Proof. The Ising weight is log-supermodular: each ferromagnetic pair interaction has nonnegative mixed difference and each field has zero mixed difference. We recall why summing out spins preserves this property. It suffices to check a two-coordinate square after summing one other binary coordinate. Denote the eight unsummed weights by \(a,b,c,d,e,f,g,h\), where \((a,b,c,d)\) and \((e,f,g,h)\) are the two square slices, in order \((--,+-,-+,++)\). Log-supermodularity gives \(ad\ge bc\), \(eh\ge fg\), \(ah\ge bg\), and \(ah\ge cf\). Moreover \((ah)(ed)=(ad)(eh)\ge(bg)(cf)\). For positive numbers \(X,Y,U,V\), the conditions \(X\ge\max(U,V)\) and \(XY\ge UV\) imply \(X+Y\ge U+V\): assume \(U\ge V\) and use \(X+UV/X\ge U+V\) for \(X\ge U\). Applying this with \((X,Y,U,V)=(ah,ed,bg,cf)\) gives \(ah+ed\ge bg+cf\). Hence \((a+e)(d+h)\ge(b+f)(c+g)\). Nonnegative adjacent-coordinate square differences are equivalent to log-supermodularity on a Boolean cube, by summing those differences along coordinate paths. Iterating the one-coordinate elimination proves that the marginal \(p\) is log-supermodular.

For ordered \(u\le v\) and \(s\le t\), put \[\Delta(u,v;s,t)=f(u,s)+f(v,t)-f(u,t)-f(v,s)\ge0.\] If \(D=f(a,b)+f(a',b')-f(a,b')-f(a',b)\), then the exact identity \[\begin{split} L-D={}&\Delta(-,a';-,b)+\Delta(a,+;b',+)\\ &+\Delta(a',+;-,b')+\Delta(-,a;b,+) \end{split}\] proves \(D\le L\). Interchanging \(b,b'\) proves \(-D\le L\). No order between \(a,a'\) or between \(b,b'\) is required. ◻

Proof. A prescribed common spin \(\tau_v\), \(v\in F\), contributes \(K_{uv}\tau_v\) to the field at each surviving neighbor \(u\). After these fields have been included, remove the vertices of \(F\). Apply Lemma 3 to the joint marginal on \(I,J\). Restricting both terminal sets to constant configurations is exactly their separate contraction. Its two-spin marginal has unnormalized weights \[w(s,t)=\exp(C+Kst+As+Bt),\qquad s,t\in\{-1,+1\}, \qquad K=L/4\ge0.\] The constants \(A,B\) include all effective fields from the other vertices. Add external fields \(-A,-B\) at the two contracted vertices. This changes their marginal to weights proportional to \(e^{Kst}\), without changing \(K\). Both means now vanish and the covariance is \(\tanh K\). The Ding–Song–Sun inequality applies to this finite ferromagnetic graph with these arbitrary mixed-sign fields (Ding et al. 2023, Corollary 1.3). It gives \[\tanh K\le\langle\sigma_I\sigma_J\rangle_0=q_0.\] The last equality is the zero-field Edwards–Sokal identity (Edwards and Sokal 1988). Internal edges of a contracted terminal contribute constants and parallel edges have nonnegative summed coupling, so all hypotheses of that inequality are preserved. Thus \(L\le4\operatorname{arctanh}q_0\).

For fixed \(b,b'\), Lemma 3 bounds the oscillation in \(a\) of the logarithm of the conditional density ratio by \(L\). The ratio has mean one under its denominator law; its minimum is therefore at most one and its maximum at least one. The ratio lies throughout \([e^{-L},e^L]\), as claimed. Averaging a pointwise inequality proves the mixture assertion, first for one law and then for the other. ◻

Proof. The rectangle assertion is the critical FK-Ising RSW theorem (Duminil-Copin et al. 2011, arXiv version 1, Theorem 1), with planar duality for dual crossings. The domain Markov property turns an exterior bond configuration into a boundary wiring. Restore the deleted vertices of the test rectangle, restore their edges, and wire its entire outer boundary. The resulting measure stochastically dominates the conditional original bond law, so a decreasing dual event is at least as likely in the original law. The domination follows, for example, from the one-edge conditional probabilities \(p_c\) and \(p_c/(p_c+2(1-p_c))\), according as the endpoints are or are not already connected: adding edges or wiring increases these probabilities. Restoring isolated vertices first and then their edges has the same effect. The hypothesis about terminals ensures that no identification in the interior of the restored rectangle is being discarded.

For the last assertion, order the test regions. Conditioning on all edges outside the next region includes the results of all earlier tests. Applying the uniform bound and then averaging proves the product lower bound by induction. The corresponding product upper bound for simultaneous failure of tests follows in the same way. If a bounded lattice correction makes a side have bounded length, finitely many prescribed closed or open edges, using the same one-edge probabilities, supplies a uniform replacement test. ◻

Proof. An annulus of fixed radius ratio separating the two terminals contains a dual circuit with probability at least \(c_1(c)>0\). Cover a middle subannulus by a fixed number of overlapping rectangles, require dual crossings in both directions of their overlap squares, and use Lemma 6. The crossings join to an actual dual circuit, which separates the terminals even after their separate internal wiring. Extra common pins are deleted in the FK comparator; restoring them only increases connectivity. Thus \(q_0\le1-c_1(c)\), giving the first assertion by Lemma 4.

Between \(B_{2w}\) and \(B_{R/2}\), choose annuli with radius ratio, say, four and disjoint test edges. There are at least \(c_2\log(R/w)-c_3\) such annuli. A connection must avoid the dual circuit in each, so Lemma 6 gives \(q_0\le C_1(w/R)^{\delta_1}\). The fixed annulus between the terminals also gives \(q_0\le1-c_4\). On \([0,1-c_4]\), \(\operatorname{arctanh}q\le C_2q\). This proves the logarithmic bound. The total variation and expectation bounds follow by integrating the density difference.

For different domains, condition on the exterior neighbor spins of a square contained in the common chart. Each local law is a mixture of the same finite-volume conditional laws, so the mixture assertion applies. Finite-volume approximation gives the plane and cylinder statements; the bounds are uniform in the approximating volume. Finally, applying the same Markov kernel preserves pointwise domination of measures. ◻

Proof. First fix a common finite local conditional graph. In its zero-field FK comparator restore any vertices and edges missing from the full square test rectangles. Restoration increases connectivity. The decreasing dual tests in Lemma 6 therefore give the same upper bound on \(q_0\) as before. Prescribed physical-boundary vertices are among the common pins: delete them in the comparator before this restoration. No identification is added in a test rectangle’s interior. Lemma 4 proves the asserted pointwise bounds for this conditional graph.

For different volumes or continuations, condition at the edge of the common chart. Both laws are mixtures of the same local kernels, with possibly different mixing measures. Apply the mixture assertion of Lemma 4. The restoration argument bounds the auxiliary connection probability; the common conditional kernel is what permits the comparison between volumes.

For a cover of the observed set, reveal its patches in a fixed order and factor each configuration probability by the chain rule. In the comparison for the next patch, previously revealed patches are common pins. They can lie anywhere off the two terminals. Each conditional ratio has its stated bound, and multiplication gives the product constant. For a cover of the source, telescope the changes in its patches; all other source patches are common pins. Apply Remark 5 to place the small patch in the inner terminal. Mixtures over source assignments and common randomizations preserve the resulting bounds. ◻

Proof. For an edge with endpoints in the same one of \(G\cup U\) and its complement, the two replica energies add to twice the energy of their common orientation. For an edge between these sets they cancel. This proves (8) and the conditional independence. Edges with both endpoints pinned contribute constants independent of \(G\), so cause no exception to the factorization.

Every FK cluster in \(G\cup U\) avoiding \(U\) has an independent fair orientation. On each such cluster replace the pair of opposite signs by two copies of the first replica’s sign. The first replica is unchanged; the second replica’s collection of independent fair cluster signs has been negated, so its conditional marginal is unchanged as well. Clusters meeting \(U\) retain the pair \((+,-)\), and agreement sites retain equal signs. This proves both the ordering and (9). The auxiliary set \(G\) need not equal the disagreement set after this operation: the latter is precisely the union of clusters meeting \(U\). ◻

Proof. Use the FK representation with edge parameters \(p_e=1-e^{-2J_e}\). In descriptions of sign compatibility, an actual cluster means a component formed by open edges of the original graph: virtual wires between pins do not count as open paths. Write \(k_0(\omega)\) for the number of actual clusters avoiding \(P\). Apart from a constant, the all-plus bond weight is \[ \prod_{e\in E(P\cup H)}p_e^{\omega_e}(1-p_e)^{1-\omega_e} 2^{k_0(\omega)}. \tag{12}\] This is the usual FK law with all pins wired. Conditional on the bonds, clusters touching a pin have sign plus, and the other clusters have independent fair signs.

Let \(\mathcal F_Q\) be the event that each actual cluster avoiding \(P\) contains an even number of vertices of \(Q\). Conditional sign averaging gives \[ \mathbb E_{P,+,H}\prod_{x\in Q}\sigma_x =\phi_{P,+,H}(\mathcal F_Q). \tag{13}\] The event \(\mathcal F_Q\) is increasing. Indeed, merging two unpinned clusters whose parities are even preserves their even parity, whereas merging any cluster with a pinned cluster removes its parity restriction. Adding an initially isolated vertex outside \(Q\) adds a cluster of even parity zero. Thus FK monotonicity under edge addition, increases of edge parameters, and increases of the boundary wiring proves the first assertion. Pinning an additional spin plus is covered by the same argument: its cluster ceases to have a parity restriction and the FK wiring increases. Positive fields, when needed, may equivalently be represented by edges to one additional plus vertex.

For mixed signs, put \[\mathcal D_\tau= \{\text{no actual cluster contains two pins with opposite signs}\}.\] The mixed-sign bond weight is exactly (12) multiplied by \(\mathbf1_{\mathcal D_\tau}\). In particular, its law is \(\phi_{P,+,H}(\,\cdot\mid\mathcal D_\tau)\). This identity also handles edges between opposite pins: such an edge must be closed. The event \(\mathcal D_\tau\) is decreasing and has positive probability. Conditional sign averaging in the mixed law has absolute value at most \(\mathbf1_{\mathcal F_Q}\). Positive association of the wired FK law therefore gives \[\left|\mathbb E_{P,\tau,H}\prod_{x\in Q}\sigma_x\right| \leq \phi_{P,+,H}(\mathcal F_Q\mid\mathcal D_\tau) \leq \phi_{P,+,H}(\mathcal F_Q),\] which proves (10).

For the last assertion, summing the spin at a newly added vertex gives the exact identity \[ g_{P,\tau,H}(v):= \frac{Z_{P,\tau}(H\cup\{v\})}{Z_{P,\tau}(H)} =2\mathbb E_{P,\tau,H} \cosh\!\left(\sum_{u\in P\cup H}J_{vu}\sigma_u\right). \tag{14}\] Zero couplings may be omitted from the sum. If \(N\) is its neighbor set, the integrand has the expansion \[ \cosh\!\left(\sum_{u\in N}J_{vu}\sigma_u\right) =\prod_{u\in N}\cosh J_{vu} \sum_{\substack{Q\subseteq N\\ |Q|\ {\rm even}}} \prod_{u\in Q}\tanh J_{vu}\prod_{u\in Q}\sigma_u. \tag{15}\] All coefficients are nonnegative. Under plus pins, enlarging \(H\) increases the old product expectations, adds nonnegative terms, and increases the prefactor. Hence \(g_{P,+,H}(v)\) increases with \(H\). Increasing site gains are equivalent to log-supermodularity: add the vertices of \(H_1\setminus H_2\) one at a time, comparing their gains above \(H_1\cap H_2\) and above \(H_2\), to obtain \[Z_{P,+}(H_1\cup H_2)Z_{P,+}(H_1\cap H_2) \geq Z_{P,+}(H_1)Z_{P,+}(H_2).\] Finally, apply (10) term by term in (15) to prove (11). ◻

Proof. By Lemma 15, both factors in (8) with plus pins are log-supermodular functions of \(G\). Complementation exchanges unions and intersections and hence preserves this property. Their product is log-supermodular, so the reference set law is positively associated. Conditional on \(G\), the FK law is positively associated and increases stochastically with \(G\). Indeed, one may first add isolated vertices and then increase their edge parameters from zero. Thus conditional expectations of increasing functions of \((G,\omega)\) increase with \(G\). The decomposition of a covariance into its conditional covariance and the covariance of its conditional expectations proves joint positive association.

The mixed law has the density \(r(G)\) displayed above. By (11), the quotient \(Z_{A,\tau}(H)/Z_{A,+}(H)\) decreases as \(H\) grows. Therefore \(r\) increases with \(G\). The conditional law on the left side of (16) has density proportional to \(r\mathbf1_{\mathcal E}\) against the associated reference. That density is increasing, and association gives \[\frac{\mathbb E_+[f r\mathbf1_{\mathcal E}]} {\mathbb E_+[r\mathbf1_{\mathcal E}]} \geq\mathbb E_+[f].\] This argument requires neither association of the mixed law nor preservation of a lattice condition after increasing-event conditioning. ◻

Proof. By Lemma 16, it suffices to compare the plus-agreement reference with \(\mathbb P_Q^{++}\). Condition its exterior auxiliary set \(G\setminus Q\). Fix \(v\in Q\) and the membership of all other sites of \(Q\). Write \(D\) for the full disagreement vertices excluding \(v\), including the pins \(U\), and \(H\) for the full agreement vertices excluding \(v\), including the pins \(A\). The conditional odds for membership of \(v\) are the quotient of two site gains: \[ \frac{\mu_+(v\in G\mid G\setminus\{v\})} {\mu_+(v\notin G\mid G\setminus\{v\})} =\frac{g_{U,+,D\setminus U}(v)} {g_{A,+,H\setminus A}(v)}. \tag{18}\] Here \(g\) is the gain in (14).

The numerator is at least the gain on the induced disagreement set in \(Q\): adding the exterior disagreement vertices, their edges, and their plus pins only increases it. For the denominator, first pin every exterior agreement vertex plus. The parity argument in Lemma 15 and the nonnegative expansion (15) show that this increases the gain. Then restore every missing exterior neighbor of \(Q\) as a plus pin, again increasing the gain. The resulting denominator is exactly the agreement-site gain for the box-alone model: partition factors involving only the now-pinned exterior cancel in the site ratio. Thus the odds in (18) are at least the corresponding odds of \(\mathbb P_Q^{++}\) at every fixed interior set configuration.

Equivalently, the density of the reference interior-set law relative to the box-alone set law is increasing: the ratio increases under every single-site addition. The box-alone set law is associated by Lemma 16. Tilting it by this increasing density therefore proves stochastic domination of the sets.

Given a full auxiliary set, conditioning the exterior FK bonds induces some wiring on its part in \(Q\). This wiring dominates the free boundary condition of the box-alone disagreement-bond system; its orientation has no disagreement pins in \(Q\) or on its exterior. The free FK law on an induced set also grows with that set. First perform this bond comparison and then the set comparison just obtained. Averaging over the exterior set proves the claim for every increasing \(f\). ◻

Proof. In the original independent pair put \(Y_x=(\eta_x-\xi_x)/2\). There are no disagreement pins. Conditional cluster signs give \[ \mathbb P_Q^{++}(x\longleftrightarrow y) =\mathbb E[Y_xY_y] =\tfrac12\operatorname{Cov}_Q^+(\sigma_x,\sigma_y). \tag{20}\] More generally, \[ \mathbb P_Q^{++}(x\longleftrightarrow y, x'\longleftrightarrow y') \leq\mathbb E[Y_xY_yY_{x'}Y_{y'}]. \tag{21}\] Indeed, if both pairs connect, all auxiliary clusters contain an even number of these four indexed factors. Conditional sign averaging of the product is the indicator of this even-parity event, with the additional requirement that all factors lie in the auxiliary set. This explanation also proves (21) when indices repeat.

Set \[X=\sum_{x\in L_1}\sum_{y\in L_2} \mathbf1_{\{x\longleftrightarrow y\}}.\] The bulk connected-correlation lower bound in Lemma 9 gives \(\mathbb EX\geq c_1w^{7/4}\). By (21), Cauchy–Schwarz, and the fourth line moments in Lemma 11, \[\mathbb EX^2 \leq\mathbb E\left[ \left(\sum_{L_1}Y_x\right)^2 \left(\sum_{L_2}Y_y\right)^2\right] \leq C_1 w^{7/2}.\] The moments for \(Y\) follow from those for either replica by the triangle inequality in \(L^4\). Paley–Zygmund therefore gives constants \(c_2,p_2>0\) such that \(\mathbb P(X\geq c_2w^{7/4})\geq p_2\).

Let \(L\) be the largest number of sites of \(L_2\) in one auxiliary cluster. Then \[L^2\leq\sum_{\mathcal C}|\mathcal C\cap L_2|^2, \qquad \mathbb E\sum_{\mathcal C}|\mathcal C\cap L_2|^2 =\mathbb E\left(\sum_{L_2}Y_x\right)^2 \leq C_2w^{7/4}.\] Choose \(K\) so large that \(C_2/K^2<p_2/2\). With probability at least \(p_2/2\), both \(X\geq c_2w^{7/4}\) and \(L\leq Kw^{7/8}\) hold. Every site of \(L_1\) reaching \(L_2\) contributes at most \(L\) to \(X\). On that intersection at least \((c_2/K)w^{7/8}\) sites reach \(L_2\), proving (19). Although the event involving \(L\) was used to establish its probability, the event in (19) itself is increasing. ◻

Proof. Consider the four strips \[[-4w,4w]\times[2w,4w],\quad [-4w,4w]\times[-4w,-2w],\quad [2w,4w]\times[-4w,4w],\quad [-4w,-2w]\times[-4w,4w].\] For the top strip, extend the long coordinate to \([-5w,5w]\) and put agreeing plus boundary conditions on this extended rectangle. Apply Lemma 18 to vertical segments at \(x=-4w\) and \(x=4w\), each with its \(y\) coordinate in \([5w/2,7w/2]\). The segments are in its bulk. Their connection stays inside the extended rectangle, since the auxiliary disagreement set cannot contain its agreeing boundary pins. A subpath of this connection gives a left-to-right crossing of the unextended top strip. Proposition 17 transfers its positive lower bound to the doubled model on \(B_{16w}\). The same construction applies to the other three strips.

The four crossing events are increasing, so their intersection has probability at least the product of their four lower bounds. On this intersection, adjacent crossings intersect inside their corner overlap: one crosses that overlap in one coordinate and the other in the perpendicular coordinate. The union contains a circuit surrounding the central square. One way to see the last assertion is to join successive intersection points along the four crossings, erase portions lying beyond the adjacent corner overlaps, and obtain a closed walk in the annulus with winding number one about the origin. Removing closed subwalks decomposes it into simple circuits; at least one retains nonzero winding number. All its edges lie in the four strips. This proves the claim. ◻

Proof. Translate the picture so that \(z=0\). Choose a translate \(L'\) of \(L\) by \(6w\) in a perpendicular coordinate. Both segments lie in fixed bulk subsets of \(B_{16w}\), while \(L'\) lies outside \(B_{4w}\). Under its doubled plus-boundary law, Lemma 18 gives positive probability that at least \(c w^{7/8}\) sites of \(L\) reach \(L'\) by paths using bonds internal to \(B_{16w}\). Intersect this increasing local event with the increasing circuit event of Lemma 19. Association makes the probability of their intersection at least a positive constant \(p\).

If \(\mathbb P(U\longleftrightarrow b_0)=0\), the claim is immediate. Otherwise condition on that increasing event and use Proposition 17. The circuit and the many-sites event still occur jointly with conditional probability at least \(p\). Every path from \(U\), which is outside \(B_{16w}\), to \(b_0\subseteq B_w\) meets the surrounding circuit. Every counted path from \(L\subseteq B_w\) to \(L'\) also meets it. By planarity all the counted sites are therefore connected to \(U\). The conditional expected number of such sites is at least \(pc w^{7/8}\). Multiplying by the probability of the conditioning event proves (22). ◻

Proof of Proposition 13. Use the monotone coupling of Lemma 14. Let \(\eta_J\geq\xi_J\) be its cut configurations. Put \(k(j)=\mathbb E[h\mid\sigma_J=j]\), where the conditional expectation uses the common target-side continuation. The spatial Markov property and the separating property imply that this function does not depend on which source configuration was prescribed. Consequently \[\Delta h=\mathbb E[k(\eta_J)-k(\xi_J)].\]

In this telescope, the other cut spins are common pins for the pointwise comparison. The local mass estimate is applied to the original coupling, whose counting charts remain pin-free.

Interpolate between the two cut configurations one patch at a time, using any deterministic ordering of their partition. Intermediate configurations are legitimate cut configurations; spins at all other patches are fixed for each comparison. Corollary [fd:patch-ratio] bounds the oscillation of \(k\) in one patch by \(C\|h\|_\infty(w/s)^\delta\). It applies with these arbitrary other cut spins as common pins. If the two sampled cut configurations agree on the patch, the corresponding summand is zero. By (9), disagreement occurs somewhere in \(b_i\) exactly when \(U\longleftrightarrow b_i\). Therefore \[ |\Delta h|\leq C\|h\|_\infty(w/s)^\delta \sum_i\mathbb P(U\longleftrightarrow b_i). \tag{23}\]

Proposition 20, applied with scale \(\ell_i\asymp w\), and the overlap bound on the counting segments give \[\sum_i\mathbb P(U\longleftrightarrow b_i) \leq C w^{-7/8}\sum_i\sum_{x\in L_i} \mathbb P(U\longleftrightarrow x) \leq C K w^{-7/8}\sum_{x\in J} \mathbb P(U\longleftrightarrow x).\] The last sum is \(\tfrac12s^{7/8}\Delta M_J\) by (9). Substitute it in (23) to obtain (6). All local constructions use the same fixed aspect ratios and overlap constants, independently of \(w/s\). ◻

Proof. For an ordered random pair of sources, apply (6) to each pair and integrate. The magnetic differences are nonnegative. The triangle inequality bounds the absolute value of the integrated target difference by the integral of its absolute value, proving the first assertion.

For the signed assertion put \(m=\mathbb E_\mu q_+=\mathbb E_\mu q_-\). If \(m=0\) there is nothing to prove. Otherwise let \(\mu_\pm=m^{-1}q_\pm\mu_S\) be probability laws on source configurations. Order positivity says \(\mu_+(F)\geq\mu_-(F)\) for every increasing set \(F\). These laws admit a coupling \((s_+,s_-)\) with \(s_+\geq s_-\). For completeness, this finite transport fact follows by sending mass \(\mu_-(x)\) from each lower configuration \(x\) to upper configurations \(y\geq x\) with capacities \(\mu_+(y)\). For a set \(B\) of lower configurations, the required cut inequality is \[\mu_-(B)\leq\mu_-(\mathord\uparrow B) \leq\mu_+(\mathord\uparrow B),\] so the finite max-flow/min-cut criterion transports all the mass. Apply the random-source assertion to this coupling and multiply by \(m\). The Gibbs continuation changes the resulting difference of source integrals into \(\mathbb E_\mu[qh]\) and \(\mathbb E_\mu[qM_J]\) respectively, which proves (24). ◻

Proof. Restrict an ordered coupling to \(S_0\) and then use the common Gibbs continuation. Corollary 22 in its random-source form applies. Taking \(h\) to be an arbitrary target-event indicator proves (25). All differing pins must be on the source side of \(S_0\); the common-continuation hypothesis explicitly excludes a changed pin in the target component. ◻

Proof. There are finitely many starting edges and finitely many triangles. At a step, the next triangle is determined by the previously exposed zero segment and the fixed diagonal convention. Its spins determine the continuation. Hence the event that a specified reveal transcript occurs depends only on the spins recorded in that transcript. In the finite Gibbs density, conditioning on such a transcript fixes precisely those spins and leaves the original interaction weights on all remaining variables. This proves the stopping assertion directly, including the case of a random examined set.

The trace stops at the second contour and examines only adjacent triangles, which proves the location assertion. Although the initial reveal contains an entire side of the band, the algorithm deciding a strand within the band uses only the spins adjacent to the starting contour and in the band itself. The target-facing datum is determined by these traces and their local signs. No spins in the rest of the initially revealed side are needed to compute it. ◻

Proof. We give the deterministic continuity argument on the probability-one event of simple, disjoint loops and local finiteness at every positive diameter. For a continuous real function on a compact parameter interval, its local maximum and minimum values form a countable set: each is the maximum or minimum on an interval with rational endpoints contained in a witnessing neighborhood. Apply this to the square radius along each loop. There are countably many loops, so the union of exceptional levels is countable for each collection. Fubini’s theorem permits a deterministic full-measure set of radius choices for the random collection. At the selected levels there is no one-sided contact or constant interval.

Only finitely many excursions have diameter greater than a prescribed \(a>0\). Indeed, only finitely many loops have that diameter; on each, uniform continuity of a parametrization gives a positive minimum parameter length for such an excursion. Their parameter intervals are disjoint. A stopped strand has diameter at least the positive distance between the contours, so there are finitely many stopped strands. Their tips are distinct. The two stopped ends of the same deep excursion could have the same tip only if that tip were a one-sided contact at the second level, which was excluded.

Distinct excursion endpoints on the starting contour are also distinct. Endpoints on different loops cannot coincide by disjointness. A shared endpoint for two excursions of one simple loop would give a one-sided level contact. Excursion interiors miss the starting contour, and simplicity and disjointness exclude all other intersections.

Each visible returning chord deletes the open interval of the starting contour which is cut off from the target component. Visible deletion intervals have disjoint interiors: in a pair of nested deletions the inner chord is screened and is not visible. A small chord deletes a small interval. To see this, join its endpoints by the short arc of the starting square. As the chord diameter tends to zero, uniform continuity of the inverse parametrization of that square makes this arc small. The resulting small Jordan pocket cannot contain the target beyond the second contour, which is separated by a fixed positive distance. The erased interval must therefore be this short arc, not the complementary long arc. This argument works also on the exterior side; equivalently, compactify the exterior by a fixed inversion away from the band.

Parametrize the starting contour and replace every deleted interval by a parametrization of its chord with the same endpoints. Retain the original parametrization elsewhere. At an accumulation of replacement intervals, the interval diameters and chord diameters tend to zero, by the preceding finiteness and small-pocket facts. The resulting map is consequently continuous. It is injective: chord interiors miss the starting contour and one another, their endpoints are distinct, and retained points of the original contour remain distinct. A continuous injection of the compact circle into the plane is a Jordan parametrization. This proves, in particular, that infinitely many small returning chords do not create an accumulating pinch. Adding the finitely many visible stopped strands gives the asserted base and slits.

For continuity, first choose \(a>0\) outside the countable set of excursion diameters and retain the finitely many excursions of diameter greater than \(a\). Uniform convergence of the parent loops gives convergence of these excursions, their endpoints, and their first hits of the second contour. Here is the role of the generic levels. A compact subarc strictly on one side of a level stays there under a small uniform perturbation. Arbitrarily near an excursion endpoint on its other side in parameter time, the limiting loop has points on the opposite side of the level; otherwise the endpoint would be a one-sided contact. These points bracket the perturbed endpoint between arbitrarily nearby parameter values. The same argument at the cutoff brackets its first hit. Splitting, merging, or losing a retained excursion would contradict one of these bracketing properties.

The circular order of the finitely many retained endpoints is stable. Their visibility is stable as well. An additional chord of small diameter cannot screen a retained chord: its deleted pocket has small diameter and cannot contain that retained chord. After matching the retained pieces, match the remaining intervals of the bases in their circular order. All discrepancies on these intervals have diameter tending to zero as \(a\downarrow0\). This constructs parametrizations with uniformly small distance. Include the finitely many slit roots in the matching; their stopped strands already converge by the same bracketing argument. In particular the fixed band width and the generic second level give eventual equality of the finite stopped-strand list, with its cyclic root and tip order. Only the returning chords require the subsequent limit \(a\downarrow0\). The recording map \(R_\delta\) adds an error of \(O(\delta)\) and ensures that the lattice records have the exact contour incidences required by the data space.

At the lattice level, a returning chord replaces an interval whose two surviving side signs agree. The only sign changes in a traversal of the visible prime boundary occur at stopped tips. Thus microscopic returning chords introduce no additional alternations between tips. When slits are present, their oriented sides determine the signs on all intervening base arcs, so these signs converge by Lemma 27.

To recover the constant sign when there are no tips, we use one further property of nested \(\mathrm{CLE}_3\): a deterministic interior point is almost surely surrounded by infinitely many nested loops (Schramm et al. 2009, sec. 1, before Proposition 1). Their diameters tend to zero by local finiteness. Choose such a point on the starting contour. A sufficiently small surrounding loop misses the second contour and has a returning excursion into the fresh side. If this chord is hidden, any chord screening it has diameter bounded away from zero by the small-pocket argument. There are only finitely many such screening chords, so one is visible. Its oriented parent loop supplies the constant side sign. That loop and its finitely many ancestors converge with their orientations by Lemma 27. This proves sign convergence also in the no-slit case. The additional fixed-point nesting property is used only for this color anchor, not for the preceding deterministic construction of the Jordan base.

The deterministic argument applies simultaneously to any fixed finite list. Lemma 27 and the continuous mapping theorem finish the proof. ◻

Proof. Subdivide the base at its roots and add further vertices as needed so that the base and attached arcs form a finite simple plane graph. Theorem 1 of (Schaefer 2021), with “isomorphic” defined there by a homeomorphism of the plane, gives a polygonal embedding obtained by an ambient homeomorphism. In particular, the free tip of an attached arc has a neighborhood in which it is a straight half-edge and can be prolonged a short distance in the complement. This supplies no common metric modulus. The Jordan–Schoenflies plane extension theorem (Moise 1977, sec. 10, Theorem 4) also supplies a collar of the base. We describe the compatibility construction for all the arcs. Prolong each slit from its free tip to a surrounding curve on the fresh side, choosing the prolongations disjoint. This can be done successively: an arc attached at just one endpoint to a boundary does not separate the adjacent open region, so its free endpoint can be connected to the surrounding curve without crossing the other attached arcs. A connecting path can be simplified to a simple arc. After one prolongation has been chosen, cut along it and repeat in the appropriate Jordan region. Choose distinct endpoints on the surrounding curve in the cyclic order of the roots. This preserves disjointness at each step.

The prolonged arcs cut the collar into finitely many Jordan cells. Map their boundary arcs, compatibly at common edges, to the sides of rectangular cells in a circular strip, sending each marked tip to an interior point of the corresponding straight side. Apply the same plane extension theorem in each cell and glue the extensions. The glued map and its inverse are continuous because there are finitely many closed cells and their boundary maps agree. This gives the required chart. When there are no slits the initial collar suffices. The surrounding curve and all prolongations can be chosen in a neighborhood of the finite graph disjoint from the target; equivalently take a sufficiently small regular neighborhood after the successive extensions. Compactness and the positive distance to the target provide the required final clearance. ◻

Proof. Use a Schoenflies parametrization of a closed annular collar of the outline. Uniform continuity permits a sufficiently thin collar and a cyclic subdivision into small quadrilateral strips, with small open overlaps between consecutive strips and with nonconsecutive strips disjoint. Take marked points away from the overlaps. In each open overlap choose a joining point. Join successive points by polygonal arcs in the intervening open strip, erasing loops. Within an overlap, truncate the two neighboring arcs before their first intersection and join them by a simple polygonal bridge. The overlaps are disjoint, so this operation is local. This gives a Jordan polygon following the prescribed cyclic order. All polygonal endpoints used in this construction are in open sets; no straight approach to an arbitrary point of a rough cell boundary is required. At a marked strip insert a short horizontal or vertical segment and put its mark there. Smooth corners, if desired, away from the marks.

Choose transverse crosscuts in the small overlaps to subdivide the old and new collars, respectively. They give corresponding Jordan cells above and below the old outline and the new polygon. Their outer boundary maps are the identity; on their middle boundaries map the old arc onto the new arc and match the marked points. Choose agreeing maps on the corresponding transverse crosscuts. The plane extension theorem extends these maps cell by cell. The extensions glue to a homeomorphism equal to the identity outside the collar. Corresponding old and new cells lie in the same small strip and its adjacent overlaps, so displacement is bounded by that neighborhood’s diameter. These diameters were made arbitrarily small. The inverse has the same displacement bound. Disjoint collars permit the finite simultaneous construction. ◻

Proof. We spell out the boundary control because unmarked kernel convergence alone would not identify the cap marks. It suffices to prove the assertion along an arbitrary null sequence of positive parameters \(q_n\to0\). In the proof all statements uniform in \(q\) concern this sequence, with its index suppressed.

Connectivity of the filled complements. In the straightened chart the filled complementary component is a base region with finitely many thin rectangular teeth. Two nearby points in one straight tooth can be joined by a segment or a two-segment path of diameter at most a fixed multiple of their distance. The same is true in the base region. Near a junction, the finitely many fixed polygonal templates have a finite local quasiconvexity constant, which is unchanged on scaling. Nonincident pieces away from their junctions have fixed positive separation. Consequently there are \(C,t_0>0\), independent of \(q\), such that points at distance \(t<t_0\) in the same model filled complementary component can be joined there by an arc of diameter at most \(Ct\).

Let \(H\) be the fixed homeomorphism from the model to the physical chart. Write the polygonalization or smoothing map as \(J_q\), so the physical complementary components are images under \(H_q=J_q\circ H\). The maps \(J_q\) and \(J_q^{-1}\) tend uniformly to the identity. Thus \(H_q\) and \(H_q^{-1}\) have common moduli of continuity \(\mu\) and \(\nu\) on the relevant compact neighborhoods: uniform convergence controls the tail of the sequence, and continuity controls its finite head. Two physical points at distance \(t\) in the same complementary component can be joined there by an arc of diameter at most \[ \theta(t)=\mu(C\nu(t)),\qquad \theta(t)\longrightarrow0. \tag{32}\] The fixed other complementary component in the annular case has the same property by its Jordan parametrization, enlarging \(\theta\) if necessary. We require this property for the filled components, not for their boundary curves; the latter need not be uniformly locally connected when a ribbon narrows.

Interior uniformization. For the classical disk and annulus uniformization theorems, including uniqueness of the annular radius ratio, see (Papamichael 2008, Theorems 1.2.1 and 1.3.1). The labeled complementary components converge in Hausdorff distance, and compact subsets of \(D\) are eventually contained in \(D_q\). The normalized kernel theorem gives interior convergence for disks (Carmona and Fedorovskiy 2024, Definition 2.8 and Theorem 2.9). Use the modulus convention \(m(A(r,R))=(2\pi)^{-1}\log(R/r)\). For annuli, write the source annulus as \(A(r_q,1)\), preserving the boundary-component labels; existence of these uniformizers follows from (Papamichael 2008, Theorem 1.3.1). A fixed smooth essential annulus compactly contained in \(D\) is eventually contained in every \(D_q\). Also every \(D_q\) is contained in a fixed bounded annulus whose inner disk lies in the inner complementary component. Both inclusions are essential: the intervening annuli separate the same two complementary components. Monotonicity of annular modulus (Beliaev 2015, Proposition 4.3.5 and Section 4.3.4) therefore gives \[0<r_-\le r_q\le r_+<1.\] To identify a subsequential limit \(r_q\to r\in(0,1)\), write the inverse uniformizers as \(g_q:D_q\to A(r_q,1)\) and the forward maps as \(F_q=g_q^{-1}\), with the inverse normalized at a fixed \(z_*\in D\). Take simultaneous locally uniform normal limits \(g\) on \(D\) and \(F\) on \(A(r,1)\). The inverse maps have winding number one on a fixed essential curve and are bounded away from zero. Their limit cannot be constant, since a nonzero constant has winding number zero. The maximum and minimum principles give \(r<|g|<1\) on \(D\). Consequently local convergence in the identity \(F_q(g_q(z))=z\) gives \(F(g(z))=z\) for every \(z\in D\), so \(F\) is nonconstant.

We must also exclude values on the limiting slit boundary. For every finite \(a\in\mathbb C\setminus D\), Hausdorff convergence of the labeled filled complementary components supplies \(a_q\in\mathbb C\setminus D_q\) with \(a_q\to a\). The functions \(F_q-a_q\) are zero-free. Hurwitz’s Theorem and the nonconstancy of \(F\) imply \(F(w)\ne a\) for every \(w\in A(r,1)\). Thus \(F(A(r,1))\subset D\). Applying local convergence now to \(g_q(F_q(w))=w\) gives \(g(F(w))=w\). The two limits are therefore conformal inverses onto their domains. The modulus of \(D\) is unique, which identifies \(r\); normalization identifies the maps. Every subsequence has these same limits, proving interior convergence of the moduli and maps.

Boundary equicontinuity. Let \(F_q\) denote a forward map from the unit disk or the source annulus. Disk uniformization and the Jordan boundary-extension theorem are recalled in (Papamichael 2008, Theorems 1.2.1–1.2.2). For each positive \(q\) the annular target boundaries are also Jordan; the annular extension preserving boundary-component labels is (Papamichael 2008, Theorem 1.3.2). Thus each map extends continuously to its boundary. Its image lies in a fixed bounded set, and univalence gives a uniform area bound \[\int |F_q'(z)|^2\,dA(z)\le A_0.\] Fix a source boundary point \(\zeta\). Choose \(R>0\) uniformly smaller than the source collar width and the distance of the source normalizing point from the boundary. For \(0<r<R\), the circular arcs \(C_t=\{|z-\zeta|=t\}\) in the source domain, \(r<t<R\), are crosscuts with both endpoints on the same boundary component. If \(\ell_q(t)\) is the length of \(F_q(C_t)\), Cauchy–Schwarz and polar integration give \[\int_r^R \frac{\ell_q(t)^2}{t}\,dt\le 2\pi A_0.\] For some \(t\in(r,R)\) it follows that \[ \ell_q(t)\le s(r):= \left(\frac{2\pi A_0}{\log(R/r)}\right)^{1/2}. \tag{33}\] The image endpoints can be joined in their common filled complementary component by a simple arc of diameter at most \(\theta(s(r))\). This arc meets the image crosscut only at its endpoints. Their union is a Jordan curve in a ball of radius \(s(r)+\theta(s(r))\) about one endpoint. For small \(r\) the ball excludes \(z_*\), whose distance from all target boundaries is bounded below. In the annular case take it also smaller than the separation of the complementary components. The source cap cut off by \(C_t\) does not contain the source normalizing point. Its image must be on the small bounded side of this Jordan curve: the other component contains \(z_*\), and the curve meets the domain only in the crosscut. Therefore \[ \operatorname{diam}F_q\bigl(B(\zeta,r)\cap\text{source domain}\bigr) \le 2\bigl[s(r)+\theta(s(r))\bigr]\longrightarrow0 \tag{34}\] uniformly in \(q\) and \(\zeta\). Interior convergence and this estimate prove uniform convergence on the closures. For annuli use the radial maps between \(A(r,1)\) and \(A(r_q,1)\); since \(r_q\to r\in(0,1)\) these maps and their inverses have uniformly bounded Lipschitz constants and tend uniformly to the identity.

Marks. A tip of a finite simple slit has exactly one prime end above it. In the chart, a small disk about a tip is cut along a segment ending there; nested crosscuts around the endpoint give its unique approach class. A root can have two preimages in the prime boundary, but no sign change is placed at a root. If \(b_q\) is a cap mark, any limit of \(F_q^{-1}(b_q)\) maps to the designated tip by uniform convergence on the closures. Uniqueness of its prime-end preimage identifies the limit. An additional mark on the base away from the roots has the same uniqueness property because the base is Jordan and no slit is incident there. Distinct prescribed marks give distinct limiting prime ends, so all sign arcs remain nondegenerate.

Spin means. Theorem 1.3 of (Chelkak et al. 2021), with the convergence convention of its Definitions 3.1–3.2, applies to the fixed smooth or polygonal domains here and to the limiting marked disk or annulus. For a slit domain, boundary arcs are interpreted on the prime boundary through the conformal pullback in its Definition 3.6. The preceding mark argument preserves the number of boundary components, the cyclic arc order, and the nondegeneracy of every sign arc. The rough-boundary local normalizations of its Lemma 3.34 cancel in the correlation ratios in the proof of Theorem 1.3. Its bulk convergence is uniform on compact sets. Continuity of the continuum correlations under the marked convergence above follows directly by a diagonal argument: if Equation (31) failed, choose an offending sequence of continuum outlines and then a lattice approximation to each so fine that its normalized mean is within the reciprocal of the sequence index of that continuum mean. Choose the approximations also sufficiently close in the marked topology. The combined lattice sequence approximates the limiting marked domain, contradicting the same bulk convergence theorem. This proves Equation (31), including uniformity over \(K\). ◻

Proof. Condition separately on all spins outside \(C\). A boundary neighbor cannot be free in both models, since its edge to \(C\) would put it in that common free component. Each of the three alternatives gives \(\sigma_0(y)\le\sigma_1(y)\) for every pair of exterior assignments consistent with the pins. Thus at each \(x\in C\) the induced boundary field in model \(0\) is at most that in model \(1\). Their internal couplings agree and are ferromagnetic. A monotone coupling of the finite heat-bath chains on \(C\), or equivalently the ferromagnetic boundary monotonicity, gives domination for each pair of exterior assignments. Mixing them proves the marginal assertion. ◻

Proof. An edge between free vertices cannot cross a traced zero segment: the traversed triangle containing such a crossing would have caused both endpoints to be inspected. The same observation applies to the open part of an edge whose last endpoint is examined. If that endpoint was inspected in a traversed triangle, append an approach of length \(O(\delta)\) in that triangle, on the side of the zero segment carrying its spin sign. Affine interpolation gives a constant vertex sign on each side of that segment. If instead it belongs to the initial revealed side, the edge lies within \(O(\delta)\) of the starting polygon; approach the accessible arc of that polygon directly. Every initial sign switch produces a zero crossing of the starting contour and is explored by the algorithm. Hence no additional switch is present on a retained base arc away from its roots. In either case this constructs an approach to the realized wall from the target-accessible side, carrying the examined endpoint’s sign.

Outside the node neighborhoods, use the chart of Lemma 34. The limiting edge lies in a strip disjoint from all nonincident graph pieces. Uniform parametrized convergence places the perturbed strand in a narrow middle substrip and makes it a crosscut between the two plugged ends. A simple crosscut separates the long sides even if it repeatedly backtracks. The approach just described cannot change from one side to the other without crossing this wall. Its sign is consequently the sign of that accessible face. The positive physical separation after the chart and node neighborhoods are fixed absorbs the \(O(\delta)\) detour. Only near a stopping tip could the short approach continue around an untraced end; these are exactly the excluded neighborhoods. ◻

Proof. For a contour assignment \(j\), sum all spins strictly between \(C\) and \(W\), keeping \(j\) on \(C\) and \(b\) on \(W\). Denote this exterior partition weight by \(A_b(j)\). Use the same allocation of energies along \(C\) in every law. The quotient \[w_b(j)=\frac{A_b(j)}{A_+(j)}\] does not involve any prescribed spin or interaction strictly inside \(C\). Factoring the Gibbs density at \(C\) proves the stated common-weight identity, including its separate normalization for each \(\nu_i\).

Consider the Ising law on the collar alone with the spins on both contours free. Its joint marginal on \((\sigma_C,\sigma_W)\) is proportional to \(A_b(j)\). For any \(j,j'\), the quantity \[\log w_b(j)-\log w_b(j') =\log A_b(j)+\log A_+(j') -\log A_+(j)-\log A_b(j')\] is therefore a log-odds rectangle. Lemmas 3 and 4 bound its absolute value by \(4\operatorname{arctanh}q_0\). The fixed square collar bounds this quantity by a geometric constant, exactly as in Corollary 7. Exponentiating proves the oscillation bound on \(w_b\). Energies involving only wall spins multiply a partition weight by a scalar and hence do not affect this bound.

Rescale \(w_b\) so that \(1\le w_b\le K\). Put \(d=\|\nu_1|_{T\cup C}-\nu_2|_{T\cup C}\|_{\mathrm{TV}}\). For an event \(A\) of these joint spins, the layer-cake formula gives \(|\nu_1(w_b\mathbf1_A)-\nu_2(w_b\mathbf1_A)|\le Kd\) and \(|\nu_1(w_b)-\nu_2(w_b)|\le Kd\). Hence \[\left| \frac{\nu_1(w_b\mathbf1_A)}{\nu_1(w_b)} -\frac{\nu_2(w_b\mathbf1_A)}{\nu_2(w_b)} \right|\le 2Kd.\] Take the supremum over \(A\) and then restrict to \(T\). ◻

Proof of Lemma 33. By the preceding common-weight reduction, take the distant wall constant in the outward case. Keep the source and test contours fixed as above. We construct the barrier and the two mean bounds used in Equation (30). The choices will be made in the order in which their uniformity is needed: node neighborhoods, comparison outlines, erosion contour, and finally the datum tolerance and lattice mesh.

Node neighborhoods and their mean error. Use the chart of Lemma 34. Choose disjoint small closed neighborhoods \(P_1,\ldots,P_k\) of the roots and tips. Include one auxiliary node on the base if needed to cut its circular strip into finitely many coordinate strips. Each neighborhood is the image of a small rectangle in the chart. Choose a smaller rectangle strictly inside each one, so that every incident ideal edge is monotone in its local strip coordinate until it leaves the larger rectangle. The incident edges use separate prescribed strips; the nested rectangles leave positive slack for the joins constructed below. Compact parts of nonincident arcs have positive separation.

In a plugged law, fix every lattice spin in these neighborhoods to the same assignment, in every law under comparison, overwriting any old prescription there. Denote this operation by a superscript \(P\). These plugs will close the channels between displaced boundaries. Their assignments need not agree with the old side signs.

For a single plug and any of the laws \(\lambda\) considered below, Corollary [fd:patch-ratio] and the moment bound give \[ |\mathbb E_{\lambda}M_J^\delta -\mathbb E_{\lambda^P}M_J^\delta| \le \omega(\operatorname{diam}P),\qquad \omega(t)\longrightarrow0. \tag{35}\] The estimate is uniform in all other pins and for sufficiently fine meshes. Indeed, the contracted connection probability in Lemma 4 is symmetric in its terminal sets. Its small-patch bound controls the density on the remote contour \(J^\delta\) when the assignment in \(P\) changes. If that density ratio is \(R_P\), then \(e^{-b_P}\le R_P\le e^{b_P}\), where \(b_P\to0\) as the physical plug diameter tends to zero. Hence \[\left|\mathbb E_{\lambda}[(R_P-1)M_J^\delta]\right| \le (e^{b_P}-1) \left(\mathbb E_{\lambda}(M_J^\delta)^2\right)^{1/2}.\] When some spins in \(P\) were free, first condition on all their values, apply the uniform assignment comparison, and average. To overwrite old pins, first remove their prescriptions while retaining all vertices and edges, and use the same conditional comparison. Pins outside \(P\) are common. No free collar is needed around the plug, whereas \(J\) retains its fixed free collar and therefore has a uniform moment bound. Sum (35) over the finitely many plugs. Choose their sizes so that the sum is less than a small number \(a>0\), for every law used in the proof. We will choose \(a\) according to the fixed transmission constant and the requested total-variation error.

Regular upper and lower boundaries. We next construct deterministic comparison laws indexed by a small chart displacement \(q>0\). On a one-sided base arc, move a plus face towards the fresh side and a minus face away from it by amounts comparable to \(q\). On a slit, move a thin ribbon entirely towards the plus side by a distance comparable to \(q\), keeping the respective plus and minus face signs. Its thickness is a small fixed fraction of \(q\). These choices give the upper outline. Reverse both moves for the lower outline. Join the pieces near roots and tips using fixed scaled polygonal templates, with one sign-switch mark on each cap and no neck narrower than a fixed fraction of the template scale. For sufficiently small \(q\), all joins lie in the smaller node rectangles. Regularly approximate each outline by Lemma 35, within a still smaller fraction of its clearances. Let \(\nu_q^{\mathrm u}\) and \(\nu_q^{\mathrm l}\) be the unmodified Gibbs laws, with the same distant wall in the annular case.

Lemma 36 makes their continuum one-point functions converge to the same limit as \(q\downarrow0\). For each fixed \(q\), Theorem 1.3 of (Chelkak et al. 2021) gives convergence of the normalized lattice means uniformly on \(J\). Since \(J^\delta\) has \(O(\delta^{-1})\) vertices, \[ \lim_{q\downarrow0}\limsup_{\delta\downarrow0} |\mathbb E_{\nu_q^{\mathrm u}}M_J^\delta -\mathbb E_{\nu_q^{\mathrm l}}M_J^\delta|=0. \tag{36}\] Choose \(q\) and its regular approximations so that this gap is small for all sufficiently fine meshes. This choice is made after the plug sizes, and keeps every join strictly inside its plug.

The erosion contour and its free neighborhood. Now choose \(0<\rho\ll q\). At chart distance \(\rho\) on the fresh side, put an outline parallel to the base and around both sides and the tip of each slit. Prescribe the face signs, with one switch on each cap. This is a Jordan contour. A sufficiently accurate regular approximation gives \(\Gamma_d\). Choose its switches on grid-aligned straight portions. For fine mesh, a simple lattice cycle following it, with a boundary layer if necessary, has the prescribed cyclic arc order and separates the target from the old boundary.

After \(\rho\) and this approximation are fixed, the new layer has positive physical separation from the ideal datum. Choose the datum neighborhood and mesh small enough to preserve it. The recorded lattice datum uses the map \(R_\delta\) from Section 7; undoing that map moves its curves by only \(O(\delta)\) in the physical plane. Uniform continuity of the fixed chart and its inverse therefore turns a sufficiently small recorded-data tolerance, followed by a sufficiently fine mesh, into the required chart tolerance for the actual wall. Absorb this displacement and the examined triangles’ \(O(\delta)\) thickness in the remaining clearance. The entire new layer is then free in the original conditional law.

In the inward case the same choice preserves more than the layer. Let \(D\) be the Jordan base interior with its attached slits removed. The outline wraps around every slit, so \(\overline{\operatorname{Int}\Gamma_d}\subset D\). It has a thin exterior collar whose union with this closed interior is a compact set \(C\Subset D\). Uniform parametrized convergence of the base preserves its winding number about every point of \(C\), uniformly because \(C\) has positive distance from the base. The nearby slits stay away from \(C\) as well. Thus \(C\) lies in the accessible domain of every sufficiently close realized datum. Nonvisible traces and hidden pockets lie in its blocked complement. All examined vertices are on the initial side or within \(O(\delta)\) of the traced boundary. It follows that the entire component kept inside \(\Gamma_d\), together with a positive exterior collar, is free of old pins, uniformly over the chosen datum neighborhood.

Incidence ordering for every nearby transcript. Let \(\mu^0,\mu^+,\mu^{\mathrm f}\) be the three laws in the statement. We compare their plugged versions to the same two plugged regular laws. Figure 1 shows the local geometry. First consider a slit strip outside the node plugs. Give the old slit plus on its left and minus on its right, and put it at chart coordinate \(x=0\). The actual perturbed strand lies in \(|x|<\xi\), where \(\xi\ll\rho\). The upper ribbon is near \(x=-q\), the erosion plus layer near \(x=-\rho\), and the erosion minus layer near \(x=\rho\).

Parametrized convergence also controls visits to the strip ends. For each ideal edge, its part between the smaller node rectangles lies in a strip separated from all nonincident edges. On the matched perturbed edge, select its last exit from the first smaller rectangle before its first entry into the second. This gives a simple crosscut. The ideal edge is monotone near each node, so closeness of the parametrizations places these exit and entry parameters near the corresponding ideal parameters. The positive gap between the smaller rectangle and the actual plug therefore keeps the entire discarded prefix and suffix inside their respective plugs. For example, in a local edge coordinate with smaller-rectangle exit at \(s=2h\) and plug exit at \(s=3h\), a tolerance \(e<h/3\) puts every point of the discarded prefix at \(s\le2h+2e<3h\).

Retain the entire selected crosscut, including any later visits to an outer plug. Every portion of the original edge outside the actual plugs is then retained. Turns and repeated plug visits do not change the crosscut’s separation of its two long sides: filling the plugs can remove components but cannot join the two sides. Its orientation identifies the accessible face signs, as in Lemma 38. This argument applies to every slit and to each base arc between consecutive nodes. There are only finitely many edges, so one tolerance works for all of them. Taking the mesh below the remaining physical slack prevents a lattice edge from passing between a wall end and its plug.

The upper ribbon and this old-wall crosscut, joined through the wholly pinned root and tip plugs, close the channel between them. That channel cannot meet the target component of the sites free in both laws. Additional examined pieces cannot reopen it. The common target component can consequently see only the plus face of the upper ribbon or the minus face of the old wall. In the fully eroded tested law, the latter frontier may instead be the erosion minus layer. In the plus-only tested law, its extra plus pins lie in the screened channel. The same upper comparison therefore works for all three laws. Lemma 38 identifies each accessible examined-neighbor sign with the appropriate wall face; the recorded-data displacement was already absorbed into its physical slack. At plug incidences the signs agree by construction.

For a base arc the verification is one-sided. If its sign is plus, the upper outline is nearer the fresh side, so the upper model supplies a plus pin at the common component’s frontier. If its sign is minus, the tested wall is nearer and supplies a minus pin. The sign-changing joins are inside the common plugs. Thus every incidence satisfies Lemma 37, and \((\mu^j)^P\preceq(\nu_q^{\mathrm u})^P\) on the common target component for \(j\in\{0,+,\mathrm f\}\).

For the lower comparison move the ribbon to the minus side. The accessible incidences are the lower minus face or the tested plus face. Additional tested plus pins reinforce this order; the erosion minus pins lie in the screened channel. The base-arc argument is reversed in the same way, and the plugs again give identical signs. Applying the incidence criterion proves the full sandwich \[ (\nu_q^{\mathrm l})^P \preceq (\mu^j)^P \preceq (\nu_q^{\mathrm u})^P, \qquad j\in\{0,+,\mathrm f\}. \tag{37}\]

Mean bounds and the completed law. Removing the plugs changes each mean by less than \(a\). Thus (37) places every \(\mathbb E_{\mu^j}M_J^\delta\) between the two unmodified regular means with error at most \(2a\) at either end. Each of the two ordered mean differences in (30) is at most the absolute regular mean gap plus \(4a\). Choose \(a\) first and then \(q\) so that Equation (36) and this error make the sum of target total-variation changes less than \(\varepsilon\). All later choices of \(\rho\), regularization, neighborhood, and mesh preserved these bounds. No spin-scaling limit in a realized transcript’s irregular domain has been used.

Finally, the completed layer separates the target from every old examined vertex. Conditional factorization of the Gibbs weights therefore leaves exactly the law with the chosen new boundary and the unchanged distant wall. It is independent of the transcript. The argument conditions on the full transcript, including the actual auxiliary radii when they are random. Omitted initial spins and nonvisible traces lie on the blocked side of the new barrier. A stopping radius producing no visible tip adds no boundary piece. Thus the same completed law works even when the datum does not record its numerical radii. Averaging over these radii or other omitted information preserves the two total-variation bounds. ◻

Proof. Choose \(\widehat\Gamma\) as a sufficiently thin outward offset of \(\Gamma\), matching the finite sign arcs. Put the new pins there before changing the old ones. The full old layer disconnects the kept volume, so these farther pins are initially harmless. We retain the same \(V^\delta\) throughout the argument; in particular we do not first discard a free strip inside the old layer.

Use the construction of Lemma 33 without slits. Put small identical plugs around the finitely many sign switches, make upper and lower regular outlines by moving plus and minus arcs in their respective ordered directions, and compare by Lemma 37. The test contour lies strictly inside \(\Gamma\), with a common free collar. As the offset tends to zero, both unmodified envelope means have the same limit by Lemma 36; the plug errors tend to zero by Corollary [fd:patch-ratio]. This controls the means for the original law, the law with only the old minus pins remaining, and the law with all old pins removed. The two actual removals are ordered, so Corollary 23 proves the claimed target estimates.

Fix the sufficiently small offset after obtaining these estimates. Its physical distance from \(\Gamma\) is positive, and mesh rounding preserves a fixed fraction of it. The kept volume is inside \(\Gamma\); the effective boundary pins are on the farther contour, and the original pins are farther away, beyond the available collar. This proves the final buffer assertion. ◻