Proof. Consider sign plus first. Fix \(B=-1\) on both face layers of the radius-\(2r\) regular contour. By 3 this lowers every increasing event. The exact cut law identifies its interior with the one-spin minus-boundary law in the cited annulus theorem. Bounded changes in the chosen contour and integer radius are absorbed by the fixed slack in (10).

Each edge of the occupied \(Y\) circuit lifts to two honeycomb edges. Both are double-\(B\) plus: the two occupied endpoints prescribe all incident face spins. The resulting closed walk stays a bounded distance from the site circuit and has the same winding about the inner ball. Split the walk at repeated vertices and discard zero-winding pieces; one obtains a simple double circuit with that winding. The \(Y\) lattice and the face-center lattice are translates with the same graph norm, so all the displacement errors just used are bounded independently of \(r\). For sufficiently large \(r\) the resulting circuit satisfies (10) and has the asserted enclosure.

Choose, for example, \(R=4\), increasing the lower threshold on \(r\) to accommodate all face stars and contour layers. The comparison takes place inside the prescribed empty patch, so arbitrary exterior \(B\) conditioning leaves it valid. Global sign reversal proves the minus case. Adding pins of the requested sign can only increase the circuit probability by 3. ◻

Proof. Take a polygonal curve at norm radius \(3n/2\). Choose \(r\) to be a sufficiently small fixed multiple of \(n\), and cover this curve cyclically by annuli as in (10). Put their centers on the curve, including its corners, with successive distances between \(1.4r\) and \(1.6r\). Integer rounding preserves these inequalities with slack for large \(n\). Choose \(r/n\) small enough that the annuli and their radius-\(Rr\) comparison patches lie strictly between radii \(n\) and \(2n\).

The circuits in neighboring annuli must intersect. Their enclosed inner balls overlap, and each inner ball protrudes outside the other’s outer ball, since \(\varepsilon<1/50\). Disjoint Jordan curves have disjoint or nested interiors, and these two geometric properties exclude both possibilities. At a common honeycomb vertex, two double paths of the same sign join with all three incident face spins of that sign. Choose intersection points for consecutive pairs and join them along the intervening circuits. The resulting closed walk stays \(O(r)\) from the center polygon and is homotopic to it in the complement of the radius-\(n\) ball. Extracting a simple circuit with nonzero winding gives the required circuit between radii \(n\) and \(2n\).

There are only a fixed number of small annuli. Their events are all increasing in the order toward \(t\), so [lem:spin-fkg,lem:local-circuit] give a positive lower bound for their intersection. A fixed sufficiently large \(K\) keeps every comparison patch away from the opposite boundary condition. The constants do not depend on \(n\). ◻

Proof. First consider a small regular annulus whose larger cut lies in the buffer. Delete any favorable pins in that cut. This can only lower the probability of the desired event in the order toward \(t\). Fix the opposite sign on both sides of the larger deterministic cut. FKG lowers the desired probability again, and the exact cut law identifies the remaining problem with 7. This gives a uniform local lower bound under every allowed exterior conditioning.

We explain the geometric gluing, with all comparison patches inside the buffer. Cover the center curve of the tube or collar by finitely many small regular annuli. Choose their common inner radius much smaller than the tube width. Consecutive centers may be separated by, for example, \(3/2\) times that radius in the graph-distance norm. Their inner balls intersect, and each inner ball protrudes beyond the other’s outer ball. Two surrounding circuits in these annuli must therefore intersect: disjoint Jordan curves would have disjoint interiors or nested interiors, and the two geometric conditions exclude these possibilities. Their union is connected and follows the prescribed tube. Around a closed center curve, its outer boundary contains a simple circuit enclosing the hole; the annuli are small enough that their union does not fill that hole. At a common dual vertex, double paths of the same sign have all three adjacent spin sites of that sign, so the gluing does not lose the double-path property. Bounded lattice rounding is absorbed by the positive geometric clearances.

All the small annulus events for one sign are increasing in the same order. FKG multiplies their local lower bounds, giving a positive constant depending only on the finite cover. For separated buffers, condition successively on the configurations outside the next buffer; the preceding lower bound applies under each such conditioning. This also permits different signs in different buffers.

Finally choose a fixed scale ratio large enough that the enlarged buffers of successive annular slots are disjoint. There are at least \(c_2\log(L/r)-C_2\) slots. At each trial, conditional on the previously examined slots, success has probability at least \(c_3>0\). Failure in all slots has probability at most \((1-c_3)^{c_2\log(L/r)-C_2}\), which is (11). The finitely many bounded lattice scales can be absorbed in its prefactor. ◻

Proof. For (i), use two separated polygonal collars at norm radii, for example, \(1.3n\) and \(1.7n\). Their comparison buffers fit strictly between radii \(n\) and \(2n\) and avoid the fixed boundary layers and equality edges. 8 gives a uniformly positive probability of a double-\(B\) plus circuit in the inner collar and a double-\(B\) minus circuit in the outer collar. Different signs cause no association issue: reveal the outer collar and apply the local conditional bound in the disjoint inner buffer. A \(B\) wall separates these opposite-sign circuits. No wall can cross either double circuit, since all three face spins at each of its vertices agree. At least one wall loop therefore surrounds the inner circuit and lies inside the outer one, proving (i).

For (ii), put \(z=0\). If a honeycomb circuit surrounds \(\Lambda_s\), its inner incident faces contain a triangular face-center circuit surrounding the real norm ball of radius \(s-b\), for a universal \(b\). To see this, follow the inner incident faces along the contour. Successive centers are equal or adjacent, and the resulting closed walk tracks the contour within bounded distance. Its edges lie on the inner side. Splitting at repeated vertices gives a simple circuit with the same nonzero winding about the smaller ball. The same observation applies to a double circuit whose face spins have one fixed sign.

Let \(E\) be the event in (12), and let \(E'\) also require that its outermost surrounding loop changes \(W\) and its second outermost changes \(B\). The independent coloring gives \[ \mathbb P_L(E')=\tfrac14\mathbb P_L(E). \tag{13}\] On \(E'\), the outer loop is double-\(B\) plus: no surrounding \(B\) wall lies outside it. The faces immediately inside the next loop have \(B=-1\). Consequently there is a simple minus face-center circuit enclosing the real norm ball of radius \(\rho n-b\). Other loops cannot alter these incident signs without intersecting one of the two selected loops.

We select an enclosing plus cut using only \(B\). Extend the flat spins as plus outside \(L\), and start with all exterior faces reached. Traverse face adjacency unless the crossed honeycomb edge, including its endpoints, is strictly inside \(L\) and is double-\(B\) plus. A deterministic queue queries the \(B\) values needed to decide each attempted traversal, and retains the full query record. Complete this reachable closure. If the center is unreached, let \(U\) be its component in the ordinary face-adjacency graph of all unreached faces. Each face adjacent to \(U\) but outside it is reached, and each edge separating the two sets is an eligible strictly interior double-plus edge. The reached set is connected to the exterior, so \(U\) has no holes. At a honeycomb vertex only three faces meet; the boundary of \(U\) therefore has degree two at every vertex and is a single simple circuit \(C\).

Every original boundary face is reached across an ineligible contour edge. Moreover, every queried face in \(U\) was tested from a reached neighbor across a blocked edge, and hence is a plus inner-layer face of \(C\). There are no other queried sites or original boundary constraints in \(U\). Each completion consistent with the record reproduces the same search and the same cut. Across \(C\) both \(B\) values are plus, so coherence places no restriction on \(W\) there. Thus 5 gives exactly the plus-\(B\), free-\(W\) cut law in \(U\), even after all unobserved exterior variables are summed out. On \(E'\) the outer selected loop blocks the search, and \(C\) lies on or outside that loop and encloses the minus face circuit. The search does not condition on \(E'\) or on any loop color.

For a fixed realized \(C\), let \(A_C\) be the event of a simple minus face-center circuit inside \(C\) enclosing the real norm ball of radius \(\rho n-b\). This is a decreasing event. In a sufficiently large regular free spin domain, prescribing plus on both layers of this fixed \(C\) gives its cut law and decreases \(A_C\), by [lem:spin-fkg,lem:double-cut]. Its probability in the stopped interior is therefore at most its probability in that free domain. We now bound the latter probability uniformly in \(C\).

Choose an exterior face incident to \(L\) with center \(u\) satisfying \(|u|_g=n+O(1)\). Fix \(M>10(R+2)\), with \(R\) from 6, and choose an integer \(m\) such that \((1-p)^m\le1/8\). About \(u\) test the annuli of that lemma at scales \[d_i=\left\lfloor\frac{n}{20M^i}\right\rfloor, \qquad i=0,\ldots,m-1.\] Test from large to small, revealing the \(B\) spins in each tested annulus and its incident face stars. The radius-\(Rd_{i+1}\) comparison patch lies strictly inside the hole of every previously tested annulus for large \(n\). Hence each trial has conditional success probability at least \(p\), and the probability of no success is at most \(1/8\). Take the free comparison domain large enough to contain \(L\) and all these patches with their bounding layers.

On success, the incident-face construction gives a simple plus face-center circuit around \(u\), enclosing the real norm ball of radius \((1-2\varepsilon)d_i\) and contained in the ball of radius \((2+2\varepsilon)d_i\). Set \[\rho=1-\frac{1}{200M^{m-1}}.\] For all sufficiently large \(n\), the enclosed ball about \(u\) overlaps the radius-\((\rho n-b)\) ball about zero with linear slack. Indeed the smallest tested radius is \(n/(20M^{m-1})+O(1)\), whereas \((1-\rho)n=n/(200M^{m-1})\). The plus circuit does not surround zero, since its containing ball has radius at most \((2+2\varepsilon)n/20\) and is centered at distance \(n+O(1)\) from zero. A minus circuit counted by \(A_C\) lies inside \(C\), and hence inside \(L\); it cannot surround the exterior face center \(u\). The two circuits consequently have overlapping interiors with neither containing the other. They must intersect. In the planar triangular face graph, such an intersection contains a common spin site, impossible for a plus and a minus circuit.

It follows that \(\mathbb P_{\rm free}(A_C)\le1/8\) for every realized \(C\). On \(E'\) the search succeeds and \(A_C\) occurs. Averaging over its full records therefore gives \(\mathbb P_L(E')\le1/8\). Equation (13) yields \(\mathbb P_L(E)\le1/2\). Taking \(c_0\) to be the smaller of \(1/2\) and the constant in (i) proves the proposition. All lower thresholds on \(n\) are imposed after the fixed constants \(R,p,M,m,\rho\) have been chosen. ◻

Proof. Apply the preceding common-lower-law coupling and search for a plus circuit in a stack between the inner ball and the deterministic outer cut. Its failure probability is bounded by (11). On success the full pair configurations agree in the inner ball. This bounds the total variation distance between any two sufficiently large free-volume laws, and proves the Cauchy property on each cylinder. It gives the same bound with one constrained law and one expanding free volume; then let the latter volume tend to the plane. Constants do not depend on the constraints outside the empty ball.

Translated or reflected exhausting volumes give the same limit by this comparison, proving the spatial symmetries. The finite laws have the stated spin symmetries. Formula (8) determines increments on every fixed path, and its vanishing curl persists in the limit. It therefore defines the plane height modulo one additive integer.

To check positivity of a finite pin event, enclose all its sites and paths by a double-\(B\) cut. Such a cut has positive probability by 8. In its finite interior, take \(W\) everywhere plus and choose \(B\) with the prescribed values and the cut value. This is a coherent extension and has positive conditional counting probability. The exact cut law proves the assertion. Uniform finite-volume estimates pass to this positive-probability plane conditioning. ◻

Proof. Fix sign \(t\) on a deterministic double layer \(\Gamma\) in the collar. Let \(q\) be the probability of \(A\) in its interior law, retaining \(R\) and the internal \(E_0\) constraints. Exact cutting makes \(q\) the same for \(\mu\) and \(\nu\). FKG in the order toward \(t\) gives, for either law \(\lambda\in\{\mu,\nu\}\), \[ \lambda(A)\leq\lambda(A\mid B=t\text{ on }\Gamma)=q. \tag{14}\] In a thinner collar strictly inside \(\Gamma\), an exterior stopping search finds a double circuit of sign \(t\) with probability at least \(c_1>0\), by 8. Its constant is uniform over \(R\) and all remote constraints: the search buffers contain none of these pins or \(E_0\) edges. Conditional on the search, the inner law is the corresponding closer-cut law. In the deterministic-\(\Gamma\) law, conditioning on this closer sign-\(t\) cut can only increase \(A\), while preserving \(R\). Its probability of \(A\) is therefore at least \(q\). Consequently \[ c_1q\leq\lambda(A)\leq q, \qquad \lambda\in\{\mu,\nu\}. \tag{15}\] Comparison through this common \(q\) proves the lemma. The retained pins may be extremely rare; their probability never enters the inequalities. The stopping exploration has not inspected the strict interior, which is essential for the lower bound. ◻

Proof. First handle the equality requirements for \(W\). Temporarily ignore all \(B\) pins. For a collection of \(m\) connected paths, let \(E\) denote constancy of \(W\) along each path and let \(L\) require all labels to be plus. Conditional on \(B\), a consistent pattern of labels has the same probability as the all-plus pattern, and an inconsistent pattern has probability zero. Summing over the at most \(2^m\) patterns gives \[ \mathbb P(L)\leq\mathbb P(E)\leq 2^m\mathbb P(L). \tag{17}\] Apply 11 with the colors interchanged to the successive all-plus requirements on the obstacles. Each conditional factor is comparable to the corresponding one-obstacle factor: the conditioning concerns only other, separated obstacles. The all-plus probability thus factors up to constants. Equation (17), for the joint family and for its individual members, gives the same conclusion for their constancy events. Denote the resulting equality edges by \(E_0\).

Now condition on all these equalities and impose the \(B\) requirements obstacle by obstacle. On each obstacle first impose its plus pins, then its minus pins. At either stage compare its conditional factor with the same stage in the one-obstacle equality problem using 11. At the minus stage the plus pins already imposed on that obstacle belong to \(R\) and are retained in both cut laws. All previous pins on other obstacles and all other equality paths lie outside \(U_i\), so the two laws have exactly the same retained constraints inside \(U_i\). Multiplication compares the joint conditional \(B\)-pin probability with the product of the one-obstacle conditional probabilities.

Finally restore all the required \(W\)-plus labels. Conditional on \(B\) and \(E_0\), their probability is between \(2^{-\sum_i m_i}\) and one; the same statement applies in each one-obstacle problem. These bounded factors, together with the preceding two comparisons, prove (16). The entire proof takes place in finite volumes with uniform constants. The plane assertion follows from 10 and positivity of each finite pin event. ◻

Proof. Consider the enlarged positive dyadic intervals \(I_k=[2^kM-4M,2^{k+1}M+4M]\), and put \(p_k=\mathbb P(Y\in I_k)\). Choose a class carrying at least \(p_k/m_*\) of this mass. Pair its indicator on \(I_k\) with its indicator on \([-M,M]\). Both rounded output statistics belong to \(I_{k-1}\): the half-width error is at most \(2M+M/2+M<4M\). Apply the product midpoint inequality with both output arrays equal to the indicator of \(\mathcal S\cap\{Y\in I_{k-1}\}\). After division by \(\abs{\mathcal S}^2\), it gives \[ p_{k-1}^2\geq c p_k/m_*. \tag{20}\] The admissibility assumption is needed only where the two input indicators are nonzero; no output class restriction occurs.

For \(k\geq4\) these enlarged intervals have bounded overlap. Hence, in an initial range of bounded length depending only on \(c,m_*\), some \(p_{k_0}\) is at most \(c/(2m_*)\). Write \(b=m_*/c\). Iterating (20) gives \(b p_{k_0+j}\leq2^{-2^j}\). The bounded size of the initial range absorbs the earlier terms, so \(p_k\leq C\exp(-c'2^k)\) for all \(k\). Repeat with negative intervals and sum the resulting tail bounds. Integrating the tail proves the moment assertion. ◻

Proof. Let \(\rho<1\) and \(c_0>0\) be the constants in 9(ii). Choose \(\eta>0\) small enough that \(\rho(1+\eta)<1\), with slack for bounded lattice rounding at large distances. Divide \([d,2d]\) into a bounded number of intervals with ratio at most \(1+\eta\). Expose surrounding loops from the exterior, and consider the first one whose graph inradius lies in one such interval \([d', (1+\eta)d']\). If it exists, its interior has the empty-exterior loop law. If its graph inradius is \(n\), then \(n\le(1+\eta)d'\). Every further loop with graph inradius at least \(d'\) surrounds \(\Lambda_{\lfloor d'\rfloor}(x)\). For sufficiently large \(d'\), \[\lceil\rho n\rceil<\lfloor d'\rfloor,\] because \(\rho(1+\eta)<1\). Thus two such loops imply the exact event in 9(ii), in the same graph convention. Their conditional probability is at most \(1-c_0\). Reapply this argument after each successive pair of exposed loops. The count in the interval is dominated by one plus twice a geometric random variable with a fixed success parameter. A bounded number of such bounds handles \([d,2d]\).

At bounded distances, disjoint loops approaching that close must use distinct vertices in a fixed finite ball, so their number is bounded deterministically. This handles the rounding exceptions. Every application of the upper bound follows exposure of a surrounding loop, so the flat loop description in 2 suffices for the stated domains.

Finally, triangular graph distance and Euclidean distance are comparable by fixed constants, and a contour is within bounded distance of its incident face centers. A Euclidean closest-approach shell \([d,2d]\) is therefore covered by a fixed finite number of graph-inradius shells, with bounded small-radius exceptions. Summing the graph-shell moment bounds proves the Euclidean assertion. This metric comparison is made only after the strict graph-radius contraction has been proved. ◻

Proof. First put zero on the boundary. Conditional on all loop geometry, independent centered jump signs give \[ \mathbb E[H(x)H(y)] =\mathbb E\bigl[\#\{\hbox{loops surrounding both }x\hbox{ and }y\}\bigr]. \tag{22}\] Let \(s=\max(1,\mathop{\mathrm{dist}}(x,y))\), for \(x,y\) in the smearing ball. Their boundary distances are comparable to \(n\). A loop surrounding both points and approaching \(x\) within distance \(2d\ll s\) supplies a wall arm from scale \(O(d)\) to scale of order \(s\). The contour changes only one spin color along its entire length. A surrounding double circuit of that color blocks the arm: at a vertex of the double circuit all three adjacent spins have that color’s fixed value, so its wall cannot cross there.

Flat height boundary conditions fix both colors on a connected boundary chain. For either color used as the first spin, constancy of the other color on this chain is an \(E_0\) requirement; its single label does not change the first marginal. The buffers between scales \(d\) and \(s\) lie away from the boundary. Thus 8, for either possible wall color, bounds the arm probability by \(C(d/s)^\alpha\). Together with the second moment in 14, Cauchy–Schwarz bounds the expected common-loop count in this shell by \(C(d/s)^{\alpha/2}\). Summing the small shells is finite. Each shell from scale \(s\) up to scale \(O(n)\) contributes at most a constant. There are no larger enclosing shells, since an enclosing loop cannot surround a fixed boundary site. Consequently \[ 0\leq\mathbb E[H(x)H(y)]\leq C\bigl(1+\log(n/s)\bigr). \tag{23}\]

Summing (23) against \(\abs{w_xw_y}\) yields \[ \mathbb E\Bigl(\sum_xw_xH(x)\Bigr)^2 \leq C\bigl(1+\log(n/a)\bigr). \tag{24}\] Indeed the part \(\log(n/a)\) is controlled by the total weight. For the remaining singularity, graph balls contain \(O(r^2)\) sites, so the bounds in (18) give a uniform average of \(\log(a/\max(1,\mathop{\mathrm{dist}}(x,y)))\). The one-site contribution is bounded by \(Ca^{-2}\log a\). This calculation also covers signed weights.

Chebyshev’s inequality gives a fixed central probability on the scale \(M=C(1+\sqrt{\log(n/a)})\), chosen also to dominate the rounding error. The height arrays with zero boundary have admissible rounded midpoints, so 13 with one class proves (21) in the flat case.

For bounded boundary data, compare with the zero-boundary height shifted by \(-\lceil K\rceil\) and \(+\lceil K\rceil\). A nonnegative smearing is stochastically trapped between the corresponding shifted smearings. Its two tails are therefore bounded by those of the zero-boundary problem, with an additional deterministic error at most \(A\lceil K\rceil\). Split signed weights into positive and negative parts and use the triangle inequality in \(L^p\). This proves the stated bound. The proof uses only the flat loop representation and the incident-star convention, and hence also applies to the asserted dual-cut and skeleton boundaries. ◻

Proof. Search from the exterior for a double-\(B\) circuit in an annulus between \(Ka\) and \(2Ka\) about the support, where \(K\) is a fixed sufficiently large constant. 8 gives probability at least \(c>0\), uniformly in \(a\) and in a still larger free volume. The search does not inspect the interior. Its inner height trace lies in one pair of consecutive integers by 5. Subtract the lower integer of that pair. Conditional on the trace, the interior is a bounded-boundary height problem. Its boundary distance from the support is comparable to \(a\), and \(K\) may be chosen to meet the small-fraction requirement in 15. That proposition bounds the second moment of the shifted smearing by a constant. The shift does not alter the original smearing, since its total is zero. We have therefore obtained an unconditional central probability \(\mathbb P(\abs Y\leq M)\geq c'>0\) for a constant \(M\).

The free pair increment law is the uniform anchored-height law, as noted after 10. Its arrays have admissible rounded midpoints and bounded rounding error. Apply 13 in this finite volume, with the anchor outside the searched circuit. This gives the bound for every \(p\), uniformly in that volume and in \(a\). For each fixed lattice smearing, let the volume expand and use 10; the bounds pass to the limit. The constants are uniform in \(a\) and the listed symmetries, proving the assertion. ◻

Proof. On every oriented edge the Gray increment gives the identity \[ d(H+BW/2)=W\,dB. \tag{26}\] Let \(J\) be one connected component of \(B\)-changing edges, and put \(\omega_J=\mathbf 1_JdB\), extended by zero to other edges. A triangle has either no \(B\)-changing edge or two incident ones in the same component. Thus \(\omega_J\) has zero curl and has a potential \(\psi_J\) on the simply connected ambient graph. On the vertices of \(J\) this potential agrees with \(B\) up to an additive constant. Every other vertex can be connected to \(J\) by a path with no edge of \(J\) before its endpoint, and so its potential continues one of the two values on \(J\). The potential has exactly two values, differing by two.

No component of \(B\)-changing edges can cross a double-\(B\) cut. Between the two cuts, the sum of the potential changes of all such components, from the outer cut to the inner one, is the total change of \(B\), namely \(2\) or \(-2\). At least one component \(J\) in the collar therefore has a nonzero change. Its potential is constant on the inner region and on the outer region, with a difference of two. Normalize it to zero outside the outer cut.

The collar is free of \(E_0\) and pinned \(W\) sites. Consequently the \(W\)-equality component containing \(J\) consists precisely of its vertices, and its common value \(w\) is an independent fair sign by 3. Flip this sign and leave all other \(W\) components unchanged. Formula (26) gives, at every vertex not in \(J\), \[H'-H=-2w\psi_J,\] when the height is anchored outside the outer cut. The correction from \(BW/2\) vanishes at these vertices. Thus the height change is zero outside and is \(+4\) or \(-4\) throughout the inner region. The spin pins outside the collar are untouched. The two statistic values differ by \(4\abs\lambda\), so a fair choice between them has variance \((4\lambda)^2/4=4\lambda^2\). ◻

Proof. If the obstacle family is empty, use 16 directly. For a nonempty family, begin in a finite free volume and anchor the height at a singleton on obstacle 1 with its prescribed unshifted value, so \(t_1=0\). The conditional pair law becomes uniform counting of admissible anchored height arrays satisfying (27). Partition them by the parities of \(t_2,\ldots,t_m\). In a pair of arrays in the same class, \((t_i+t_i')/2\) is an integer. Both rounded midpoint arrays therefore satisfy the same interval requirements with this new offset: the interval endpoints are integral. The edge bounds and the anchor are preserved. The rounding error of \(Y\) is bounded by \(A/2\). We verify the remaining central-mass hypothesis of 13.

First retain W-constancy as \(E_0\) and temporarily drop its plus labels. Around the test find a double-\(B\) cut by an exterior stopping search. Around each non-anchor obstacle find two nested double-\(B\) cuts of opposite signs in its pin-free collar. Choose all these search buffers disjoint. 8 gives their joint event \(\mathcal A\) probability bounded below, conditional on the \(B\) pins and \(E_0\). Restoring the bounded number of W-plus labels loses at most the fixed factor explained after 5. Thus \[ \mathbb P(\mathcal A\mid\hbox{all pins})\geq c_1>0. \tag{29}\]

The test search has not revealed its strict interior. Conditional on its trace, its height boundary lies in one consecutive pair, and there are no interior pin constraints. Subtract that pair’s lower value. The zero total removes the shift, and 15 supplies a uniform conditional second moment. If necessary cover the test support by a fixed number of smaller balls to meet that proposition’s small-radius condition. For a sufficiently large fixed \(M\), Chebyshev gives \[ \mathbb P(\mathcal A,\abs Y\leq M\mid\hbox{all pins})\geq c_2>0. \tag{30}\] Here we conditioned on the searches and their exterior traces, not on all the \(B\) values inside the test disk.

Now expose all \(B\) values and use 17 separately in each non-anchor obstacle collar. Choose a contributing component measurably from \(B\). Its \(W\) label is an independent fair bit, unaffected by the required plus labels because the collar has no pinned W sites or \(E_0\) edges. Flipping it changes \(t_i\) by an odd integer and changes no other offset. The anchor, every other obstacle, and the whole test neighborhood lie outside \(U_i\), hence outside this collar’s outer cut. The flip consequently preserves \(Y\), all other offsets, all pins, and \(\mathcal A\). Distinct collars give distinct independent bits.

Conditional on \(B\) and all remaining W-component labels, these switches act transitively on the \(2^{m-1}\) parity vectors, preserve probability, and preserve the event in (30). Each parity class thus receives at least \(2^{-(m-1)}c_2\) of that central probability. This is (19), with a number of classes independent of the mesh. 13 proves every moment bound.

Finally let the free volume expand at each fixed mesh. The finite pin event has positive plane probability by 10; cylinder convergence therefore passes to its conditional law. The uniform moment bounds pass as well, proving (28) in the plane. ◻

Proof. Anchor at obstacle 1. Equal offsets make all the offsets zero, so the admissible height arrays now satisfy fixed integer interval constraints. Their rounded midpoints are admissible without a parity partition. On the asserted cut event, the trace lies in a consecutive pair and the conditional interior is free of further constraints. The fixed buffer bounds its distance from the test above and below, uniformly in the arrangement. Covering the support by a fixed number of smaller balls if necessary, 15 therefore gives the uniform central probability in (30). Apply 13 with one class. This proof uses neither the separate obstacle collars nor bounds on moments outside the equal-offset event.

For the sufficient condition, if \(\varepsilon=\mathbb P(\mathcal E^c\mid\hbox{pins})\), then the conditional cut probability after adding \(\mathcal E\) is at least \((c_0-\varepsilon)/(1-\varepsilon)\). This is bounded below for all sufficiently small \(\varepsilon\). Exterior measurability ensures that the conditioning has not changed the free interior once the trace is fixed. Finite-volume approximation justifies the counting argument exactly as in 18. ◻

Proof. The phase law does not depend on its anchor: replacing \(o\) by another site multiplies each \(U_\theta\) by a unit complex number determined by the increments, which preserves its conditional Haar law. The same observation under translations and reflections gives the phase law the spatial symmetries of the pair law. We may therefore put the phase anchor on \(\ell\).

First take a finite reflection-symmetric free volume. Conditional on the spin values on its intersection with \(\ell\), the configurations on the two sides are independent and reflected-identical. Indeed, no nearest-neighbor edge jumps across a row of sites. A lift on the row is fixed by its spin values and one anchor value on the row. If phases are present, additionally condition on their values at that anchor. The phases on each side are then determined by the increments on that side and these row values, so conditional independence persists. Thus \[\mathbb E\big[\overline{\Theta_\ell F}F\big] =\mathbb E\big[\lvert\mathbb E[F\mid\text{row data}]\rvert^2\big]\ge0.\] The plane limit gives the assertion for cylinder functions. Functions in ordinary \(L^2\) obtained by approximation are included because \(\lvert\langle F,G\rangle_\ell\rvert\le \|F\|_{L^2}\|G\|_{L^2}\).

For clarity, take \(\ell=\{y=0\}\), with the upper half-plane chosen. Write a site as \((m+n/2,n\sqrt3/2)\). Reflection about \(y=0\) sends its integer coordinates to \((m+n,-n)\), and reflection about \(y=\sqrt3/2\) sends them to \((m+n-1,2-n)\). Both preserve \(\mathbb T\). Their composition is the pure normal translation \(a=(0,\sqrt3)\); no tangential correction is required. Let \(T\) be translation by \(a\) on cylinder functions. Stationarity gives \(\langle TF,G\rangle_\ell=\langle F,TG\rangle_\ell\). Moreover, \(\langle F,TF\rangle_\ell\) is the reflection form of \(TF\) at the halfway row, since reflecting \(TF\) in that row gives \(\Theta_\ell F\). It is therefore nonnegative.

To prove boundedness before taking the quotient, put \(a_n=\|T^nF\|_\ell\). Reflection Cauchy–Schwarz and the preceding symmetry imply \(a_n^2\le a_{n-1}a_{n+1}\) for \(n\ge1\). Ordinary Cauchy–Schwarz and stationarity give \(a_n\le\|F\|_{L^2}\) for every \(n\). A bounded nonnegative log-convex sequence cannot have \(a_1>a_0\); if \(a_0=0\), the same inequality gives \(a_1=0\). Consequently \(\|TF\|_\ell\le\|F\|_\ell\). It follows that \(T\) descends to the quotient and extends to the stated positive self-adjoint contraction \(S\). Tangential translation commutes with reflection, preserves the form, and has an inverse with the same properties. It is therefore unitary and commutes with \(S\). Rotation gives the other lattice directions, and rescaling gives the mesh statement. ◻

Proof. Let \(V_u\) be the ordinary unitary translation operators on the plane probability space, and let \(E(dp)\) be their joint spectral resolution. The increment cocycle satisfies \[\xi_{u+v}=\xi_v+V_v\xi_u, \qquad (V_v-I)\xi_u=(V_u-I)\xi_v.\] On a spectral neighborhood where \(e^{ip\cdot v}-1\ne0\), this identity expresses every \(E(dp)\xi_u\) as \(e^{ip\cdot u}-1\) times the same spectral vector. Equivalently, on that neighborhood define \[C(dp)=\frac{\langle\xi_v,E(dp)\xi_v\rangle_{L^2}} {|e^{ip\cdot v}-1|^2}.\] The cocycle identity makes these definitions agree on overlaps. The two basis directions cover the punctured torus, so they define a locally finite measure there and give [eq:increment-spectral] for the nonzero spectral part.

Let \(P_0=E(\{0\})\). The invariant part of the cocycle is additive, hence \(P_0\xi_u=A\cdot u\) for a random vector \(A\in L^2\). Take a smooth zero-integral function \(f\) with nonzero first moment \(m=\int x f(x)\,dx\), and smear the unit-mesh affine height against \(L^{-2}f(x/L)\). Affine interpolation reproduces linear functions exactly, so the invariant part of this statistic is \(L A\cdot m\). Its \(L^2\) norm is bounded uniformly in \(L\) by 16, since orthogonal projection cannot increase that norm. Two choices of \(f\) with independent first moments imply \(A=0\). This also rules out cancellation of a tilt by other spectral components: those components are orthogonal to the invariant subspace.

The representation determines \(C\) locally by any nonvanishing lattice difference. The symmetries of the increment law therefore give the asserted symmetries of \(C\). ◻

Proof. Choose two real smooth zero-integral functions \(f_1,f_2\) whose Fourier transforms have no common zero on \(1\le|\zeta|\le2\). For example, take the two first derivatives of a smooth bump whose transform does not vanish on that annulus. Let \(a^{(r,i)}_u\) be the affine interpolation weights for \(r^2f_i(rx)\). Uniformly on the fixed annulus, \[\sum_u a^{(r,i)}_ue^{ir\zeta\cdot u} \longrightarrow \widehat f_i(-\zeta).\] This follows directly by interpolating the smooth plane wave; its error on a unit triangle is \(O(r^2)\), uniformly for \(\zeta\) in a compact set. Consequently the sum of the squared absolute values of these two transforms is bounded below on \(r\le|p|\le2r\), for all sufficiently small \(r\). Apply [eq:increment-spectral] and the uniform \(L^2\) bound of 16. The remaining range of \(r\) is covered by local finiteness of \(C\). Summing over dyadic annuli proves the last assertion. ◻

Proof. Let \(F=H(e_1)-H(0)\), supported on the reflection row. In the reflection Hilbert space, let \(U\) be horizontal unit translation and \(S\) the positive normal transfer from 20. For \(j\ge0\), \[\mathbb E\big[F\,V_{ne_1+j(0,\sqrt3)}F\big] =\langle F,U^nS^jF\rangle_\ell.\] The joint spectral resolution of the commuting operators \(U,S\) represents this as \(\int e^{ink}\lambda^j\,\gamma(dk,d\lambda)\), with \(0\le\lambda\le1\). Reflection in the row gives the same formula with \(|j|\) for negative \(j\). Write \(\lambda=e^{-\sqrt3s}\), including \(s=\infty\) at \(\lambda=0\). The vertical Fourier coefficients of \(Q_s\) are \(e^{-\sqrt3s|j|}\). Uniqueness of Fourier coefficients of finite measures therefore identifies the ordinary increment spectral measure as the mixture of \(Q_s\)’s with mixing measure \(\gamma\). By 21, that increment measure is \(|e^{ik}-1|^2 C'(dk,dl)\). Division on \(k\ne0\) proves [eq:cauchy-mixture] and the local finiteness assertion.

Since \(Q_s\) has total mass one, the left side of [eq:weighted-nu] equals \(\int_{0<|k|<\kappa}|k|\,C'(dk,dl)\). Near the physical origin this is finite by 22, because \(|k|\le|p|\). The rest consists of finite mass away from the physical origin, including the alias contribution. This proves the assertion without any horizontal-band estimate on \(\nu\). ◻

Proof. Put \(a=|k|\). The bounded analytic function \[z\longmapsto\left(\frac{a+iz}{a-iz}\right)^2\] on the upper half-plane has boundary real part \(\chi(a,l)\). The Poisson formula at \(z=is\) gives, for the unperiodized Cauchy density \(p_s(l)=s/\{\pi(s^2+l^2)\}\), \[\int_\mathbb R\chi(a,l)p_s(l)\,dl =\left(\frac{a-s}{a+s}\right)^2.\] To bound periodization uniformly, put \(f_a(l)=1-\chi(a,l)=8a^2l^2/(a^2+l^2)^2\). Then \(\int_\mathbb Rf_a(l)\,dl=4\pi a\), and \(f_a(l)\le 8a^2/l^2\) away from zero. Also \[\sup_{s>0}\sup_{|l|\le P/2}\sum_{n\ne0}p_s(l+nP)<\infty.\] For this last bound, split the sum at \(|n|\) comparable to \(s/P\): the near terms are bounded by \(1/(\pi s)\), and the remaining square tail is summable. Comparing the integrals of \(f_a\) over the period and over the line now gives an \(O(a)\) error; the unfolded tail contributes only \(O(a^2)\). The cases \(s=0\) and \(s=\infty\) follow directly from their definitions. ◻

Proof. Integrate [eq:angular-poisson] over \(\varepsilon<|k|<\kappa\). The total error is bounded independently of \(\varepsilon\) by [eq:weighted-nu]. Replacing the projected measure by the physical measure in the disk \(|p|<\kappa\) incurs another bounded error: the omitted physical region and the aliases have finite mass. Thus it suffices to bound above \[\int_{\varepsilon<|p|<\kappa,\ |k|>\varepsilon}\chi(k,l)\,C(dp).\] The integral over the full radial annulus vanishes. Indeed, the sum of the fourth angular harmonic over rotations by multiples of \(\pi/3\) is zero, and \(C\) is invariant under those rotations. Write \(E_\varepsilon=\{|k|\le\varepsilon,\ \varepsilon<|p|<\kappa\}\) for the removed endcap. Then \[\int_{\varepsilon<|p|<\kappa,\ |k|>\varepsilon}\chi\,dC =-\int_{E_\varepsilon}\chi\,dC \le\int_{E_\varepsilon\cap\{\chi<0\}}|\chi|\,dC.\] Negativity requires \(|l/k|<1+\sqrt2\), so the last integral is confined to \(\varepsilon<|p|\le K_2\varepsilon\). A fixed number of the annuli in [eq:annulus-mass] bounds it uniformly. This gives the required upper bound even if the positive part of the endcap integral is large. Since \(D\) is nonnegative, monotone convergence gives [eq:dispersion-defect].

For [eq:horizontal-band-mass], on \(r\le|k|\le2r\) the part with \(s\notin[r/2,4r]\) has \(D\ge1/9\), and hence bounded mass. For \(s\in[r/2,4r]\), a fixed positive fraction of \(Q_s\) lies in \(|l|\le2r\). [eq:cauchy-mixture] bounds that part of \(\nu\) by a constant times \(C'\{r\le|k|\le2r,\ |l|\le2r\}\). This is bounded by 22 and the finite alias mass. ◻

Proof. Rotate the region by a lattice rotation of angle \(\pi/3\). In the resulting coordinates \((k',l')\), both \(|k'|\) and \(|l'|\) are comparable to \(r\). At fixed \(k'\), the original condition \(|k|\le\varepsilon\) restricts \(l'\) to an interval of length \(O(\varepsilon)\). On this interval the density of every \(Q_s\), \(0<s<\infty\), is at most \(K/r\): its central Cauchy term is bounded by \(1/(2\pi|l'|)\), and the periodized tails are uniformly bounded. The \(s=0\) atom is outside this interval; the uniform law satisfies the same bound. The relevant horizontal band has bounded \(\nu\) mass by [eq:horizontal-band-mass]. Since projecting the original local measure only adds nonnegative alias mass, [eq:cauchy-mixture] proves the estimate for small \(r\). The remaining compact range of radii is treated identically, using finiteness of \(\nu\) on horizontal bands away from zero. Finally let \(\varepsilon\downarrow0\). ◻

Proof. Fix a compact horizontal band \(a\le|k|/\delta\le b\), where \(0<a<b\). Rescale the mixing variables to \((k/\delta,s/\delta)\). Their masses are uniformly bounded by [eq:horizontal-band-mass]. The integral of \(D\) over the original band tends to zero by [eq:dispersion-defect]. On the rescaled band the function \(D\) is bounded below away from zero outside any fixed neighborhood of \(s/\delta=|k|/\delta\), including at \(s/\delta=\infty\). Therefore every limiting mixing measure is supported on that graph.

On compact vertical sets, the rescaled periodized Cauchy kernel converges to the ordinary Cauchy kernel; the folded density error after rescaling is \(O(\delta)\). Thus, for a nonnegative measure \(\eta\) in the horizontal coordinate, the limiting physical measure on \(k\ne0\) has the form \[ M(dk,dl)=\frac{|k|}{\pi(k^2+l^2)}\,\eta(dk)\,dl. \tag{42}\] Here aliases introduce no extra term. Their preimages shrink to nonzero points of the triangular reciprocal torus but, because \(|k|/\delta\ge a\), exclude those points themselves. Finite measures give vanishing mass to such shrinking punctured neighborhoods, even when there is an atom at the alias.

[eq:limiting-cauchy] says that \(|p|^2M(dp)\) is locally invariant under vertical translations on \(k\ne0\); it does not assume that \(\eta\) has a density. Apply the same argument after the lattice rotations. At every nonzero point, at least two of the three tangential coordinates are nonzero, and the corresponding normal directions are nonparallel. Both constant directional derivatives of the distribution \(|p|^2M\) vanish in a neighborhood of that point. It is consequently a constant multiple of Lebesgue measure there. One can see this by convolving locally with a smooth approximate identity: the resulting smooth function has both independent derivatives zero. The constants agree on overlapping neighborhoods, and the punctured plane is connected. This proves [eq:spectral-subsequence], including on the axes. Finiteness and nonnegativity of \(c\) follow from 22 and positivity. ◻

Proof. The supports of the \(w_j\)’s lie in \(2^{-j-2}<|k|<2^{-j}\) and have bounded overlap. Replacing the physical cutoff integral \(I_j\) by \(\int w_j(k)\chi(k,l)\,C'(dk,dl)\) has errors whose absolute values sum to a finite number. Indeed, near the physical origin the measures and cutoff agree, whereas the remaining physical and alias contributions have finite mass; use the bounded overlap. For all sufficiently large \(j\), [eq:angular-poisson] then gives \[\int w_j\chi\,dC'=\int w_j(k)D(k,s)\,\nu(dk,ds)+e_j, \qquad |e_j|\le K\int |w_j(k)|\,|k|\,\nu(dk,ds).\] [eq:dispersion-defect,eq:weighted-nu] and bounded overlap prove absolute convergence. The finitely many remaining bands cause no difficulty.

Put \(\varepsilon_N=2^{-N}\). Telescoping, and using \(q(k)=1\) on the support of \(q(4|p|)\), gives \[\sum_{j=0}^{N-1}I_j =\int q(4|p|)\chi(k,l)\bigl[1-q(k/\varepsilon_N)\bigr]C(dp).\] Subtract the identically zero radial integral with bracket \(1-q(|p|/\varepsilon_N)\). The result is \[ \sum_{j=0}^{N-1}I_j =\int q(4|p|)\chi(k,l) \bigl[q(|p|/\varepsilon_N)-q(k/\varepsilon_N)\bigr]C(dp). \tag{45}\] The same right side tends to \(\sum_jI_j\) for arbitrary \(\varepsilon\downarrow0\). To check this, choose a dyadic \(\varepsilon_N\) comparable to \(\varepsilon\). The difference of the two horizontal cutoffs is bounded and supported on a band \(c\varepsilon<|k|<C\varepsilon\). The preceding Poisson comparison bounds its signed integral by the integrals of \(D\) and \(|k|\) over that band, plus finite outer and alias mass on a shrinking band excluding \(k=0\). All three tend to zero. This is also a useful reason for retaining the integrable defect, rather than only a subsequential limiting formula.

Now take any sequence along which \(C_\varepsilon\) has the limit \(c\,dp/|p|^2\). After rescaling, the bracket in [eq:cutoff-difference] vanishes on \(|p|\le1/2\). Its tails are uniform: on a dyadic annulus \(R<|p|<2R\), \(R>10\), it is supported on \(|k|<1\), whose rescaled mass is at most \(K/R\) by 26. Summing gives an \(O(1/R)\) tail bound. Vague convergence may therefore be used in the whole integral. Polar coordinates yield \[\begin{align*} \sum_{j\ge0}I_j &=c\int_0^{2\pi}\cos(4u) \int_0^\infty\bigl[q(r)-q(r|\cos u|)\bigr]\frac{dr}{r}\,du\\ &=c\int_0^{2\pi}\cos(4u)\log|\cos u|\,du =-\frac\pi2c. \end{align*}\] For the middle equality, change variables in the second radial integral, first with finite cutoffs. The absolute radial integral is bounded by a constant times \(1+|\log|\cos u||\), which is integrable in \(u\). For the last equality, use the fourth Fourier coefficient of \(\log|1+e^{2iu}|\), obtained by expanding from inside the unit circle; the constant \(-\log2\) in \(\log|\cos u|\) has zero pairing with \(\cos4u\). The left side is independent of the subsequence, so all coefficients agree. The final integral also proves independence of \(q\). ◻

Proof. Uniqueness of all vague subsequential limits was proved in 28; relative compactness then gives full vague convergence. To pass to tests, let \(A_{\delta,f}(p)=\sum_u a_{\delta,u}(f)e^{ip\cdot u}\) be the transform of the affine interpolation weights. Their exact total mass is zero. On compact sets of \(\zeta\), \(A_{\delta,f}(\delta\zeta)\to\widehat f(-\zeta)\) uniformly. Moreover, for any fixed integer \(N\ge3\), in a fixed neighborhood of zero, \[ |A_{\delta,f}(p)|\le K_N \min\left\{\frac{|p|}{\delta}, (1+|p|/\delta)^{-N}\right\}. \tag{47}\] The first bound follows from zero total mass and \(\sum_u|a_{\delta,u}|\,|u|=O(\delta^{-1})\). For the second, view \(a_{\delta,u}\) as the sample at \(u\) of the smooth test convolved with the compact nodal hat. Each lattice difference gains a factor \(\delta\); \(N\) summations by parts give the estimate. On the complement of the neighborhood the same argument gives \(O(\delta^N)\), since at least one basis difference \(|e^{ip\cdot e_i}-1|\) is bounded below.

Using 22, the squared-transform integral over \(|p|<\varepsilon\delta\) is \(O(\varepsilon^2)\), and that over \(R\delta<|p|\) within the neighborhood is \(O(R^{-2N})\). The remaining part tends to zero by local finiteness and the \(O(\delta^N)\) bound. On the intermediate annulus, vague convergence and the uniform transform limit apply. Polarization proves the first equality in [eq:plane-covariance]. Its second equality is the Fourier representation of the planar inverse Laplacian on zero-integral tests, whose kernel is \((2\pi)^{-1}\log(1/|x-y|)\). Height inversion symmetry makes all these zero-total plane statistics have mean zero. Uniform bounds in [eq:smearing-transform-bound] prove the assertion for varying tests.

It remains to prove strict positivity. Choose two nonnegative smooth tests of mass one supported on disjoint disks, and let \(f\) be their difference. By 8, with probability bounded below uniformly in mesh, two nested opposite-sign \(B\) double cuts surround one disk and avoid the other. The collar contains no pins. The component switch of 17 then supplies a fair \(W\)-component label which changes \(H_\delta(f)\) by exactly four when flipped. Conditional on all other labels its variance is four. Averaging yields a fixed positive lower bound for \(\mathbb EH_\delta(f)^2\). The covariance limit just proved forces \(c_*>0\). Its finiteness follows already from the annular mass bound. ◻

Proof. The squared norm is \(\mathbb E[(\Theta_{\ell_\delta}X_\delta)X_\delta]\). By 29, it converges to the inverse-Laplacian pairing of \(\Delta\phi\) and its reflected copy, multiplied by \(v\). That pairing is \(\int\nabla\phi\cdot\nabla(\phi\circ\theta)\), and is zero because the supports are disjoint. Reflection positivity makes these squared norms nonnegative. Auxiliary phases do not change the expectation of a statistic which does not use them. Converging placements are covered by the varying-test assertion of 29. The conclusion concerns the reflection norm; the ordinary variance of \(X_\delta\) need not vanish. ◻

Proof. The inverse images of \(B=+\) in \(\mathbb Z\) are the disjoint adjacent pairs \(\{4t,4t+1\}\); those of \(B=-\) are \(\{4t-2,4t-1\}\). Along a connected chain with constant \(B\), the height cannot move from one such pair to another, since the distance between successive pairs is three. The nearest-neighbor height constraint therefore propagates one pair along the entire chain.

For an upper window adjoining exact height \(0\), this pair is \(\{0,1\}\). If it adjoins exact height \(-1\), its first vertex must have height \(0\), again selecting \(\{0,1\}\). For a lower window, height \(-1\) directly selects \(\{-2,-1\}\), and an adjoining exact height \(0\) forces the first window vertex to be \(-1\). At a marked transition the exact residues \(0\) and \(-1\) select the unique adjacent branches. Starting from \(v_*\) and following the connected boundary therefore gives (62) everywhere, including every later exact-value arc. This proves an integer statement, independent of any choice of representatives modulo four.

The spin-height bijection and the Gibbs property in 2 identify the conditional interior law. No edge from an interior vertex reaches past the pinned vertex boundary. The upper trace dominates \(g_\delta\), and the lower trace is dominated by it. Height comparison, applied to each trace and then averaged, gives the two stochastic orders and an ordered coupling. Comparison with constant boundary heights \(-2\) and \(1\), whose centered laws are sign-symmetric, bounds the means. The same comparison and 15 give the asserted moment bounds. ◻

Proof. Let \(P=\bigcap_iP_i\) and \(J=\prod_i p_i\). Directly from the definition, \[ \|V\|_{\vartheta}^{2} =\frac{\mathbb E[\mathbf 1_{P\cap\vartheta P} \overline{\vartheta A}\,A]}{J^2}. \tag{67}\] There are \(2m\) obstacles in this expression. The strict separation from the row separates the original obstacles from their reflected copies; their individual probabilities are \(p_1,\ldots,p_m,p_1,\ldots,p_m\). Consequently 12 gives \[ \mathbb P(P\cap\vartheta P)\le C\prod_{i=1}^m p_i^2=CJ^2. \tag{68}\] For bounded \(A\) this proves the claim. Otherwise write \(A=BY\), with \(B\) bounded and \(Y\) the zero-total insertion. 18, applied separately to \(Y\) and \(\vartheta Y\) in the full reflected obstacle arrangement, gives \[\mathbb E[|Y\,\vartheta Y|\mid P\cap\vartheta P] \le\mathbb E[|Y|^2\mid P\cap\vartheta P]^{1/2} \mathbb E[|\vartheta Y|^2\mid P\cap\vartheta P]^{1/2}\le C.\] Combining this with (68) cancels the squared denominator in (67). The estimates invoked have uniform constants on the indicated compact families. Unit-modulus phases do not alter them. ◻

Proof. We prove the high-group version. Choose \(\eta>0\) so small that moving this group by \(-\eta n\) still leaves a fixed positive projection gap. Put \(\eta_\delta=N_\delta \tau_\delta\to\eta\) with \(N_\delta\in\mathbb N\), and use the translated arrangement at \(-\eta_\delta\) as the initial arrangement. Choose a lattice reflection row in its gap, tending to a row strictly in the limiting gap. Reflect and conjugate the normalized low-group factor to obtain a positive-side Hilbert vector \(A_\delta\); denote the normalized high-group factor at \(-\eta_\delta\) by \(B_\delta\). Individual obstacle probabilities are invariant under these lattice translations and reflections. By 34, \[ \|A_\delta\|_{\vartheta},\ \|B_\delta\|_{\vartheta}\le C. \tag{69}\]

For \(\mathop{\mathrm{Re}}z>-\eta_\delta\) define \[ f_\delta(z)= \big\langle A_\delta, S_\delta^{(z+\eta_\delta)/\tau_\delta}B_\delta\big\rangle_{\vartheta}. \tag{70}\] If \(S_\delta=\int_{[0,1]}\lambda\,dE_\delta(\lambda)\), the power in this formula means \[S_\delta^w=\int_{(0,1]}e^{w\log\lambda}\,dE_\delta(\lambda), \qquad \mathop{\mathrm{Re}}w>0.\] Its value on the zero spectral subspace is zero. It is holomorphic in this open half-plane and has operator norm at most one. Holomorphy follows, for example, by dominated differentiation on compact subsets, using boundedness of \(\lambda^a|\log\lambda|^k\) for \(a>0\). Thus (69) makes \(f_\delta\) uniformly bounded. For small \(\delta\) all these functions are defined on the common half-plane \(\mathop{\mathrm{Re}}z>-\eta/2\). This open half-plane contains zero in its interior.

At every lattice parameter \(t=m\tau_\delta\ge0\) the exponent in (70) is the positive integer \(m+N_\delta\), so the transfer interpretation gives exactly \(f_\delta(t)=M_\delta(t)\). There is no assertion here about the spectral power at the initial boundary point \(-\eta_\delta\). Cauchy’s estimate on compact subsets gives a uniform derivative bound, so replacing a real parameter by its nearest lattice parameter changes \(f_\delta\) by \(o(1)\).

Along any subsequence, the normal-family theorem supplies a further subsequence converging locally uniformly to a holomorphic function \(f\). The hypothesis makes \(f\) zero on \(I\). The identity theorem then makes it zero throughout the common half-plane, in particular at zero. Every subsequence has this property, proving the claim. Choose the filled neighborhoods strictly inside the two separated projection regions. Rigid motion preserves their within-group separations, while moving the groups apart preserves their projection gap. The initial negative backoff is smaller than that gap. Reflected copies likewise have disjoint filled neighborhoods by their positive row margin. These observations verify the geometric hypotheses of the side norm bounds throughout the motion. Convergent variations of the other placements preserve those bounds and the common half-plane. Replacing \(n\) by \(-n\) proves the low-group version. ◻

Proof. We first prove (73). Include each representative phase factor with its obstacle. Move all supports apart by the following sequence of one-sided translations. Each translation is across a strict projection gap and moves the entire selected side rigidly.

First move \(T\) by \(a_+e\) and \(B\) by \(-a_-e\), successively. These are legitimate moves across the original horizontal gaps. Choose the positive distances sufficiently large that \(T\) is above \(K_0\) in \(u\)-projection and \(B\) is below it in that projection. Together with (72), the two combined families \[Q_+=T\cup R,\qquad Q_-=B\cup L\] are now strictly above and below \(K_0\) in \(u\)-projection.

Next move \(Q_+\) by \(bu\) and \(Q_-\) by \(-cu\), successively. Choose \(b\) large enough that all of \(Q_+\) is strictly above \(K_0\) in vertical coordinate, and strictly below \(K_0\) in \(u'\)-projection. This is possible because \(u\cdot e=1/2\) and \(u\cdot u'=-1/2\). Then choose \(c\) large enough that all of \(Q_-\) lies strictly above \(K_0\cup Q_+\) in \(u'\)-projection. Each of these moves increases its separating \(u\)-gap. No separation within either combined family changes.

Finally move \(Q_-\) by \(du'\). Its \(u'\)-projection is already separated from every other support, so this is another legitimate one-sided move. Because \(u'\cdot e=1/2\), a sufficiently large \(d\) puts all of \(Q_-\) strictly above \(K_0\) vertically. Thus every factor other than \(X_\delta\) is now strictly above \(K_0\). All choices can be made in open ranges with positive margins; every finite choice leaves a fixed finite arrangement. The successive strict projection gaps also keep different moving families disjoint, while distances inside each moving family are preserved. In particular, all individual arc collars remain available. 1 illustrates the three directions.

At the final placement put a horizontal reflection row between \(K_0\) and all the other supports. Reflection Cauchy–Schwarz and 34 bound the normalized expectation by \[C\|X_\delta\|_{\vartheta}.\] The norm tends to zero by 30. That corollary remains valid in the phase-augmented Hilbert space, since \(X_\delta\) itself does not involve phases and its reflection inner product is unchanged by their addition.

For every fixed admissible \(a_+,a_-,b,c\), this proves vanishing for \(d\) in a sufficiently large open interval. Apply 35 to remove the last move. The same argument then applies for every sufficiently large \(c\) with \(a_+,a_-,b\) fixed, so it removes the \(-cu\) move. Remove \(bu\), \(-a_-e\) and \(a_+e\) in that order, each time using an open interval of allowed large values. This is successive one-variable analytic continuation; no simultaneous motion through surrounding pins is involved. It proves (73) at the original placement.

To impose \(E\), note that on \(P\) \[D_i=\frac{H(v_i)-g_i-H(v_1)+g_1}{4}=t_i-t_1\in\mathbb Z \quad(2\le i\le m).\] For \(s=(s_2,\ldots,s_m)\in[0,2\pi]^{m-1}\), \[ e^{i\sum_{i=2}^m s_iD_i} =\prod_{i=2}^m e^{-is_i(g_i-g_1)/4} Z_{s_i/4}^{(i)}(v_i) \overline{Z_{s_i/4}^{(i)}(v_1)}. \tag{75}\] Use an independent phase field for each \(i\) on the right. Every Haar factor cancels with its conjugate. Hence this is an identity of functions of the original increments, including for unbounded relative lifts. Fourier orthogonality gives exactly \[ \mathbf 1_E=(2\pi)^{-(m-1)} \int_{[0,2\pi]^{m-1}}e^{i\sum_{i=2}^m s_iD_i}\,ds \quad\text{on }P. \tag{76}\] For each fixed \(s\), the normalized expectation tends to zero by (73). Its absolute value is bounded uniformly in \(s\) and \(\delta\): indeed \[\frac{\mathbb E[|X_\delta F_\delta|\mathbf 1_P]}{J_\delta} \le \frac{\mathbb P(P)}{J_\delta}\, \|F_\delta\|_\infty\, \mathbb E[|X_\delta|\mid P]\le C\] by [prop:pin-product,prop:obstacle-moments] at this fixed gap geometry. Dominated convergence in (76) proves (74). ◻

Proof. First replace each requirement \(W=+\) by the requirement that \(W\) be constant on that subpath, and call the resulting edge equalities \(E_0\). If there are \(M\) prescribed subpaths, then, conditional on \(B\) and \(E_0\), requiring all their component labels to be plus has probability between \(2^{-M}\) and one. The number \(M\) is fixed by the original windows and does not grow as gaps shrink.

Consider one gap. In the annulus from scale \(Ca\) to scale \(r\), the only pins are \(B\) pins of the desired window sign. There are no \(E_0\) edges there. The favorable-pin version of 8, used in a logarithmic collection of separated annular slots, therefore gives a double circuit of that sign surrounding the gap except with probability \[C(a/r)^\beta+o_\delta(1).\] This estimate holds before any offset equality has been imposed. The two adjoining pinned boundary arms leave the neighborhood, so a surrounding dual circuit crosses each of them. At a crossing both incident primal sites have the window sign. Consecutive edges of a double circuit also share a connected chain of sites of that sign. Thus the circuit joins the two pinned arms through a connected constant-\(B\) chain. The adjacent-pair argument of 33 forces the same lifted pair on both arms. Their offsets are therefore equal.

The gap adjacency graph of the remaining boundary arcs is a cycle, hence connected. Circuits around all the gaps force every \(t_i\) to agree. A union bound over the fixed number of gaps, followed by restoration of the \(W=+\) labels at cost at most \(2^M\), proves (77), with for example a constant multiple of \((a/r)^\beta\) as the error. In particular \(\mu_{\delta,a}(E)\) is bounded away from zero for small \(a\) and then small \(\delta\).

For the coupling assertion first work with \(E_0\) and fill only one gap. All the new \(B\) pins have the same sign. The \(B\) marginals before and after adding them are ordered, by 3. Use the stopping coupling of 3, based on 5, exploring outwards from the gap and coupling the queried values in this order. In the more adverse law a double circuit of the added sign occurs before the search leaves the window neighborhood with failure probability at most \(C(a/r)^\beta+o_\delta(1)\), by the same favorable-pin estimate. On success it is a common double circuit in the two laws. All changed constraints are on its gap side; no \(E_0\) edge crosses it. The conditional law on its other side is therefore identical. Sample that side identically, including its \(W\) components.

Do this successively at the finitely many gaps. Searches stay in disjoint window neighborhoods. The resulting coupling matches the joint increments in every test disk and all the \(W\) labels that must subsequently be checked, except with probability tending to zero with \(a\). This last point uses that the label sites are outside the single-color window neighborhoods. If two laws can be coupled with error \(\varepsilon\) on variables including an event \(L\), and both give \(L\) probability at least \(c>0\), their restrictions conditioned on \(L\) differ in total variation by at most \(2\varepsilon/c\). Indeed, for any event \(A\) the numerators \(\mathbb P(A\cap L)\) differ by at most \(\varepsilon\), as do the two denominators, and division gives the stated bound. Applying this with \(c=2^{-M}\) restores all the required plus labels. It proves the claimed approximation between the unconditioned-offset gapped law and the filled law. Conditioning the former additionally on \(E\) changes any marginal by at most \(\mu_{\delta,a}(E^c)\). This proves (78), after enlarging \(\epsilon(a)\).

It remains to check the uniform moments under equal offsets. Fix an annular buffer around \(K_0\), with fixed inner and outer radii and positive clearance from \(K_0\). Choose its filled interior compactly contained in \(D\) and disjoint from all gap neighborhoods, so the buffer and its interior contain no pins. Conditional just on the pins, a double \(B\) cut in this buffer has probability at least a fixed \(c_0>0\). This follows from the local cut estimate and the bounded cost of the plus labels. By (77), its probability under additional conditioning on \(E\) is at least \(c_0/2\) for all sufficiently small \(a\) and then small mesh. Moreover \(E\) is measurable outside this cut: relative heights between the representatives can be evaluated on paths in the connected exterior of a disk containing the buffer. Such paths avoid the unrevealed interior of the cut. The usual outside-in cut exploration therefore leaves a Gibbs interior with its bounded-range trace, even after \(E\) is imposed.

All offsets are now common. Anchoring one representative makes the interval restrictions fixed integer intervals, so rounded midpoints require no parity classes. Together with the fixed buffer, these observations verify the hypotheses of 19; it gives the asserted bounds uniformly as the gaps close. The filled law has bounded boundary data by 33, and 15 gives its bounds. This argument does not estimate the exceptional unequal-offset law. ◻

Proof. At fixed gaps, 12 gives \(\mathbb P(P)/J_\delta\ge c_a>0\). By 37, for all sufficiently small fixed \(a\), \(\mathbb P(E\mid P)\ge1/2\) at small mesh. Division of (74) by \(\mathbb P(P\cap E)/J_\delta\) therefore gives \[ \lim_{\delta\downarrow0} \mathbb E_{\mu^{=}_{\delta,a}}[X_\delta F_\delta]=0. \tag{80}\] The positive number \(c_a\) may tend to zero as \(a\downarrow0\); it is used only with \(a\) fixed.

Let \(X^{(T)}=\max(-T,\min(X,T))\) and put \(M=\sup_\delta\|F_\delta\|_\infty\). The total variation bound in 37 applies to \(X^{(T)}F_\delta\), which is a bounded function of the joint test increments. The same lemma, at any \(p>1\), bounds the truncation errors under the equal-offset and filled laws by \(C_pM T^{1-p}\). Consequently \[\limsup_{\delta\downarrow0} \left|\mathbb E_{\mu_{\delta,0}}[X_\delta F_\delta]\right| \le 2MT\epsilon(a)+C_pM T^{1-p}.\] First send \(a\downarrow0\) and then \(T\to\infty\). This proves (79). For bounded continuous functions of smooth tests, distributional convergence and the same uniform integrability pass the expectation to a subsequential limit.

For completeness, here is a quantitative choice of the local geometry in the last assertion. Translate \(x\) to the origin. Take \(K_0\subset B(0,\rho/1000)\) and seek levels in \((\rho/40,\rho/20)\) and \((-\rho/20,-\rho/40)\). Choose spectator radii so small that the total length of their vertical projections is less than half the length of either candidate interval; a bound \(\rho/(1000(q+1))\) on each radius suffices. There is then a nonempty open subinterval of each candidate interval avoiding all spectator disks. Regular values are dense, so a sufficiently dense regular grid supplies both levels, with positive margins.

Every boundary point in the resulting middle band is outside \(B(0,\rho)\). The same is true of every middle spectator disk by assumption. Since \(|y|<\rho/20\) there, each such point has \(|x|>\sqrt{1-1/400}\,\rho\). Its \(u\)-projection \((\sqrt3/2)x+y/2\) consequently has magnitude greater than \(\rho/2\), with the sign of \(x\). A connected middle arc or disk cannot change that sign. Thus (72) holds with ample margin, whereas all projections of \(K_0\) have magnitude at most \(\rho/1000\). Small gaps and lattice approximation errors preserve the strict inequalities. This proves the geometric assertion. ◻

Proof. Apply the anchored comparison with the high interval on the two clean portions of the base arc running from just inside the innermost comparison strip to beyond the outermost strip. If a central exception is present, the two portions are separated by a low central portion. The other skeleton sites receive the low interval. The transition buffers are disjoint from all the strips and their surrounding buffers. The bounds in (83) give (87).

Before imposing the complementary transition labels, every pin in a buffer used for a \(t=s\) double circuit is favorable. By 8, there is a constant \(a>0\) such that, conditional on all first-color spins outside a block buffer, a pair of such double circuits in slightly narrower versions of (88) has probability at least \(a\). Choose \(M\) so that the block buffers are disjoint. Successive revelation of these buffers gives the same lower bound conditional on the preceding outcomes. Indeed, for \(\lambda>0\) and the success indicators \(I_i\), \[\mathbb E\exp\!\left(-\lambda\sum_{i=1}^N I_i\right) \le (1-a+ae^{-\lambda})^N.\] It follows that \(\sum I_i\ge aN/2\) except with probability \(C e^{-cN}\). Requiring at most four complementary labels multiplies this failure bound by at most \(2^4\).

Each double circuit meets the high part of the skeleton: the original boundary has a segment from the inner box to the exterior of the outer box. The adjacent faces along the double circuit all have the same first color and, by continuation from that intersection, their lifts lie in the anchored pair \([k-1,k]\). The corresponding closed primal walk stays within a bounded number of lattice spacings of the dual circuit and has winding number one about the inner box. It contains a simple surrounding site circuit. This produces the preliminary circuits in the synthetic height. The event that at least a given number of pairs exist is increasing in the height, so (87) transfers its probability to the actual law. ◻

Proof. If two surrounding simple cycles have at least two common vertices, their union is two-vertex-connected. After deleting any one vertex, the remaining part of each cycle is connected and the two parts still share a vertex. Its complementary face containing the inner box and its unbounded face therefore have simple cycle boundaries. These boundaries are the inner and outer envelopes. They use only good vertices of the original cycles and stay in the same strip. If there is only one common vertex, the two cycles cannot cross just once. Since both enclose the inner box, their interiors are nested. Disjoint surrounding cycles are nested as well. Iterating the envelope operation in the finite strip gives the extremal circuit, regarded without an orientation.

For a specified good circuit \(C\), whether it is the inner extremal circuit can be decided from its closed inner side in the strip. A competing circuit that improved it while partly leaving that side would have an inner envelope with \(C\) that is itself a better good circuit entirely on that side. Thus the absence of a better circuit has no dependence on unseen heights outside \(C\). The outer statement is identical with the directions reversed. Equivalently, these statements specify stopping sigma-fields: on the event that the selected circuit is \(C\), the recorded heights and the selection event are measurable using the deterministic set consisting of \(C\) and its tested side. One may reveal that entire set, including bounded pockets. Failure to find a circuit requires only the strip itself.

For the selected pairs, let \(V_i\) be the actual free sites between \(C_i^-\) and \(C_i^+\). Reveal the unsuccessful strips, the tested sides of successful strips, the selected circuits, and then every site outside \(\bigcup_i V_i\). Call the resulting sigma-field \(\mathcal G\). On every realization of the selected sets, the preceding measurability statement says that the event specifying these sets places no condition on the heights in \(\bigcup_i V_i\) beyond their recorded boundary values. The finite Gibbs specification therefore applies. Distinct \(V_i\) have no interacting edge after their separators are fixed, giving conditional independence. Touches with the original boundary may divide \(V_i\) into several bounded faces; the same argument applies to their union. ◻

Proof. Translate the attachment location to the origin. Take \(W_0\) to be a small fixed fraction of \(\ell\), and later take \(d\) to be a smaller fixed fraction of \(W_0\). The two preliminary circuits are outside this box. The unobstructed part above the boundary tolerance belongs to one bounded face of \(S\); in particular, a site \(x\) close to \((0,3d)\) is free in that face.

Synthetic boundary and its mean. On \(S\), prescribe the high interval \([k-1,k]\) on a base subarc near the origin, of half-width approximately \(W_0/4\), and the low interval \([k-3,k-2]\) elsewhere. Choose its endpoints so that the high part covers horizontal width \(W_0/8\), lies within horizontal half-width \(W_0/3\), and avoids all tiny replacement attachments. A replacement attached within the high portion receives the high prescription throughout. There are only two transition labels. This construction is feasible for all sufficiently small replacement diameters and gives the synthetic lower comparison \(K\) in (87).

Let \(A\) be the event that a primal path of heights at least \(k\) joins this high patch to depth \(d\) inside the attachment box. This is an increasing height event, so it is enough to give it positive probability in the synthetic law. We shall first use the mean at \(x\) to detect which side of the boundary-entering contours contains \(x\), and then show that the high side forces \(A\).

Between radii of order \(d\) and a small fixed fraction of \(W_0\), annular buffers centered on the projected boundary contain only favorable \(t=s\) pins. The annular stack estimate gives, except with probability \[ e(d/W_0)\longrightarrow0, \tag{90}\] a double \(t=s\) circuit surrounding \(x\) and meeting the high patch. The same bound, multiplied by a fixed factor, holds after the two complementary labels are required. Choose such a cut using only the first color. Its inner face chain belongs to the anchored pair \([k-1,k]\), and both transition labels lie outside it. No component of \(t\)-changing edges crosses the double cut. Thus the complementary labels in its interior, including its inner cut layer, are all free. Flipping these labels preserves their conditional law. It reverses every interior height increment and swaps the two values of the anchored cut pair, hence sends \(K\) to \(2k-1-K\) there. The conditional mean at \(x\) is therefore \(k-1/2\).

We also need a bound on the mean on failure, uniform in the lattice size. The component potentials in the proof of 17 give this bound. On each component of \(t\)-changing edges, \(dt\) extended by zero has a potential with two values differing by two. If \(t=B\), the identity \[d(K+BW/2)=W\,dB\] expresses the height difference from the transition anchor to \(x\) as an endpoint term of absolute value at most one, plus the component labels times their potential differences. Each potential difference has absolute value at most two. If \(t=W\), use \(d(K-BW/2)=-B\,dW\) instead. The unprescribed labels have conditional mean zero, and at most two components have prescribed labels. Since the anchor has value \(k-2\) or \(k-1\), this proves \[ \bigl|\mathbb E[K(x)\mid t,\text{transition labels}]-k\bigr|\le C_1 \tag{91}\] for every first-color configuration, with an absolute constant \(C_1\). Combining the symmetry on success with this bound gives \[ \mathbb EK(x)=k-\tfrac12+O(e(d/W_0)). \tag{92}\] All these arguments take place in a finite free volume around the construction, so no infinite-volume height or absolute anchor is required.

Low lateral barriers. Around the boundary locations with horizontal offsets \(\pm2W_0/3\), use annuli with inner radius several times \(d\) and outer radius less than \(W_0/8\). All local first-color pins are now favorable to \(t=-s\). With failure probability again tending to zero with \(d/W_0\), each side contains a double \(t=-s\) circuit meeting the low skeleton. Its face paths have the anchored low values and include a path from the low skeleton to above depth \(3d\), inside the horizontal band of width \(W_0/4\) about that side location. Write \(L\) for the event that both such lateral barriers exist. They have height at most \(k-2\) in the synthetic problem. The estimates survive the same two complementary labels. We may increase the lower end of the annular range by a fixed factor when a little more vertical clearance is needed.

For the planar separation below, continue each barrier along its local low skeleton arc and then through the exterior to a line below the boundary band. Uniform parametric approximation keeps this continuation in a slightly enlarged lateral strip; a tiny replacement attaches to the same local base arc. The skeleton portion is outside the high patch and is low throughout. The exterior portion is only a geometric continuation, which a contour lying in the face cannot cross. Thus the two barriers separate the high patch laterally all the way through the boundary band.

The mean detects a side of the open contours. Condition on the synthetic trace on the whole skeleton \(S\). In its bounded face containing \(x\), follow every dual level path at threshold \(k-1/2\) that enters through a skeleton edge. At each crossed edge, the adjacent heights are exactly \(k-1,k\). On entering a triangle, those two values force its third value to be one of \(k-1,k\), and hence determine the next crossed edge. The exploration only reveals the adjacent values of the paths being followed. Exhaust all boundary crossings in this way, including crossings on either side of a skeleton edge bordering a slit in the face.

Conditional on these open paths and their lips, their occurrence imposes only the indicated fixed vertex values. Any admissible completion with those values has the same open paths, by the degree-two rule and exhaustion of the boundary crossings. Unseen closed contours remain unrestricted. The remaining free components therefore have their Gibbs laws.

Cut the face along the exposed open paths. Their lips have known sides: the \(k\) lip is high and the \(k-1\) lip is low. Together with the threshold signs of the skeleton trace, these sides label every resulting geometric cell high or low. Here is why these labels are consistent even when the skeleton boundary has slits. A closed threshold contour in this face cannot enclose a skeleton site, since the connected skeleton continues outside the contour and cannot cross it. It cannot enclose part of an exposed open path either: that path continues to the skeleton and contours do not intersect. In the full threshold-sign picture, one may therefore erase the closed contours, filling each outermost enclosed region with its exterior sign, without changing any recorded skeleton or lip sign. The remaining sign is constant in each cell cut out by the open paths. This proves consistency; it is a deterministic explanation, not an additional revelation or conditioning of the closed contours.

The free primal Gibbs components left by the exploration lie in these geometric cells; removing the revealed lip vertices may divide a cell further. In a low cell every adjacent fixed value is at most \(k-1\). In a high cell every adjacent fixed value is at most \(k\): the high skeleton values and the high lips are exactly \(k\), and all other fixed values satisfy the same ceiling. Let \(U\) be the event that \(x\) belongs to a high cell, including the case that it lies on a revealed high lip. A revealed low lip belongs to the low case.

Comparison with a constant ceiling, including the deterministic case when \(x\) lies on a revealed lip, yields \[ \mathbb EK(x)\le k-1+\mathbb P(U). \tag{93}\] Together with (92), this gives \(\mathbb P(U)\ge1/2-O(e(d/W_0))\). A closed high island around \(x\) does not suffice for \(U\). Such islands remain inside their unrestricted Gibbs cells, and comparison controls the cell’s mean rather than each of its height values.

The correct boundary lip reaches the box. The high lip of every exposed open path is a connected primal chain at height \(k\). Its endpoints are skeleton sites at height \(k\), so they lie on the designated high patch, including any tiny replacements carrying that patch’s prescription. No endpoint lies on a preliminary circuit or on another low part of the skeleton.

We claim that \(U\cap L\subseteq A\). Suppose instead that \(L\) occurs but \(A\) does not. Follow a high lip from either of its endpoints on the high patch. It cannot cross a low lateral barrier: these are primal paths of height at most \(k-2\), whereas the lip has height \(k\). It cannot leave through the exterior side of the face. To pass above a barrier it would first have to reach depth \(d\), already giving \(A\). Consequently every exposed open path stays between the two barriers and below depth \(d\). This statement concerns all the open paths, since every one of their high lips starts on the designated high patch.

The site \(x\) is at depth \(3d\). Staying above depth \(2d\), it can be joined to one lateral barrier without crossing any exposed open contour. Following that barrier down to its low skeleton endpoint also crosses none: all its sites have height at most \(k-2\). The resulting route puts \(x\) in the cell with the low label inherited from that endpoint. This contradicts \(U\). The boundary and the tiny replacements stay in a much shallower band and do not obstruct the deep portion of this route; the barrier itself supplies the continuation through that band. Fixed margins between depths \(d\) and \(3d\) absorb lattice rounding.

Since \(\mathbb P(L^c)=O(e(d/W_0))\), the inclusion proves \[\mathbb P(A)\ge\mathbb P(U)-\mathbb P(L^c)\ge\tfrac12-O(e(d/W_0)).\] The mean calculation was not conditioned on the lateral barriers. Choose \(d/W_0\) once so that this bound is at least, say, \(1/4\). The event \(A\) asks only for a superlevel path, even though the synthetic construction produced an exact-height-\(k\) lip. It is therefore increasing, and (87) gives the same positive lower bound for the actual height.

Finally, in the actual comparison every usable skeleton site in the local box is at least \(k\); the other skeleton sites there are artificial. A tiny replacement is attached to the local base arc, and that arc, with its parametric slack at the box margins, is connected through open sites to the exterior below. For example a path from below stopped at its first encounter with the base boundary reaches such an open site. Thus the attachment has the exterior continuation claimed in the lemma. ◻

Proof. Fix the realized middle problem. Its actual boundary has the lower bounds recorded after (89). The two attachments of 42 have uniform positive probabilities, and their events are increasing in the actual height.

We first record a bulk path estimate under this same conditional law. For a box with a fixed relative annular buffer inside the free middle, compare from below with constant height \(k-2\) on its bounding skeleton. The latter comparison may be realized by fixing both spins of that height on \(S\). Connectedness of \(S\) fixes a common flat lift on every relevant bounded face. Its height contours are closed loops: a contour cannot cross the flat skeleton or enclose just part of a connected skeleton. Given the unoriented loops, all their jump signs are independent fair signs. For use of the first-spin circuit estimates, first enforce constancy of the second spin along \(S\). This is an \(E_0\) constraint outside the bulk buffers. Since \(S\) is connected, specifying its second-spin value costs one fair component label and does not change the first-spin marginal.

Apply 8 in a fixed number of thinner concentric buffers, requesting alternating first-spin signs on five double circuits. All these buffers are inside the free region. Successive conditional estimates give a positive probability, uniform in mesh and exposed actual heights, for this event in the flat comparison. Between each successive pair there is a first-spin wall loop surrounding the inner box. Hence there are at least four nested height loops in the allocated annulus.

Retain only the event that there are at least four enclosing unoriented loops in the allocated annulus, forgetting the first-spin event used to lower-bound its probability. This event and the choice of four consecutive enclosing loops depend only on the full unoriented geometry. Conditional on that geometry, the height just outside the first chosen loop is \(k-2\) plus a sum of other fair loop signs. It is therefore symmetric about \(k-2\) and independent of the four chosen signs. With probability at least \(2^{-5}\) it is at least \(k-2\) and all four signs are positive. The inner face path of the fourth loop is then at least \(k\). Its closed primal walk has winding number one and contains a surrounding circuit, with only bounded mesh displacement. Thus a fixed bulk annulus has a uniformly positive probability of a circuit at height at least \(k\). Height order transfers this to the actual middle law.

These bulk circuits give paths along any fixed polygonal tube with positive width and clearance. To see the deterministic gluing, place small annuli along the tube so that consecutive inner disks overlap and each protrudes beyond the other annulus’s outer disk. Two surrounding circuits cannot then be disjoint or nested, so they intersect. A finite string supplies the desired connected crossing. Positive association for heights bounds the probability of the simultaneous increasing circuit events below by the product of their positive lower bounds.

Use such a tube to cross each attachment box horizontally near depth \(d/2\), extending a little beyond its sides. Any rising attachment from the boundary band to depth \(d\) intersects this crossbar. Join the left and right crossbars by tubes going upward near horizontal locations \(\pm2\ell\), across at depth \(2\ell\), and downward on the right. Tube widths are much smaller than \(d\), and all their bulk boxes have clearance from the skeleton. The whole construction lies between \(Q_\ell\) and \(\partial Q_{5\ell}\); see 2.

Apply height positive association in the actual conditional Gibbs problem to the two attachment events and the finite collection of bulk circuit events. This gives a fixed lower bound \(a_1>0\) for their intersection. The exterior continuations of the attachments can be joined below the boundary, around the bottom of \(Q_\ell\), through artificial sites and good local boundary sites. The resulting closed open walk follows a rectangle around \(Q_\ell\) and has winding number one, so it contains an open surrounding simple cycle. The tubes avoid the preliminary circuits; in particular, the construction has not declared any of their possibly sub-\(k\) usable values artificial. Every artificial site used here was already outside the original region or in a removed boundary patch, which proves the upgrade. ◻

Proof of 39. Let \(Z\) be the number of successful preliminary blocks. 40 gives \(\mathbb P(Z<c_0N)\le C_0e^{-c_0N}\). Use the actual-height stopping sigma-field \(\mathcal G\) of 41. Conditional on it, the retained middle regions have independent Gibbs laws, and 43 gives success probability at least \(a_1\) in each. Consequently \[\mathbb P(\text{no upgraded block}) \le C_0e^{-c_0N}+(1-a_1)^{c_0N} \le Ce^{-cN}.\] Every upgraded circuit surrounds \(Q_r\) and lies in \(Q_R\) by (82). This proves (84). Applying the argument to \(-H\) proves the reversed assertion. ◻

Proof. Every probabilistic lower bound in the proof is uniform in the specified trace; only its displayed one-sided bounds were used. It may therefore be applied to each conditional trace. A boundary-attached replacement keeps the skeleton connected and preserves the full-neighborhood condition for the retained free sites. For fixed scales, sufficiently small replacements stay in the shallow boundary tolerance and do not meet the bulk tubes. The transition points of the synthetic prescriptions can be chosen away from all attachment neighborhoods, as explained after (87). The number of complementary labels is consequently still at most four in the preliminary construction and at most two in each attachment construction. These are the only label costs in the proof. The geometric approximation threshold may depend on the finite family, but none of the block probability constants does. ◻

Proof. We first identify the bracket means. Fix a point \(p\) on an open arc of value \(k\), a center \(x\) near \(p\), and its testing ball of radius \(\kappa d(x)\). Choose finitely many final arch scales surrounding this ball in a small clean neighborhood of the nearest boundary projection of \(x\). Work initially at finite mesh with a fixed grid and windows sufficiently small for the construction below, and write \(Y_{\delta,x}\) for the chosen bracket’s unit average in the ball. Two searches will isolate the ball from the exceptional window values and then from the remote boundary data.

For each window choose a clean censor neighborhood on its constant arc, of value \(k_i\). Choose these neighborhoods disjoint, away from the marks, small relative to the final arch scales, and small enough to miss the testing ball. Conditional on the original bracket trace, all its values are at least \(k_i-2\), and those in this censor neighborhood outside its window are at least \(k_i\). Enclose the entire discrete window and its adjacent sites in the inner comparison box, with a positive margin before sending the mesh to zero. Theorem 39 supplies an arch at level \(k_i\) surrounding this box. For a fixed grid there are only finitely many windows; shrinking them relative to their censor neighborhoods makes the total failure probability as small as desired.

On success choose the innermost arch by exploring from the window side, as in 41. Discard the patch between the window and this arch, and retain the side away from the window. This retained side contains the testing ball and has not been examined by the search. Conditional on the revealed patches and separator traces, its free heights have their Gibbs law. No original exceptional vertex is adjacent to a retained free site. The new usable separator sites carry their actual heights, which are at least \(k_i\); the global lower bounds are also preserved. Thus the first searches replace the windows by small boundary pieces with the required one-sided data. They do not condition on the preliminary revelations used only to prove the arch probability estimate.

Apply 44 at the chosen final scales in the retained domain. The censor patches are small enough for its replacement hypothesis. In the chosen clean neighborhood every usable original or replacement boundary site has height at least \(k\). With high probability there is an arch at level \(k\) separating the testing ball from the remote boundary. Choose the outermost such arch by searching from the remote side. This time the unexamined region is the side containing the testing ball, between the final arch and the clean boundary arc. Its usable boundary consists of sites on the arch and any original or censor pieces it contains, all at height at least \(k\). Height comparison with constant data therefore gives conditional mean at least \(k\) for \(Y_{\delta,x}\).

Let \(E_\delta\) denote success of both stages, and expose only the discarded patches, the remote side of the final arch, and their usable traces. On \(E_\delta\) this information determines the event and leaves the testing region Gibbs. For any \(q>1\) the preceding conditional comparison gives \[ \mathbb E(Y_{\delta,x}-k)\ge -\norm{Y_{\delta,x}-k}_{L^q} \mathbb P(E_\delta^c)^{1-1/q}. \tag{100}\] The norm is bounded by the unconditional comparable-depth smearing estimate, whose limit form is (96). No moment estimate after conditioning on censor success is needed.

For the upper bound start afresh from the original bracket trace. Its global values are at most \(k_i+2\), and its clean values in the \(i\)th censor neighborhood are at most \(k_i\). Use decreasing censor arches and then a decreasing final arch in the same two stopping directions. The testing region now has usable boundary at most \(k\), giving the matching upper estimate, with the same Hölder bound on the failure contribution. This separate construction is necessary because the lower censor arches carry no upper bound on their revealed heights.

We make explicit the uniformity that permits the grid limit. Given an error tolerance, first choose finitely many final arch blocks in a sufficiently straight part of the limiting smooth arc. They are available for every sufficiently shallow testing center. For each fixed grid, choose the outer censor sizes so small that the robust comparison applies in these blocks; the synthetic interval endpoints in that comparison avoid all the tiny replacements. By 44, the final block estimates do not incur a factor for the number of replacements. Next decrease the window sizes so that all censor searches succeed with the desired total error, and before that take the mesh and boundary approximation errors to zero. The resulting bound on the mean depends on the final blocks and the error tolerance, and not on the grid. It thus survives window limits followed by grid refinement. This is exactly the order of limits used to define \((H^-,H,H^+)\). Nearest projections and the uniform smooth-arc estimates allow tangential as well as normal approaches to \(p\).

Since each bracket mean is harmonic, radial averaging recovers its value at \(x\). The two bounds just proved show that its boundary trace is \(k\) on the open arc. Together with (98), bounded harmonic uniqueness gives \(m^-=m^+=m\). The two marked points do not change this uniqueness: for a bounded harmonic function with zero trace elsewhere, compare its absolute value with its supremum times the harmonic measure of arbitrarily small boundary neighborhoods of the marks. At any fixed interior point these harmonic measures tend to zero, since a smooth domain has a Poisson kernel and single points have zero arclength.

The order now identifies the fields, rather than merely their means. For every nonnegative test \(f\), \(H^+(f)-H^-(f)\ge0\) and its expectation is zero. Thus it vanishes almost surely. Apply this to a countable dense family of nonnegative tests and use continuity of distributions. The ordered fields agree as distributions almost surely, proving the first assertion and transferring the bracket identities to \(H\).

For the mixed assertion no censoring is needed in the true law. Choose deterministic outer arch neighborhoods that do not meet the supports of the other observations. First suppose \(F\ge0\) is measurable outside those neighborhoods. On the lower good event \(E_-\), choose the outermost lower arch from the exterior. The event, the selected trace, and \(F\) are measurable in the revealed exterior; the Gibbs property gives \[\mathbb E[(Y_x-k)F;E_-]\ge0.\] The upper search separately gives an event \(E_+\) on which the reverse inequality holds. These two searches need not share a sigma-field. For \(q>1\), their failure contributions satisfy \[ -\norm{(Y_x-k)F}_{L^q}\mathbb P(E_-^c)^{1-1/q} \le \mathbb E[(Y_x-k)F] \le \norm{(Y_x-k)F}_{L^q}\mathbb P(E_+^c)^{1-1/q}. \tag{101}\] All norms here are bounded by Hölder and the smearing estimates. For signed \(F\), apply the bounds to its positive and negative parts; they are exterior measurable and have the same moment control. One uses the actual exterior stopping search at this step, without retaining any of the preliminary revelations used to prove the arch probability estimate. The result follows first at finite mesh, then for limit moments by 45 and truncation.

When other observations approach different boundary points, fix that finite set first and take the outer neighborhoods smaller than their mutual distances. The observations remain exterior measurable as their depths decrease. Estimate (96) bounds their moments uniformly, so the same proof gives (99) in this joint limiting regime. ◻

Proof. Write \(Y_i(t)\) for the \(i\)th summand. Uniform comparable-depth moments give \(\mathbb E|Y_i(t)-k|^2\le C\). For \(i\ne l\), apply (99) with exterior factor \(Y_l(t)-k\): \[\mathbb E[(Y_i(t)-k)(Y_l(t)-k)]\longrightarrow0.\] Expanding the square, taking the depth limit at fixed \(N\), and then using the diagonal bound proves the first estimate in (102). Minkowski’s inequality gives the second from (96), uniformly in \(N\). For \(p\le2\) use the \(L^2\) bound; for \(p>2\) interpolate \(L^2\) with an \(L^q\) bound for any \(q>p\). A diagonal choice of finite depths as \(N\) increases gives the claimed reference sequence. In particular, we have not asserted that one comparable-depth average, or a fixed finite collection of them, becomes deterministic. ◻

Proof. The identity already transferred to the true law says \[ \mathbb E[H(\Delta\phi)F]=0 \tag{104}\] for bounded products of localized increment functions satisfying the separation conditions of 38. In particular, each factor may be a bounded continuous function of \(H(q)\), where \(q\) has integral zero and is supported in a small disk. Moment bounds and truncation extend the identity to products of finitely many such \(H(q)\)’s whenever the same geometric conditions hold. The disks may overlap one another.

Fix distinct points \(x_0,x_2,\ldots,x_j\). Take a small closed disk about \(x_0\) with a larger concentric neighborhood compactly contained in \(D\) and disjoint from the spectator points. Choose spectator neighborhoods and tests \(f_i\) therein, and write \(c_i=\int f_i\). All these neighborhoods can be made as small as needed below. We will approximate each absolute observation \(H(f_i)\) by constants and local zero-total observations supported a fixed positive distance from the Laplace insertion.

Choose a unit smooth average \(\rho_i\) near \(x_i\), and a clean boundary subarc accessible without entering a fixed larger disk about \(x_0\). Such access is always possible. A path in \(D\) that meets the excluded small disk can be detoured through a concentric annulus compactly contained in \(D\); thus removing this disk does not block a route to the boundary subarc. For finitely many reference endpoints in 47, join \(\rho_i\) to each endpoint by a finite chain of unit smooth averages \[\rho_{i,n,0}=\rho_i,\ \rho_{i,n,1},\ldots, \rho_{i,n,L_{i,n}}=\rho_{x_n(t),\kappa d(x_n(t))}.\] Consecutive supports lie in one disk of arbitrarily small prescribed maximum diameter. Construct this by a sufficiently fine subdivision of a path in its open tube, reducing the radii as necessary near the boundary. All chain disks avoid the fixed neighborhood of \(x_0\); different chains need not avoid one another. There is the exact identity \[\begin{align*} H(f_i)={}&c_i k+H(f_i-c_i\rho_i) +\frac{c_i}{N}\sum_{n=1}^N\sum_{l=0}^{L_{i,n}-1} H(\rho_{i,n,l}-\rho_{i,n,l+1}) +c_i(A_{N,t}-k). \tag{105}\end{align*}\] Every nonconstant term before the last has total integral zero and is supported in one small disk. The final term tends to zero in every required \(L^p\), with depth first and then \(N\) increasing.

We check the separation geometry uniformly in the number of chain links. Translate coordinates so that \(x_0=0\), and suppose all chain disks are outside \(B(0,R)\). Choose \(\ell\ll R\), and support \(\phi\) in \(B(0,\ell/10)\). Available lower and upper levels lie in \((-2\ell,-\ell)\) and \((\ell,2\ell)\), respectively. Cap the diameter of every chain disk, and of each initial spectator neighborhood, by \(\ell/(100j)\). Expand a product of the approximations on the right side of (105), omitting the last error term. Each expanded term contains at most \(j-1\) disks: it selects at most one local observation from each original spectator factor. Thus the total length of the forbidden vertical projections is less than \(\ell/100\) in either available level interval, regardless of the number or length of the routes. Choose regular levels from the dense union of grids outside these projections and with positive clearance.

Any selected disk in the resulting middle band is outside \(B(0,R)\) and has vertical coordinate of size at most \(2\ell\). It consequently lies well to the left or right of the insertion, also in projection along \(u=(\sqrt3/2,1/2)\), because \(\ell\ll R\). The boundary pieces in the middle band have the same property by the interior clearance of \(x_0\). These are exactly the geometric conditions of 38. Each finite expanded term therefore satisfies (104); the levels can depend on that term. There is no requirement that a single pair avoid every disk in all the routes at once.

For each finite approximation the expanded sum is finite, so its expectation against \(H(\Delta\phi)\) is zero. The approximated spectator variable differs from \(H(f_i)\) by \(c_i(A_{N,t}-k)\); in particular its higher moments stay bounded independently of the number of links. Hölder’s inequality now passes to the \(L^p\) reference limit in the unexpanded product. We obtain \[\mathbb E\bigl[H(\Delta\phi)\prod_{i=2}^jH(f_i)\bigr]=0.\] This proves distributional harmonicity in the distinguished variable on a neighborhood of every collision-free configuration. Product tests suffice by density in the smooth tests on a product neighborhood. Repeating with each distinguished variable gives \(\Delta_{x_i}T_j=0\) on \(\mathcal C_j\). Their sum is the ordinary elliptic Laplacian on \(\mathbb R^{2j}\), so distributional harmonic regularity gives a smooth representative there. The separate identities then hold classically. ◻

Proof. Radially mollify every variable at the common radius \(r\) and denote the result by \(T_j^{(r)}\). For centers in \(K\) and sufficiently small \(r\), the smearing bounds and Hölder give \[ |T_j^{(r)}(x_1,\ldots,x_j)| =\left|\mathbb E\prod_{i=1}^j H(\rho_{x_i,r})\right| \le C_{K,j}(1+|\log r|)^{j/2}, \tag{108}\] even if some centers coincide. When \(d_*>2r\), the averaging balls are disjoint and separate harmonicity gives \(T_j^{(r)}=T_j\) exactly by successive applications of the mean-value property. Taking \(r\) to be a fixed small fraction of \(d_*\), with a fixed upper cutoff depending on \(K\), proves (106). Its right side is integrable on \(K^j\): it is bounded by a constant times one plus the sum over pairs of powers of \(|\log|x_i-x_l||\), and such powers are integrable in two dimensions.

The collision neighborhood \(\{d_*\le2r\}\cap K^j\) has volume \(O_{K,j}(r^2)\). By (108), the integral of \(|T_j^{(r)}|\) over that neighborhood tends to zero. The integral there of the off-diagonal function tends to zero by its local integrability. Off that neighborhood the two functions agree. Thus \(T_j^{(r)}\) converges in \(L^1\) on compact sets to the extended off-diagonal function. It also converges distributionally to the original \(T_j\), by the approximate-identity property. The two are therefore the same distribution; no additional term supported on a diagonal remains.

For the boundary claims, choose fixed, disjoint, small radial unit averages \(\rho_i\) at the spectator points. For \(x\) sufficiently near the boundary, its comparable-depth averaging ball is disjoint from all of them. Separate mean value gives \[T_j(x,y_2,\ldots,y_j) =\mathbb E\left[H(\rho_{x,\kappa d(x)}) \prod_{i=2}^jH(\rho_i)\right].\] Hölder and (96) bound this uniformly in \(x\), including approaches to the marks. At a clean point, (99) gives its limit as \(k\mathbb E\prod_{i=2}^jH(\rho_i)\), which equals the right side of (107) by mean value once more. ◻