fixed additive constant. ◻

Proof. If \(e\) uses changed charts, first hold those charts fixed and compare it, by Lemma 34 and the common-closure rule, to the corresponding image of its standard representative. Apply Lemma 31 only to this buffered representative to return to the nominal charts. Finally compare the nominal unsplit representative to the inserted system of Lemma 35, using its common complement. Thus, on the product of endpoint and auxiliary state laws, \[i_e^\varepsilon=\sum_j i_j^\varepsilon+o_{L^1}(1).\] Each chain of comparisons is finite with its pattern fixed. Narrow bands, small positive conditioning probabilities and the length of a finite deformation chain may enter the rate of this vanishing error; they do not enter the short-template constant.

Average over every auxiliary state, leaving \(\sigma_s,\sigma_t\) fixed. Only the fixed first and last insertions still have a nonconstant part. Define \[\begin{split} \phi_s^{X,+}(\sigma_s) &=\mathbb E_\mu[i_{\rm first}^\varepsilon\mid\sigma_s] -\mathbb E_\mu i_{\rm first}^\varepsilon,\\ \phi_t^{X,-}(\sigma_t) &=\mathbb E_\mu[i_{\rm last}^\varepsilon\mid\sigma_t] -\mathbb E_\mu i_{\rm last}^\varepsilon. \end{split}\] Conditional expectation followed by centering is an orthogonal projection in \(L^2\), so the fixed short-completion bounds give the stated norms. The definitions use the nominal fixed insertions; this is why neither the long route nor the allowed chart changes affect them. Conditional Jensen preserves the comparison’s \(o_{L^1}(1)\) error.

Let \(J\) be the short pieces and \(T\) the straight ones, with oriented center displacements \(d_j\). Their total displacement is \(z_t-z_s\). For each fixed \(j\in T\), Proposition 32 gives \(\mathbb E_\mu i_j^\varepsilon-a_h^X d_j\to0\). Put the sum of these finitely many deterministic deviations into \(\rho_e^\varepsilon\), together with the preceding comparison error. Define the remaining deterministic term by \[b_e^\varepsilon=\sum_{j\in J} \bigl(\mathbb E_\mu i_j^\varepsilon-a_h^X d_j\bigr).\] There are at most \(C(1+k)\) such pieces. Each has cochain norm at most \(C_{\rm short}h\), displacement \(O(h)\) and the same bounded scalar \(a_h^X\). Enlarging \(C_{\rm short}\) once gives the bias bound in (24). In particular a large total length of straight legs creates no length-proportional bias: their limiting means have already been included exactly in the affine displacement. ◻

Proof of Lemma 35. All lengths until the last rescaling are normalized. Let \(D_{\rm st}\) be the lower length threshold for a straight transport. Continue the two stubs at each terminal along opposite east and west spokes beyond all local structures. Set \(\delta=2\pi/16\). To make a turn \(\Theta\), including its bounded directional adjustment and integer full turns, write \[|\Theta|=N\delta+\alpha,\qquad N\in\mathbb Z_{\ge0},\quad 0\le\alpha<\delta.\] Choose a large constant \(D_2\) and take \(D_1=D_2\). Starting at radius \(D_1\), use \(N\) polygonal legs of angular increment \(\delta\) in the sign of \(\Theta\), increasing radius by \(D_2\) at each leg. If \(\alpha>0\), append one leg of angle \(\alpha\), also increasing radius by \(D_2\). The second strand is the image under rotation by \(\pi\). Both finish with radial outward continuations.

At \(s\) choose the final used spoke toward \(t\). At \(t\) choose its final used spoke, viewed outward, back toward \(s\); the connecting arc traverses this latter spiral inward. In the integer convention of Lemma 34, the outward constructions therefore use \[\Theta_s=\theta_s+2\pi n_s,\qquad \Theta_t=\theta_t-2\pi n_t, \qquad |\theta_s|+|\theta_t|\le C.\] This specifies both used strands and their unused opposite companions.

We verify the geometric margins, including a last partial leg. A leg from radius \(r\) to \(r+D_2\) with angular increment \(0<\gamma\le\delta\) has length at least \(D_2\), since \[|(r+D_2)e^{\mathrm i\gamma}-r|^2 =D_2^2+2r(r+D_2)(1-\cos\gamma).\] Its direction relative to its initial radial ray is \[\frac\gamma2+\arctan\left( \frac{2r+D_2}{D_2}\tan\frac\gamma2\right).\] This lies between \(0\) and \(\pi/2+\delta/2\); relative to the final radial ray the incoming direction lies between \(0\) and \(\pi/2\). Thus all bends, including radial joins and a partial last leg, have absolute turning angle at most \[\beta_* =\pi/2+\delta/2<\pi.\] Reflection gives the same assertion for the opposite sign of turn.

For a full leg its radius at angle \(0\le\theta\le\delta\) is \[ F_r(\theta)= \frac{\sin\delta} {\sin(\delta-\theta)/r+\sin\theta/(r+D_2)}. \tag{25}\] Uniformly for \(r\ge D_2\), \[\cos\delta\,r\le F_r(\theta)\le r+D_2,\qquad |\partial_\theta F_r(\theta)|\le C_\delta r,\qquad \partial_rF_r(\theta)\ge\cos(\delta/2).\] For the last inequality put \(A=\sin(\delta-\theta)\) and \(B=\sin\theta\); Cauchy–Schwarz gives \[\partial_rF_r =\sin\delta\frac{A/r^2+B/(r+D_2)^2} {(A/r+B/(r+D_2))^2} \ge\frac{\sin\delta}{A+B}\ge\cos(\delta/2).\] In the sixteen fixed angular sectors, two distinct full legs in one sector have starting radii differing by at least \(8D_2\): a return of one strand is offset by sixteen steps and the opposite strand by eight. Their radial graphs are therefore separated by at least \(8D_2\cos(\delta/2)\). For legs in adjacent sectors, the endpoint radii at the common ray differ by at least \(8D_2\), unless they are consecutive legs of one strand.

These radial gaps give Euclidean clearance \(c_0D_2\) between nonconsecutive full legs, for a fixed \(c_0>0\). Indeed, if two points were within \(cD_2\), their radii would be comparable and their angular separation at most \(C cD_2/r\). For small \(c\), their sectors must be equal or adjacent. Compare their radii on a common ray, or on the intervening step ray for adjacent sectors. The angular derivative bound in (25) changes the radii by only \(C'cD_2\), contradicting the preceding radial gaps.

A partial last leg need not have a uniform angular derivative bound. Instead let \(P_{r,\alpha}(\theta)\) be its radial graph. For \(0\le\theta\le\alpha<\delta\) it lies outside the hypothetical full leg: \[P_{r,\alpha}(\theta)\ge F_r(\theta).\] To check this, write \[\frac1{P_{r,\alpha}(\theta)} =\frac{\cos\theta}{r} +\frac{\sin\theta}{r} \frac{r/(r+D_2)-\cos\alpha}{\sin\alpha}.\] For \(0<a<1\), the derivative of \((a-\cos\gamma)/\sin\gamma\) is \((1-a\cos\gamma)/\sin^2\gamma>0\). This proves the comparison. Every earlier nonincident leg in the same or an adjacent sector is on the inner side of the hypothetical full leg, with the established gap. The close-points argument thus uses only the angular derivatives of that hypothetical leg and the earlier full leg, not of the partial one. The simultaneous partial legs of the two strands lie in opposite sectors and are separated on the scale \(D_2\).

The initial radial continuations have radius at most \(D_1=D_2\). All turning legs after the first have radius at least \(2D_2\cos\delta>D_2\), giving a gap of order \(D_2\); the first is incident to its own initial spoke and angularly separated from the opposite one. At the outer end, each radial continuation begins at the largest vertex radius, whereas all earlier legs except its last incident leg have radius at least \(D_2\) less. The opposite final leg is separated angularly. The central stub neighborhoods retain their fixed buffers, and the turning legs lie beyond them when \(D_2\) is sufficiently large.

The number of legs and bends is \(O(1+|\Theta|)\), and their maximum radius is \(O(1+|\Theta|)\). Thus the two terminal turning regions lie in disks of radii \(Ch(1+k)\) after rescaling. Under the separation assumption they are disjoint. Join the two used final spokes by their straight gap. The unused spokes point away from this gap; continue them outward and join them by a remote arc outside the two turning disks and the used gap. Its narrow band gives a complementary connector avoiding all subsequent insertions.

We next fix the short corner experiments. Let \(S\) bound the reference copy and its fixed stub structures, and choose \(E_0\) large compared to \(S\). At a corner \(p\) with incoming and outgoing unit directions \(u_-,u_+\), place aligned reference copies at \(p-E_0u_-\) and \(p+E_0u_+\). The non-reversal bound gives their separation at least \(2E_0\cos(\beta_*/2)\). Outside their stub exit squares, place a narrow strip around the intervening two-leg path, with brief straight continuations matching the exits. Round the corner within a fixed small neighborhood if desired. The angle bound leaves positive margins from both copies, their back stubs and adjacent straight transports. A standalone completion joins the two unused back stubs outside a regular neighborhood of this bounded configuration, still in a ball of radius \(C E_0\).

The bend angles range over the compact interval \([-\beta_*,\beta_*]\). A buffered experiment at one angle persists in a neighborhood of it: move the terminal copy together with both its stubs by a small rigid motion, taper this motion to the identity in a disjoint collar and deform the strip slightly. Its fixed continuation corridors keep their margins, so Lemma 23 supplies a common positive bound in this neighborhood. A finite subcover gives common bounds on normalized size and density over all corner angles. Proposition  28 now supplies one \(C_{\rm short}\). Include the fixed first and last insertions and bounded straight-size templates in the same finite collection of compact families.

Finally take \(D_2=D_1\) much larger than \(E_0,D_{\rm st}\), the central local scales and \(c_0^{-1}\). The corner neighborhoods are disjoint. Place copies before and after every bend as just described; after reserving their bounded neighborhoods, all remaining aligned gaps have length exceeding \(D_{\rm st}\). The first and last fixed insertions fit on the initial radial parts. Increasing the constant in the terminal separation condition ensures the same bound for the middle gap. Narrow strips along all gaps give the inserted simple system. The unchanged remote complementary connector also closes the unsplit representative.

There are only \(O(1+k)\) short pieces, each with displacement \(O(h)\). No linear bound on the total arclength is asserted or needed: the spiral’s full legs can have total length of order \((1+k)^2\) in normalized units. For each nominal terminal pair, type pair and pair of integers, fix this construction once. In the support of allowed chart changes it uses only fixed local stubs, with no return of remote pieces. The auxiliary placements and straight lengths can therefore be chosen before those changes. Grid terminal pairs and twist pairs are countable, the menu is finite, and the scale list is countable. Hence the required straight lengths are countable and may be included in the simultaneous extraction of Proposition 32. ◻

Proof. The complementary portion of the rim makes a fixed large departure. Consequently, if two points of \(K\) are sufficiently close, the portion between them in \(K\) has small diameter; otherwise both directions would violate one of the imposed constraints. More precisely, for a fixed \(j\), closeness at scale \(\delta_j\) forces this portion to have diameter at most \(4\eta_j\), after requiring \(\delta_j\le\eta_j\). For tests initially formulated at vertices, use the buffered test and mesh smaller than a fixed fraction of each of the finitely many used \(\delta_j\)’s to obtain the same implication for points on the polygonal arc. The harmless factor four can be absorbed by starting with a correspondingly smaller sequence \(\eta_j\).

Here is a finite-net argument that does not assume length bounds. Fix a displacement scale \(d>4\eta_j\). Starting at the first endpoint, stop successively when the curve first reaches distance \(d\) from the preceding stop, also inserting the marked times. Apart from a bounded number of stops adjacent to inserted marks, different stop starts are \(\delta_j\)-separated: an intervening displacement \(d\) rules out their closeness by the preceding inverse-diameter property. Packing in the fixed disk bounds the number of stops. Every portion between stops is contained in a \(d\)-ball about its start. Joining the stops by segments and rounding their positions to a sufficiently fine finite grid gives a finite marked Fréchet net, with error tending to zero as \(d\to0\).

Choose a rapidly Cauchy subsequence in these nets and align successive parametrizations by increasing homeomorphisms matching the marked times. The aligned curves converge uniformly to a continuous curve. If two of its times have the same image, the intervening portion has arbitrarily small diameter by the inverse constraints, and hence is constant. Every point fiber is thus an interval. Collapse these constant intervals. The quotient is a nontrivial compact connected metrizable linear order, with its order topology: order rays have open preimages, and compactness identifies the quotient topology. Such an order is an interval, as is seen by embedding a countable dense suborder in an interval and completing the order. The induced parametrization is injective and continuous, so the resulting arc is simple. A nondecreasing parametrization with constant intervals can be uniformly approximated by increasing homeomorphisms; with finitely many distinct marks this can be done separately between consecutive marks. More explicitly, after choosing the quotient parametrization so that its nondecreasing map \(q\) fixes the normalized mark times, the maps \((1-t)q+t\,\mathrm{id}\), \(0<t<1\), are increasing homeomorphisms fixing those times and converge uniformly to \(q\). Thus the convergence is in the asserted marked Fréchet topology.

The radial requirements keep the marks distinct. For example, if the ends of \(\Delta\) approached one another, the path through the core would make a fixed departure in one direction and the complementary path would do so in the other, contradicting an inverse constraint. The same argument treats the endpoints of \(K\) and every pair of nonadjacent marks; adjacent marks have the radial separation furnished by the core and the two circles. In particular the limit is not constant.

Pass also to a Hausdorff subsequence for \(E\). Connectedness is preserved under Hausdorff limits of nonempty compact connected sets. Before the limit, all visits to \(B(0,a_3)\) belong to \(\Delta\), and \(a_3>100b_2\); this gives the assertion about \(\overline B(0,4b_2)\) with a margin. If a complementary-rim point approached a point of \(K\) in \(B(0,6L)\), both intervening rim portions would pass through a terminal hit at radius \(10L\). A fixed inverse-diameter constraint excludes this approach. It excludes a limiting intersection as well. The added circle at radius \(50L\) is irrelevant in these inner regions.

Finally take any sequence of limiting profiles. Approximate its \(j\)th member closely by a prelimit profile satisfying at least the first \(j\) constraints, with the same finite trial data after passing to a subsequence. The argument just proved gives a convergent subsequence of these approximants, and therefore of the limiting profiles. This proves compactness of the limiting family. ◻

Proof. Extend \(K\) to a proper line by outward rays at its two distinct endpoints on \(\partial B(0,10L)\). The rays are disjoint and meet the closed disk only at those endpoints. In the one-point compactification of the plane, this proper line together with infinity is a Jordan curve on the sphere. Apply the sphere Jordan–Schoenflies extension theorem (Moise 1977, Theorems 10.2–10.3, p. 71) to a correspondence from the compactified horizontal axis that fixes infinity. Choose the correspondence in the given longitudinal order and match the two complementary sides to preserve orientation. The extension fixes infinity and hence restricts to an orientation-preserving plane homeomorphism \(\Psi\) taking the ordered axis to this line. Prescribe its longitudinal parametrization in the marked order, with \(p\) at \(u=0\) and all of \(\Delta\) strictly within \(|u|<1\). Assign \(u=-16\) and \(u=16\) to the first extensions of \(\Delta\) in the two directions that reach radius \(2L\). The central axis segment through these parameters lies in \(\overline B(0,2L)\); by continuity, for some \(d>0\), \(\Psi([-16-d,16+d]\times\{0\})\subset B(0,3L)\). This compact segment has positive distance from \(E\). Its parameter interval contains the chart interval \([-12,12]\), the later cuts at \(u=\pm11\), and their longitudinal buffers. The rest of the line follows the remaining parts of \(K\) and, beyond its endpoints, the prescribed outward rays to infinity. Outside the marked middle interval the line misses \(\overline B(0,4b_2)\).

For sufficiently small \(\tau>0\), the chart \[\Phi(u,v)=\Psi(u,\tau v),\qquad (u,v)\in[-12,12]\times[-3,3],\] lies in \(B(0,4L)\) and misses \(E\); its section at zero is strictly inside \(B(0,a_1)\). We describe the state and obstacle disks to ensure the prescribed exact slab intersections. In straight coordinates take nested polygonal Jordan neighborhoods approximating the preimages of the disks of radii \(2b_2\) and \(3b_2\), inside the preimage of \(B(0,4b_2)\). They contain the preimage of \(\overline B(0,b_2)\) with a margin and avoid a thin strip about the axis for \(|u|\ge2\). Adjoin thin horizontal rectangles with longitudinal intervals \([-3,3]\) and \([-8,8]\), respectively, and fill bounded complementary components. The polygons and rectangle widths can be chosen strictly nested and with transverse intersections of boundaries. Their unions have connected interiors and disjoint simple polygonal boundary circuits; filling the holes leaves a single outer boundary. They are therefore nested Jordan disks.

Choose \(3\tau\) smaller than these rectangle half-widths. Beyond the specified vertical cuts the remaining thin axis strip escapes to infinity without meeting either union. Hole filling consequently does not change its intersections with this strip. Their images \(O,H\) have precisely the slab intersections \(|u|\le3\) and \(|u|\le8\) required in Section 4. They miss \(E\): before filling they miss this connected set, which reaches the outer circle, and hence \(E\) lies in the unbounded complementary component. In physical coordinates the unions lie in \(B(0,6L)\), and their filled hulls remain there because the exterior of that disk is connected. Fix stubs and a small positive chart-change tolerance as in Sections 4 and 5. Decrease the tolerance to preserve all compact window margins; in particular nominal coordinates in the terminal windows change by less than \(0.1\). The support of the ambient change is allowed inside \(B(0,50L)\).

It remains to justify exact fitting, rather than only close tracking. Consider simple profiles converging to the nominal profile in the marked topology. In \(\Psi\)-coordinates take the subarc from its last hit of \(u=-11\) before the core mark to its first hit of \(u=11\) after the mark. For close enough profiles this is a proper left-to-right crosscut of a fixed thin rectangle. Its interior lies between the vertical sides, and it stays away from the horizontal sides. Fréchet convergence controls its order and backtracking, so these crosscuts converge in parametrized boundary distance to the axis crosscut.

There are homeomorphisms of this rectangle converging uniformly to the identity that take its midline, in order, to the new crosscuts. Here is the extension argument. Split the rectangle along the old midline and the new crosscut. On the boundaries of the two old half-rectangles prescribe correspondences to the two new halves, using the matching crosscut parametrizations, the identity on the horizontal sides, and piecewise linear increasing height maps on the vertical sides. These maps agree on the common cut and converge uniformly to the identity. Each new half is a Jordan domain, and its parametrized boundary converges uniformly to that of the old half. Choose a fixed interior point of each old half, which belongs to every sufficiently close new half, and normalize the conformal maps at that point with positive derivative. Radó’s uniform convergence theorem for Jordan domains with Fréchet-convergent boundaries (Radó 1923, secs. 1–2, pp. 182–183) gives convergence of these maps on the closed disk. Write them as \(F_n,F\), and write \(h_n\) for the prescribed boundary correspondence. The circle homeomorphisms \(\chi_n=F_n^{-1}h_nF\) converge uniformly to the identity: for \(\zeta\in\partial\mathbb D\), \[|F(\chi_n(\zeta))-F(\zeta)| \le\|F_n-F\|_\infty+\|h_n-\mathrm{id}\|_\infty,\] and the inverse of the fixed closed Jordan chart \(F\) is uniformly continuous. Extend these circle homeomorphisms radially by \(re^{\mathrm i\theta}\mapsto r\chi(e^{\mathrm i\theta})\); these are disk homeomorphisms with the same uniform convergence. Conjugation by the conformal charts gives the desired half-rectangle homeomorphisms, with the prescribed exact values on the common cut and uniform convergence to the identity. The cut parametrizations can match all three interior marks: interpolate the small changes of their terminal parameters by increasing piecewise linear maps fixing those marks. Glue them along the cut. Interpolate the small vertical-side height changes to the identity in thin exterior collars, keeping the top and bottom fixed. The result extends by the identity to a plane homeomorphism. After conjugating by \(\Psi\), its support can be kept in \(B(0,50L)\) and its displacement tends uniformly to zero.

The portions of the ordered arc omitted by this crosscut correspond, for close enough profiles, to nominal axis positions with \(|u|>10.5\). The original compact margins separate them from the changed obstacle and the central track through \(|u|\le9.5\). The unwanted set \(E\) is separated there as well. Thus every sufficiently close simple profile admits the required change within any prescribed positive tolerance. First obtain this conclusion on a neighborhood, and then choose a smaller neighborhood whose closure among simple profiles is contained in it. This gives the stated closed-limit property and the strict prelimit tracking requirements.

Cover the compact family of limiting profiles by finitely many such open neighborhoods. If no finite initial constraint list forced membership in their union, one could choose a profile outside the union satisfying the first \(j\) constraints for every \(j\). Lemma 38 would give a subsequential limit inside the union, contradicting openness. Thus one finite list suffices. The construction uses only normalized profile geometry and the finite trial data, so the same finite menu works at every sufficiently small \(h\). ◻

Proof. For either color, the preceding local tests have the following deterministic implication on patch agreement and good mesh: every eligible marked rim vertex within distance one of a grid center fits some qualified choice unless that center is flagged. The trial part of its limiting flag probability is (26). For fixed \(L\) all other costs can be made arbitrarily small. The order of choices is essential. Choose \(M,n\) first. Allocate a summable inverse-test budget, which determines the finite profile menu and its finite constraint list. For that menu fix the preparation corridors and their positive lower bounds, then truncate ranks, and finally choose the two propensity thresholds. Every choice is in normalized units and independent of \(h\). Admission itself need not stabilize here.

Tile the plane by blocks of side \(Lh\). A block is bad if its enlargement by distance \(2Lh\) contains at least \(c_0L\) flagged grid centers, where \(c_0>0\) is a sufficiently small absolute constant. There are \(O(L^2)\) candidate centers in that enlargement, so the first-moment bound gives \[ \limsup_{\varepsilon\to0}\mathbb P(\text{a specified block is bad}) \le CL\bigl[Cn^2(CM^{-1-c})^{n-2}+\zeta\bigr], \tag{29}\] where \(\zeta\) is the sum of the adjustable additional costs. Since \(L=30M(10^4M)^{n-1}\), first taking \(M\) large and then \(n\) large makes the contribution \(L n^2(CM^{-1-c})^{n-2}\) as small as desired. Choose \(\zeta\) afterward. The dependence range of these block events is an absolute constant in block units, including the independently assigned preparation seeds.

An eligible subarc of diameter at least \(r_h\) without a qualified pin forces a connected chain of bad blocks of length at least \(c r_h/(Lh)\). Choose two points of the subarc at distance at least \(r_h/2\) and follow the portion between them. Since \(r_h/(Lh)\to\infty\), at least one of these endpoints lies outside the \(2Lh\) enlargement of each visited block. From a visit to the block, follow the subarc toward that endpoint until it first leaves the enlargement. Even if the visit only clips a corner of the block, this portion travels distance at least \(2Lh\). On fine mesh, its initial passage over distance \(Lh\) supplies vertices within distance \(h\) of at least \(c_0L\) distinct grid centers, all still inside the enlargement. Every such center is flagged by the deterministic implication above. Thus every visited block is bad. The visited blocks connect the two chosen endpoints; erasing repetitions gives a chain of distinct neighboring bad blocks, whose length is at least \(c r_h/(Lh)\) by their separation.

There are exponentially many block chains of a specified length, and only \(O(h^{-2})\) possible starting blocks in the fixed patch. From any chain of length \(k\) choose at least \(k/C_0\) blocks whose flag-data regions are pairwise disjoint. Their bad events are independent under the full-plane marked Poisson law with the local seeds. If the bound in (29) is sufficiently small, exponential path counting therefore gives an upper bound of the form \[C_Rh^{-2}\exp\{-c r_h/(Lh)\}\longrightarrow0.\] At each fixed \(h\) the chain length and all local scale lists are finite; take the microscopic limit before this estimate. Discarding the vanishing patch-agreement and mesh failures completes the proof. ◻

Proof. For every nonnegative measurable function \(g\), positivity of a kernel and disintegration give \[\begin{align*} \int\lambda(\mathrm d\omega)Kg(\omega) &\le C\int\mathbf 1_H(Z(\omega))\nu(\mathrm d\omega)Kg(\omega)\\ &=C\int_H\nu^Z(\mathrm dz)\, \frac{1}{p(z)}\int\mathbf 1_W(\omega')g(\omega') \nu_z(\mathrm d\omega')\\ &\le(C/q)\int g\,\mathrm d\nu. \end{align*}\] No assumption is made about the conditional law of \(\lambda\) given \(Z\). Under the additional hypothesis, association of \(\pi_z\) shows that \(\pi_z(\cdot\mid W)\) stochastically dominates \(\pi_z\) in the white order. The finite ordered coupling construction in Section 6, applied on each retained-data fiber, gives a measurable white-adding kernel for this block. Independence of the complementary data lets us use the identity there: the output law is \(\pi_z(\cdot\mid W)\otimes\tau_z=\nu_z(\cdot\mid W)\). ◻

Proof. A Delaunay triangle is determined by its three vertices and emptiness of its circumdisk, apart from the null degeneracies excluded in our model. If its circumradius is at most \(\rho_\varepsilon\) and it meets a fixed envelope, its circumdisk lies within a constant multiple of \(\rho_\varepsilon\) of that envelope. This is eventually inside its kept collar. The same observation determines all retained triangles incident to its local edges. On mesh-good events all triangles needed for the ordinary local paths and their right tags satisfy this bound, by Lemma 12. This proves the first assertion without any assumption on the area of the boundaries of the kept sets.

For the second assertion, the paths in the cycle experiment lie in fixed inset layers of the embedded annular band. Outside states these layers, the tube-test footprints, and the joining seams have positive margins to the lateral boundaries of the used sector. Inside states, the relevant second-gate paths and cutoff triangles lie in a compact subset of \(O\), since the gate annuli and cutoff balls have the prescribed radial margins. The same envelopes contain the actual paths when the track fit is sufficiently close. Their exterior portions and state portions therefore have collars within the kept union, as also in Lemma 27. The other sector queries disjoint exterior restrictions. Deterministic Borel allocation of boundaries suffices; no smooth-boundary convention is needed. The first assertion now applies in both configurations. ◻

Proof. Fix a ray from the center and divide its relevant distances into dyadic bins \(d\asymp 2^jh\), up to scale \(O(R)\). In each bin a bounded number of balls of radius a small fixed fraction of \(d\) cover possible ray passages. Give each ball a buffered outer ball of comparable radius, all within the bulk patch. A passage off the ray followed by a quarter turn leaves one of these outer balls, and so gives a monochromatic crossing of its fixed-ratio annulus. Successive full turns contributing to a net argument change can be selected by first hits of successive levels of a lifted argument. Their portions between the relevant passage and departure times are disjoint on the simple rim; trimming with the fixed margins yields vertex-disjoint annular crossings on fine mesh.

On crossing-good geometry an opposite-color circuit in such a test annulus has probability at least a fixed \(q>0\). Hence one monochromatic crossing has probability at most \(1-q\), and conditional BK bounds the probability of \(k\) disjoint crossings by \((1-q)^k\). There are \(O_R(h^{-2})\) centers and \(O(1+|\log h|)\) bins per center. At each fixed \(h\) the geometry exceptions for this finite list vanish as \(\varepsilon\to0\). A union bound with \(k=C_R(1+|\log h|)\) leaves probability tending to zero as \(h\to0\). Summing over the bins bounds the net number of turns by \(C_R(1+|\log h|^2)\), and the incomplete initial and final turns add only a constant. ◻

Proof. We construct one comparison for all qualifying arcs of each pattern in the finite deterministic cover. Write \(\xi\) for the original input, with law \(\mathsf P\), including the bulk/full-plane coupling and all assigned local preparation seeds. Let \(E_0\) be the common input event on which patch agreement and the covering and winding conclusions hold, intersected with the original mesh and continuation-geometry conditions of all patterns. These additional conditions have total failure probability \(o(1)\) at fixed \(h\). The covering exception will be paid once, through \(\mathsf P(E_0^c)\).

Fix a placed connector \(e:s\to t\), a complementary connector \(c\), and their endpoint choices. Let \(\mathscr C_e(\xi)\) be the collection of all qualifying ordered rim arcs of this pattern, including their assigned preparation outcomes. Put \[A_e=\{\mathscr C_e(\xi)\ne\varnothing\},\qquad A_{e,0}=A_e\cap E_0.\] We retain the original input throughout and work with the unnormalized restriction \(\mathsf P|_{A_{e,0}}\); its mass is never divided by the probability of a fit. We describe the construction for black arcs; for white arcs exchange the colors.

Proof. Let \(\mathcal G=\mathcal G_{\varepsilon,h}\) be the common event on which the preceding qualified-pin assignments, density, winding bounds, finite-cover conclusions, and simultaneous actual-arc comparisons hold, together with bulk-patch agreement. Choose the finite-cover budget tending to zero with \(h\). The preceding results give \[\limsup_{\varepsilon\to0}\mathbb P(\mathcal G^c)\le q(h), \qquad q(h)\longrightarrow0.\] This event is distinct from the local usable-state events \(G_s\). Extend the adaptive coefficient assignment arbitrarily off \(\mathcal G\); all restricted counts and errors below are set to zero there. We restrict the original probability measure by \(\mathbf 1_{\mathcal G}\), without dividing by \(\mathbb P(\mathcal G)\). Any additional mesh tests at the finitely many levels used below have failure probability tending to zero at fixed \(h\) and fixed depth.

All discrepancies occur in a fixed compact neighborhood of \(\mathop{\mathrm{supp}}f\). Tile it by \(h\)-squares and dyadically refine to \(s_j=2^{-j}h\). At every level intersect the squares with the deterministic partition recording membership in all candidate cutoff balls, with fixed disjoint Borel boundary conventions. A square meets only a bounded, menu-dependent number of these balls: their radii are fixed multiples of \(h\), and their centers lie on the \(h\)-grid. Thus there are at most a fixed number of atoms and children per square. The partitions are deterministic, regardless of which pins were selected.

For a fixed rim call an occupied atom mixed if two of its tag centers have different assigned numerical coefficients. Stop unmixed atoms and refine mixed ones. We first establish the geometric implication that makes this refinement summable. A mixed atom has a nonzero coefficient involved, so its tags are near the support. Neither of two differently labeled tags can be in any cutoff ball assigned to that rim. Membership in every candidate ball is constant throughout the atom, and an assigned ball near the support assigns its entire rim mass a single coefficient. Thus if one tag belonged to it, both would have that coefficient. This argument concerns assigned balls; it asserts nothing about unused candidates.

Choose a differently labeled tag with nonzero coefficient. Since it is outside all assigned cutoffs, it lies on a traversal chunk between consecutive pins \(p_i,p_{i+1}\). The other tag cannot lie on that same chunk. In one direction along the rim between the two visits the path must therefore pass \(p_i\), and in the other direction it must pass \(p_{i+1}\). This remains true if the second coefficient is zero, or if its visit belongs to another excursion. Each pin core is within \(h\) of its center, whereas the first tag is outside both corresponding radius-\(mh\) cutoffs. Its distance from each core is at least \((m_{\min}-1)h-o(h)\), with \(m_{\min}>100\). The two visits in a common \(s_j\)-square are at distance \(O(s_j)\). They thus have two distinct departures of order \(h\), as illustrated in Figure 5.

For sufficiently fine fixed levels, Lemma 16 gives the local necessary near-return arm event from scale \(O(s_j)\) to a fixed multiple of \(h\). It can be tested inside, for example, \(B(y,10h)\) about the square center \(y\), taking the outer radius of the arm test below \(8h\). The same macroscopically reaching rim supplies a one-arm event through a separated outer annulus, from scale \(32h\) to a fixed \(\delta>0\), whose retained queries can be confined outside \(B(y,24h)\). The numerical constants can be changed to fit the fixed buffers. At fixed \(h,j\) the mesh exceptions vanish and these tests use disjoint deterministic marked-Poisson restrictions. For each color \(b\), the precise necessary-event inclusion is \[\begin{aligned} &\{\mathcal G;\ \text{atom mixed for some color-}b\text{ rim}\} \\ &\qquad\subset E_{{\rm in},b}(s_j,h)\cap E_{{\rm out},b}(h,\delta) \ \cup\ \{\text{mesh failure}\}. \end{aligned}\] The events on the right do not select a rim. We multiply their ordinary probabilities, not probabilities conditioned on adaptive labels or on the good event. The near-return gain and the one-arm bound give, after decreasing \(c>0\) to a common value, \[\mathbb P(E_{{\rm in},b}\cap E_{{\rm out},b}) \le C(s_j/h)^{2+c}h^c.\] The two colors cost only a factor two. At the bounded number of coarse levels where these numerical buffers do not fit, use the one-arm estimate alone.

On \(\mathcal G\), let \(N_j\) count the geometric atoms to be considered at level \(j\) because their parent is mixed for at least one relevant rim. At level zero count the occupied atoms in use. Each atom is counted once, regardless of how many rims visit it. There are \(O_R(s_j^{-2})\) candidate squares, with a bounded number of atoms and children each. Define \(\widehat N_j=\mathbf 1_{\mathcal G}N_j\), with value zero off \(\mathcal G\). For \(j\ge1\), apply the preceding inclusion to the mixed parent at scale \(s_{j-1}=2s_j\); its bounded number of children and the factor two in scale change only the constant. Summing the ordinary arm probabilities yields \[ \limsup_{\varepsilon\to0}\mathbb E\widehat N_j \le C_Rh^{-2+c}2^{-cj}. \tag{36}\] The same restricted bound at level zero follows from one arm.

Choose \(f_D\) to be the value of \(f\) at the center of the level-\(j\) square containing \(D\). Define the nonnegative square sum \[ B_j=\sum_{D,\lambda}\left( |X_\lambda(\mathbf 1_D)|^2+ s_j^{-2}|X_\lambda(\mathbf 1_D(f-f_D))|^2\right). \tag{37}\] The sum includes every deterministic atom at this level and every circuit-component label \(\lambda\). Since these are disjoint deterministic masks, \(|f-f_D|\le C_fs_j\), and all masks have support in a fixed compact region, the Bessel bound (5) and the local triangle moments give \[\limsup_{\varepsilon\to0}\mathbb E\widehat B_j\le C_{f,R}, \qquad \widehat B_j=\mathbf 1_{\mathcal H_\varepsilon}B_j.\] Here \(\mathcal H_\varepsilon\) is the color-independent geometry event of bulk-patch agreement, and is the whole space in the full-plane setting. We have \(\mathcal G\subset\mathcal H_\varepsilon\). The full deterministic atom and label sum defining \(B_j\) is never restricted according to the adaptive coefficients; its geometric indicator is allowed in the conditional Bessel bound. The bound is uniform in \(j\) when that finite level is fixed before the microscopic limit.

For one rim \(P\) let \(\mathcal S_j(P)\) be its stopped atoms at level \(j\), and write \(\lambda(P)\) for its circuit-component label. On such an atom its constant coefficient error relative to \(f_D\) has magnitude at most \(C_fr_h\). The other error uses \(f-f_D\). Because \(s_j\le h\le r_h\), Cauchy–Schwarz gives, on \(\mathcal G\), \[\begin{align*} &\left|\text{level-}j\text{ contribution to } X_P(f)-\mathcal A_X(P)\right|\\ &\quad\le C_fr_h\sqrt{\#\mathcal S_j(P)} \left[\sum_{D\in\mathcal S_j(P)} \bigl(|X_{\lambda(P)}(\mathbf 1_D)|^2+ s_j^{-2}|X_{\lambda(P)}(\mathbf 1_D(f-f_D))|^2\bigr)\right]^{1/2}\\ &\quad\le C_fr_h\sqrt{\widehat N_j\widehat B_j}. \end{align*}\] After extending the level error by zero off \(\mathcal G\), the last bound holds everywhere and bounds the maximum over all rims. In particular no extra number-of-rims factor appears: \(\widehat N_j\) counts geometric atoms, while the second factor already sums over all field labels. Taking expectation and applying Cauchy–Schwarz once more, (36) and (37) bound the limiting upper expectation by \[C_{f,R}r_hh^{-1+c/2}2^{-cj/2}.\] This is summable over levels, with total \(O_{f,R}(r_hh^{-1+c/2})\).

To justify stopping, fix \(h\) and a finite depth \(J\ge1\). A mixed level-\(J\) atom has a mixed parent for the same rim, so it is itself counted by \(\widehat N_J\) on \(\mathcal G\). Therefore \[\limsup_{\varepsilon\to0} \mathbb P(\mathcal G\cap\{\text{some level-}J\text{ atom is mixed}\}) \le C_Rh^{-2+c}2^{-cJ}.\] On \(\mathcal G\) when no terminal atom is mixed, the stopped atoms through depth \(J\) account for the whole discrepancy. The preceding finite sum is bounded by its geometric series, independently of \(J\). For each \(t>0\), Markov’s inequality and the last display give \[\begin{aligned} &\limsup_{\varepsilon\to0} \mathbb P\left(\max_P|X_P(f)-\mathcal A_X(P)|>t\right)\\ &\qquad\le q(h)+C_Rh^{-2+c}2^{-cJ} +\frac{C_{f,R}}{t}r_hh^{-1+c/2}. \end{aligned}\] For a prescribed tolerance \(\tau_h\downarrow0\), choose a finite \(J=J(h)\) making the middle term at most \(\tau_h\), before taking \(\varepsilon\to0\). Only finitely many mesh tests are needed at that fixed \(h,J\), so their failure terms vanish in this microscopic limit. Finally let \(h\to0\): both \(q(h)+\tau_h\) and \(r_hh^{-1+c/2}=h^{c/2-\kappa}\) tend to zero. This proves the claim. ◻

Proof of Proposition 46. Since \(V=U+Q\), the cross terms in the following real part cancel exactly. By (5) and the triangle area identity, \[\begin{align*} \mathbb E\Re\sum_\lambda(V_\lambda(f)-2Q_\lambda(f)) \overline{V_\lambda(f)} &=\mathbb E\sum_\lambda\bigl(|U_\lambda(f)|^2-|Q_\lambda(f)|^2\bigr)\\ &=\sqrt3\,\mathbb E\sum_t\mathop{\mathrm{Area}}(t)|f(z_t)|^2 \longrightarrow\sqrt3\int_\mathbb C|f(z)|^2\,\mathrm d^2z>0. \end{align*}\] The final approximation follows from fine mesh and the diameter moments. On the other hand, (39) writes the left side as \[(1-2\mathop{\mathrm{Re}}a)\,\mathbb E\|V(f)\|^2+o(1).\] The norm is nonnegative and has bounded expectation. A nonpositive coefficient is incompatible with the strictly positive limit. Therefore \(1-2\mathop{\mathrm{Re}}a>0\), as claimed. ◻

Proof. On fine mesh in a fixed collar of \(\partial D\), record the successive Voronoi cell sites met during one traversal of \(\partial D\) and join successive sites by primal edges. A fixed polygon meets cell switches generically almost surely. Equivalently, one can use walks through the nearby triangles. This produces a closed primal walk, with sites and edges uniformly close to the boundary position at the corresponding traversal times. Its winding about a fixed interior point is one.

Extract return loops and contract them to obtain the oriented simple-cycle decomposition of this walk. Winding is additive, so retain a cycle with positive winding about the interior point, retaining the original cyclic times of its edges. Every dropped interval of times has endpoints whose physical positions tend together. Boundary simplicity then forces its parameter length to tend to zero, unless it tends to a whole turn. The latter possibility would put all retained times, and thus the retained cycle, in a shrinking neighborhood of one boundary point, contradicting its positive winding about the fixed interior point. Hence the retained cycle follows the boundary in its order, and marked vertices can be selected at the desired parameters. Finite deterministic tie-breaking makes the choice measurable. Filling the cycle by its interior triangles gives the required disk. The mesh and local-determination estimates ensure agreement on interior compacts. Taking a deterministic sequence of decreasing tolerances gives the claimed convergence in probability. ◻

Proof. Start with the uniform edge-pattern ensemble having oddness at \(t_*\) and \(m\), and move the star defect to \(n_i\) as in the proof of (2). If the star has degree one, the loop count is independent of the chosen port. If it has degree three, the extra loop count is the indicator that the exterior partner of \(m\) is port \(i\): joining the other two star ports creates exactly that extra loop. Degree three has probability \(1/4\). Indeed, toggling a fixed defect path bijects to the even ensemble, whose four local even patterns at the star are equiprobable. The port-independent terms cancel in the weighted sum, and the factor four in (45) cancels this probability. Thus \(F_m^\eta\) is the mean partner label conditional on degree three, proving the triangular convex-combination assertion.

For a fan probe \(s_i u\), split \(s_i\) into two exterior vertices, fanned respectively to the two successive subarcs, both containing \(u\), and add the triangle formed by these two vertices and \(u\). Remove \(t_*\) and prescribe alternating colors, in one fixed convention, on the four exterior vertices now present. Leave all original disk spins free. The four boundary edges joining consecutive exterior vertices are switches. Their disagreement strands have the preceding conditional pairing law.

Here is the parity correspondence explicitly. The two half-edge values at the old probe, with oddness there, have exactly one occupied half. They become the two spoke values entering the added triangle, which has even parity because its split exterior boundary edge is occupied. The other three occupied exterior boundary edges are the three forced switches at the old degree-three star. This gives a bijection from the conditional edge-pattern ensemble to parity-even disagreement data on all triangles of the augmented disk, with the four boundary switches fixed. Triangle parity gives cycle parity, so the data recover colors up to a global flip. Fixing the alternating exterior convention removes that flip, leaving exactly uniform independent colors on all original disk vertices.

The four switches have a noncrossing pairing. The split switch can therefore be paired only with port \(i\) or port \(i+1\), proving the side constraint. For \(i=0\) at \(u\approx b\), prescribe the exterior vertices along \(ab\) and \(cd\) to be black, and the other two to be white. Connecting the two black exterior vertices is equivalent to a black crossing in the original disk: any connection between the two black fans contains a path of original black vertices between their respective arcs, and conversely such a path joins the fans. Pairing the split switch with port \(1\) gives this connection along the neighboring black sites. The other pairing joins the opposite white pair, and the two possibilities are exclusive by planarity. This proves the stated coefficient identity. ◻

Proof. We use the ordinary-wall representation in (45). A nonisolated selected component has a rim using the exterior edge \(s_{i-1}s_i\), followed by a path of original disk sites between attachments to the two different boundary arcs of those fans. The third exterior vertex has the opposite color and cannot lie on this rim. The disk path can degenerate to the common endpoint of its two arcs. The component indicator tests the component side of the rim, including its rim edges: an open dual edge belongs to that component exactly when it is incident to an untouched triangle on the filled disk side. An isolated selected component contributes zero at the probes in use.

For two nearby bulk probes, different indicators force a selected-rim visit between them. Such a visit gives a monochromatic path of original disk sites from their neighborhood to a fixed positive distance, since the disk portion reaches the boundary. Near a boundary point off the three marks, compare a fan probe first with its attached disk vertex. Unless that vertex lies on the rim, following its incident fan edge to the vertex tests the same side; if it does lie on the rim, the required visit already occurs. Uniform closed-disk Jordan charts join nearby disk positions tending to the same boundary point by paths of vanishing diameter inside the closed approximating disk. Thus a difference again forces a nearby visit by the disk portion of the selected rim. Its two attachments lie on different marked arcs and cannot both approach this nonmark boundary point. One direction of the disk path therefore reaches a fixed positive distance among original disk sites. The exterior fan edges do not replace this arm.

More precisely, on a fixed compact set off the marks choose a positive distance \(\rho\) valid for these continuations. For probes in a small neighborhood of diameter \(s\), choose a deterministic enlarged neighborhood containing their short comparison paths on good geometry. Let \(q_{s,\rho}(\eta)\) be the conditional probability of a monochromatic arm of either color from that neighborhood to distance \(\rho\), under the full-plane independent site colors. Then, for every pair of admissible geometry-selected probes in that neighborhood, the same test gives \[ |F_m^\eta-F_{m'}^\eta| \le4q_{s,\rho}(\eta). \tag{46}\] There is no union bound over the chosen edges. The disk and chart control needed for this statement follows along every deterministic sequence of good marked Jordan approximations by closed Jordan convergence; mesh control transfers edges to their sites. Taking expectations and using the ordinary one-arm estimate gives an upper limit tending to zero as \(s\downarrow0\). The geometry errors vanish first at fixed \(s\).

Choose near-position representative edges at a countable dense set of interior points, and fan representatives at a countable dense set of each open boundary arc. These choices exist with vanishing location error on good geometry. Boundedness permits a diagonal subsequence of their annealed expectations. Estimate (46) extends the resulting function continuously and identifies the same limit for all geometry-selected probes converging to any such point. The triangle and side constraints pass to this limit by Proposition 48. All choices are made after the universal bulk subsequence, so the transport conclusions remain valid. ◻

Proof. For a smooth compactly supported bulk test \(f\), orthogonality against the closing star, with its reversed labels, gives the exact conditional identity \[\mathbb E_{\mathrm{sp}}[w_*U_{\lambda_*}(f)]=0.\] Indeed, at every bulk triangle the weighted sum \(\sum_j\omega^jF_{m_j(t)}^\eta\) is zero by (2). The selected nonisolated rim reaches the exterior and therefore leaves every sufficiently small fixed test collar. An isolated selected pair tags only exterior triangles. Apply (17), partitioning \(f\) into local disk tests when needed. Its residual at the selected label tends to zero in probability and in mean: its absolute value is bounded by the full residual \(\ell^2\) norm, which is bounded in \(L^2\) by (5). Since \[U=(1-a)V-(Q-aV),\qquad a\ne1,\] we obtain \(\mathbb E[w_*V_{\lambda_*}(f)]\to0\). On high-probability geometry events the discrete disk agrees with the full triangulation throughout the test collars; the conditional identities hold on each such graph. The energy moments remove the omitted geometry.

Using (45) and \(\sum_jd_{tj}=0\), we have therefore proved \[ \mathbb E\left[\sum_t f(z_t)\sum_jd_{tj}F_{m_j(t)}^\eta\right] \longrightarrow0. \tag{47}\] The edge coefficients and the conditional observable both depend on the point geometry, so their product cannot be replaced by a product of expectations. We instead express the edge coefficients as a random measure and prove that this measure has a deterministic limit.

Group the two contributions of each shared edge \(e=t\cap t'\). If \(d_{t,e}\) is its clockwise side vector for \(t\), set \[\mu_\varepsilon=\sum_e d_{t,e}\bigl(f(z_t)-f(z_{t'})\bigr)\delta_{m_e},\] where now \(m_e\) denotes the physical midpoint of the side. This weight is independent of which adjacent triangle is called \(t\). All edges involved are buffered inside the disk. Let \(T_\varepsilon(B)=|\mu_\varepsilon|(B)\). Because \(f\) is smooth, \[|d_{t,e}|\,|f(z_t)-f(z_{t'})| \le C_f\bigl(\mathop{\mathrm{diam}}(t)^2+\mathop{\mathrm{diam}}(t')^2\bigr).\] Consequently, for each sufficiently small fixed square \(B\) of side \(s\) in a compact bulk neighborhood, \[ \limsup_{\varepsilon\to0}\|T_\varepsilon(B)\|_2\le C_fs^2. \tag{48}\] The triangles incident to an edge assigned to \(B\) lie in a fixed enlargement on fine mesh; the exceptional contribution is controlled by their higher moments.

For a deterministic smooth function \(G\), midpoint integration on clockwise triangle boundaries gives \[\mu_\varepsilon(G) \longrightarrow -\int_D f\,\mathrm dG\wedge\mathrm dz =\int_DG\,\mathrm df\wedge\mathrm dz \quad\text{in }L^1\text{ of the geometry}.\] To check the approximation, the midpoint error on a triangle is \(O_{f,G}(\mathop{\mathrm{diam}}(t)^3)\). The sum of these errors tends to zero in \(L^1\), using the maximum mesh diameter times the bounded diameter-square sum. Stokes’ theorem, the clockwise orientation, and area approximation give the displayed sign. Replacing \(f(z_t)\) by \(f\) in each triangle has the same vanishing cubic-error bound. The resulting limiting complex measure is \[\mu=\mathrm df\wedge\mathrm dz,\] which has a bounded compactly supported density.

We need the stronger conclusion for fixed-square indicators. If \(B\) is such a square, choose a smooth \(g_\tau\) with \(0\le g_\tau\le1\) agreeing with \(\mathbf 1_B\) outside a boundary strip \(U_\tau\) of width \(O(\tau)\). Then \[|\mu_\varepsilon(B)-\mu(B)| \le |\mu_\varepsilon(g_\tau)-\mu(g_\tau)| +T_\varepsilon(U_\tau)+|\mu|(U_\tau).\] Cover \(U_\tau\) by \(O(\operatorname{perimeter}(B)/\tau+1)\) squares of side \(O(\tau)\). The first-moment consequence of (48), after the microscopic limit, makes the expected variation on this strip \(O_f(\operatorname{perimeter}(B)\tau+ \tau^2)\). The limiting variation tends to zero there as well. First take \(\varepsilon\to0\), then \(\tau\downarrow0\), obtaining \[ W_\varepsilon(B):=\mu_\varepsilon(B)\longrightarrow w(B):=\mu(B)\quad\text{in }L^1. \tag{49}\]

Now fix a deterministic tiling of the test neighborhood by squares of side \(s\), and choose a conditional representative \(F_B^\eta\) on an edge near the center of each square. The same ordinary arm test in (46) controls every edge assigned to \(B\): \[|F_e^\eta-F_B^\eta|\le4q_B(\eta),\qquad \limsup_{\varepsilon\to0}\mathbb Eq_B\le C(s/\rho)^c, \qquad 0\le q_B\le1.\] Here \(\rho>0\) is fixed by the distance of the support neighborhood to the boundary. Thus \(\mathbb Eq_B^2\le\mathbb Eq_B\), and without an independence assumption Cauchy–Schwarz gives \[\limsup_{\varepsilon\to0}\mathbb E[T_\varepsilon(B)q_B] \le C_{f,\rho}s^{2+c/2}.\] Summing over \(O(s^{-2})\) squares bounds the mean replacement error by \(C_{f,\rho}s^{c/2}\). For fixed \(s\), omitted geometry contributes a vanishing mean by boundedness of the observable and the variation moments.

At this fixed tiling, (49) can be multiplied by the correlated representative because \(|F_B^\eta|\le1\): \[\left|\mathbb E[F_B^\eta W_\varepsilon(B)]-w(B)\mathbb EF_B^\eta\right| \le\mathbb E|W_\varepsilon(B)-w(B)|\longrightarrow0.\] Lemma 49 identifies \(\mathbb EF_B^\eta\to F(\operatorname{center}(B))\). Hence (47), followed first by \(\varepsilon\to0\) at fixed tiling and then by \(s\downarrow0\), yields \[ \int_D F\,\mathrm df\wedge\mathrm dz=0 \qquad(f\in C_c^\infty(D)). \tag{50}\] Since \(\mathrm df\wedge\mathrm dz=2\mathrm i\,\partial_{\bar z}f\,\mathrm d^2z\), this is the distributional Cauchy–Riemann equation. Local mollification and continuity of \(F\) show that \(F\) is holomorphic. ◻

Proof. On each open source arc, Schwarz reflection across the unit circle extends \(G\) holomorphically across the boundary. Near each point of that arc, \(G\) is nonzero, so \(u=\log|G|\) is harmonic, negative on the disk side, and zero on the circle. The Hopf boundary lemma and the Cauchy–Riemann equations give \[\frac{\mathrm d}{\mathrm dt}\arg G(e^{\mathrm it}) =\left.\frac{\partial}{\partial r}\log|G(re^{\mathrm it})|\right|_{r=1} >0.\] The proper target arc has a bounded real argument lift. Thus the boundary argument on each open source arc is strictly increasing and has finite one-sided limits at its endpoints.

We prove that the two limits at a source mark agree, without assuming interior continuity there. Rotate that mark to \(1\) and denote the limits as \(t\uparrow0\) and \(t\downarrow0\) by \(L_-\) and \(L_+\), respectively. Let \(g(e^{\mathrm it})\) be the known boundary values, with arbitrary values assigned at the three marks. For fixed \(z\in\mathbb D\), Cauchy’s formula on a circle of radius \(R>|z|\) is \[G(z)=\frac1{2\pi}\int_{-\pi}^{\pi} \frac{G(Re^{\mathrm it})}{1-(z/R)e^{-\mathrm it}}\,\mathrm dt.\] As \(R\uparrow1\), the numerator converges away from the three marks and is uniformly bounded, while the denominator stays uniformly away from zero for this fixed \(z\). Dominated convergence therefore gives \[G(r)=\frac1{2\pi}\int_{-\pi}^{\pi} \frac{g(e^{\mathrm it})}{1-re^{-\mathrm it}}\,\mathrm dt, \qquad 0<r<1.\] For a fixed small \(\alpha>0\), write \(\ell_r=\log(1/(1-r))\). The logarithmic primitive of the kernel gives \[\int_0^\alpha\frac{\mathrm dt}{1-re^{-\mathrm it}} =\frac{\ell_r}{\mathrm i}+O_\alpha(1),\qquad \int_{-\alpha}^0\frac{\mathrm dt}{1-re^{-\mathrm it}} =-\frac{\ell_r}{\mathrm i}+O_\alpha(1).\] For example, a continuous branch of \(\mathrm i^{-1}\log(e^{\mathrm it}-r)\) is a primitive on either interval. Moreover, the kernel is bounded in modulus by \(C/(1-r+|t|)\) there. Replacing \(g\) by its corresponding one-sided limit therefore gives an \(o(\ell_r)\) error: for every \(\delta>0\) choose a smaller fixed interval where \(|g-L_\pm|<\delta\), use the logarithmic kernel bound on that interval, and note that the complementary fixed interval has a bounded integral as \(r\uparrow1\). Then let \(\delta\) tend to zero. The part away from the mark is also bounded. Consequently, \[ G(r)=\frac{L_+-L_-}{2\pi\mathrm i}\log\frac1{1-r} +o\!\left(\log\frac1{1-r}\right). \tag{51}\] Boundedness of \(G\) forces \(L_+=L_-\). The common value belongs to the two adjacent target arcs, whose intersection is their prescribed common endpoint; hence \(G\) has the required continuous circle trace \(g\).

To obtain continuity for all interior approaches, apply the same radial-circle limiting argument to the Poisson formula for the bounded harmonic real and imaginary parts of \(G\). They are the Poisson extensions of the continuous circle trace \(g\). The Poisson extension of continuous circle data is continuous on the closed disk. In particular, this proves full continuity at the marks, not merely radial or nontangential convergence.

On each open arc the boundary argument increases strictly between the two prescribed consecutive endpoints. The boundary curve therefore traverses the circle once in positive order. For any \(w\in\mathbb D\), continuity on the closed disk gives uniform convergence of \(G(re^{\mathrm it})-w\) to \(g(e^{\mathrm it})-w\) as \(r\uparrow1\). The winding number is one for all sufficiently large \(r\). The argument principle gives exactly one zero of \(G-w\) in \(|z|<r\), counted with multiplicity. Letting \(r\uparrow1\) proves that every \(w\) has exactly one preimage, of multiplicity one. Thus \(G\) is a conformal bijection. ◻

Proof. Choose an orientation-preserving plane Jordan–Schoenflies homeomorphism \(\Phi\) taking \(\overline D\) to \([0,1]^2\), with the target arcs \(ab\) and \(cd\) going to the left and right sides. For fixed \(0<\delta<1/8\), define in these coordinates the harder and easier rectangles \[R^-_\delta=[-\delta,1+\delta]\times[\delta,1-\delta], \qquad R^+_\delta=[\delta,1-\delta]\times[-\delta,1+\delta].\] The first rectangle lengthens a horizontal crossing while restricting its height; the second shortens the crossing and allows more height. Figure 7 displays these two comparisons. Approximate their inverse images under \(\Phi\) by deterministic polygonal marked quadrilaterals, in their boundary order and much more accurately than the buffer margins. For each of the two polygons use the discrete disks from Lemma 47. Their original vertices inherit the full-plane free colors, so the crossing results already proved apply to them.

Keep all these choices fixed while taking \(\varepsilon\to0\). Work on the high-probability event of the needed disk approximation and mesh control in a fixed compact neighborhood. Both \(\Phi\) and \(\Phi^{-1}\) are uniformly continuous on the relevant compact sets. Thus any fixed coordinate margin is preserved by taking the physical polygonal and mesh errors sufficiently small. No Lipschitz estimate for either chart is required.