Proof. The right side of (8) is \(0\), \(1\), or \(-1\), and has the required residue in (6). A triangle cannot have both spins nonconstant: each nonconstant binary spin changes on two triangle edges, and these two pairs of edges must intersect. If \(B\) is constant on a triangle, (8) is \(-B\,dW/2\) there; if \(W\) is constant, it is \(W\,dB/2\). Its circulation therefore vanishes. Every cycle of the triangulated disk is a sum of its triangle boundaries, so the increments integrate to \(H\). The four choices of the anchor residue give four equally weighted spin representations of each anchored increment configuration: cyclic rotation of the states in (6) preserves all increments and the weight. Finally each disagreeing pair contributes exactly one factor \(x\). Fixing the connected trace fixes the lift on the whole skeleton, and the remaining edge factors are precisely those of the height Gibbs specification. ◻

Proof. Every interacting cell meeting both sides has its honeycomb center on \(\gamma\). Its open prescription fixes every incident first spin to \(s\), and its factor in (9) is the constant \(q_y\) for every choice of its second spins. All remaining factors involve sites on one side only. Equality edges do not cross by assumption. The product weight and its partition sum thus factor. In particular, complementary components cannot cross the circuit. After specifying the complete inner height layer, the spin lift and the edge weights in Lemma 3 give the claimed height specification. No first-spin agreement event without the open-cell prescriptions has been used in this factorization. ◻

Proof. Monotonicity follows term by term. For positive coordinates the second assertion follows by differentiating in \(\log a_e\). For distinct edges \(e=uv\) and \(f=st\), \[\frac{\partial^2\log Z}{\partial\log a_e\,\partial\log a_f} =\mathop{\mathrm{Cov}}(\mathbf 1\{\sigma_u\ne\sigma_v\}, \mathbf 1\{\sigma_s\ne\sigma_t\}) =\tfrac14\mathop{\mathrm{Cov}}(\sigma_u\sigma_v,\sigma_s\sigma_t)\ge0.\] The last inequality is a special case of the second Griffiths inequality (Kelly and Sherman 1968); we recall its random-cluster proof (Fortuin et al. 1971, 102). Put \(J_e=-\frac12\log a_e\ge0\) and \(p_e=1-e^{-2J_e}=1-a_e\). Expanding \[e^{J_e\sigma_u\sigma_v} =e^{J_e}\bigl((1-p_e)+p_e\mathbf 1\{\sigma_u=\sigma_v\}\bigr)\] over edges gives a joint spin–bond law whose bond marginal on \(\omega\subset E\) has weight \[2^{k(\omega)}\prod_{e\in\omega}p_e \prod_{e\notin\omega}(1-p_e),\] where \(k(\omega)\) counts all connected components, including isolated vertices. This weight satisfies the lattice condition. Indeed, if \(\mathcal C(A)\) is the binary cycle space of the edge set \(A\), then \(\mathcal C(A)\cap\mathcal C(B)=\mathcal C(A\cap B)\) and \(\mathcal C(A)+\mathcal C(B)\subset\mathcal C(A\cup B)\). Taking dimensions and using \(k(A)=|V|-|A|+\dim\mathcal C(A)\) proves supermodularity of \(k\); the remaining edge factors are modular. The FKG theorem (Fortuin et al. 1971) therefore associates increasing bond events. Conditional on \(\omega\), the spins are independent fair signs on its components. Consequently \(\mathbb E(\sigma_u\sigma_v)=\mathbb P(u\leftrightarrow v)\), while the conditional expectation of \(\sigma_u\sigma_v\sigma_s\sigma_t\) is one precisely when each component contains an even number of the four listed vertices, counted with multiplicity, and is zero otherwise. In particular \[\mathbb E(\sigma_u\sigma_v\sigma_s\sigma_t) \ge\mathbb P(u\leftrightarrow v,\ s\leftrightarrow t) \ge\mathbb P(u\leftrightarrow v)\mathbb P(s\leftrightarrow t).\] This proves the required covariance inequality, also when the edges share vertices. The cases \(p_e\in\{0,1\}\) follow by continuity. Integrating the nonnegative mixed derivatives over coordinate rectangles gives (11); continuity of the polynomial \(Z\) includes its zero-coordinate cases. ◻

Proof of Proposition 6. We first sum the second spins by splitting their walls according to the first-spin sign. Lemma 7 will give the two-law inequality for this marginal. We then restore the statuses.

Fix a first-spin configuration \(b\). A binary edge set is a second-spin wall precisely when it has even intersection with every triangle boundary. Indeed this condition makes its mod-two integral independent of the path from a fixed anchor, since triangle boundaries generate the cycle space. There are exactly two spin labelings for each such wall. Every allowed wall edge has homogeneous \(b\) endpoints. Split these edges according to their common sign. A homogeneous triangle has only one sign class. A mixed triangle has just one homogeneous edge, which its even rule forces to be absent. Thus none of the triangle equations couples the two sign classes. Each sign restriction of an allowed wall is itself a wall, and arbitrary walls supported in the two allowed sign sets can be combined. This proves factorization without an additional global compatibility condition.

For an unobserved cell let \(r_y(b)=1\) if \(b\) is homogeneous there and \(r_y(b)=q_y\) otherwise. Define \(a_h^+(b)\) as follows: it equals \(x\) on edges of unobserved homogeneous-plus cells, equals \(1\) on observed open-plus cells, and is zero on all other edges. Set it to zero also on \(E_0\). Define \(a_h^-(b)\) with minus in place of plus. Summing the unobserved statuses and then the second spins gives the first-spin weight \[ w_h(b)=\frac{c_h}{2}\, \prod_{y\text{ unobserved}}r_y(b)\, Z(a_h^+(b))Z(a_h^-(b))\, \mathbf 1\{b\text{ satisfies its fixed sites}\}. \tag{12}\] Here \(c_h>0\) is the product of the observed-cell factors; these are constants because their first spins are fixed. The factor \(1/2\) is exact: each \(Z\) counts two global signs whereas the actual second-spin sum counts only two, not four. In particular \(E_0\) deletes wall coordinates and does not impose a relation between the two global signs. Formula (12) would in general fail on a graph with a hole if the missing cycle constraints were omitted, explaining our ambient triangulation convention.

Let \(h_-\le h_+\) be ordered histories. For any \(b\) in the lower support and \(c\) in the upper support, \(b\wedge c\) belongs to the lower support and \(b\vee c\) to the upper support, also when pins occur in just one law. On an unobserved cell, homogeneous-plus eligibility at the meet is the intersection of the input eligibilities; at the join it contains their union. Therefore, coordinatewise, \[\begin{align*} a_{h_-}^+(b\wedge c)&\ge a_{h_-}^+(b)\wedge a_{h_+}^+(c),& a_{h_+}^+(b\vee c)&\ge a_{h_-}^+(b)\vee a_{h_+}^+(c),\\ a_{h_-}^-(b\wedge c)&\ge a_{h_-}^-(b)\vee a_{h_+}^-(c),& a_{h_+}^-(b\vee c)&\ge a_{h_-}^-(b)\wedge a_{h_+}^-(c). \end{align*}\] On observed cells the plus vectors are already ordered and the minus vectors reverse-ordered; hence the same inequalities hold there. Deleting the common coordinates \(E_0\) preserves all four inequalities. Furthermore \[r_y(b\wedge c)r_y(b\vee c)\ge r_y(b)r_y(c).\] To check this last assertion, comparable restrictions are unchanged by sorting. For incomparable restrictions both inputs are mixed, whereas each output has weight at least \(q_y\). A homogeneous input is extremal and cannot be lost. Applying monotonicity and (11) to each sign in (12) now proves \[ w_{h_-}(b\wedge c)w_{h_+}(b\vee c) \ge w_{h_-}(b)w_{h_+}(c). \tag{13}\] Observed-cell constants cancel. This also proves the lattice condition within either first-spin law.

For clarity, zero weights and unequal supports cause no invocation of an unstated strictly positive version of Holley’s criterion (Holley 1974). Extend the weights by zero to the whole first-spin cube and put \(\widetilde w(b,0)=w_{h_-}(b)\), \(\widetilde w(b,1)=w_{h_+}(b)\). The two within-layer inequalities and (13) say that \(\widetilde w\) satisfies the lattice condition on the cube times \(\{0,1\}\). The nonnegative-weight FKG theorem (Fortuin et al. 1971) applies. For increasing \(f\), its positive covariance with the last coordinate says exactly \(\mathbb E_{h_-}f(B)\le\mathbb E_{h_+}f(B)\). The same theorem gives association of each first-spin law. This proves the needed extension to sorted supports.

It remains to restore statuses. Given \(b\), each unobserved homogeneous cell is eligible to open with its sign; mixed cells are necessarily closed. Let \(S_+\) and \(S_-\) be the two open subsets of eligible cells. Wall factorization now gives independent weights for these subsets. Apart from constants their respective weights are \[ \prod_{y\in S_s}q_y \prod_{y\in I_s\setminus S_s}(1-q_y)\, Z\bigl(\mathbf 1\{e\text{ lies in an open cell of sign }s\} \mathbf 1\{e\notin E_0\}\bigr), \tag{14}\] where \(I_s\) is the eligible set and observed open cells of sign \(s\) are included as fixed allowed cells in the last factor. Different cells have disjoint edge sets. The Bernoulli factor is modular; the \(Z\) factor is log-supermodular in the cell set by (11). Thus each subset law is associated. The same two-law calculation shows stochastic increase under enlarging \(I_s\) or the fixed allowed cells: on a common subset cube the sorted allowed-edge sets contain the coordinatewise intersection and union, and the Bernoulli factors cancel. Ineligible cells are held closed. The preceding nonnegative-weight argument again handles these deterministic restrictions.

Increasing \(b\) enlarges plus eligibility and shrinks minus eligibility. Hence \(S_+\) increases and \(S_-\) decreases in inclusion order, exactly the status order \(-<0<+\). Association of the minus block is also association in reverse inclusion order, since the covariance of decreasing functions equals that of their negatives. The two blocks are independent. If \(F,G\) are increasing functions of \((B,\tau)\), conditional association gives \(\mathbb E(FG\mid B)\ge\mathbb E(F\mid B)\mathbb E(G\mid B)\); the two conditional expectations are increasing in \(B\). Association of \(B\) completes the joint association claim. An ordered coupling of the first spins followed by ordered conditional status kernels gives the two-law assertion.

If \(q_y=1\), a homogeneous unobserved cell is open deterministically, and an observed homogeneous closed cell is infeasible. All displayed inequalities remain valid on feasible supports, either directly or by continuity as \(q_y\uparrow1\); no division by \(1-q_y\) is used. Finally an extra open-plus cell in the upper law has the maximum possible status and first-spin values. Querying that same block in the lower law always gives ordered values and retains its correct distribution. This is observation in the other law, not imposition of the upper event on it. The open-minus case is symmetric. ◻

Proof. At a joint record \(R\) let \(\mu_i^R\) be the two specified own-history conditional laws. Select a common query block \(Q=Q(R)\) and couple its marginals by an ordered kernel \(\kappa_R\), available from Proposition 6. For each possible pair of answers \((a_1,a_2)\) recursively complete with the respective conditional laws \(\mu_i^R(\cdot\mid Q=a_i)\). This is a finite recursion: every nonterminal query adds a new cell or site. At a terminal node with the same remaining specification, use one draw from that common law. At any other terminal node use any coupling of its two remaining laws.

Backward induction proves the marginal invariant. For a completed observable \(f\) in coordinate \(1\), the expectation from the node \(R\) is \[\sum_{a_1,a_2}\kappa_R(a_1,a_2) \mu_1^R(f\mid Q=a_1) =\sum_{a_1}\mu_1^R(Q=a_1)\mu_1^R(f\mid Q=a_1) =\mu_1^R(f).\] The second coordinate is identical. Applying the same induction starting at any reached node gives the assertion conditional on the whole joint record, not merely on an individual history. Independent randomness used to select a query can be put in the record and conditioned on first. No conditioning of a previously sampled full-field coupling is involved. ◻

Proof. The primitive comparison takes place in a small filled regular disk whose entire larger comparison cut and interior fit inside the buffer. In that disk consider a smaller regular annulus. Delete favorable pins in the filled disk, which can only decrease the probability of the favorable increasing event. Prescribing the outer cut open with the adverse sign decreases it again by Proposition 6. There are no other constraints inside this small outer cut. Lemma 4 and Theorem 5 give a positive lower bound in the resulting deterministic regular problem. The same comparison applies after any allowed outside history. For a minus request reverse all first spins and the status order.

For gluing, use regular annuli of inner radius \(\rho\) and outer radius \(2\rho\), with a small fixed geometric slack. Consecutive centers are separated by a distance strictly between \(\rho\) and \(2\rho\) in the triangular graph norm. Their inner balls intersect, and each inner ball has a point outside the other’s outer ball. Any two corresponding surrounding circuits must therefore intersect: disjoint Jordan curves with overlapping interiors would be nested, contradicting the latter property. Intersecting good honeycomb paths of the same sign join. Finitely many such annuli give a connected path following a prescribed tube. A closed chain following a polygonal Jordan curve gives a closed walk with the same winding number about its protected target, and hence a surrounding simple circuit. Choose the annuli small enough that their larger filled comparison disks stay in the buffer. Thus the target annulus may enclose arbitrary pins in its hole: none lies in these small comparison disks. Association multiplies their fixed positive probabilities. A general fixed collar is covered by a finite such chain. Rounding moves paths by only \(O(\delta)\).

For requests in disjoint buffers expose the previous successful constructions only in those buffers and condition on their full allowed histories. The next buffer still has the same uniform bound, regardless of the previous sign. For a stack choose \(k\ge c_1\log(L/r)-c_2\) separated slots. Conditional on all previously inspected slots, failure in the next slot has probability at most \(1-c_3\). Thus the total failure probability is at most \((1-c_3)^k\), which is (15). Slots of bounded lattice size are absorbed into \(C\); no estimate at a sublattice scale is asserted.

Finally a prescription of open-plus cells is an extreme upper constraint. Proposition 6 shows that it increases every increasing ambient event. This conclusion does not require the event to lie inside the prescribed contour. If an ambient favorable path runs from that contour to an interior target, the segment after its last intersection with the contour lies inside and connects its prescribed first-spin layer to the target. This observation requires no regularity or interior corridor for the contour itself. ◻

Proof. Inside the pin-free radius-\(L\) ball choose a deterministic regular open-minus cut of radius comparable to \(L\). Let \(\mu^-\) be the original law with this extra prescription. It is below the original law \(\mu\). Query the cut cells also in \(\mu\), and use the ordered online kernels thereafter. Search inward in separated annular slots for a good plus circuit in the lower law. By Lemma 10, the failure probability is \(C(r/L)^\alpha\). A plus cell in the lower law is plus also in the upper law, so the selected circuit is common. The exploration geometry just proved leaves identical inner conditional problems, with no equality edges or pins. Sample them once, including their free second-spin component labels. The inside law of the deterministic minus cut is independent of all the original exterior constraints. Thus every eligible \(\mu\) is within \(C(r/L)^\alpha\) of this common comparison law on the small ball.

For two free ambient volumes containing a radius-\(L\) ball this gives a total-variation distance at most \(2C(r/L)^\alpha\) on the radius-\(r\) ball. The finite-state local marginals are Cauchy along any exhaustion. Their consistent limits define a plane law. Comparing any given finite ambient law first with a much larger free volume and then taking its limit gives the claimed bound, with a changed constant. Apply the same argument to translated, reflected, rotated, or spin-rotated exhaustions to obtain the stated invariances. The augmentation itself singles out one partite class, so only its corresponding lattice symmetries are asserted; the unaugmented interaction (7) has all lattice symmetries.

For additional plus pins search outward for a good plus circuit in the lower law; every such circuit is common. On its exterior the deterministic constraints agree and the stopped specifications are identical. The upper law can therefore use the same exterior sample. For additional minus pins reverse the order and the signs. The slot bound gives \(Ce^{-ck}\). The correctness of all marginal conditionings follows from the finite recursion of Lemma 9, followed only then by volume limits. ◻

Proof. The local comparison proof applies uniformly to the retained open-cut law. To see this directly, put a deterministic regular open-minus circuit inside \(B(z,L)\) at a radius comparable to \(L\). Adding this prescription lowers the first-color/status law by Proposition 6. The old boundary prescriptions are outside this regular circuit. Search inward for a common open-plus circuit and use the same annular stack and exact inside sampling as in Proposition 11. The error is \(C(r/L)^\alpha\), independently of the old circuit and its exterior record. The law inside the deterministic minus circuit is independent of that record. Comparing it with the free-plane law gives the stated conditional total-variation bound, with a changed constant.

For each record choose a maximal coupling of its retained marginal with the same second marginal \(\nu_r\). On failed records use that second marginal and any coupling. On these finite local state spaces the usual maximal-coupling formula gives a measurable kernel. The conditional second-marginal identity in the statement proves independence from \(\mathcal R\), and averaging the mismatch bound proves the final estimate.

For a stopped exploration from Lemma 9, assume there are no additional pins or equality edges inside the cut and no equality edge crosses it. The geometric invasion above records only the prescribed first-color cut layer on the retained side. Exterior second-spin labels, including those needed to read absolute heights along exterior paths, reveal no retained component label: open crossing cells have constant factors and no equality edge crosses the cut. Lemma 4 therefore verifies the retained-law hypothesis for precisely these records. Retained second-spin labels must remain unobserved. ◻

Proof. We first establish a single-sign comparison under arbitrary existing first-spin pins and second-spin equalities. Consider an obstacle \(K\) with an empty collar, and an event \(A\) adding only first-spin pins of sign \(s\) on \(K\). There can be previous pins of either sign inside \(K\), as well as equality edges there. Choose a deterministic outer good cut \(\gamma_s\) of sign \(s\) in the collar. Let \(p_*\) be the probability of \(A\) in the law with this additional cut. By the exact-cut lemma, \(p_*\) depends on the previous inside constraints and on the chosen outer cut, but not on any remote constraints. Order gives \[ \mathbb P(A)\le p_*. \tag{17}\] For minus pins this is the same assertion in reversed order.

An exterior search finds a closer good cut \(C_s\) of sign \(s\) in a strictly inner collar with probability at least \(c_0>0\), uniformly over all existing constraints. Conditional on any stopped successful record, the remaining conditional probability of \(A\) is exactly its probability inside \(C_s\), with all previous internal constraints retained. In the deterministic outer-cut law, conditioning on this particular closer favorable cut increases the event \(A\). Its inside law is the same by Lemma 4; hence the conditional probability is at least \(p_*\). Averaging over successful records proves \[ c_0p_*\le\mathbb P(A)\le p_*. \tag{18}\] In particular probabilities under two different remote constraint sets are comparable whenever their previous internal constraints coincide. This argument retains opposite pins inside the obstacle; it does not erase them when the next sign block is considered.

Write \(E_i\) for the event that each of the specified second-spin subgraphs in obstacle \(i\) is constant, and \(L_i\) for the event that their labels are all plus. Initially impose no first-spin pins. Apply (18) with \(W\) as first color to the events \(L_i\), obstacle by obstacle. Each conditional factor, given earlier obstacles, is comparable with its unconditional factor, because those constraints are remote and the same reference cut can be used. Therefore \[\mathbb P\Bigl(\bigcap_i L_i\Bigr)\asymp\prod_i\mathbb P(L_i).\] The component-label estimate gives, both jointly and separately, \[2^{-m}\mathbb P\Bigl(\bigcap_i E_i\Bigr) \le\mathbb P\Bigl(\bigcap_i L_i\Bigr) \le\mathbb P\Bigl(\bigcap_i E_i\Bigr),\qquad 2^{-m_i}\mathbb P(E_i)\le\mathbb P(L_i)\le\mathbb P(E_i).\] Consequently \(\mathbb P(\bigcap_iE_i)\asymp\prod_i\mathbb P(E_i)\).

Now regard all \(E_i\) as the common equality set \(E_0\) for the original first color. Let \(A_i\) be its prescribed first-spin event. Add the plus pins in obstacle \(1\), then its minus pins, then the two blocks in obstacle \(2\), and so on. At each step (18) compares the conditional probability with the same factor in the individual law conditioned on \(E_i\) and on earlier sign blocks in this obstacle. Remote \(E_j\) and earlier remote first-spin pins disappear behind the reference cut. Previous internal plus pins remain when the minus block is added. There are at most \(2k\) factors, so multiplication gives \[\mathbb P\Bigl(\bigcap_i A_i\,\Bigm|\,\bigcap_i E_i\Bigr) \asymp\prod_i\mathbb P(A_i\mid E_i).\] Combining with the equality comparison gives \(\mathbb P(\bigcap_i(A_i\cap E_i))\asymp \prod_i\mathbb P(A_i\cap E_i)\). Finally restoring all the actual plus labels costs a factor between \(2^{-m}\) and \(1\), even after the arbitrary first-spin pins and statuses have been specified. The same holds individually with \(m_i\). Since \(P_i=A_i\cap L_i\), this proves (16). Every comparison was finite-volume and uniform. Positive probability of the finite pin events permits passage to the plane limit. No division by a mesh-dependent lower bound for a pin probability occurred. ◻

Proof. For an oriented edge, direct use of (8) gives \[ d(H+BW/2)=W\,dB. \tag{19}\] Whenever \(B\) changes, \(W\) is constant on that edge and its endpoints belong to the same closed-cell component. For such a component \(C\) define an edge form \(\omega_C\) equal to \(dB\) on its changing edges and zero on the other edges. On any triangle the two changing edges, if present, share a vertex and consequently belong to the same component. Their contributions cancel. Thus \(\omega_C\) has zero circulation and, by simple connectedness, has a potential \(\psi_C\).

On \(C\) one has \(\psi_C=B+\text{constant}\), since this is true along each closed-cell or equality edge of a component path. Every site outside \(C\) can be joined to a first site of \(C\) by a path whose preceding edges have zero \(\omega_C\). A changing last edge would put both endpoints in \(C\), so it too has zero form. Hence every value of \(\psi_C\) is a value attained on \(C\), and its range has diameter at most two. In particular its nonzero differences are \(\pm2\).

Open cuts prevent these components from crossing either circuit. Choose a path in the collar from the exterior-side layer of the inner circuit to the interior-side layer of the outer circuit. The difference of its first-spin endpoints is \(\pm2\). Since \(\sum_Cd\psi_C=dB\), at least one component meeting this path contributes a difference \(\pm2\). Such a component is confined to the collar; its potential is constant on each connected protected inner and outer region. Choose the first such component in a fixed deterministic order. No label of this component is prescribed. Let its label be \(w\in\{-,+\}\). Flipping it changes the right side of (19) by \(-2w\,d\psi_C\). Neither endpoint spin changes in the protected regions, so their relative height shift is \(-2w(\pm2)=\pm4\). Anchor the lift in the protected exterior to fix its absolute value there. The flip is an involution preserving its conditional probability because the label is fair.

The chosen component and circuits are unchanged by this operation, since they depend only on \((B,\tau,E_0)\). A switch changes an inner offset \(4t\) to \(4(t\pm1)\) and therefore reverses its parity. Switches in disjoint collars act on disjoint components and commute. Every orbit has \(2^j\) equally weighted elements for \(j\) collars and visits each parity vector exactly once. This conclusion remains true after imposing any checks invariant under the switches; it asserts an orbit identity, not independence after arbitrary checks. The statements about tests follow from their supports and total weights. If a central event involves absolute offsets and is not invariant, one must enlarge its bounded range to contain the orbit before applying this identity. Finally apply Lemma 10 successively in two disjoint nested collars to require opposite signs, with the given constraints away from both buffers. Swapping the colors is a symmetry of the Gray representation. ◻

Proof. Write the edge potential as \[V(t)=\begin{cases}-(\log x)|t|,&t\in\{-1,0,1\},\\ +\infty,&\text{otherwise}. \end{cases}\] This is a convex extended-real potential on the integers. For two admissible arrays \(h,g\), sorting their endpoint values gives on each edge \[V((h\wedge g)_v-(h\wedge g)_u) +V((h\vee g)_v-(h\vee g)_u) \le V(h_v-h_u)+V(g_v-g_u).\] For completeness, if the endpoint orders agree this is equality. If they disagree, write the four ordered endpoint values; sorting replaces two crossing intervals by two nested pairings and cannot increase the sum of their lengths. The new edge differences lie between the two old ones, so their absolute values remain at most one. Since \(-\log x\ge0\), the displayed inequality follows, including \(x=1\). Ordered traces put the meet in the lower support and the join in the upper support. Multiplying the edge inequalities proves the within-law lattice condition and the two-law inequality. Heights are bounded in this finite problem by the trace bounds plus graph distance to the trace. FKG and the product-with-\(\{0,1\}\) argument used in Proposition 6 therefore give association and ordered comparison on a finite lattice, with inadmissible arrays assigned zero weight.

On each triangle integer Lipschitz heights take at most two consecutive values. Its dual honeycomb vertex consequently has either zero or two changing edges. Constant trace prevents such edges from ending on the bounding skeleton. They thus form mutually vertex-disjoint loops. A loop lying in a bounded face of the connected skeleton cannot enclose any piece of that skeleton: a connection from that piece to the bounding skeleton would cross the loop. Every choice of one of two jump signs for each loop therefore yields exactly one admissible height field with the specified constant trace. Its weight is \(x\) to the total loop length, independent of all the signs. Conditional jumps are independent and fair, and the marginal geometric weight is \(2\) per loop times \(x\) per occupied edge.

For a fixed exposed loop, the loops strictly inside cannot touch its vertices. The edge and loop factors split between inside and outside; the inside boundary face layer has a constant height. Selecting the next enclosing loop by an exploration depending only on already exposed outside geometry does not inspect a strict inside edge. Summing the product weights over completions, or equivalently conditioning the constant inner height layer, proves the same factorization at each stopping step. This is the asserted empty-exterior law and applies without regularity of the exposed loop. Finally the ambient realization in Lemma 3, with constant pair pins on the connected skeleton, gives the same flat height problems used above. ◻

Proof. A constant shift reduces to boundary height zero. Conditional on their unoriented geometry, the level loops have independent fair height increments. Expose enclosing loops from outside inward, using the stopping rule in Lemma 15. If the first loop counted by \(N_d(z)\) exists, its unexplored interior has the empty-exterior loop law with a constant inner trace. Its distance from \(z\) is at most \(2d+O(1)\). Every further counted loop surrounds \(B(z,d/2)\), for all sufficiently large \(d\). The factor \(1/2\) provides clearance from all rounding and incident-face conventions.

We first prove a uniform obstruction in any such remaining domain. Suppose an exterior search finds a good open-plus \(B\) contour \(C\) strictly inside its boundary, surrounding \(B(z,d/2)\). Choose the outermost such contour by the authorized-edge invasion following Lemma 9. The search exposes only exterior history and the prescribed open-plus cut data. In particular it reveals no strict interior cell and leaves the exact cut law of Lemma 4 inside \(C\). Moreover \[ d/2\leq\mathop{\mathrm{dist}}(z,C)\leq2d+O(1). \tag{22}\] There is a number \(\kappa>0\), independent of \(C\), such that under this interior law a \(B\)-plus path joins its prescribed inner layer to \(B(z,d/2)\) with probability at least \(\kappa\).

Here are geometric and conditional details of this last assertion. Fix a sufficiently small constant \(\epsilon>0\), and choose a closest point \(p\) of \(C\) to \(z\). Realize the same interior law in a new free ambient triangulation containing \(B(z,10d)\), conditioned only on the same open-plus cells and first-spin sites of \(C\). Exact cutting justifies this replacement even for an arbitrarily nonconvex \(C\). In this ambient graph request a good plus circuit in an annulus about \(p\) with radii \(\epsilon d\) and \(2\epsilon d\). Since \(C\) visits the inner ball and surrounds \(B(z,d/2)\), it must exit the outer ball and intersect this circuit. Next request a chain of annular plus circuits of these same radii, from \(p\) to an annulus wholly inside \(B(z,d/2)\). The centers may follow a straight segment in the ambient plane. Choose successive spacings strictly between \(\epsilon d\) and \(2\epsilon d\), with fixed slack. The inner balls of consecutive annuli overlap and each protrudes outside the other’s outer ball. Consequently their surrounding circuits intersect: disjoint circuits would have either disjoint interiors, contradicting the overlap, or nested interiors, contradicting that protrusion. Equation (22) bounds the number of requests by a constant depending only on \(\epsilon\). Bounded lattice displacements are absorbed by the slack.

Lemma 10 and positive association give a fixed positive probability for their intersection in the free ambient law. All the data imposed on \(C\) are favorable for this increasing event, so Proposition 6 gives the same lower bound after conditioning on \(C\). The connected circuit chain contains a path from \(C\) to the terminal circuit inside the disk. Take its last intersection with \(C\) before that endpoint. The remaining path stays inside \(C\), since otherwise planar separation would force another intersection. Its adjacent plus face chain therefore joins the prescribed inner layer to the disk. This proves the asserted lower bound \(\kappa\). In particular this connection precludes a deeper \(B\) wall surrounding the disk. The ambient chain was allowed to leave \(C\); no interior corridor of prescribed width was assumed.

To turn this obstruction into a bound on the number of loops, let \(N\) be the number of further qualifying enclosing loops in the remaining flat problem. Closest approach is nonincreasing under nesting, so the qualifying loops are consecutive in the enclosing nesting order until their closest approach falls below \(d\). Conditional on all unoriented geometry, write the height residues along them as a nearest-neighbor walk on \(\mathbb Z/4\mathbb Z\), with independent fair signs. From every starting residue there is a prescribed block of four signs that contains a \(W\) wall with unchanged \(B=+\), followed inward by a \(B\) wall. For example, starting from residues \(0,1,2,3\), respectively, the initial strings \(++\), \(--\), \(---\), \(+++\) have that property; complete each to four signs arbitrarily. Every chosen four-sign block has probability \(1/16\). Hence, on \(\{N\geq k\}\), the conditional probability of seeing this ordered pair of walls among the first \(k\) loops is at least \[1-\varepsilon_k, \qquad \varepsilon_k=(15/16)^{\lfloor k/4\rfloor}.\] The statement follows by conditioning successively on the starting residue of each block; independence between block-success events is not required.

Call this ordered-wall event \(\mathcal A\). A \(W\) wall with \(B=+\) is a good open-plus contour, because every cell at which \(W\) is nonconstant is necessarily open. Thus \(\mathcal A\) implies existence of the exterior-searched outermost plus cut above and of a \(B\) wall strictly inside it surrounding the disk. The preceding obstruction, conditioned on every possible stopped cut, yields \[(1-\varepsilon_k)\mathbb P(N\geq k) \leq\mathbb P(\mathcal A)\leq1-\kappa.\] Choose a fixed integer \(k\) with \(\varepsilon_k\leq\kappa/2\) and put \(\rho=(1-\kappa)/(1-\kappa/2)<1\). The probability of continuing for \(k\) further qualifying loops is at most \(\rho\).

After exposing these \(k\) loops, the last one’s interior is again an empty-exterior flat problem of exactly the same type, with closest boundary distance at most \(2d+O(1)\). Restarting conditionally gives \[\mathbb P\{N_d(z)\geq1+mk\}\leq\rho^m\qquad(m\geq0).\] This is a stopping recursion; it asserts no independence of successive geometric shells. It proves (21) for sufficiently large \(d\). For bounded \(d\), each counted loop visits a fixed-size ball and the loops have disjoint honeycomb vertices, giving a deterministic bound on their number. Finally, integration of the exponential tail gives every stated moment. ◻

Proof. The independent centered loop orientations give the exact identity \[ \mathbb E[H(z)H(w)] =\mathbb E\#\{\text{level loops surrounding both $z$ and $w$}\}. \tag{25}\] Fix a dyadic closest-approach shell \([d,2d)\) about \(z\) with \(d\leq b/100\). A loop surrounding both sites and entering \(B(z,2d)\) must also leave a ball about \(z\) of radius comparable to \(b\); otherwise it could not enclose \(w\). It supplies a wall arm across the intervening annuli. Along the whole loop exactly one of \(B,W\) changes. A good circuit of that color cannot be crossed by its wall, since the circuit’s open cells have constant first spin.

Use the wall color as the first spin. The zero-height skeleton fixes that first spin and imposes constancy of the second spin along a connected skeleton. Represent the latter constancy by \(E_0\) edges. Its actual constant label costs exactly \(1/2\), independent of the first spins and statuses, and hence does not change their marginal. All arm annuli lie away from the skeleton, since \(b\leq2a+1\) and \(c_0\) is small. The logarithmic circuit stack from Lemma 10, applied for each of the two possible wall colors, consequently gives \[ \mathbb P\{\text{such a common loop exists in the shell}\} \leq C(d/b)^\alpha. \tag{26}\] If \(M_d\) counts common loops in this shell, then \(M_d\leq N_d(z)\) and \(M_d=0\) outside the event in (26). Cauchy–Schwarz and Lemma 16 imply \[\mathbb EM_d\leq (\mathbb EN_d(z)^2)^{1/2} \mathbb P(M_d>0)^{1/2} \leq C(d/b)^{\alpha/2}.\] Loops at distance below one give a deterministic bounded contribution by disjointness. The sum over the small dyadic scales is bounded. Each of the \(O(1+\log(n/b))\) remaining shells contributes at most a constant. There are no shells beyond scale \(O(n)\): a nearest skeleton site is outside every enclosing loop, so the segment from \(z\) to that site meets the loop within distance \(n+a+O(1)\). These facts and (25) prove (23).

For completeness the average of the singularity is uniformly bounded, including its diagonal. Triangular-lattice balls have at most \(C(1+r)^2\) sites. For fixed \(z\), the layer-cake formula gives, when \(a>1\), \[\begin{align*} &\sum_w |w_w|\log^+\frac{a}{\max(1,|z-w|)}\\ &\qquad=\int_0^{\log a} \sum_{\max(1,|z-w|)<ae^{-t}}|w_w|\,\,\mathrm dt \leq C A a^{-2}\int_0^{\log a}a^2e^{-2t}\,\,\mathrm dt \leq C A. \end{align*}\] For \(a=1\) the sum vanishes. Sum (23) against \(|w_zw_w|\), separate the constant term \(\log(n/a)\), and use this bound and \(\sum|w_z|\leq A\). This proves (24) for arbitrary signed weights. ◻

Proof. First let \(m=1\). Write \(A=\sum f\) and \(B=\sum g\). If either is zero there is nothing to prove. Otherwise let \(\nu_0=f/A\) and \(\nu_1=g/B\), and couple them monotonically: take their quantile functions at the same uniform point of \((0,1)\). Let \(\pi\) be this coupling, and let \(\nu_-,\nu_+\) be its pushforwards by the floor and ceiling midpoint maps. The entropy theorem just cited states \[\mathcal H(\nu_-)+\mathcal H(\nu_+) \le\mathcal H(\nu_0)+\mathcal H(\nu_1), \qquad \mathcal H(\nu)=\sum_z\nu(z)\log\nu(z),\] for finitely supported probability measures on \(\mathbb Z\), with zero terms omitted. Every pair charged by \(\pi\) has \(f(s)g(t)>0\); the pointwise hypothesis therefore makes both corresponding output weights positive. All logarithms below are thus evaluated on finite positive supports. Integrating the logarithm of the pointwise hypothesis and then using the entropy inequality gives \[\begin{align*} \log A+\log B &\le\sum_z\nu_-(z)\log u(z)+\sum_z\nu_+(z)\log v(z) -\mathcal H(\nu_0)-\mathcal H(\nu_1)\\ &\le\left(\sum_z\nu_-(z)\log u(z)-\mathcal H(\nu_-)\right) +\left(\sum_z\nu_+(z)\log v(z)-\mathcal H(\nu_+)\right)\\ &\le\log\sum_z u(z)+\log\sum_z v(z). \end{align*}\] The last inequality is Jensen’s inequality applied to \(\sum\nu\log(a/\nu)\) on the positive support of each probability measure; adding any unused nonnegative weights only enlarges the sum. Exponentiating proves the scalar statement without modifying zero values or assuming that a positive approximation preserves its premise.

For the induction step write \(s=(s',a)\) and \(t=(t',b)\). Fix \(s',t'\) and apply the one-dimensional theorem in \(a,b\). For the partial sums \(F(s')=\sum_a f(s',a)\), and likewise \(G,U,V\), its conclusion is \[F(s')G(t')\leq U(m_-(s',t'))V(m_+(s',t')).\] The induction hypothesis in the other \(m-1\) coordinates proves the claim. No factorization of the functions is used. A finite array is extended by zero to \(\mathbb Z^m\); the premise must still be checked on every pair of its nonzero input supports. ◻

Proof. One fixed height bounds every coordinate by its graph distance from that vertex, so the array is finite. For an edge \(ij\) put \(a=s_i-s_j\) and \(b=t_i-t_j\). If \(c,d\) are its differences in the two rounded arrays, and \(\varepsilon_i\) is the parity of \(s_i+t_i\), then \[c=\frac{a+b-\varepsilon_i+\varepsilon_j}{2},\qquad d=\frac{a+b+\varepsilon_i-\varepsilon_j}{2}.\] Thus \(c+d=a+b\), \(|c-d|\leq1\), and \(c,d\) are its two nearest integer halves. In particular \(c,d\in\{-1,0,1\}\) and \[|c|+|d|=|a+b|\leq|a|+|b|.\] Each fixed integer coordinate is preserved exactly. Summing over edges and using \(x\leq1\) proves (27), including \(x=1\). Each rounded coordinate differs from its average by at most \(1/2\), which proves (28) even for signed weights. ◻

Proof. For integers \(j\geq3\) define \[I_j=[(2^j-4)M,(2^{j+1}+4)M],\qquad p_j=\mu(Y\in I_j).\] For \(j\geq4\) choose a class carrying at least \(p_j/J\) of the mass of \(I_j\), and pair its tail arrays with its central arrays from (29). Every output has its statistic in \[[(2^{j-1}-7/2)M,(2^j+7/2)M]\subset I_{j-1};\] indeed the central average contributes at most \(M/2\) and rounding at most \(A/2\leq M\). Apply Theorem 18 to \(\mathcal W\) restricted to the chosen tail and central sets, and to two copies of \(\mathcal W\) restricted to \(\{Y\in I_{j-1}\}\). Dividing by \(Z^2\) gives \[ p_j\leq (J/c)p_{j-1}^2. \tag{31}\]

Put \(K_*=J/c\geq1\), \(N=\lceil4K_*\rceil\), and \(L=3+N\). For \(j\geq4\) these intervals have overlap at most two: the upper endpoint of \(I_j\) is strictly below the lower endpoint of \(I_{j+2}\). Thus \(\sum_{j\geq4}p_j\leq2\), and among \(j=4,\ldots,L\) there is \(j_0\) with \(K_*p_{j_0}\leq1/2\). Iteration of (31) yields \[K_*p_{j_0+r}\leq2^{-2^r}\quad(r\geq0), \qquad p_j\leq K_*^{-1} \exp\{- (\log2)2^{j-L}\}\quad(j\geq L).\] Repeat the argument for \(-Y\), with the same \(L\). If \(t\geq2^L\) and \(k=\lfloor\log_2t\rfloor\), the cores \([2^jM,2^{j+1}M]\subset I_j\) for \(j\geq k\) cover the positive tail, and their negative counterparts cover the negative tail. Writing \(q_0=\exp\{-(\log2)2^{k-L}\}\leq1/2\), we obtain \[\mu(|Y|>tM) \leq\frac2{K_*}\sum_{r\geq0}q_0^{2^r} \leq4q_0 \leq4\exp\{- (\log2)2^{-L-1}t\}.\] For \(0\leq t<2^L\) the last bound exceeds one. This proves the tail claim with \(a_*=(\log2)2^{-L-1}\). Finally \(\mathbb E|Y|^p=\int_0^\infty pt^{p-1}\mu(|Y|>t)\,\,\mathrm dt\) proves the moment claim. This proof explicitly uses the unconditional central mass in each class; a central bound for their union would not suffice. ◻

Proof. For zero trace, Lemma 17 and Chebyshev’s inequality supply central probability at least \(1/2\) with \(M=C_A(1+\sqrt{\log(n/a)})\), increased if necessary to exceed \(A/2\). Lemma 19 verifies all midpoint hypotheses for the actual weighted height measure. Apply Lemma 20 with one class to prove (32) for zero trace.

For general trace, put \(K_0=\lceil K\rceil\). Height order from Lemma 15 traps the law between the flat laws with traces \(-K_0\) and \(K_0\). For nonnegative weights, the corresponding smearing laws are the zero-trace smearing shifted by \(\pm K_0\sum_uw_u\). Both tails are therefore bounded by the zero-trace tails with an additive displacement at most \(AK_0\). This gives (32). Split arbitrary signed weights into their positive and negative parts; each satisfies (20), and the triangle inequality in \(L^p\) finishes the proof. On a cut, fix the entire actual inner trace first, so its interior is an ordinary weighted height problem by Lemma 4. The same argument applies conditionally. ◻

Proof. Work first with the free spin pair in the finite ambient graph. Choose a fixed large constant \(L_0\) with \(1/L_0<c_0/4\), and search from outside for a good \(B\) circuit in an annulus with radii \(L_0a\) and \(2L_0a\). Choose \(C_0\) large enough that all search buffers fit in the ambient graph. Lemma 10 gives this stopped-cut event probability at least \(c_1>0\). Its trace is contained in two consecutive integers. Conditional also on the complete inner trace, subtract the lower of these integers. The interior is now an ordinary height problem with trace in \(\{0,1\}\), and its distance from \(z_0\) is between fixed multiples of \(L_0a\). Proposition 21 therefore gives a uniform conditional second-moment bound \(C_A\). The random subtraction changes no zero-total smearing. Chebyshev’s inequality and then averaging over the search show, for some fixed \(M\geq A/2\), \[ \mathbb P(|Y|\leq M)\geq c_1/2. \tag{34}\] If the finitely many smallest lattice scales require enlarging the annulus to accommodate rounding, use a fixed minimal annulus radius; its ratio to \(a\geq1\) remains bounded and the same constants suffice.

To amplify this central bound, use the correct finite height measure. For any chosen anchor \(o\), the four cyclic rotations of the Gray states preserve every increment and every interaction weight, and act transitively on the pair state at \(o\). Consequently the unconditioned free pair has exactly the same increment law as its version with state \((+,+)\) at \(o\). By Lemma 3, the latter is the measure on integer \(1\)-Lipschitz arrays with \(H(o)=0\) and density proportional to \(x^{\sum_{ij}|H(i)-H(j)|}\). Thus \(Y\) in this anchored measure has the central bound (34). This argument obtains the cut central bound before fixing the anchor, and hence introduces no extra anchor constraint into the stopped interior law.

The anchored arrays form one midpoint-admissible class by Lemma 19. Applying Lemma 20 proves (33) in finite volume. At fixed lattice weights, enlarge that volume. The increment convergence from Proposition 11 and the uniform bounds pass the estimate to the plane; higher moments also give convergence of every fixed finite moment. All constants above are independent of the location and radius of the smearing. ◻

Proof. Choose two disjoint nested annular collars around \(\mathop{\mathrm{supp}}f_1\) inside \(U_1\), avoiding \(\mathop{\mathrm{supp}}f_2\). Prescribe an outer good \(B\)-plus cut and an inner good \(B\)-minus cut. Successive applications of Lemma 10 in these separated buffers give their joint probability at least \(c_2>0\), uniformly in \(R\) large enough for rounding. Make the choices by the stopped searches following Lemma 9; the event and the selected cuts depend on first spins and cell statuses, and do not prescribe any second-spin labels.

By Lemma 14, conditional on those variables there is a second-spin component in the intervening collar whose fair label flip changes every height in the inner protected region by \(4\) in absolute value relative to the exterior of the outer cut. Choose the component measurably using a fixed ordering, and condition on all its other labels. The cuts, their event, and all weights of \(Y_R\) are unchanged by this flip. Exact unit mass in the first disk and zero weight in the collar mean that the two values of \(Y_R\) differ by \(4\). A fair pair of real numbers at distance \(4\) has variance \(4\) and second moment at least \(4\). Averaging over the cut event yields \(\mathbb EY_R^2\geq4c_2\) in finite volume. The fourth-moment bound in Proposition 22 allows passage of these second moments to the plane. This proves the claim. ◻

Proof. Take first a reflection-symmetric finite free volume, chosen so that its intersection with the row is connected and contains the observables under consideration. No edge of the triangular lattice jumps across a row of sites. Conditional on the row spins, the two strict sides therefore have independent, reflected-identical laws: every original edge weight belongs to a single side together with its row boundary. Row spins determine relative row heights. If phases are present, condition also on their values at one row anchor. On each side the other phases are determined from those anchor phases and the increments on that side. Their conditional Haar law at the row anchor is independent of the spins, even if the original anchor \(o\) was elsewhere. Consequently \[\mathbb E\big[\overline{\theta_\ell F}\,F\big] =\mathbb E\left[\left|\mathbb E[F\mid\text{row spins and row phases}]\right|^2\right] \ge0.\] The plane limit gives the same inequality for local functions. Ordinary \(L^2\) approximation is also allowed, since \(|\langle F,G\rangle_\ell|\le\|F\|_2\|G\|_2\). Changing the anchor or translating the configuration multiplies each \(U_s\) by a unit complex number determined by the increments. Conditional Haar measure is unchanged. This proves stationarity and all the lattice reflection symmetries also for the augmented law.

For the operator statement take \(\ell=\{y=0\}\) and the upper half-plane. A site has coordinates \((m+n/2,n\sqrt3/2)\). Reflections in \(y=0\) and \(y=\sqrt3/2\) send its integer coordinates respectively to \((m+n,-n)\) and \((m+n-1,2-n)\). Thus both are site-row reflections, and their composition gives the pure normal translation \(a=(0,\sqrt3)\). Let \(T\) denote translation of observables by \(a\). Stationarity and \(\theta_\ell T=T^{-1}\theta_\ell\) imply \[\langle TF,G\rangle_\ell=\langle F,TG\rangle_\ell.\] Furthermore \(TF\) is supported above the halfway row \(y=\sqrt3/2\), and reflection of \(TF\) in that row is \(\theta_\ell F\). Reflection positivity at that row gives \(\langle F,TF\rangle_\ell\ge0\).

Boundedness must be proved before taking the quotient. Put \(a_n=\langle T^nF,T^nF\rangle_\ell^{1/2}\). Symmetry and reflection Cauchy–Schwarz yield \[a_n^2=\langle T^{n-1}F,T^{n+1}F\rangle_\ell \le a_{n-1}a_{n+1}\quad(n\ge1), \qquad a_n\le\|F\|_2.\] If \(a_0=0\), the first inequality gives \(a_1=0\). Otherwise an increase \(a_1>a_0\) would, by log-convexity, force geometric growth, contradicting the second inequality. Hence \(\|TF\|_\ell\le\|F\|_\ell\). The operator descends to the quotient and extends to a bounded symmetric positive operator on its completion, hence to a positive self-adjoint contraction \(S\). Tangential translation preserves the form and has an inverse with the same property. It therefore gives a unitary \(U\); \(U,U^{-1}\) commute with \(S\). Rotation by \(\pi/3\) and \(2\pi/3\), and translation of the base row, give exactly the same construction in all three directions. ◻

Proof. Let \(V_u\) be ordinary unitary translation on the plane probability space. Lemma 60, applied to the two basis translations, constructs their scalar covariance measures \(\mu_{a,b}(\,\mathrm dp)\) on \(\widehat\mathbb T\). Write \(P_0\) for the common invariant projection. All increments belong to \(L^2\), since \(|\xi_u|\) is bounded by the graph distance from zero to \(u\). The cocycle identity gives \[\xi_{u+v}=\xi_v+V_v\xi_u, \qquad (V_v-I)\xi_u=(V_u-I)\xi_v.\] Put \(d_u(p)=e^{ip\cdot u}-1\). The polynomial covariance identity in Lemma 60 transfers the cocycle relation to the scalar measures. On a patch where \(d_v\ne0\), it gives \(\mu_{\xi_u,\xi_w}=\overline{d_u}d_w \mu_{\xi_v,\xi_v}/|d_v|^2\). Division is made first on compact subpatches where the denominator is bounded below. Thus define there \[C(\,\mathrm dp)= \frac{\mu_{\xi_v,\xi_v}(\,\mathrm dp)} {|e^{ip\cdot v}-1|^2}.\] The definitions agree on overlaps. The two choices \(v=e_1,e_2\) cover the punctured torus; on every compact subset their denominators are bounded below on a finite cover. This proves local finiteness and (39) for the part orthogonal to \(P_0\).

The remaining projected cocycle is additive: \(P_0\xi_u=A\cdot u\) for some random vector \(A\in L^2\). For a smooth compactly supported \(f\) of integral zero, use the exact face weights for \(R^{-2}f(z/R)\) at unit mesh. Their first lattice moment is \[\sum_u u\int_{u+V}R^{-2}f(z/R)\,\mathrm dz =R\int zf(z)\,\mathrm dz+O(1),\] where the deterministic error is bounded because \(|u-z|\) is bounded on \(u+V\). Orthogonal projection and Proposition 22 bound the \(L^2\) norm of the dot product of \(A\) with this vector uniformly in \(R\). Divide by \(R\) and let \(R\to\infty\). Choosing two such \(f\) with independent first moments proves \(A=0\). Orthogonality rules out any cancellation between this tilt and other spectral components. Uniqueness of the local construction of \(C\) gives its stated symmetries.

For the annular estimate choose a real smooth bump \(b\) with integral one and support so small that \(\widehat b\) has no zero on \(|p|\le2\). The tests \(f_1=\partial_1b\) and \(f_2=\partial_2b\) have integral zero, and \(|\widehat f_1|^2+|\widehat f_2|^2\) is bounded below on \(1\le|p|\le2\). Let \(b^{(r,j)}_u\) be the unit-face weights for \(r^2f_j(rz)\). Their absolute sums are bounded and their largest absolute values are \(O(r^2)\); their supports have radius \(O(r^{-1})\). For \(1\le|p|\le2\), \[\sum_u b^{(r,j)}_ue^{irp\cdot u}\longrightarrow\widehat f_j(-p) \quad\text{uniformly as }r\downarrow0.\] Indeed replacing the face center by a point in its face changes the exponential by \(O(r)\) uniformly here. Apply Proposition 22 and (39) to the two weights. Their squared transforms bound below a positive constant on \(r\le|p|\le2r\), proving the first assertion in (40) for small \(r\). Local finiteness handles any remaining compact range of radii. Summation over dyadic annuli gives the second assertion. ◻

Proof. Take \(F=\xi_{e_1}\), an observable on the horizontal reflection row. Apply the scalar construction of Lemma 61 to the commuting \(U,S\) from Proposition 25. It gives a finite positive measure \(\gamma(\,\mathrm dk,\,\mathrm d\lambda)\) on the unit circle times \([0,1]\) such that, for every \(m\in\mathbb Z\) and \(j\ge0\), \[\mathbb E[F\,V_{me_1+j(0,\sqrt3)}F] =\langle F,U^mS^jF\rangle_\ell =\int e^{imk}\lambda^j\,\gamma(\,\mathrm dk,\,\mathrm d\lambda).\] Reflection in the base row gives the same expression with \(|j|\) for negative \(j\). Put \(\lambda=e^{-\sqrt3s}\), interpreting \(\lambda=0\) as \(s=\infty\). The \(j\)th Fourier coefficient of \(Q_s\) is \(e^{-\sqrt3s|j|}\), including both endpoints. Uniqueness of Fourier coefficients of finite measures on the rectangular torus identifies the ordinary spectral measure of \(F\) with this mixture. By (39) that measure is \(|e^{ik}-1|^2 C'(\,\mathrm dk,\,\mathrm dl)\). Dividing by this strictly positive factor on \(k\ne0\) gives (42) and local finiteness of \(\nu\) there. Since \(Q_s\) has total mass one, the left side of (43) equals \(\int_{0<|k|<\kappa}|k|\,C'(\,\mathrm dk,\,\mathrm dl)\). Near the physical origin this is finite by (40) and \(|k|\le|p|\); its remaining preimage has finite mass by local finiteness on the physical torus away from zero. This includes a neighborhood of \(a_*\) and does not require any estimate on horizontal bands near zero.

For the physical line take a compact segment of \(k=0\) in a small punctured disk and rotate it through \(\pi/3\). Its coordinates become \((k',l')=(-\sqrt3l/2,l/2)\): both are nonzero, and the image is a nonvertical graph away from \(l'=0\). For each fixed \(k'\) its normal section is a singleton outside zero. Every \(Q_s\) assigns zero mass to this singleton: \(Q_s\) is nonatomic for \(s>0\), including \(s=\infty\), while \(Q_0\) is supported at zero. Applying (42) and then positivity of the pushforward gives zero mass to the rotated segment, hence to the original segment. Countably many segments cover the punctured line. For the alias, rotation sends \((0,P)\) to \((-\pi,P/2)\). On the rectangular torus this point has nonzero tangential and nonzero normal coordinates. The same mixture argument gives zero mass to that point. Projection can only add mass, so invariance under rotation yields \(C\{a_*\}=0\). In particular this argument does not identify the two boundary representatives of the rectangle on the physical torus. Rotating the entire construction proves the assertions for the other row directions. ◻

Proof. Put \(a=|k|\). The analytic function \(((a+iz)/(a-iz))^2\) is bounded in the upper half-plane, has boundary real part \(\chi(a,l)\), and has real value \(((a-s)/(a+s))^2\) at \(z=is\). The Poisson formula therefore gives the exact line identity \[ \int_\mathbb R\chi(a,l)\frac{s\,\mathrm dl}{\pi(s^2+l^2)} =\left(\frac{a-s}{a+s}\right)^2\quad(0<s<\infty). \tag{47}\] Here the Poisson formula applies to a bounded harmonic function with continuous boundary values, so no boundary growth condition is missing. To estimate periodization, let \(f_a(l)=1-\chi(a,l)=8a^2l^2/(a^2+l^2)^2\). Direct integration gives \[\int_\mathbb Rf_a(l)\,\mathrm dl=4\pi a, \qquad 0\le f_a(l)\le\frac{8a^2}{l^2}\quad(l\ne0).\] If \(p_s(l)=s/\{\pi(s^2+l^2)\}\), then \[M_P:=\sup_{s>0,\ |l|\le P/2}\sum_{n\ne0}p_s(l+nP)<\infty.\] Indeed for \(s\le P\) use \(|l+nP|\ge P(|n|-1/2)\) and sum the square tail; for \(s>P\) the \(O(s/P)\) terms with \(|n|\le2s/P+1\) are each at most \(1/(\pi s)\) and the remaining terms have a summable \(s/(nP)^2\) bound. The difference between the periodized and line integrals of \(f_a\) is at most \[4\pi M_Pa+ \int_{|l|>P/2}f_a(l)p_s(l)\,\mathrm dl \le4\pi M_Pa+32a^2/P^2.\] This proves (44) uniformly for finite positive \(s\). For \(s=0\) it is exact; for \(s=\infty\) use the uniform density and \(\int f_a\le4\pi a\).

Integrate (44) over \(\epsilon<|k|<\kappa\). Its error is bounded independently of \(\epsilon\) by (43). Replacing the projected integral by the physical integral inside \(|p|<\kappa\) changes it by a bounded amount: the remaining physical region, including the other sheet, has finite mass away from zero. Thus it suffices to bound above \[\int_{\epsilon<|p|<\kappa,\ |k|>\epsilon} \chi(k,l)\,C(\,\mathrm dp).\] The integral over the full radial annulus is exactly zero, since \(\sum_{j=0}^5\cos(4(\arg p+j\pi/3))=0\) and \(C\) is rotation invariant. In the deleted cap \(|k|\le\epsilon\), the negative part of \(\chi\) is confined to \(|l|<(1+\sqrt2)|k|\), hence to \(\epsilon<|p|<K_2\epsilon\) for an absolute constant \(K_2\). A fixed number of annular bounds in (40) controls that negative part. Deleting the positive part can only lower the integral. We have obtained an upper bound independent of \(\epsilon\). Since \(D\ge0\), monotone convergence proves (45). This use of a negative cap requires only annular bounds, not the quantitative endcap estimate proved below.

On \(r\le|k|\le2r\), if \(s\notin[r/2,4r]\) then \(D(k,s)\ge1/9\); (45) bounds the mass of this part. For \(s\in[r/2,4r]\), the central, unperiodized part of \(Q_s\) gives \[Q_s\{|l|\le2r\}\ge\frac2\pi\arctan(1/2)>0\] when \(r\) is small. Equation (42) therefore bounds the remaining \(\nu\) mass by a constant times \(C'\{r\le|k|\le2r,\ |l|\le2r\}\). The physical-origin preimage lies in a fixed number of annuli of scale \(r\), and the alias preimage has bounded finite mass. This proves (46). ◻

Proof. Rotate the indicated cap through \(\pi/3\). Both new coordinates \(k',l'\) have magnitude comparable to \(b\). At fixed \(k'\), the inequality \(|k|\le h\) restricts \(l'\) to an interval of length \(4h/\sqrt3\). For example \(0.8b\le|k'|\le1.8b\) and \(0.4b\le|l'|\le1.1b\) suffice when \(h<b/10\). The central Cauchy density is bounded there, uniformly in \(s>0\), by \(1/(2\pi|l'|)\le K/b\); its folded tails are bounded by \(M_P\). The uniform endpoint obeys the same bound, and the \(s=0\) atom misses the interval. The relevant horizontal band has bounded \(\nu\) mass by (46), enlarging by finitely many dyadic bands if necessary. Apply (42) and note that projection adds nonnegative mass to the rotated physical cap. This proves (48). Applying the bound with \(b=\delta b_0\) and \(h=\delta h_0\), then sending \(h_0\downarrow0\), proves the last assertion for every vague limit; continuous cutoff majorants justify passage to such limits. ◻

Proof. Fix \(0<a<b<\infty\) and work on the rescaled horizontal band \(a\le|k|/\delta\le b\). Rescale the mixing variables to \((k/\delta,s/\delta)\). Their masses are bounded by finitely many instances of (46). Their integral of \(D\) tends to zero by (45), since the unscaled bands shrink to zero while excluding it. On this band the scale-invariant function \(D\) is bounded below away from zero whenever \(s/\delta\) stays outside any fixed neighborhood of \(|k|/\delta\); this includes \(s/\delta=\infty\). Thus every limiting mixing measure is concentrated on the graph \(s=|k|\) in the rescaled coordinates. More explicitly, mass outside \(|s/|k|-1|<\varepsilon\) is bounded by a constant depending on \(\varepsilon\) times the integral of \(D\), and tends to zero. This also gives tightness in the rescaled \(s\) coordinate.

Uniformly for \(a/2\le t\le2b\) and \(l\) in a compact set, \[\delta q_{\delta t}(\delta l) =\frac{t}{\pi(t^2+l^2)}+O(\delta),\] by the folded-tail bound in the proof of Lemma 28. Consequently every further extraction of the bounded mixing measures gives, on \(k\ne0\), \[ M(\,\mathrm dk,\,\mathrm dl)= \frac{|k|}{\pi(k^2+l^2)}\,\eta(\,\mathrm dk)\,\mathrm dl \tag{50}\] for some locally finite positive measure \(\eta\) in the tangential coordinate. A diagonal extraction over compact horizontal subbands makes these descriptions consistent. The other physical preimage of a bounded rescaled set shrinks to \(a_*\); its mass tends to zero by local finiteness there and \(C\{a_*\}=0\), already proved in Lemma 27. Thus (50) indeed describes the physical measure, with no alias contribution.

Multiplying (50) by \(|p|^2\) shows that the normal distributional derivative of \(|p|^2M\) vanishes on \(k\ne0\). No density assumption on \(\eta\) is being made. Repeating in the other two row directions, at every nonzero point at least two of the three tangential coordinates are nonzero. The corresponding normal directions are independent. On a neighborhood of that point the two independent directional derivatives of \(|p|^2M\) vanish. A distribution with this property is constant there: convolve on a smaller ball with a smooth approximate identity, observe that both derivatives of each convolution vanish, and pass to the limit. The constants agree on overlaps, and the punctured plane is connected. It follows that \(|p|^2M=c\,\mathrm dp\) there. This argument covers the axes by the other charts; Lemma 29 also directly excludes concentration on them. Positivity and the annular bound give \(0\le c<\infty\). ◻

Proof. Choose an even smooth \(q_0\) with \(0\le q_0\le1\), equal to one on \([-1/2,1/2]\) and zero outside \((-1,1)\). Let \(Q\) be a smooth radial cutoff supported in a sufficiently small physical disk and equal to one near zero. For \(h>0\) put \[ J(h)=\int Q(p)\chi(k,l) [q_0(|p|/h)-q_0(k/h)]\,C(\,\mathrm dp). \tag{51}\] Each integrand vanishes near zero and has compact support in the physical chart, so this is a finite integral. We first show that \(J(h)\) has a full limit as \(h\downarrow0\). Split its bracket as \[[q_0(|p|/h)-1]+[1-q_0(k/h)].\] The first term has integral zero by sixfold rotation, since its other factors except \(\chi\) are radial and it vanishes near zero. For the second term replace the physical cutoff \(Q\) by the rectangular strip \(0<|k|<\kappa\) in (42). The resulting error has a limit by dominated convergence: in a sufficiently small physical neighborhood of zero the cutoffs and centered coordinates agree; all remaining contributions come from a finite measure away from that origin, including the alias neighborhood. Their multipliers are bounded and converge pointwise as \(h\downarrow0\). On \(k=0\) the factor \(1-q_0(k/h)\) is identically zero.

For the strip contribution set \(A(k,s)=\int\chi(k,l)Q_s(\,\mathrm dl)\). Equations (44), (45), and (43) give \[|A(k,s)|\le D(k,s)+K|k|, \qquad \int_{0<|k|<\kappa}|A(k,s)|\,\nu(\,\mathrm dk,\,\mathrm ds)<\infty.\] Dominated convergence applied to \(\int(1-q_0(k/h))A(k,s)\nu(\,\mathrm dk,\,\mathrm ds)\) proves that the strip contribution has a limit. Thus \[ L:=\lim_{h\downarrow0}J(h) \tag{52}\] exists, using only the summable defect and finite cutoff errors, without any assumption about a spectral scaling limit.

Now take any sequence along which \(C_h\to c\,\mathrm dp/|p|^2\). In the rescaled variable the bracket in (51) is \(q_0(|p|)-q_0(k)\), which vanishes on \(|p|\le1/2\). On a dyadic annulus \(R<|p|\le2R\) with \(R>10\), it is supported on \(|k|\le1\). Equation (48), used at physical scales \(b=hR\) and horizontal width \(h\), bounds that mass by \(K/R\). Summing dyadic annuli gives a uniform \(O(1/R)\) tail, up to the edge of the fixed support of \(Q\). Vague convergence on the remaining compact annulus therefore gives \[\begin{align*} L&=c\int_{\mathbb R^2}\frac{\chi(k,l)}{|p|^2} [q_0(|p|)-q_0(k)]\,\mathrm dp \\ &=c\int_0^{2\pi}\cos(4u) \int_0^\infty[q_0(r)-q_0(r|\cos u|)]\frac{\,\mathrm dr}{r}\,\mathrm du =-\frac\pi2c. \tag{53}\end{align*}\] For completeness, for \(t>0\) the inner radial integral with \(t=|\cos u|\) equals \(\log t\): integrate first from \(\varepsilon\) to \(M\), change variables in the second term, then use that \(q_0=1\) near zero and vanishes for large argument. Its absolute integral is bounded by \(K(1+|\log t|)\), which is integrable in \(u\), justifying Fubini. The angular value can be checked without a Fourier-series limit: \[\int_0^{2\pi}\cos(4u)\log|\cos u|\,\mathrm du =4\int_0^{\pi/2}\cos(4u)\log(\cos u)\,\mathrm du =4\int_0^{\pi/2}\sin^2u\cos(2u)\,\mathrm du =-\frac\pi2.\] The middle equality follows by integration by parts; the boundary term \(\sin(4u)\log(\cos u)\) vanishes at both ends. Since \(L\) was defined by a full limit, (53) fixes the same \(c\) for every subsequence. Subsequence compactness then gives full vague convergence. ◻

Proof of Proposition 24. For the exact area weights define \(A_{\delta,f}(p)=\sum_uw_{\delta,f}(u)e^{ip\cdot u}\). On every fixed compact set of \(\zeta\), \[ A_{\delta,f}(\delta\zeta)\longrightarrow\widehat f(-\zeta) \quad\text{uniformly}. \tag{54}\] Indeed the defining sum is the integral of \(f(z)\) times the exponential at its face center. Replacing that center by \(z\) changes the exponential by \(O(\delta|\zeta|)\) on each face. The total integral is exact, so no lattice-cell area factor remains.

We record bounds that also control frequencies outside a fixed rescaled annulus. For every integer \(N\ge1\), in a fixed small physical neighborhood of zero, \[ |A_{\delta,f}(p)|\le K_N \min\left\{\frac{|p|}{\delta}, (1+|p|/\delta)^{-N}\right\}, \tag{55}\] and on the complement of that neighborhood it is \(O_N(\delta^N)\). The first bound follows from exact zero total and \(\sum_u|w_{\delta,f}(u)|\,|u|=O(\delta^{-1})\). For the other bounds, let \(\Delta_e\) denote a unit lattice difference in a basis direction. Translation of faces gives \[\sum_u|\Delta_e^Nw_{\delta,f}(u)| \le\int_{\mathbb R^2}|\Delta_{\delta e}^N f(z)|\,\mathrm dz \le\delta^N\|\partial_e^N f\|_{L^1}.\] Discrete summation by parts therefore bounds \(|1-e^{ip\cdot e}|^N|A_{\delta,f}(p)|\) by this last quantity. Near zero at least one of the two basis multipliers is at least \(a|p|\); away from zero on the reciprocal torus at least one is bounded below on a finite compact cover. Together with \(|A_{\delta,f}|\le\|f\|_1\), these estimates prove (55). Their constants are uniform for a bounded smooth family with a common compact support.

Equations (39) and (40) now allow passage to the full Fourier integral. For example the contribution of \(|p|<\epsilon\delta\) to a variance is \(O(\epsilon^2)\) by the first bound in (55) and dyadic summation. The contribution of \(R\delta<|p|<r_0\) is \(O_N(R^{-2N})\); the remaining compact part of the torus contributes \(O_N(\delta^{2N})\) by local finiteness. On \(\epsilon\delta\le|p|\le R\delta\) use Lemma 31 and (54). Let first \(\delta\downarrow0\), then \(\epsilon\downarrow0\) and \(R\to\infty\). This gives the variance limit \(c\int|\widehat f(p)|^2|p|^{-2}\,\mathrm dp\); polarization gives the bilinear formula. The same estimates prove the assertion for smoothly converging tests.

The distribution \((2\pi)^{-1}\log(1/|z|)\) is the fundamental solution of \(-\Delta\) on the plane. Its ambiguity by a constant is irrelevant to zero-integral tests. Fourier inversion with the stated convention thus gives the second line of (37) and the normalization \[ v=(2\pi)^2c,\qquad \sigma(x)=\sqrt v. \tag{56}\] Finiteness of \(c\) was already proved. Lemma 23 supplies two fixed smooth nonnegative unit averages on separated disks whose difference \(f\) has \(\inf_{\delta\text{ small}}\mathbb EH_\delta(f)^2>0\). The covariance limit therefore rules out \(c=0\). All constructions have used only the fixed plane law at weight \(x\); there is no domain or approximation parameter in \(v\). ◻

Proof. All Laplace tests have integral zero. Reflection in a site row permutes the hexagonal faces exactly, so \[\theta_{\ell_\delta}X_\delta =H_\delta\bigl(\Delta(\varphi_\delta\circ\theta_{\ell_\delta})\bigr)\] as zero-total increment observables. Their supports, including the finitely many faces meeting the test supports, lie strictly on opposite sides once \(\delta\) is small. The tests and their reflections converge smoothly on fixed compact sets. By Proposition 24, the limit of their covariance is \[\frac{v}{(2\pi)^2}\int |p|^2\widehat\varphi(p) \overline{\widehat{\varphi\circ\theta_\ell}(p)}\,\mathrm dp =v\int\nabla\varphi(z)\cdot \nabla(\varphi\circ\theta_\ell)(z)\,\mathrm dz=0,\] because the two supports are disjoint. Reflection positivity makes the finite-mesh quantity a squared seminorm. The ordinary second moments are bounded by Proposition 22, or by Proposition 24. Adjoining independent Haar variables does not change the reflection norm of the phase-independent \(X_\delta\). Finally apply reflection Cauchy–Schwarz. Only the reflection seminorm vanishes; an ordinary variance need not do so. ◻

Proof. Associate to each successive edge of the honeycomb boundary cycle its incident interior face. At a trivalent boundary vertex, the two such faces are either the same, or meet across the third edge. Joining their centers therefore gives the required walk after constant steps are removed. This also proves the \(O(\delta)\) position bound. Every face adjacent to the boundary appears. An edge path leaving the enclosed face set first passes through such a face, proving separation.

One can deform each boundary edge to the corresponding center path in the union of its incident faces and the stars of its endpoints. This deformation never meets a nonboundary interior face center. Consequently the original cycle and the center walk have the same winding number around every free interior center. In particular those centers lie in bounded complementary components of the walk.

Use the uniformly convergent parametrizations in Definition 1, and parametrize the center walk in the corresponding traversal order. The position error still tends to zero uniformly. Normal projection is uniformly continuous on a smaller closed tubular collar, so the projected walks converge uniformly to the corresponding degree-one parametrization of the boundary circle. They have degree one for fine mesh by homotopy in that circle. For any fixed \(b\), uniform continuity of the limiting parametrization gives \(\omega(b)>0\) such that a projected excursion of diameter \(b\) needs at least \(\omega(b)\) units of parameter time, once the uniform error is smaller than \(b/4\). Disjoint excursion intervals therefore number at most \(1/\omega(b)\), after normalizing the parameter circle to length one. Finally lift the projected walk to the real line. Its net change in one traversal is \(L\). Before a first visit to the right endpoint of a lifted proper arc, take its last visit to the left endpoint. The intervening subwalk stays in that interval and crosses it. Discrete endpoints introduce only \(o(1)\) errors. This proves the final assertion. ◻

Proof. A retained component contains a traversal segment beginning next to a removed interval and reaching distance at least \(\epsilon\) from \(\mathcal C\), without visiting a removed vertex. Indeed use a maximal portion of the original traversal outside the removed set that visits a deep vertex. Its projection travels a distance at least \(\epsilon/2-o(1)\). For distinct graph components these portions have disjoint time interiors, because a shared traversal vertex would join the components. Lemma 33 bounds their number. The same proof, using any fixed smaller constants in the two thresholds, covers rounding at endpoints. It puts no bound on the number of short excursions confined to the windows. ◻

Proof. The integer inverse image of \(B=+\) is the union of the pairs \(4t+\{0,1\}\), whereas that of \(W=+\) is the union of \(4t+\{-1,0\}\). A nearest-neighbor chain of constant first color cannot leave its lifted pair, since different pairs have distance at least three. Connectedness of \(S_{0,\delta}\) and its zero anchor therefore give precisely (59); second-color plus pins select the zero endpoint of the pair. This argument establishes the integer branch, rather than merely its residue modulo four.

Lemma 33 separates every free site from exterior interactions. After conditioning on the skeleton heights, the nearest-neighbor weights factor over the interior components and the exterior; factors between fixed sites affect only the trace probability. Lemma 3 identifies each interior factor with the weighted height law. Applying Lemma 15 to each fixed trace, and then averaging the resulting kernels, proves (60). All these traces have absolute value at most one, so Proposition 21 gives the last assertion. In particular the upper mean is between zero and one, and the lower mean between minus one and zero, by comparison with the flat laws. ◻

Proof. Consider one intervening projected arc. Every retained graph component touches the boundary of one of the deleted intervals: otherwise it would be a component of the original connected walk with no neighbor outside. More explicitly, a maximal unremoved traversal interval containing a vertex at depth \(2a\) starts or ends at depth \(a+o(1)\). It gives a pinned chain from that endpoint to depth \(2a-o(1)\), hence a crossing of the projected marker at depth \(3a/2\). There is also a retained component crossing the whole intervening arc. The last-left-visit, first-right-visit construction in Lemma 33 gives such a chain; since \(a\) is much smaller than the arc length, it crosses both endpoint markers. Thus every component reaches at least one marker, and one component joins the two markers.

Fix an endpoint marker \(s_*\) and its spatial point \(z_* =\gamma(s_*)\). All crossings of that marker, however numerous, occur in a ball \(B(z_*,r_\delta)\) with \(r_\delta\to0\). This follows from uniform convergence into the normal collar; an edge crossing the projected coordinate adds only an \(O(\delta)\) error. Choose a fixed outer radius \(\rho\ll a\) such that \(B(z_*,2\rho)\subset U_i\) and the ball avoids all second-color pins and equality edges. Each of the relevant pinned chains enters the inner ball and leaves the outer one, because its endpoints have projected distances at least \(a/2-o(1)\) from \(s_*\). At these short scales arclength and Euclidean distance along the \(C^2\) boundary are uniformly comparable.

First fix \(r>0\), and then take \(\delta\) small enough that \(r_\delta<r\). By Lemma 10, a stack of favorable good circuits between radii \(r\) and \(\rho\) fails with probability at most \(C(r/\rho)^\alpha\). Apply this estimate keeping all second-color constraints first as equality edges \(E_0\). They avoid this annulus, and every first-color pin in it is favorable. Restoring the actual plus labels costs at most \(2^{M(\mathcal L,\epsilon)}\) by Lemma 34 and the component-label construction of Section 2. A good surrounding circuit joins all the pinned chains through its constant first-color boundary layer. Every joined chain therefore has the same actual lifted pair. This costs one circuit event at the marker, regardless of the number of chains.

There are finitely many markers. After the mesh limit send \(r\) to zero to make all their failure probabilities vanish. The traversing component ties together the two marker pairs of each group, proving (64). Translations and reflections preserve this geometric argument; apply the buffer estimate afresh in their images. Only spins and increments enter \(C_i\), so no reflection invariance of the particular cell augmentation is being assumed. ◻

Proof. Work first in a finite simply connected free ambient triangulation, large enough to contain all neighborhoods and collars. On \(P_{\mathrm{loc}}\), anchor \(H(v_1)=0\). The constrained height arrays are exactly those satisfying, for integers \(t_i\) with \(t_1=0\), \[ \begin{aligned} H(v)&\in4t_i+I\quad &&\text{on all first-color pins in group }i,\\ H(v)&=4t_i&&\text{on its second-color pins}, \end{aligned} \qquad I=\begin{cases}\{0,1\},&\text{upper bracket},\\ \{-1,0\},&\text{lower bracket}. \end{cases} \tag{67}\] The forward implication is the definition of the checks. Conversely these constraints impose both colors at their prescribed sites and one pair on each whole group. Simple connectedness and the Gray-code lift exclude additional winding constraints. The array weight is exactly the product of the original edge weights.

Partition these arrays by \(\kappa=(t_2\bmod2,\ldots,t_m\bmod2)\), so there are at most \(2^{m-1}\) classes. If \(H,H'\) belong to the same class, put \(s_i=(t_i+t_i')/2\in\mathbb Z\). On a first-color pinned site, each of \(\lfloor(H+H')/2\rfloor\) and \(\lceil(H+H')/2\rceil\) belongs to \(4s_i+I\); on a double pin both equal \(4s_i\). Thus both rounded outputs satisfy (67), although their parity classes can differ from the input class. The anchor stays zero. The weighted midpoint verification in Section 3 establishes the edge-weight premise of Theorem 18. What remains is a positive central mass in every input class.

In the test neighborhood require one good cut surrounding the test, with a fixed positive separation from its support. In the collar of each nonanchor obstacle require two nested good cuts of opposite signs of the first color in \(U_i\setminus V_i\), enclosing the entire check support. All these searches have disjoint fixed buffers free of equality edges and second-color labels. Lemma 10, successive conditioning, and the bounded label cost give a joint conditional probability at least \(c_0>0\) under \(P\). Expose the test cut from the exterior and choose the collar cuts by their corresponding stopped explorations. Lemma 37 implies that intersecting this event with \(P_{\mathrm{loc}}\) still has probability at least \(c_0/2\) under \(P\) for fine mesh.

The pin checks can be read within the \(U_i\), outside the entire test search region. Conditional on the exterior record and on the actual height layer of the test cut, its inside is the unconstrained weighted height problem with a consecutive-pair trace. Subtract its possibly random common offset. Since \(\int f=0\), this does not change \(Y_\delta\). Proposition 21, using a finite partition into smaller smearing balls if necessary, gives a fixed \(M_0\) and a conditional probability at least \(1/2\) of \(|Y_\delta|\le M_0\). Consequently a central event \(\mathcal A\), including all selected cuts and checks and the condition \(|Y_\delta|\le M_0\), has \[ \mathbb P(\mathcal A\mid P_{\mathrm{loc}})\ge c_1>0, \qquad \mathcal A\subset\{|Y_\delta|\le M_0\}. \tag{68}\]

Lemma 14 supplies in each nonanchor collar a weight-preserving involution shifting all its inner heights by \(+4\) or \(-4\), relative to the exterior. The relevant component is chosen from the first-color, status, and cut record, without using its fair second-color label. Its flip preserves that record. All spins on the protected inner and outer regions, all pin labels, and all local checks remain unchanged. It also preserves \(Y_\delta\) and its test-cut record: the entire test search neighborhood lies outside \(U_i\), in the protected exterior of the switch. The disjoint collars give commuting involutions; the \(i\)th one flips precisely the parity of \(t_i\). These involutions act on the augmented spin and status measure. The event \(\mathcal A\) records only the selected cuts, the checks, and the displayed bound on \(Y_\delta\); it does not fix the extra exterior heights used to estimate its probability. It is therefore invariant under the involutions. Each orbit has one equally weighted member for every parity vector, so \[ \mathbb P\bigl(|Y_\delta|\le M_0,\ \kappa=k \mid P_{\mathrm{loc}}\bigr) \ge 2^{-(m-1)}c_1\quad\text{for every }k. \tag{69}\] Marginalizing the statuses preserves this identity for the height parity classes. The orbit calculation does not assert independence after conditioning on the checks. When \(m=1\) no collar switch is needed.

The face-area weights of \(f\) have bounded total absolute mass, and coordinatewise rounding changes the associated linear statistic by at most half that mass. Increase \(M_0\) to absorb this bound. Apply Lemma 20 with (69) and the parity partition. It gives (66), uniformly in the ambient volume. Exhaustion at each fixed mesh gives the plane assertion; Fatou’s lemma transfers the moment bounds. With no obstacles use Proposition 22. The construction uses only fixed positive margins and finitely many requests, proving the stated uniformity and the finite-collection version by Hölder’s inequality. ◻

Proof. Let \(J\) denote the product of the probabilities on this side. Squaring the norm gives \[J^{-2}\mathbb E\left[\overline{\vartheta A}A\, \mathbf 1_{P_{\mathrm{loc}}\cap\vartheta P_{\mathrm{loc}}}\right].\] The reflected arrangement consists of two copies of every obstacle, all with disjoint filled neighborhoods. The strict row margin separates the two copies. Their individual probabilities are the same \(p_{i,\delta}\), so (65) bounds the probability of the ordinary doubled pin event by \(CJ^2\). The checked event has no larger probability. If \(A\) contains an insertion \(Y\), Proposition 38 in this doubled arrangement and Cauchy–Schwarz bound \(\mathbb E[|Y\,\vartheta Y|\mid P_{\mathrm{loc}}\cap \vartheta P_{\mathrm{loc}}]\) by a constant. Bounded factors and unit-modulus phases contribute only their uniform bounds. The factor \(J^2\) cancels exactly, proving the assertion. ◻

Proof. Choose \(\eta>0\) smaller than the actual projection gap between the enlarged supports. Back the high family up by \(\eta_\delta=N_\delta b_\delta\to\eta\), where \(N_\delta\) is a positive integer. There is still a fixed gap; choose a site row in it, converging to a line in the limiting gap. Let \(B_\delta\) be the normalized high-side factor in this backed arrangement and let \(A_\delta\) be the reflected conjugate of the normalized low-side factor. Lemma 39 gives \(\|A_\delta\|_\vartheta,\|B_\delta\|_\vartheta\le C\). Individual \(p_{i,\delta}\) are invariant under lattice translation and reflection, so their product is the same throughout the motion.

For \(\Re z>-\eta_\delta\) set \[ f_\delta(z)=\int_{[0,1]} \lambda^{(z+\eta_\delta)/b_\delta}\, \gamma_{A_\delta,B_\delta}(\,\mathrm d\lambda), \tag{72}\] where the finite cross measure is constructed in Lemma 61, with its circle variable integrated out. For \(\Re w>0\) use \(\lambda^w=e^{w\log\lambda}\) on \(\lambda>0\) and give it value zero at \(\lambda=0\). Corollary 62 proves that this is holomorphic and bounded by \(\|A_\delta\|_\vartheta\|B_\delta\|_\vartheta\). Thus \(|f_\delta|\le C^2\), and all fine meshes have the common domain \(\Re z>-\eta/2\), which contains zero as an interior point.

At \(s_\delta=m b_\delta\ge0\), the exponent in (72) is the positive integer \(m+N_\delta\). The transfer interpretation therefore gives the exact equality \(f_\delta(s_\delta)=M_\delta(s_\delta)\). This never evaluates a zeroth spectral power at the kernel. Cauchy’s estimates give a uniform derivative bound on compact subsets of the common half-plane, so converging lattice roundings sample the same limiting function. Every sequence has a subsequence on which \(f_\delta\) converges locally uniformly to a bounded holomorphic function. It vanishes on \(I\) by hypothesis, hence throughout the half-plane by the identity theorem, and in particular at zero. Subsequence extraction proves the claim for the original sequence.

Rigid translation preserves distances inside each family, and moving the families apart increases their projection gap. The preliminary backoff was smaller than that gap. Thin filled neighborhoods and check paths can therefore be kept on their own sides at every step; their reflected copies have the same separation. These observations verify the side-norm hypotheses, including for convergent variations of the other placements. Reversing \(n\) proves the low-family version. ◻

Proof. Translate \(z_0\) to zero for this geometric calculation. In each interval \((\rho/40,\rho/20)\) and \((-\rho/20,-\rho/40)\) choose \(q+1\) regular levels with mutual distances exceeding \(\rho/(80(q+1))\). They exist because regular values are dense and each candidate interval has length \(\rho/40\). Their union is the grid. A spectator’s closed vertical projection has length less than \(2\rho/(1000(q+1))\), so it meets at most one level in either candidate family. At least one level in each family therefore avoids all \(q\) disks, with a positive margin. These two levels strictly bracket \(K_0\).

Every boundary point and middle-strip spectator point is outside \(B(0,\rho)\) and has \(|y|<\rho/20\). Thus \(|x|>\sqrt{399/400}\,\rho\), and \(|(\sqrt3/2)x+y/2|>\rho/2\), with the sign of \(x\). A connected middle boundary arc or spectator disk cannot change this sign. It consequently belongs wholly to \(L\) or \(R\), whereas the \(u\)-projections of \(K_0\) have absolute value at most \(\rho/1000\). Opening small intersection gaps and choosing sufficiently thin filled neighborhoods preserves these strict inequalities. The construction of the grid preceded the choice of the disks, proving the last assertion. ◻

Proof. Figure 2 records the successive separating projections. Keep \(X_\delta\) fixed and move all other factors, including each check and each phase with its obstacle. First translate \(T\) by \(a_+e\) and \(B\) by \(-a_-e\). Each is an entire extreme side across an existing horizontal gap. Choose these two distances sufficiently large that \(T\) is strictly above \(K_0\) and \(B\) strictly below it in \(u\)-projection. With (74), the families \(Q_+=T\cup R\) and \(Q_-=B\cup L\) are then strictly above and below \(K_0\), respectively, in that projection.

Translate \(Q_+\) by \(bu\), with \(b\) sufficiently large that it is vertically above \(K_0\) and below \(K_0\) in \(u'\)-projection. Both requirements are possible because \(u\cdot e=1/2\) and \(u\cdot u'=-1/2\). Translate \(Q_-\) by \(-cu\), with \(c\) so large that it is above \(K_0\cup Q_+\) in \(u'\)-projection. Each of these moves increases an existing \(u\) gap. Finally translate \(Q_-\) by \(du'\). It is an entire extreme side across a strict \(u'\) gap, and \(u'\cdot e=1/2\), so large \(d\) puts it vertically above \(K_0\) as well. At every move, within-family distances stay fixed and between-family projection gaps increase. Thus no filled obstacle neighborhood meets another obstacle or the insertion neighborhood. The required thin local paths remain available.

All distances can be chosen in open ranges, in the order \(a_+,a_-,b,c,d\). The translations can be made exactly on the lattice. If \(a_1=(1,0)\) and \(a_2=(1/2,\sqrt3/2)\) generate \(\mathbb T\), then \[\sqrt3 e=-a_1+2a_2,\qquad \sqrt3 u=a_1+a_2,\qquad \sqrt3 u'=-2a_1+a_2.\] Round each physical distance to a multiple of \(b_\delta\). The five rounding errors are \(O(\delta)\); all the finitely many strict margins therefore survive. Site reflection rows in these directions have spacing \(b_\delta/2\), so a row can also be chosen in every fixed gap.

At the final placement there is a horizontal site row separating \(X_\delta\) from every other factor. Reflection Cauchy–Schwarz, Lemma 39, and Proposition 32 bound the absolute value of the normalized expectation by \(C\|X_\delta\|_\vartheta\to0\). Adding phases does not change this norm of the phase-free insertion.

For every fixed admissible \(a_+,a_-,b,c\), the last assertion holds for \(d\) in a nonempty open interval of sufficiently large values and for every lattice rounding. Lemma 40 therefore removes the \(du'\) move. The resulting assertion holds for every sufficiently large \(c\) with earlier distances fixed, so the same lemma removes the \(-cu\) move. Continue in the order \(bu\), \(-a_-e\), \(a_+e\). At each stage later distances may depend on the already fixed earlier ones; the open-interval assertion is exactly what the analytic lemma needs. This proves (75) at the original placement. Empty families simply omit their moves. ◻

Proof. For each whole gap use a logarithmic stack between radii \(Ca\) and \(r_0\), retaining the second-color prescriptions first as equality edges \(E_0\). All first-color pins in this stack are favorable and there is no equality edge crossing it. A surrounding good plus circuit joins the traversing first-color arms on its two sides. The arms exist by the degree-one traversal in Lemma 33; they run from the inner gap neighborhood to beyond the chosen outer collar. Together with the local checks, such a circuit identifies the lifted pairs of the neighboring groups. The stack failure probability is at most \(C(a/r_0)^\alpha\). Restoring the boundedly many second-color plus labels costs at most a fixed factor \(2^M\). There are finitely many gaps, and their successive connections join all groups cyclically. Lemma 37 thus proves (80), with a bound of this power form plus terms whose mesh limsup is zero.

For comparison with the filled bracket, keep exactly the same second-color equalities \(E_0\) in both laws. Add the missing favorable first-color pins in one gap and use the ordered stopped coupling of Proposition 6. Search outward from the changed region for a common good plus circuit in the same stack. Except with probability \(C(a/r_0)^\alpha\), such a cut is found before the search meets any equality edge or label prescription. The only changes are on its near side, so the correct stopped kernels agree on its far side by Lemma 4. Sample those remaining variables identically. This matches the spins and increments in all test disks, and also the second-color data needed to read every prescribed plus label. The exterior of the small gap ball contains paths within each test disk, so the matching of pair variables there indeed matches its full local increment information.

Repeat for the finitely many gaps, including all discarded shallow components inside the changed balls. The conditional laws at every stage use the common \(E_0\) and their own correct query marginals. After this coupling restore the label event. Under each equality law its probability is at least \(2^{-M}\). If the joint records match the label data outside an exceptional event of probability \(r\), then for any event of the test data its two label-conditioned probabilities differ by at most \(C2^M r\). This follows by comparing the two joint numerators and the two label-event denominators separately. Thus the bounded label cost preserves the vanishing total-variation error. Finally conditioning the gapped law further on \(P_=\) changes its total variation by at most \(\mathbb P(P_=^c\mid P)\), which proves (81). No estimate here divides an additive error by the probability of the rare first-color pin event: all couplings already have those conditional laws as their marginals.

It remains to prove (82); the separated estimate (66) alone does not provide a uniform constant as the obstacle gaps close. Fix one cut-search annulus around the test support, independent of \(a\), away from the boundary collar. Under \(P\) it has a favorable good cut with conditional probability at least \(c>0\), uniformly in \(a\): use the buffer bound with all remote equalities retained and then the same bounded label cost. By (80), for all sufficiently small \(a\) its intersection with \(P_=\) has conditional probability at least \(c/2\) in mesh limsup. More precisely choose \(a\) so that \(\omega(a)<c/4\), and then take the mesh sufficiently small.

The common-offset event can be read outside this whole search disk. Indeed its local checks use the boundary neighborhoods, and one can join all representatives by paths in the complement of the disk. Expose the cut from outside. Conditional on the exterior and its height trace, the interior is again the unconditioned weighted height problem with a consecutive-pair trace. Subtracting its offset and applying Proposition 21 provides a central interval \(|X_\delta|\le M_1\) with positive conditional mass, where \(M_1\) is independent of \(a\).

In the anchored array description of \(P_=\) every offset is zero. Thus the constraints are just the fixed intervals \(I\) and fixed singletons \(\{0\}\) in (67). Both rounded midpoints of any two such arrays are admissible, without a parity partition. Lemma 20, with the preceding uniform central mass and bounded rounding error, proves (82). Ambient exhaustion is justified as in Proposition 38. The filled brackets have the corresponding uniform moments by Lemma 36. ◻

Proof. At each fixed sufficiently small \(a\), equations (65) and (80) give \(\liminf_{\delta\downarrow0}\mathbb P(P_=)/J_\delta>0\). Divide (79) by this quantity to obtain \(\mathbb E_{\mu_{\delta,a}^{=}}[X_\delta F_\delta]\to0\). Let \(T_M(t)=\max(-M,\min(t,M))\). The second moments in (82) and in the filled bracket imply \[\mathbb E[|X_\delta-T_M(X_\delta)|]\le C/M\] in both laws, uniformly in small \(a\) after the mesh limit. Total variation in (81) therefore yields \[\limsup_{\delta\downarrow0} |\mathbb E_{\mu_{\delta,0}}[X_\delta F_\delta]| \le 2C_F M\omega(a)+2C_F C/M.\] First take the mesh limit at fixed \(a\) as displayed, then let \(a\downarrow0\), and finally let \(M\to\infty\). This proves (61). Weak convergence of the local test statistics, continuity of their bounded factors, and the same moment bound pass the identity to subsequential bracket limits.

The window size \(\epsilon\) has stayed fixed throughout. The boundary estimates in Section 6 will control the effect of shrinking these windows, and Section 7 will then transfer the identity to the true field. In particular no separated-obstacle constant was used uniformly when the gaps collided; the independent common-offset estimate was the ingredient that permitted their closure. ◻

Proof. Choose shortened cores of the prescribed arcs, leaving disjoint transition neighborhoods between consecutive cores. The image of an appropriate consecutive portion of the boundary traversal gives a connected graph for each core. Compact limiting cores of opposite types are separated, so their images neither intersect nor have an edge between them at sufficiently small mesh. Any portion not yet assigned is in one of the transition neighborhoods. Repeated visits inside a core belong to that same connected core image. Any repeated visit that could affect assignment outside the cores is localized to a transition neighborhood: otherwise two traversal parameters a fixed positive distance apart would converge to the same point of the Jordan curve. Uniform continuity of its inverse parametrization excludes this possibility.

Grow the connected core sets in the finite walk graph. For example, give each unassigned vertex to a nearest core in graph distance, breaking ties in a fixed order, and use a shortest path to that core as its parent. Each resulting part is connected. Each component of the still unassigned graph had a neighboring core, since the entire walk graph is connected; hence the procedure assigns every vertex. The growth stays in the transition neighborhoods until it meets an already assigned core. In particular it never changes a prescribed core or its type. Merge neighboring parts of the same type. Added separator pieces are assigned low and joined to the low parts they meet. Their clearance from the high patch makes this consistent. Every added piece attaches to an existing low part, so it creates no new part; it may merge existing low parts. In particular no cost is charged for the number of real components of an added circuit.

Pin \(W=+\) or \(W=-\) on the resulting parts. Their adjacency graph is connected. On each edge of a spanning tree joining opposite-sign parts, choose one corresponding skeleton edge and prescribe \(B=-\) there. Its two endpoints already have opposite \(W\) signs, so coherence forces their \(B\) signs to agree. There are at most \(M\) such labels. Anchor the lift by \(K(o)=-1\) at a plus endpoint of one of these edges. Along a connected constant-\(W\) part the lifted pair is unchanged; a labeled transition changes \(-1\) to \(-2\). Propagation along the spanning tree therefore gives (84). Extra contacts between parts create no further choice of lift: the ambient lift is single valued. The configuration \(B\equiv-\) shows that all the imposed conditions are simultaneously feasible. The separate flat comparison below fixes both spins and does not use this construction.

Conditional on \(W\) and its statuses, the \(B\) components have fair independent labels. Requiring at most \(M\) of them to have label minus has conditional probability at least \(2^{-M}\), even if several requests refer to the same component. This proves the last assertion. The height Gibbs property follows on conditioning on the whole trace and integrating out the statuses. On each high part the synthetic trace is at most zero, and on each low part it is at most \(-2\). Lemma 15 gives the claimed comparison on every retained component. ◻

Proof. Two eligible cycles have inner and outer envelopes made of edges in their union. To see this, if they share at least two vertices, their union is two-vertex-connected. Indeed, after deleting any vertex, each cycle is connected and at least one common vertex remains. The face of this union containing the inner box and its unbounded face have simple cycle boundaries. They give the required envelopes. The inner box is connected and disjoint from the cycles, so it lies in a single face. If the cycles have no common vertex or only one, the requirement that both surround the box makes them nested. In every case the envelopes stay in the original strip and surround its inner box. Finitely many repeated envelopes give the least and greatest Jordan interiors. Geometric interiors, not just sets of enclosed lattice vertices, determine this order.

For a proposed innermost circuit \(\gamma\), the assertion that it is selected is equivalent to its being open and to the nonexistence of an eligible open circuit with strictly smaller interior contained in its interior. A nonnested competitor would give such a circuit by the inner-envelope operation. The equivalent assertion reads only the sites on or inside \(\gamma\) in the strip. The outer case is identical with the order reversed. Deterministic tie conventions for a traversal of the selected cycle do not reveal additional sites. An unsuccessful strip can be read in its entirety.

All these selection events are consequently measurable in the discarded regions and the separating traces. The nearest-neighbor height density factors over the remaining components, proving the Gibbs and independence claims. Only the goodness test of a genuinely exterior site is deterministic. Every selected real site retains its actual observed height in this factorization. ◻

Proof. Choose \(s/b\) small, to be fixed at the end of the proof. Apply Lemma 45 with one high patch as stated and the rest of the skeleton low. The separators are low and meet the low base part, so the number \(M\) of transition labels is bounded by an absolute constant. Work first with this synthetic height \(K\) in a large free ambient triangulation containing the local boxes. Any sufficiently large finite volume suffices: after conditioning on the full skeleton trace, the interior height law is independent of that ambient volume. All estimates below are uniform in this choice. Let \(z\) be a lattice site at distance \(O(\delta)\) from \((0,3s)\).

Proof. Compare from below with the flat trace \(-2\) on the connected bounding skeleton. Realize this flat problem by constant spin pins in the ambient graph. In an allocated annulus around a tube ball, ask for at least three nested good first-color circuits with alternating signs, using disjoint buffers. The conditional buffer estimates give positive probability. Constancy of the other color on the connected skeleton is first imposed as equality. Its single component label is fair conditional on the first color and its statuses. Prescribing that label therefore leaves the circuit-search probability unchanged. Between successive opposite-sign circuits there are enclosing first-color walls, so there are at least two enclosing height loops in the allocated annulus. The loops lie wholly in the free bulk.

Let \(E\) be the event that at least two enclosing height loops lie wholly in the allocated annulus. This event depends only on the unoriented geometry and contains the alternating-color circuit event, so its probability has the same positive lower bound. Condition now only on unoriented geometry in \(E\). Choose two consecutive enclosing loops in the annulus by a deterministic rule, ordered from outer to inner. All loop jumps are independent fair signs under the flat boundary, by Lemma 15. The height outside the first chosen loop is \(-2\) plus a symmetric sum of other signs, and hence is at least \(-2\) with probability at least \(1/2\). Independently, both chosen jumps are positive with probability \(1/4\). With probability at least \(1/8\) their inner lip therefore supplies a surrounding site circuit of height at least zero. Each loop’s two site lips have constant heights: at every trivalent vertex on the loop the third edge is unoccupied, joining the same-height sites along each lip. Loops not enclosing the target ball cannot change this lip height. A surrounding site circuit can be extracted from its connected site walk.

The allocated annuli can be chosen so that successive surrounding site circuits must intersect. For example take equal inner radii \(a\), outer radii \(2a\), and successive center distance \(3a/2\). The enclosed inner disks overlap, whereas either inner disk has a portion beyond the other outer disk; disjoint surrounding circuits could consequently be neither unnested nor nested. After small fixed changes to the radii for lattice rounding, planarity forces an intersection. Finite strings of these annuli give the desired tube paths. Each surrounding-height-circuit event is increasing, so height association multiplies their lower bounds, first in the flat problem and then in the actual problem by height comparison. Only finitely many balls are needed for the fixed tube network. ◻

Proof. We give the increasing construction. All choices of relative widths below are fixed once and for all, depending at most on \(x\).

Proof. The bracket trace belongs to \([-1,1]\) and equals zero away from the finitely many exceptional windows. Choose pairwise disjoint fixed outer neighborhoods of their centers, missing \(\mathop{\mathrm{supp}}f\). Given \(N\), choose \(N\) separated positive scales in each neighborhood and then take \(\epsilon\) sufficiently small that its exceptional window lies inside the innermost box. Finally take the mesh small with this geometry fixed.

Condition on the bracket trace. Proposition 49 gives increasing level-zero arches around all windows except with probability at most \(JC e^{-cN}\), where \(J\) is their fixed number. Select innermost arches by testing from the exceptional side. On simultaneous success every retained component meeting \(\mathop{\mathrm{supp}}f\) has nonnegative usable trace: each new real separator site has height at least zero and the old visible base trace is zero. By height comparison the conditional mean on each such component is nonnegative, and hence so is their sum against a nonnegative \(f\). Thus, using the unconditional bracket moment bound of Proposition 21, \[\mathbb EH_\delta^{\pm,\epsilon}(f) \ge -\|H_\delta^{\pm,\epsilon}(f)\|_2 (JC e^{-cN})^{1/2} \ge -C_f e^{-cN/2}.\] The same argument with decreasing arches gives the upper bound. In particular no bound conditioned on arch failure is needed. The moment constant is uniform over the original traces in \([-1,1]\). Since \(N\) is arbitrary, this proves (92) for nonnegative tests. Splitting a smooth signed test into a difference of nonnegative smooth compactly supported tests proves the general case. ◻

Proof. At mesh \(\delta\) put \(Y_{z,\delta}=h_\delta(\rho_z)\), and denote the corresponding lattice spectator product by \(F_\delta\). At a fixed \(z\), the distance from its lattice representative to the prescribed skeleton tends to \(d(z)\). The exact face-area weights of \(\rho_z\) have total mass one, are supported in a ball of radius \(\theta d(z)+O(\delta)\), and are bounded by a constant times \((\delta/d(z))^2\). The ratio between boundary distance and smearing radius is thus bounded above and below independently of \(z\) once the mesh is taken sufficiently small for that fixed placement. Proposition 21, with original trace zero, gives (93). The passage to the limit uses the higher moments from the same proposition. Compactness of the smooth boundary makes its constants uniform over boundary points; no estimate uniform in the approximation error divided by a moving depth is asserted before taking the mesh limit.

Choose a small fixed outer neighborhood of \(p_0\) missing all the spectator supports; for approaching spectators this is possible uniformly once their limiting distinct points have been fixed. Given \(N\), choose \(N\) fixed separated scales inside this neighborhood. For all sufficiently close \(z\) the support of \(\rho_z\) is in the inner box of this stack. Apply Proposition 49 to the true zero boundary. Select an outermost increasing arch by testing from outside and reveal its exterior and real trace. The spectator product \(F_\delta\), including its absolute height values, is measurable in that exterior record. On success, the retained region containing the testing ball has nonnegative usable trace, and hence nonnegative conditional mean for its unit nonnegative average. For any nonnegative exterior-measurable factor \(U\) it follows that \[\mathbb E[Y_{z,\delta}U] \ge -\mathbb E\big[|Y_{z,\delta}|U\mathbf 1_{\{\text{increasing search fails}\}}\big].\] The separate decreasing search gives the corresponding upper bound. These searches reveal only their tested sides; they do not condition on the intermediate constructions in the arch proof.

Apply the two bounds to \(U=(F_\delta)_+\) and \(U=(F_\delta)_-\), the positive and negative parts of the whole spectator product. If \(F\) has \(k\) factors, Hölder’s inequality and the unconditional \(2(k+1)\)-moment bounds give \[\mathbb E\big[|Y_{z,\delta}F_\delta|\mathbf 1_{\{\text{a search fails}\}}\big] \le \|Y_{z,\delta}F_\delta\|_2\, \mathbb P(\text{a search fails})^{1/2} \le C_F e^{-cN/2}.\] The constants remain bounded for comparable-depth spectators at the other fixed boundary points, by (93). In particular the estimate uses moments in the original zero-boundary law, not moments under a failure event or a random high trace. First take the mesh limit at fixed positions and scales. Then take the indicated approaches to the distinct boundary points. The resulting limsup is at most \(C_F e^{-cN/2}\). Since \(N\) was arbitrary, this proves (94).

Finally, (93) bounds the diagonal terms in \(\mathbb EZ_m^2\) by \(C/m\). Each of the finitely many off-diagonal terms vanishes as the depths tend to zero, by (94) with one spectator. This proves the first assertion of (95). Minkowski’s inequality gives its uniform \(L^p\) bounds independently of \(m\). Interpolation between \(L^2\) and any larger finite moment then proves the final claim; for \(p\le2\) use monotonicity of \(L^p\) norms. These are absolute-height references, with no lift constant left unspecified. ◻

Proof. A consequence of Proposition 21, using a finite cover of a compact set by interior disks, is \[ \sup_\delta\|h_\delta(f)\|_{L^p} \le C_{K,p}\|f\|_\infty,\qquad \mathop{\mathrm{supp}}f\subset K\Subset D,\quad 1\le p<\infty. \tag{96}\] Indeed, split the face weights by a smooth partition of unity subordinate to that cover. Each resulting weight family has bounded total variation and the required maximum weight on its fixed disk; apply the smearing estimate and the triangle inequality. The argument applies to the brackets after conditioning on their traces, whose values lie between \(-1\) and \(1\). Its constants thus do not depend on the windows. Complex tests are handled by their real and imaginary parts.

Here is the compactness argument in detail. Place \(D\) in a square torus and fix \(\chi\in C_c^\infty(D)\). For its Fourier basis \(e_k\), \(k\in\mathbb Z^2\), let \(a_{\delta,k}=h_\delta(\chi\overline{e_k})\). Since the moduli of the basis functions are constant, (96) bounds \(\|a_{\delta,k}\|_{L^p}\) independently of \(k\). For \(s>1\) and \(p\ge2\), Minkowski’s inequality gives \[\begin{align*} \bigl\|\|\chi h_\delta\|_{H^{-s}}^2\bigr\|_{L^{p/2}} &\le \sum_{k\in\mathbb Z^2}(1+|k|^2)^{-s} \|a_{\delta,k}\|_{L^p}^2 \le C_{\chi,p,s}. \tag{97}\end{align*}\] The bounds for smaller \(p\) follow by monotonicity of \(L^p\) norms. Choose \(1<s<t\). A bounded \(H^{-s}\) ball is relatively compact in \(H^{-t}\): its Fourier tail beyond frequency \(M\) has squared \(H^{-t}\) norm at most \((1+M^2)^{-(t-s)}\) times its squared \(H^{-s}\) norm, and its first finitely many coordinates lie in a bounded finite-dimensional set. Markov’s inequality and (97) therefore give tightness of \(\chi h_\delta\) in \(H^{-t}\).

Choose nested interior compacts exhausting \(D\), and smooth cutoffs equal to one on successive compacts. For any prescribed error probability, choose a compact set for the \(n\)-th cutoff with error at most that probability times \(2^{-n}\). The simultaneous localizations then lie in a relatively compact set for the local \(H^{-t}\) topology. This proves the stated local tightness. Equivalently, this topology is the compatible subspace of a countable product of the localized Sobolev spaces, with its usual metrizable topology. The same argument and a union bound treat finite joint couplings.

For fixed \(f_1,\ldots,f_k\), a monomial \(\prod_i h_\delta(f_i)^{a_i}\) has uniformly bounded \(L^{1+\epsilon}\) norm, for instance with \(\epsilon=1\), by (96) and Hölder’s inequality at sufficiently high orders. Its uniform integrability, joint convergence of the pairings, and truncation prove convergence of that mixed moment. The same argument gives moment convergence with bounded continuous functions of finitely many pairings. In particular all the preceding moment bounds pass to every limit. ◻

Proof. Fix a finite grid of regular horizontal levels and positive exceptional windows sufficiently small for the given arrangement. We pass first to the mesh limit and then to the window limit, keeping the chosen true-field marginal throughout.

Couple its lower and upper brackets with the true field in order, as in Lemma 36. First take a joint law limit along the chosen subsequence, with the windows fixed. Next shrink the windows and take a further joint limit. Proposition 52 applies uniformly throughout, so these extractions are possible and the middle marginal is always the already chosen law of \(H\). Distributional order passes to the limits because evaluation against each nonnegative smooth test is continuous. Write the resulting ordered triple as \[H^-\le H\le H^+.\] Proposition 50 and moment convergence give \(\mathbb EH^-(f)=\mathbb EH^+(f)=0\), as well as \(\mathbb EH(f)=0\). For every nonnegative smooth \(f\), both nonnegative random variables \(H^+(f)-H(f)\) and \(H(f)-H^-(f)\) thus have expectation zero. Apply this to a countable dense family of nonnegative tests on an exhaustion of \(D\). Continuity of distributions then shows \(H^-=H=H^+\) almost surely as distributions.

At fixed windows, Proposition 44 gives the insertion identity for bounded local increment factors. For bounded continuous functions of zero-total pairings it passes to the first law limit by uniform integrability of \(H(\Delta\varphi)\), and then to the window limit by the same argument. The collapsed marginal is the chosen \(H\), so this proves that identity for \(H\). Repeat for any desired finite grid; no change to the chosen marginal is involved. Finally truncate each \(H(g_\ell)\) by a bounded continuous truncation. Hölder’s inequality and moments of orders larger than the number of factors show that the expectation of the truncation error tends to zero. This proves (99). ◻

Proof. Choose distinct points \(p_1,\ldots,p_m\) on a fixed boundary arc. For small \(a>0\), put the centers a distance \(a\) along their inward normals and use bumps \(\psi_{\ell,a}\) of radius \(\theta a\), where \(0<\theta<1/4\) is fixed. The tubular neighborhood theorem for a \(C^2\) Jordan curve gives distance exactly \(a\) to the boundary for these centers when \(a\) is sufficiently small. Define \[Y_{\ell,a}=H(\psi_{\ell,a}),\qquad R_{m,a}=\frac1m\sum_{\ell=1}^m Y_{\ell,a}.\] The comparable-depth moment bound in Proposition 51 is uniform along this fixed arc. Minkowski’s inequality therefore gives \(\sup_{m,a}\|R_{m,a}\|_{L^q}\le C_q\) for each finite \(q\), after restricting to the corresponding sufficiently small depths. For fixed \(m\), the same proposition gives \(\mathbb E[Y_{\ell,a}Y_{k,a}]\to0\) when \(\ell\ne k\). Consequently \[ \limsup_{a\downarrow0}\|R_{m,a}\|_{L^2}^2 \le \frac{C_2^2}{m}. \tag{100}\] For each \(m\), choose a positive depth \(a_m\) small enough that this squared norm is at most \((C_2^2+1)/m\) and all disk diameters meet the prescribed bound. Then \(R_m=R_{m,a_m}\) tends to zero in \(L^2\). For \(p>2\), interpolate this convergence with the uniform \(L^q\) bound for any \(q>p\); for \(p\le2\), use monotonicity. This proves the assertion. The order in (100) is depth first at fixed \(m\), then \(m\to\infty\). ◻

Proof. We prove the first-variable distributional Laplace identity in a small product neighborhood of any distinct tuple; symmetry then gives all the identities. Translate the first center to the origin. Choose \(R>0\) so that \(B(0,4R)\Subset D\) and all the other centers lie outside \(\overline{B(0,4R)}\). Choose \(0<b<R/1000\) and \(0<\eta<R/(1000j)\). Fix the finite regular grid supplied by Lemma 41 for \(z_0=0\), \(\rho=R\), and \(q=j-1\). Its choice is independent of the small spectator disks and chains constructed below. Take \(\varphi\) supported in \(B(0,b)\), and take each spectator test \(f_i\), \(2\le i\le j\), in a sufficiently small neighborhood of its center, of diameter less than \(\eta/4\). We will show \[ \mathbb E\left[H(\Delta\varphi)\prod_{i=2}^j H(f_i)\right]=0. \tag{101}\] For \(j=1\) this follows from the zero mean; suppose \(j\ge2\).

Write \(c_i=\int f_i\), and choose a unit bump \(\rho_i\) in that spectator neighborhood. Take a reference \(R_{m,a}\) from the preceding lemma, at a fixed positive depth small enough that all its disks have diameter less than \(\eta/4\) and lie outside \(\overline{B(0,4R)}\). This is possible because that closed ball is a compact subset of \(D\), whereas the reference disks can be placed arbitrarily near the boundary. Every center can be joined to each reference center by a finite path in \(D\setminus\overline{B(0,R)}\). To verify this last geometric point, start with a polygonal path in the connected open set \(D\). Whenever it enters the protected disk, replace that portion by an arc in \(B(0,2R)\setminus\overline{B(0,R)}\); this annulus lies in \(D\). A small perturbation and a finite subdivision give a path with compact image and positive clearance from both the protected disk and the boundary.

Along each such path choose a finite chain of smooth unit bumps, beginning with \(\rho_i\) and ending with \(\psi_{\ell,a}\). Make the step lengths and bump radii so small that the union of two consecutive supports is contained in a disk of radius less than \(\eta\), disjoint from \(\overline{B(0,R)}\). If necessary interpolate their radii in finitely many steps near the endpoints. Telescoping yields \[H(f_i)-c_iH(\psi_{\ell,a}) =H(f_i-c_i\rho_i) +c_i\sum_{\text{chain}}H(\rho_{\mathrm{old}}-\rho_{\mathrm{new}}).\] Every summand on the right is a zero-total test in one of these small disks. Averaging this equality over \(\ell\) expresses \(H(f_i)-c_iR_{m,a}\) as a finite linear combination of such observations.

Expand the product of these \(j-1\) finite sums. Each monomial selects at most \(j-1\) spectator disks, whatever the number or length of the chains. The insertion support lies in \(B(0,R/1000)\), and every selected disk is outside \(B(0,R)\) with radius less than \(R/(1000j)\). These are precisely the hypotheses for the grid already fixed from Lemma 41. For each monomial its two selected levels give the oblique separation of the middle-strip boundary pieces and spectators needed by Proposition 44. The pair of levels may depend on the monomial; the grid does not.

There are only finitely many monomials. Choose the exceptional windows small enough to preserve all their positive margins. Lemma 53 gives zero expectation for each monomial with the insertion. Summing these exact identities gives \[ \mathbb E\left[H(\Delta\varphi) \prod_{i=2}^j\bigl(H(f_i)-c_iR_{m,a}\bigr)\right]=0. \tag{102}\] All disks, chains, tests and grid levels used to obtain this equality are finite and fixed before using the limiting insertion identity.

It remains to remove the reference. This error is estimated before the chain expansion. The algebraic difference between the product in (102) and \(\prod_iH(f_i)\) is a sum of \(j-1\) terms, each containing one factor \(c_iR_{m,a}\) and other factors of the form \(H(f_k)\) or \(H(f_k)-c_kR_{m,a}\). Hölder’s inequality and the uniform moment bounds give, for some finite \(p\) depending only on \(j\), \[\left|\mathbb EH(\Delta\varphi) \left\{\prod_{i=2}^jH(f_i) -\prod_{i=2}^j(H(f_i)-c_iR_{m,a})\right\}\right| \le C\|R_{m,a}\|_{L^p}.\] The constant does not count chain terms. Choose depths and then \(m\) as in Lemma 54; the right side tends to zero. This proves (101).

Finite sums of tensor tests are dense in the test-function space on each such product neighborhood, for example by a localized Fourier expansion with convergence of every derivative. Consequently \(\Delta_{z_1}T_j=0\) there as a distribution. Symmetry gives all \(j\) separate Laplace equations. Their sum is the ordinary elliptic Laplacian on \(\mathbb R^{2j}\); the local regularity theorem for harmonic distributions makes \(T_j\) smooth there. The separate distributional equations then hold pointwise. ◻

Proof. Let \(T_{j,h}\) be the convolution of \(T_j\) with a product of \(j\) radial unit bumps of common radius \(h\), considered on a fixed interior compact and with \(h\) small enough to remain in \(D\). By Hölder’s inequality and (98), \[ |T_{j,h}(\mathbf z)| =\left|\mathbb E\prod_{i=1}^jH(\rho_{z_i,h})\right| \le C(1+\log(1/h))^{j/2}, \tag{105}\] including at coincident centers. Whenever \(d(\mathbf z)>2h\), the product of bump supports is off the diagonals. The radial mean value property, successively in each harmonic variable, gives \(T_{j,h}(\mathbf z)=T_j(\mathbf z)\). Taking \(h\) to be the smaller of \(d(\mathbf z)/4\) and a fixed interior radius proves (103).

For completeness, a collision tube \[A_h=\{\mathbf z:d(\mathbf z)\le2h\}\] in a fixed compact set has volume at most \(C_jh^2\): use the union bound over pairs, fix all but one point in a selected pair, and integrate that point over a disk of radius \(2h\). The same bound at every radius \(s\), followed by integration of the distribution function, gives for \(q\ge0\) \[\int_{A_h}\bigl(1+\log_+(1/d(\mathbf z))\bigr)^q\,\mathrm d\mathbf z \le C_qh^2(1+\log(1/h))^q.\] Indeed for \(q>0\), integrate \(q(1+\log(1/s))^{q-1}\,\mathrm ds/s\) against the bound \(Cs^2\); the endpoint term is \(Ch^2(1+\log(1/h))^q\). This proves local integrability. It also shows that the off-diagonal function has vanishing \(L^1\) mass on \(A_h\). Equation (105) gives the same assertion for \(T_{j,h}\), since \(h^2(1+\log(1/h))^{j/2}\to0\). Off \(A_h\) the two functions agree exactly. Thus \(T_{j,h}\) converges locally in \(L^1\) to the off-diagonal function. But convolution also converges to \(T_j\) in distributions. These limits are equal. In particular neither a delta distribution nor any of its derivatives on a collision diagonal can be present. The case \(j=1\) has no collision set and follows directly from harmonicity.

To prove (104), choose disjoint small fixed radial averaging disks around \(z_2,\ldots,z_j\). For \(z_1\) near \(p\), choose its averaging radius to be \(\theta\mathop{\mathrm{dist}}(z_1,\partial D)\). The supports are pairwise separated. The harmonic mean value property in every variable identifies the displayed pointwise value with \[\mathbb E\left[H(\rho_{z_1,\theta\mathop{\mathrm{dist}}(z_1,\partial D)}) \prod_{i=2}^jH(\rho_{z_i,s_i})\right].\] Proposition 51 makes this tend to zero. For \(j=1\) the same conclusion also follows from the zero mean. ◻

Proof. For fixed parameters, the angular Fourier coefficients of a harmonic function in a punctured disk have the forms \(a_0+b_0\log r\) and \(a_nr^n+b_nr^{-n}\), \(n\ge1\), for the sine and cosine modes. Equation (103) bounds the function by a power of \(1+\log(1/r)\). Multiplying its \(n\)-th Fourier coefficient by \(r^n\) and sending \(r\downarrow0\) therefore shows \(b_n=0\) for every \(n\ge1\). After subtracting the logarithmic term, the remaining nonnegative modes are the harmonic extension of their values on any fixed smaller circle, and hence extend across the center.

On a circle of radius \(s>0\) enclosing no other spectator, the coefficient is the flux \[ L(y,\mathbf z) =-\frac1{2\pi} \int_{|x-y|=s}\partial_{n_x}T_j(x,y,\mathbf z)\,\mathrm d\ell(x). \tag{107}\] One can keep \(s\) fixed on a neighborhood of any separated parameter tuple. Parameterizing \(x=y+se^{i\theta}\), the integrand is smooth by Proposition 55; thus \(L\) is smooth. Applying a spectator Laplacian under this integral also proves its separate harmonicity. Subtract \(L\log(1/|x-y|)\) on that circle. Its values are smooth and uniformly bounded on compact parameter sets. The Poisson formula for their harmonic extension, differentiated in \(x\) on a smaller disk, yields the claimed uniform derivative bounds for \(R\). ◻

Proof. Choose two real smooth signed tests \(\alpha,\beta\) with zero integral, supported in disjoint disks, such that \[I=\iint\alpha(u)\log(1/|u-w|)\beta(w)\,\mathrm du\,\mathrm dw\ne0.\] Such tests can be obtained by taking horizontal derivatives of unit mollifiers concentrated about two distinct points on a horizontal line. As their radii tend to zero, integration by parts makes the displayed integral tend to the mixed horizontal derivative of the logarithmic kernel at these two points, which is nonzero. Fix one such pair. For a fixed point \(y_0\), define \[\alpha_r(x)=r^{-2}\alpha((x-y_0)/r),\qquad \beta_r(x)=r^{-2}\beta((x-y_0)/r).\] Let \(F=\prod_{i=3}^jH(\rho_{z_i,s_i})\), where the spectator disks are fixed, small, pairwise disjoint and separated from a fixed closed disk \(\overline{B(y_0,R_0)}\Subset D\). An empty product is one. We first prove \[ \lim_{r\downarrow0} \mathbb E[H(\alpha_r)H(\beta_r)F] =\frac{v}{2\pi}I\,T_{j-2}(\mathbf z). \tag{109}\] We will evaluate this limit in two ways. A plane copy independent of the spectators gives the right side of (109); the punctured harmonic expansion will give \(I L(y_0,\mathbf z)\). Their equality determines \(L\). For the first evaluation, the plane copy must be independent of the exterior record that determines the absolute spectator heights.

Fix \(r>0\) small enough that both inner supports lie in \(B(y_0,Cr)\) and \(\sqrt r<R_0/4\), and return to the finite mesh along the subsequence defining \(H\). Use the ambient pair representation of the true law, with both colors plus on the connected boundary skeleton and height anchored to zero at one of its sites. As in Lemma 33, this gives the required interior height law. Retain the complementary-spin boundary constraint first as edge equalities on this connected set. Conditional on the first spins and cell statuses, prescribing its single plus component label has probability exactly \(1/2\). It therefore creates no additional bias in a first-color circuit search.

Search from outside for a good first-color circuit in separated annular slots between scales \(\sqrt r\) and \(R_0/2\). Lemma 10 gives failure probability at most \(Cr^{\kappa_1}\) in the mesh-first limsup, for some \(\kappa_1>0\). The exploration always queries a cell status together with all its first-color sites, using the stopped kernels of Proposition 11. Let \(\mathcal G_\delta\) contain the stopped query record (including its success or failure outcome), the selected cut and its first-color data when it exists, and the genuinely exterior pair data needed below. Do not reveal complementary-spin values at strictly inside boundary-layer sites merely because their first-color values were queried.

We verify both the measurability and the conditional law at this stage. Each spectator disk can be joined to the zero-height anchor by paths outside \(B(y_0,R_0)\). One may choose paths in the ambient triangulation, outside the search disk, to the anchored skeleton. Their sums of the local increments from Lemma 3 give the absolute spectator heights, with the correct common lift fixed at the anchor. Alternatively, portions of paths in \(D\setminus\overline{B(y_0,R_0)}\) can detour around the interior disk and then reach the boundary layer; their fixed compact portions occur in every sufficiently fine approximation. The ambient construction requires no microscopic regularity of that last approach to the skeleton. Include all these exterior paths and the pair data on the spectator disks in \(\mathcal G_\delta\). Thus the entire discrete spectator product \(F_\delta\) is \(\mathcal G_\delta\)-measurable, as an absolute-height observation, not merely a spin observation.

Every crossing cell of the selected contour is open with prescribed first color. Its factor is constant and leaves the complementary spin unrestricted. No imposed equality edge crosses the contour. By Lemma 4, closed-cell complementary components on its two sides are disconnected. Hence revealing exterior labels reveals no strictly interior component label. The stopped query convention likewise fixes no further strictly interior first-color variables. Conditional on \(\mathcal G_\delta\), the unexamined interior is exactly the open-cut law with no inner pins or equalities. Its height lift may have an exterior-dependent additive constant; both local tests have exactly zero total and discard that constant.

The cut surrounds a ball of radius comparable to \(\sqrt r\). The retained-law and exterior-measurability verifications above allow us to apply Corollary 12. It couples the increments in \(B(y_0,Cr)\), with its connecting paths, to the fixed plane increment marginal \(\nu_\delta\) with error \(C(r/\sqrt r)^{\kappa_2}\), uniformly in each successful exterior record. For every such record choose a coupling kernel whose second marginal is this same \(\nu_\delta\). On failed searches choose an independent variable with marginal \(\nu_\delta\). These are ordinary conditional couplings on finite discrete spaces; the finite ambient volume may subsequently be exhausted using the same local comparison. In particular the resulting plane variable \(P_\delta\) satisfies \[\mathbb P(P_\delta\in A\mid\mathcal G_\delta)=\nu_\delta(A)\] for every measurable event \(A\), also on failure. This identity proves genuine independence of \(P_\delta\) from \(\mathcal G_\delta\), and therefore from \(F_\delta\). It is stronger than specifying just its unconditional marginal.

Write \(U_\delta=h_\delta(\alpha_r)\), \(V_\delta=h_\delta(\beta_r)\), and let \(U'_\delta,V'_\delta\) be the corresponding statistics of this plane copy. Matching increments in a connected disk implies equality of both statistics because their area weights have exact total zero. If \(E_\delta\) is the union of search failure and coupling mismatch, then for some \(\kappa>0\) \[\limsup_{\delta\downarrow0}\mathbb P(E_\delta)\le Cr^\kappa.\] Cauchy–Schwarz and equality off \(E_\delta\) give \[\begin{align*} &\left|\mathbb E[(U_\delta V_\delta-U'_\delta V'_\delta)F_\delta]\right| \\ &\quad\le \mathbb P(E_\delta)^{1/2} \left(\|U_\delta V_\delta F_\delta\|_{L^2} +\|U'_\delta V'_\delta F_\delta\|_{L^2}\right). \tag{110}\end{align*}\] Applying Hölder inside each norm at an order at least \(2j\), (98) bounds the first norm by \(C(1+\log(1/r))\). All fixed spectator norms are bounded. The plane zero-total moment bound in Proposition 22 bounds the second norm uniformly. Thus the mesh limsup of (110) is at most \[Cr^{\kappa/2}(1+\log(1/r)),\] which tends to zero.

For each fixed \(r\), first let the mesh tend to zero. Moment convergence treats the true field and the spectators, whereas independence factors the plane term as \(\mathbb E[U'_\delta V'_\delta]\mathbb E[F_\delta]\). Proposition 24 evaluates its first factor: \[\lim_{\delta\downarrow0}\mathbb E[U'_\delta V'_\delta] =\frac{v}{2\pi} \iint\alpha_r(x)\log(1/|x-y|)\beta_r(y)\,\mathrm dx\,\mathrm dy =\frac{v}{2\pi}I.\] The last equality uses exact zero totals to cancel the dilation constant. Separate harmonicity and radial mean value give \(\mathbb EF=T_{j-2}(\mathbf z)\). Only now let \(r\downarrow0\); the preceding error bound proves (109).

We read the same limit from (106). First average its spectator variables on their radial disks. Their harmonic mean value property, including that for \(L\) from Lemma 57, evaluates them at \(\mathbf z\). After substituting \(x=y_0+ru\), \(y=y_0+rw\), the singular term is \[\iint\alpha(u)\beta(w)L(y_0+rw,\mathbf z) \{\log(1/r)+\log(1/|u-w|)\}\,\mathrm du\,\mathrm dw .\] For every fixed \(w\), the term containing \(\log(1/r)\) is zero because \(\int\alpha=0\). This cancellation is exact even though \(L\) varies with the second colliding variable. The remaining kernel is bounded on the two disjoint supports; smoothness of \(L\) makes the remaining integral tend to \(I L(y_0,\mathbf z)\). For the regular term, subtract \(R(y_0;y_0+rw,\mathbf z)\) before integrating against \(\alpha(u)\). The uniform first-variable derivative bound from Lemma 57 bounds its contribution by \(Cr\). The left side of (109) therefore tends to \(I L(y_0,\mathbf z)\). Since \(I\ne0\), comparison proves (108). ◻

Proof. Fix the distinct \(y_i\)’s and subtract the right side of (111) from its left side as a function of \(x\). It is harmonic outside these finitely many points. The Dirichlet Green function for the negative Euclidean Laplacian has local singularity \[\frac1{2\pi}\log(1/|x-y_i|).\] Proposition 58 shows that its coefficient agrees with the logarithmic singularity of \(T_j\) at each point. Lemma 57 therefore makes the difference removable there. It is harmonic on all of \(D\). Equation (104) gives zero limit at every boundary point, as does the Dirichlet Green function on a \(C^2\) Jordan domain. Thus the difference extends continuously by zero to the boundary. The maximum principle, applied to it and to its negative, makes it identically zero. This proves (111). The base \(T_1=0\) follows from the exact height-reflection symmetry and uniform integrability.

Induction gives zero for odd moments and, for even \(j\), the sum over pairings of the products of \(vG_D\). Lemma 56 says that these are identities of the entire moment distributions, including tests crossing diagonals. The Green products are locally integrable; in each pairing their variables occur in disjoint pairs, and the logarithmic singularity is integrable in two dimensions. It follows that for every real \(f\in C_c^\infty(D)\), \(H(f)\) has exactly the centered Gaussian moments with variance \[v\iint_{D\times D}f(z)G_D(z,w)f(w)\,\mathrm dz\,\mathrm dw.\] These moments determine its law. One direct justification avoids assuming an exponential moment in advance: the nonnegative series for \(\cosh(tH(f))\), monotone convergence, and the known even moments give \(\mathbb E\cosh(tH(f))=\exp(t^2\mathop{\mathrm{Var}}(H(f))/2)\) for every real \(t\). Since \(e^{|tH(f)|}\le2\cosh(tH(f))\), the exponential moment is finite. The absolutely integrable power series for its characteristic function then yields the Gaussian characteristic function. This also covers zero variance.

Apply this conclusion to \(f=\sum_{\ell=1}^k a_\ell f_\ell\), for every real vector \((a_1,\ldots,a_k)\). All these real linear combinations are Gaussian with the stated covariance form. Their characteristic functions identify the joint vector \((H(f_1),\ldots,H(f_k))\) with that of \(\sqrt v\,\Phi_D\).

Finally choose a countable dense family of smooth tests on an interior exhaustion of \(D\). Evaluation on these tests determines a distribution and generates the Borel sigma-field of the local Sobolev spaces used in Proposition 52; this follows, for example, from their realization by countably many localized Fourier coordinates. Hence equality of the finite-dimensional laws on this family determines the random-distribution law. We chose the original subsequence arbitrarily, so every limit allowed by tightness has the same law. Tightness and uniqueness give convergence of the full sequence, for each \(t>1\) in \(H^{-t}_{\mathrm{loc}}(D)\), and therefore as distributions. Continuity of test evaluation gives the claimed simultaneous test limits, and Proposition 52 also gives their mixed-moment convergence.

The normalization is \(\sigma(x)=\sqrt v\). Its strict positivity, finiteness, and independence of \(D\) and the approximation were proved in Proposition 24. All estimates here are for fixed \(x\) in the stated closed interval, so the endpoint is included with no additional limiting argument in the parameter. This completes the proof of Theorem 2. ◻

Proof. Cap a longer periodic cylinder at each end by a triangulated disk; triangular fans suffice. Every mod-two cycle on this sphere is generated by triangular face boundaries. Consequently, for fixed \(B=b\), a possible \(W\) disagreement set satisfies the same triangle equations as in the proof of Proposition 6. On a mixed-\(b\) triangle its single homogeneous edge cannot belong to that set. Restricting to homogeneous-plus edges or to homogeneous-minus edges therefore separately preserves every triangle equation. Each restriction integrates to a spin configuration, and the wall factorization (12) holds, with its same factor \(1/2\).

For an unobserved triangle cell, \(r_y(b)\) is \(1\) or \(x^2\) according as \(b\) is constant or mixed; for an edge cell it is \(1\) or that edge’s parameter. On an unobserved homogeneous cell the corresponding allowed wall-edge parameter is \(x\), or the specified single-edge parameter; on an observed open cell it is \(1\). Equality edges have parameter zero in the wall partition sums. The cell inequality for \(r_y\) and the intersection/union inequalities for the two allowed-edge vectors are unchanged. The Ising partition-sum inequality (11) thus proves the same two-law inequality. Restoring statuses uses only disjointness of the cells’ edge sets. This proves association and conditional order, including parameters \(1\).

For the cut assertion, include among the selected triangles all cells responsible for authorizing the contour. Each crossing cell has its first spins prescribed and contributes a constant, exactly as in Lemma 4; the remaining factors separate. This proof is local and applies to a winding contour, which separates the capped sphere. The equality constraints do not cross the contour by hypothesis. Two such cuts isolate the law between them. The excluded boundary-cell factors are precisely the constants removed by the cuts. No extra winding constraint is imposed: an annular configuration satisfying the cut-layer pins extends coherently to either cap by taking the prescribed first spin constant there, and the exterior partition factors are independent of the interior second spins. ◻

Proof. An open \(W\) triangle with prescribed sign contributes the constant \(q\) for every assignment of its \(B\) spins. Represent its prescription by single-edge parameters \(1\) on its three edges and by \(W\) equality along those edges, then specify the common \(W\) label. The incident-site union of the contour cells is connected: consecutive cells along the contour share a site. Before that label is specified, global \(W\) flip pairs the two possible labels with equal weight, for every \(B\) and every permitted \(B\)-status history. Specifying its sign therefore multiplies each such marginal weight by \(1/2\). It does not change the order comparison. Apply Lemma 63 with these deterministic equality edges. ◻

Proof. Work on a common sufficiently long capped cylinder. Prescribe both \(a_0\) and \(a_1\). By the cut property, the law above \(a_0\) is unaffected by the additional prescription at \(a_1\). Releasing the old open-minus prescription at \(a_0\) increases the probability of \(A\), by Lemma 63. Next prescribe \(b_1\), which increases that probability, and discard the farther contour \(b_0\) by the cut property. This proves (112). Translate the new cylinder back up by \(d\) to obtain (113); the lattice translation preserves its class \(T\).

For (114), prescribe both \(b_0\) and \(b_1\) and then release the nearer positive prescription at \(b_0\). This decreases the probability of \(A\), by Lemma 64. The distant prescription did not affect the original law below \(b_0\), by the cut property. All simultaneous prescriptions used here are feasible: one may take the complementary spin constant; when the fixed \(W\) contour is present, take that constant to be its prescribed sign. ◻

Proof. Use the face-invasion construction after Lemma 9. Start from the lower face layer and invade across unauthorized honeycomb edges, querying a responsible cell together with all its spins of the searched color. Restrict queries to the search band; declare failure if the invasion reaches its upper edge. An authorized winding with clearance prevents that failure. The boundary against the uninvaded component toward the upper end contains a separating authorized winding contour. Retain such a simple contour, discarding contractible boundary components. Trivalence prevents a four-edge crossing at a boundary vertex.

Every queried cell touches the invaded face set. If it is not favorable, all its incident edges are unauthorized and the invasion reaches all its incident faces. Thus a queried cell meeting the retained side must be a favorable contour cell. The only first spins revealed there are the prescribed cut-layer spins; complementary-spin labels are never queried. The record’s remaining restrictions lie below the contour. The product factorization of Lemma 63, with the colors interchanged when necessary, now identifies the conditional law above it. We condition on the actual successful search record, not on an additional event in the retained region. A bounded enlargement of the band accommodates conversion between triangular-cell windings and honeycomb contours. ◻

Proof. Couple the two color occupations using opposite tests on each uniform, as in Definition 3.5 of (Glazman and Lammers 2025). Since \(q\ge1/2\), every cell belongs to at least one occupation. If there is no vertical \(B\)-occupied path, triangular-site path/circuit duality on the cylinder gives a winding of cells outside \(B\) occupation, hence inside \(W\) occupation. Adjacent occupied cells of a given color share a site, so their spin signs agree. Both a vertical path and a winding therefore have one constant sign, proving the four-way alternative.

For the last assertion, lift a path to the flat strip cover and let \(m\) be the band height. If a vertical path’s horizontal extent is at most \(5m\), one of finitely many width-\(6m\) windows contains it. Otherwise it has a horizontal subcrossing of width greater than \(4m\). A winding also has such a horizontal subcrossing. Choose horizontal centers on a net of spacing at most \(m/4\) and leave fixed slack for lattice rounding. Taking subpaths yields the local crossing tests. Their number depends only on the fixed circumference-to-height ratio. The duality argument is unchanged by bounded adjustments of the band edges. ◻

Proof. For fixed retained \((B,\tau)\), the complementary-spin sum is two to the number of components of the graph of closed cells and equality edges. Exterior components missing all ports contribute a common factor. The other exterior components only wire together a partition of the ports, merging at most \((K-1)_+\) retained components. The ratio of unnormalized weights, after removal of the common exterior factor, is therefore in \([2^{-(K-1)_+},1]\). The same bounds hold for the ratio of partition sums. Division proves the assertion. Constant complementary spins satisfy all these equalities, so the two first-spin/status laws have the same support. ◻

Proof. At most \(M\) exterior complementary-spin components meet the patch. Restrict their free labels to agree, losing a factor at most \(2^{(M-1)_+}\), and set all patch complementary spins to this common label. Choose the compatible first-spin extension. On every unprescribed cell select its open status if homogeneous, and its closed status if mixed. All its local weights are positive. If \(a(x)>0\) is the least positive member of \(x,x^2,1-x,1-x^2\), omitting zero entries, the resulting local weight is at least \(a(x)^M\). There are at most \(2^M3^M2^M\) local spin/status assignments, each of weight at most one. These bounds give a uniform lower bound for the ratio of the restricted and unrestricted conditional partition sums. In particular the argument includes \(x=1\): a prescribed closed cell must then be mixed. For an exterior marginal \(\mu\), conditioning multiplies its density by \(f/\mu(f)\), where \(c\leq f=\mathbb P(A\mid\hbox{exterior})\leq1\). This proves the last assertion. ◻

Proof. Use \(E_-=E_0\cup F\) for the lower law and \(E_+=E_0\) for the upper law in (12). For first-spin configurations \(b,c\), put \(m=b\wedge c\) and \(j=b\vee c\). On \(F\) the plus-wall vectors satisfy \[a_-^+(m)=0=a_-^+(b)\wedge a_+^+(c),\qquad a_+^+(j)\geq a_+^+(c)=a_-^+(b)\vee a_+^+(c).\] All minus-wall coordinates on \(F\) vanish, since its endpoints are common plus pins. Off \(F\) the four inequalities in the proof of Proposition 6 are unchanged. The cell factors \(r_y\) do not depend on the equality set. Thus monotonicity and log-supermodularity of \(Z\) prove the same two-law inequality for the first-spin marginals.

To restore statuses, first couple these marginals in order. In the conditional plus-cell weights (14), write \(A_-(S)\) and \(A_+(T)\) for the allowed wall-edge vectors of lower and upper open-cell sets. Their sorted vectors obey \[A_-(S\cap T)\geq A_-(S)\wedge A_+(T),\qquad A_+(S\cup T)\geq A_-(S)\vee A_+(T).\] On \(F\) the first comparison is \(0\geq0\) and the second is the inclusion of the upper open set in its union; elsewhere these are the usual cell-set comparisons. The Bernoulli factors cancel in the resulting two-law inequality. No cell containing an edge of \(F\) can be eligible for minus occupation, so the minus-status comparison is unchanged. The ordered status kernels complete the joint coupling. The nonnegative-weight argument applies also at \(x=1\). ◻

Proof. All crossings below use the induced triangular site graph. Lattice rows and boundary contours are chosen with the fixed collars of Lemma 69; that lemma absorbs the bounded corner changes. The crossing between the two displayed interior heights is measurable away from these patches for sufficiently large \(r\).

Divide \(P\) into three rhombi \(R_1,R_2,R_3\) of width \(r\). An ordinary \(B\)-plus vertical crossing in each outer rhombus has probability bounded below. Indeed, impose an adverse open-minus cut on its inner side. Conditional order and the finite corner/port comparison reduce the bound to an alternating open-arc rhombus. Reflection in its ascending diagonal, followed by reversal of \(B\), preserves that law and exchanges an ordinary plus vertical crossing with a minus horizontal crossing. Ordinary triangular-site duality gives a positive lower bound for these two symmetric alternatives. Both outer crossing events are increasing; association gives their intersection probability at least \(c_0(x)>0\). The auxiliary adverse cuts were used only to compare the two individual probabilities and are not retained.

Run the following searches in the original law, without conditioning on the two ordinary-crossing events. Explore the \(B\)-minus component attached to the left short side, and similarly the component attached to the right short side. Reveal first spins only, and declare failure if either component reaches the middle rhombus. Retain the full Peierls interface separating each explored component from the middle rhombus, including all its excursions. The intersection of the two ordinary-crossing events guarantees that each interface is confined to its outer rhombus, since its ordinary plus crossing blocks the attached minus component. Thus the successful records have probability at least \(c_0(x)\). On each such record, fill holes on the explored side and retain the region \(D_0\) between the two interfaces. Its interior is unobserved, and \[R_2\subseteq D_0\subseteq P.\] Along either lateral interface the outer first spin is minus and the inner first spin is plus. Each selected mixed triangle contributes the constant \(q=x^2\) and forces its three complementary spins to agree. Successive selected triangles share a face. Thus each full lateral interface supplies one connected \(W\)-equality chain, regardless of its length or excursions. The constants cancel after conditioning on the exploration record. Away from the bounded corner patches, the top and bottom remain open-plus cuts and impose no \(W\) equality.

Reflect in the ascending diagonal of \(R_2\). In these coordinates \[\rho(u,v)=(r+v,u-r),\qquad D=D_0\cup\rho(D_0).\] This reflection preserves the chosen cell class. For example, with \(\omega=e^{i\pi/3}\), \(\Lambda=\mathbb Z+\omega\mathbb Z\) and \(Y=\Lambda-i/\sqrt3\), its linear part is \(z\mapsto\omega\bar z\), and \(\omega\overline{(-i/\sqrt3)}+i/\sqrt3=\omega-1\in\Lambda\). The same holds for \(Z\). The inclusions above give \[\rho(D_0)\cap P=R_2,\qquad D\cap P=D_0.\] The union is a simply connected symmetric domain: its lower and upper reflected arms attach to \(D_0\) along the bottom and top sides of \(R_2\). The construction is made on the actual lattice contours with matching endpoint collars.

Put \(A=(r,0)\), \(C=(2r,r)\), \(B_0=(2r,0)\) and \(D_1=(r,r)\). Reflection fixes \(A,C\) and exchanges \(B_0,D_1\). The boundary arc \(AB_0\) surrounds the lower reflected arm, and \(CD_1\) surrounds the upper one. Any occupied site connection between these two arcs has a bottom-to-top subcrossing of \(P\) entirely in \(D_0\): take its segment after the last bottom-side visit before the first subsequent top-side visit. It therefore contains the crossing between the two interior heights in the statement. No confinement of this subcrossing to \(R_2\) is required.

On \(D\) take the excluded-boundary-cell law with \(B=+\) on its inner boundary layer and with \(W\) constant, of unspecified sign, on that layer. Denote it by \(\nu_D\). This is the equal mixture of the two laws with boundary spin pairs \((+,+)\) and \((+,-)\); global \(W\) reversal preserves all \(B\)-occupation events. For the \((+,+)\) law, occupied-site super-duality, reflection and interchange of colors give \[ \nu_D(AB_0\mathrel{\longleftrightarrow}_{\xi^{B,+}}CD_1) \geq\tfrac12. \tag{116}\] This is precisely the site-arc convention of (Glazman and Lammers 2025, Lemma 3.17): the two alternative color crossings have union of probability one, and symmetry equates their probabilities. Their disjointness is not required. The sign of a crossing incident to the corresponding boundary layer is plus.

Where the old top and bottom cuts lie in the interior of \(D\), prescribe them open-plus. These are increasing first-spin/status prescriptions, so association preserves the lower bound (116). Their open factors are constants and remove all \(W\) interaction across them. On old top/bottom pieces that lie on \(\partial D\), the boundary factors are already excluded; no additional exterior minus pins are imposed on the sites of those open cells. This convention avoids a conflict between an open-plus cell and an exterior minus spin in the same cell.

We compare the resulting law in \(D_0\) with the law left by the actual exploration. The old top/bottom pieces outside the middle rhombus lie on \(\partial D\) and hence have constant \(W\) under \(\nu_D\). In the actual law add exactly these equalities, along edges whose endpoints are already pinned \(B=+\) by the open cut. By Lemma 70, this decreases all increasing \(B\)-occupation events, with no loss depending on the number of edges. No equality is added on the old top/bottom pieces that run through the interior of \(D\), namely the middle-rhombus pieces.

After this lowering, the two laws have identical retained cell factors, first-spin pins, and long boundary equality chains. The two lateral chains, extended along their adjacent old top/bottom boundary pieces, contain every remaining complementary-spin connection to the exterior, except for the bounded transition patches. Their sites all have fixed boundary first-spin data. The remaining exterior can only identify these two chains or leave their labels separate. For fixed retained \((B,\tau)\), such an identification changes the complementary spin sum by a factor in \(\{1,1/2\}\); after normalization the two marginal densities differ by at most a factor two. Mixtures over unobserved exterior statuses preserve this bound. Applying Lemmas 68 and 69 to the finitely many additional corner components gives a uniform factor \(c_1(x)>0\).

Consequently, for every successful ordinary-interface exploration record, the original probability of the occupied crossing between the interior heights is at least \(c_1(x)/2\). Averaging over those records and using their probability at least \(c_0(x)\) proves the assertion. All modifications at the corners are compatible bounded patches and all other comparisons are monotone or involve the two connected boundary chains. Thus no bound depends on a boundary length. The argument and the patch lemma include \(x=1\). ◻

Proof. We prove the estimate uniformly after refining the two extremal paths to their exterior exploration records. The upper exploration reveals only the part of \(P_L\) above its full crossing, and the lower one only the part of \(P_R\) below its full crossing, together with the prescribed first-spin layers on the crossings. All observed statuses include all their first-spin sites. No complementary-spin label is revealed. The path version of the invasion argument in Lemma 66 gives these records. Use the actual authorized honeycomb boundary of the explored face set to realize each extremal crossing. Every queried cell meeting both sides is then an open boundary cell; its retained first spins are its prescribed plus layer. This avoids revealing an additional strip along an arbitrary interpolation of a site path. We retain the full crossings, including all their excursions near a side.

Construct a stepped reference region \(S\). Its upper portion has horizontal range \([0,6\ell]\) and vertical range \([\ell/2,5\ell/4]\); its lower portion has horizontal range \([\ell,7\ell]\) and vertical range \([-\ell/4,\ell/2]\). Move each of its two stepped sidewalls outward by a fixed number of lattice spacings, say twenty, and round to honeycomb paths. The top and bottom sides are likewise chosen as actual lattice contours. The steps lie at height near \(\ell/2\), at positive macroscopic distance from both extremal paths.

Join the actual four endpoints of the full crossings to the respective outer sidewalls through four bounded patches. The region \(D\) between the two full paths and these sidewalls is unobserved outside those patches. Indeed its upper part is below the full \(\Gamma_L\) in \(P_L\), and its lower part is above the full \(\Gamma_R\) in \(P_R\). In their overlap the two statements hold simultaneously, since the height bands are separated. The narrow strips introduced outside a side of a parallelogram were not queried by its exploration; the other exploration is separated from them in height. Thus the only extra recorded sites or cells introduced by an endpoint join lie in its bounded patch. This uses the full paths: no discarded prefix is placed in \(D\).

Prescribe the two adverse sidewalls open-minus, stopping them before the endpoint patches. They lie in the unobserved region and are disjoint from the occupied-plus crossings. By conditional order this can only decrease the probability of the desired central vertical crossing; the rarity of the prescribed walls incurs no probability factor. We next make bounded, compatible completions in the endpoint patches. Let \(h\) be the complete original record in those patches: retain every recorded first spin and status, including all first-spin sites of each recorded cell. Retain also the plus pins forced by the crossings. The new minus walls are separated from these sites by the fixed outward displacement. Choose a compatible completion \(A\) only in a bounded core, keeping an outer collar unprescribed except for the common crossing and wall pins. The fixed first-spin values are compatible; extend them across this free collar and set the complementary spin constant. On unprescribed core cells choose open status when homogeneous and closed status when mixed. Include every cell incident to a mutable core site in the patch. Thus Lemma 69 gives a uniform positive lower bound for \(A\). At \(x=1\), recorded closed cells keep their already recorded mixed first spins. Write \(h^*=(h,A)\), and keep this complete bounded history in all reference laws below. No arbitrary part of the original record is removed by a finite-energy assertion.

Let \(\nu_{S,h^*}\) be a reference law with open-plus top and bottom arcs, open-minus stepped sidewalls, and this same history \(h^*\) in the four endpoint cores. Realize the law in a filled ambient triangulation by these cell and first-spin prescriptions, with a fixed feasible completion of its outer corner cells and no complementary-spin labels. Its top and bottom are at distance at least \(\ell/4\) from the original parallelograms, so their prescribed plus layers do not meet \(h^*\). The law is positively associated by Lemma 63.

Condition this reference law on the two full paths being occupied-plus. The law in \(D\) has the same retained cell factors and first-spin pins as the original record completed by \(A\). Every other boundary cell is open and removes the complementary-spin interaction across its arc. Choose each completed core to contain its entire short endpoint join, including every non-open cell crossing the chosen boundary of \(D\) and all its first-spin sites. The boundary enters and leaves the free outer collar only along the common open crossing or wall arcs. Thus every cross-boundary cell outside the completed cores is already open. All other cross-boundary first spins and statuses are fixed in \(h^*\); only the uniformly bounded number of complementary-spin sites in those cores can communicate with the exterior. Lemma 68 compares the two retained laws with a uniform factor, independently of the exterior complementary-spin wiring. The original endpoint history \(h^*\) is kept on both sides of this comparison.

Write \(E\) for the vertical crossing of the central height interval in the statement. Its support, including the incident first-spin sites of its cells, lies in \(D\) for sufficiently large \(\ell\); the full comparison rectangle need not lie in \(D\). The bounded displacement between the occupied site path and its authorized exploration boundary is absorbed by the strict height gaps. Its horizontal window is at macroscopic distance from all actual endpoint patches, whose horizontal coordinates are near \(0,6\ell,\ell,7\ell\). The fixed corners of \(S\) are also outside this window. The lower bound for \(A\), followed by the port comparison, gives \[ \mathbb P(E\mid\hbox{original record}) \geq c_1(x)\, \nu_{S,h^*}(E\mid\Gamma_L,\Gamma_R\hbox{ occupied-plus}) \geq c_1(x)\,\nu_{S,h^*}(E). \tag{117}\] The last inequality is association: the full-path prescriptions are increasing, with every first-spin site of each path cell included. All constants are uniform in the retained bounded history \(h^*\).

It remains to bound \(\nu_{S,h^*}(E)\). In the reference law alone, prescribe two additional open-minus walls at horizontal positions \(5\ell/4\) and \(23\ell/4\), ending a bounded distance before the top and bottom plus arcs. This decreases \(E\). Complete the four new transition patches as above. These walls and patches are at macroscopic distance from every actual endpoint patch. The central region is a straight parallelogram of width \(9\ell/2\) and height \(3\ell/2\), hence aspect ratio three. Exact cut factorization off its corners and Lemmas 68–69 compare its law with the alternating open-arc law of Lemma 71. Lemma 71, with \(r=3\ell/2\) and \(\eta=11/30\), bounds below the probability of the crossing \(E\) itself: its two heights are \(-\ell/4+(11/30)(3\ell/2)=3\ell/10\) and \(-\ell/4+(19/30)(3\ell/2)=7\ell/10\). It follows that \(\nu_{S,h^*}(E)\geq c_2(x)>0\).

The additional interior minus walls were imposed only in \(\nu_{S,h^*}\), after removing the conditioning on the actual crossings in (117). They are therefore not required to be compatible with paths that might cross them. Combining the two bounds and averaging over the exterior exploration records proves the lemma. All exceptional corner adjustments involve a bounded number of cells, with free outer collars and compatible common pins as above. ◻

Proof. Start under \(P_Y\) with a thin band \(S\) centered near \(H/3\). Lemma 67 gives a vertical occupied \(B\) crossing of one sign, or a winding occupied \(W\) circuit of one sign. The sum of the probabilities of these four events is at least one.

A \(B\)-plus vertical crossing gives a \(Y\) seed by the finite-family extraction above. The slide comparison of Lemma 65 moves its test upward. A \(B\)-minus vertical crossing is reflected across the horizontal middle line of the cylinder, followed by a flip of \(B\). The resulting law is \(P_Z\), including its boundary-cell convention: horizontal reflection interchanges the two cell classes. It gives a \(Z\) plus seed near \(2H/3\), which can again be moved upward. Neither step asserts a color-preserving symmetry of a fixed \(P_Y\) law.

It remains to treat a \(W\) winding. Global \(W\) flip lets us take its sign to be plus, at a loss of at most a factor two. Search from below in a slightly enlarged band, as in Lemma 66. On a successful record the retained upper region lies above one open \(W\)-plus winding \(\gamma\) of class \(Y\) and below the original upper \(B\)-plus cut. All tests used next lie in its unobserved part.

Choose a horizontal lattice reflection axis near \(.69H\). The reflection of \(\gamma\) lies strictly above the old top, since \(2(.69)-1/3>1\) and the search band is thin. Move the top \(B\)-plus cut up to that reflected contour, now of class \(Z\). By Lemma 64 and the cut comparison, this move decreases every increasing \(B\)-plus test. The law between \(\gamma\) and its reflection is invariant under reflection followed by exchange of the two spin colors. This follows from the coherent edge weight and the exact cut factors; the transformation also transports \(Y\) tests to \(Z\) tests.

Apply band duality near the reflection axis, using fresh \(Y\) augmentation. The finite extraction gives a family of \(B\) seeds of either sign and \(W\) horizontal seeds of either sign whose probabilities have sum bounded below. Reflection and color exchange turn the \(W\) seeds into \(Z\)-oriented \(B\) seeds. For either orientation a minus seed has probability at most that of the corresponding plus seed: flipping \(B\) changes the top prescribed cut from plus to minus while retaining the one \(W\)-plus cut, and Lemma 64 orders these two laws. Consequently a fixed finite family of \(B\)-plus tests, in the two orientations, has total probability bounded below in the symmetric law, uniformly over the successful search record.

The same lower bound holds in the original law above \(\gamma\), by the comparison that moved the top upward. Its index family can be chosen independently of \(\gamma\): the search band, reflection axis, and enlarged test windows are deterministic and separated. Averaging over successful records therefore gives a positive lower bound for one of those tests under \(P_Y\). If that test uses class \(Z\), its augmentation is fresh and unobserved. Apply the slide comparison, changing the boundary class to that of the test, to move it upward and obtain an ordinary matching \(P_Z\) experiment. The same comparison moves a \(Y\) test upward within \(P_Y\).

All three branches can thus be placed near \(.71H\), with clearance between every old and new cut when their classes differ. Taking the minimum of the finitely many bounds proves the lemma. The sum bound is obtained before choosing its successful member, so no random test is selected inside a conditional probability. ◻

Proof. We first obtain a short vertical crossing in a controlled horizontal window. Use the two overlapping width-\(6\ell\) parallelograms of Lemma 72, choosing their vertical height to be about \(50m\). Their total horizontal span is less than \(500m\). Place their bottom at, or just above, the seed band. In a thin band above relative height \(40m\), request a horizontal crossing through the left parallelogram; in one below relative height \(10m\), request a horizontal crossing through the right parallelogram. These estimates follow from the original horizontal seed, upward translation by Lemma 65, and horizontal lengthening with fixed aspect ratios. Cross rectangles extending beyond the parallelogram’s sloping sides and then trim the paths to those sides. Association makes the intersection of the two requests have probability bounded below.

Let \(\Gamma_L\) be the uppermost crossing of the full left parallelogram and \(\Gamma_R\) the lowermost crossing of the full right one. A crossing in the upper thin band separates the top side from the region below that band; hence the uppermost crossing stays above relative height \(39m\), allowing lattice rounding. Similarly the lowermost right crossing stays below height \(11m\). Thus the two thin-band requests imply that the pair \((\Gamma_L,\Gamma_R)\) lies in a set \(\mathcal G\) of separated realizations in gluing order.

For each realization in \(\mathcal G\), Lemma 72 gives a vertical crossing of the central gap with conditional probability at least \(c_0>0\). The bounds \(39m\) and \(11m\) put the extremals above three quarters and below one quarter of their height-\(50m\) parallelograms. We condition only on the two extremal crossings, not additionally on the thin-band requests. Averaging gives a lower bound \(c_0\mathbb P(\mathcal G)\), and \(\mathbb P(\mathcal G)\) is at least the probability of the requests. The resulting crossing runs between relative heights \(15m\) and \(35m\) in a horizontal window of width at most \(500m\). This supplies the required local vertical-crossing estimate.

Place a thin horizontal band about relative height \(25m\), strictly between these two vertical levels. Translate the vertical-crossing window to two seam neighborhoods separated by about \(L/2\). Their widths are much smaller than \(L\). Request also two horizontal crossings in the thin band: one follows each of the two arcs between the seams, and each extends beyond the whole vertical window at both ends. These horizontal rectangles have length, for example, about \(3L/4\), so each is embedded in the cylinder. Their crossing estimates are ordinary horizontal lengthenings. All four events are increasing in the same signed \(B\)-status field. Association bounds the probability of their intersection below by a positive constant.

On that intersection both horizontal paths meet the vertical path in each seam neighborhood, by planar site crossing. In the flat cover, the first horizontal path connects seam 1 to seam 2 on its right, and the second connects seam 2 to the next periodic copy of seam 1. The connections therefore join a site path to its translate by one full period. Its projection contains a noncontractible occupied circuit. The interpolation to authorized honeycomb edges changes its position by only a bounded number of lattice spacings and yields a simple winding cut in a slightly larger band. All these bands remain below \(.73H\) because \(m/H\) is fixed and sufficiently small. ◻

Proof. Only the \(T=Z\) branch remains after Lemma 73 and (115). By Lemma 74, with probability bounded below an exterior search finds a \(Z\)-oriented \(B\)-plus winding below \(.73H\). Above a successful record the retained law is between that plus cut and the original upper plus cut. It is therefore above the free capped-sphere law for increasing \(B\)-plus tests: prescribing either plus cut is an increase by Lemma 63. Tests of class \(Y\) well above the explored band use disjoint, fresh cells, so this comparison applies to them too.

Test a thin band near \(.75H\) using class \(Y\). The free sphere law is invariant under the two spin flips and exchange of the spin colors. Band duality and finite extraction therefore imply a positive lower bound for the sum of a fixed finite family of \(Y\)-oriented \(B\)-plus seed probabilities. These spin symmetries do not require spatial translation or reflection symmetry of the caps. The lower bound holds in every retained law between the two plus cuts. Average over successful records, and then choose one member of the deterministic family. This gives a \(Y\) seed under \(P_Z\).

Finally slide the boundary cuts and change their class to \(Y\), moving the test to height near \(.77H\). Lemma 65 preserves the lower bound and gives the matching mixed law \(P_Y\). The clearance between old and new cuts guarantees that the cell families used in this comparison are edge-disjoint. Apply (115) if the seed is vertical, then horizontally lengthen to width \(4n\). Since \(.77H\sim3.85n\) and all vertical bands used here are much thinner than \(n\), the resulting path stays in \([3n,4n]\). It is the crossing in (118). ◻

Proof of Theorem 5. Proposition 75 proves the horizontal alternative in (58) of (Glazman and Lammers 2025). We now use the implications of that source’s Appendix A. The RSW estimate (56) and horizontal lengthening convert (58) into the pushing estimate (57); the comparison between rectangle and cylinder has exactly the boundary-cell convention realized by our cuts. Pushing gives the renormalization inequality (23), and its non-exponential-decay branch, selected by the exclusion argument on pp. 44–45 of (Glazman and Lammers 2025), gives equation (55). Those implications use association, domain Markov comparison, the weighted site-crossing inputs already specified, and qualitative delocalization. Their output is precisely an occupied plus circuit under an excluded-boundary-cell law with the first color minus on its boundary layer and the second color free. Lemma 4 identifies this with the law used in the statement of Theorem 5.

Our seed proof was for sufficiently large lattice scales. The finitely many remaining admissible \(n\) can be absorbed by decreasing the constant. The condition \(n\geq3/(a-1)\) implies \(\lfloor an\rfloor-n\geq3\), so a plus ring can be placed around layer \(n+1\), with plus first spins through layer \(n+2\), minus first spins on the outer prescribed boundary layer, and constant second spin. Force the finitely many ring cells open. This is a feasible positive-weight configuration for every \(x\in[1/\sqrt2,1]\). Thus each of those finite probabilities is positive. Spin flips and color exchange give the other versions of the theorem. ◻