The law and the topology

Write \[\mathbb H=\{z\in\mathbb C:\operatorname{Im}z>0\},\qquad \mathbb H_\delta=\mathbb H\cap\delta\mathbb Z^2 .\] Edges join nearest neighbors. A dimer covering is a perfect matching. To specify its infinite-volume law, start with a simple lattice polygon in the coarse grid \((0,\delta)+2\delta\mathbb Z^2\), retain the fine-grid vertices and edges in its closure, and delete one coarse boundary vertex, called the root. Uniform matchings on these finite Temperleyan graphs converge locally as the polygons exhaust \(\mathbb H\) and their roots escape to infinity. We use this standard law, equivalently specified by the inverse Kasteleyn operator vanishing at infinity [4]. The coarse-grid translate puts the lower boundary on the bottom row of \(\mathbb H_\delta\).

Take two independent matchings with this law. Almost surely their superposition consists of doubled edges and finite simple even cycles [4]; see also [7]. Delete the doubled edges and let \(\mathcal L_\delta\) be the collection of all cycles, drawn with straight edges and regarded as unrooted, unoriented loops. Nested \(\mathrm{CLE}_4\) means an ordinary nonnested \(\mathrm{CLE}_4\), followed recursively by conditionally independent copies inside every loop. The normalization is the one in which the outermost ensemble comes from a Brownian loop soup of central charge \(1\) [19].

Let \(\mathbb D=\{w\in\mathbb C:|w|<1\}\). We fix, throughout, the compactification \[F:\mathbb H\longrightarrow\mathbb D,\qquad F(z)=\frac{z-i}{z+i}.\] For continuous loops \(\gamma,\widetilde\gamma:S^1\to\overline{\mathbb D}\), put \[d_{\rm loop}(\gamma,\widetilde\gamma) =\inf_{\phi}\sup_{t\in S^1} |\gamma(t)-\widetilde\gamma(\phi(t))|,\] where \(\phi\) ranges over circle homeomorphisms of either orientation. We identify loops at zero distance; in particular a weakly monotone change of parameter, with pauses, does not change the loop. Collections omit constant loops and have finitely many members of diameter greater than any fixed positive number, with multiplicities retained. An \(\varepsilon\)-matching of two collections is a partial bijection covering every loop of diameter greater than \(\varepsilon\) on either side, with matched loops at distance at most \(\varepsilon\). The unmatched loops therefore all have diameter at most \(\varepsilon\). Convergence means that such matchings exist with \(\varepsilon\downarrow0\). All diameters and loop distances in this definition are measured after applying \(F\).

It is convenient to use a compatible metric \(\rho\) on collections. For a partial bijection, charge the distance of each matched pair and half the diameter of each unmatched loop; take the supremum of these charges, then the infimum over partial bijections. This defines \(\rho\), with the supremum of an empty family equal to zero. The triangle inequality follows by composing partial bijections: a loop whose partner becomes unmatched has half-diameter at most the sum of the two charges, since \[|\mathop{\mathrm{diam}}\gamma-\mathop{\mathrm{diam}}\widetilde\gamma| \le 2d_{\rm loop}(\gamma,\widetilde\gamma).\] Vanishing \(\rho\) identifies equal collections because there are only finitely many loops above each positive diameter. A matching of cost at most \(\varepsilon\) leaves only loops of diameter at most \(2\varepsilon\) unmatched, so \(\rho\) induces exactly the topology specified above. We use the inherited topology on the subspace of collections whose loops are contained in the open disk. This is the target space for the theorem. The larger closed-disk space will be useful during the compactness proof, before boundary contact has been excluded.

History and the remaining geometric question

Theorem 1 resolves the half-plane Temperleyan form of the double-dimer \(\mathrm{CLE}_4\) scaling-limit conjecture. Two strands of earlier work explain both its prediction and the remaining obstacle.

Exact dimer enumeration on the square lattice was developed by Kasteleyn [10] and Temperley and Fisher [20], using Pfaffian and determinant methods. Lieb [15] introduced the transfer-matrix approach. These finite algebraic descriptions provide the background for our rectangle calculation. At the continuum scale, Kenyon [11] proved convergence of appropriately normalized dimer-height fluctuations for suitable Temperleyan approximations of smooth Jordan domains to the Gaussian free field. That field limit motivates the connection with \(\mathrm{CLE}_4\), but does not by itself give convergence of loop curves: passing from a distribution-valued field to its interfaces requires additional control.

Further evidence for the interface prediction came from Kenyon and Wilson [14]. For alternating black and white boundary nodes in the half-plane, they showed that the limiting double-dimer pairing probabilities agree with those for the corresponding Gaussian-free-field contour lines. This identifies the boundary pairings, while leaving convergence of the paths open. Kenyon [12] used quaternionic weights to construct loop-homotopy observables with conformally invariant scaling limits. Dubédat [7] identified a class of these topological correlators with their nested \(\mathrm{CLE}_4\) counterparts through isomonodromic tau-functions. Basok and Chelkak [3] recovered probabilities of cylindrical events from their lamination expansion, subject to a continuum lamination-tail estimate; Bai and Wan [2] supplied that estimate for the nested ensemble. We use the precise half-plane formulation in Basok and Izyurov’s preprint [4]. It retains all nested generations. Basok and Izyurov also prove local Gaussian fluctuations and convergence of the centered nesting field, which describe further aspects of the ensemble beyond a fixed puncture record.

The distinction needed here is between those puncture records and convergence in the curve topology. A long thin excursion may surround none of the selected points. Moreover, even equality of a limiting trace and winding number one allows a path to move backward along an arc before continuing forward. We must control both effects while retaining the number of loops and their nesting. The new estimates address these geometric questions directly.

The geometric estimates and the proof

The new input consists of two geometric estimates in a corner-rooted Temperleyan rectangle. The first bounds the total number of disjoint traversals of a fixed-width slab, including multiple traversals by a single loop. It supplies both macroscopic loop counts and moduli of continuity after reparametrization. The second controls the signed crossing counts of arcs whose endpoints reach prescribed opposite sides of the slab; the sign records the direction of a crossing. On a shrinking transverse interval it excludes an arc’s nonzero count disagreeing with that of its whole loop, and simultaneous nonzero counts from arcs on distinct loops. After localization, this prevents cancellation of an excursion against a return path that finite-puncture data would fail to see.

Both estimates come from a transfer matrix counting dimer configurations column by column. A boundary state records which dimers cross the seam between columns. Complementing alternate occupancy bits turns such a state into a subset of transverse sites, called a particle state; the column transfer is an exterior power of a one-particle matrix. This uses Lieb’s transfer method [15], with sine modes and particle sectors corresponding to the open-boundary spectral analysis of Rasmussen and Ruelle [18]. We derive the needed formulas in the convention appropriate to the deleted corner and prove the cap normalization explicitly, where a cap is the part of the rectangle on one side of a chosen seam. Changing particle number by a large amount gives a quadratic spectral penalty and hence traversal tightness. A two-particle insertion supported on a short interval moves two occupied sites from one matching’s state to the other’s. Its matrix coefficients are two-by-two minors of an interval projection. Modes spread across the transverse sites make these coefficients quadratic in the interval length. To turn that signed insertion into a probability bound, we prove a pointwise positivity identity, weighting nested loop portions cut off by transverse lines, so that contributions decrease geometrically down each nesting chain. This separation of a local operator estimate from a planar positivity argument is the main technical mechanism. The estimates are proved in Sections 2 and 3.

Wilson’s algorithm and the Temperley correspondence [21, 17, 13] transfer the estimates to bounded half-plane windows, uniformly in the mesh (Section 4). That section then proves compactness by constructing parametrizations directly from the fixed-slab estimates. The passage from crossing counts to controlled parametrizations follows the method of Aizenman and Burchard [1].

Section 5 couples a subsequential path limit and its puncture records with a canonical nested \(\mathrm{CLE}_4\). It fixes each finite puncture grid before choosing a sufficiently small lattice mesh. On the resulting joint realization, a pathwise argument rules out invisible curves and identifies each visible trace. A second grid refinement then excludes backtracking along that trace. This last step recovers the traversal order that trace equality and winding number one leave undetermined.

Established inputs

We state precisely the information imported from the literature. A lamination relative to a finite set of punctures records the homotopy classes of disjoint simple loops in the punctured domain, including their multiplicities. Delete every loop surrounding at most one puncture. The remaining record will be called the reduced lamination.

The precise input is equation (4.17) of the cited theorem in the January 2, 2025 preprint version. It is uniform when the puncture tuples remain in a compact set away from collisions. We only need the weaker formulation above. For a fixed puncture set the lamination state space is countable. Convergence of its point probabilities to a probability law implies total variation convergence: first restrict to a finite set carrying nearly all limiting mass. Passing to enclosed subsets is then a measurable pushforward; we do not claim that subsets determine every homotopy class. Later refinements always fix a finite puncture configuration before taking the lattice mesh sufficiently small.

The nonnested properties and the recursive construction are established in [19]. In particular, every fixed interior point is surrounded almost surely; conditional iteration in the successive loop interiors gives infinitely many surrounding loops and avoidance of all traces. Children lie strictly inside their parents, so disjointness also persists across generations. Bounded-domain macroscopic finiteness for the entire nested ensemble is recorded explicitly in [3]. We also use the classical Temperley correspondence [13] and Wilson’s rooted spanning-tree algorithm [21], in the order-independent form of [17]. Their exact application to the finite rectangles will be stated when the localization argument is introduced.

Rectangles and traversals

After translation, rotation, and possibly reflection, the rectangle has vertices \[R_\delta=\{(\delta x,\delta y):1\le x\le m,\ 1\le y\le n\} \setminus\{(\delta,\delta)\}, \qquad n=2p+1,\] where both \(m\) and \(n\) are odd. Nearest-neighbor edges join the remaining vertices. We consider sequences for which \(\delta m\) and \(\delta n\) converge to positive finite limits. The deleted corner is on the first column. We transfer in the horizontal direction and call it the longitudinal direction. Geometric cut locations use physical coordinates; transfer calculations divide them by \(\delta\) and count columns. A seam is a line between two consecutive columns. The part of the rectangle on either side of a seam is called a cap.

Two path portions are called parameter-disjoint when their parameter interiors are disjoint; they may share endpoints. For two longitudinal cuts \(u<v\), a loop that does not visit both \(\{x\le u\}\) and \(\{x\ge v\}\) contributes zero traversals. For every other loop, follow its orientation and record its successive visits to \(\{x\le u\}\) and \(\{x\ge v\}\), suppressing repeated visits to the same side before the other side is visited. The resulting cyclic word alternates and has even length. Its length is the number of traversals of the slab by that loop. Sum over all loops to obtain \(N(u,v)\). Equivalently this is the maximal number of parameter-disjoint trips between the two cuts, in either direction. In particular any chosen family of disjoint crossing subarcs is bounded by \(N(u,v)\), even if those subarcs have excursions elsewhere. Every traversal crosses any intermediate seam, so \[ N(u,v)\le n. \tag{1}\] All cuts below may be moved by one mesh step to seams.

The proof will compare the ordinary partition function with modified counts obtained by changing the allowed matching assignments in the caps. We first develop the column transfer and its boundary normalization. The same calculation also gives the uniform summability of spectral states needed for the short-interval estimate in Section 3.

Particles and column transfer

Take black sites in the first column to have odd row indices. A seam’s intersection with a horizontal dimer is described by an occupancy bit \(b_i\in\{0,1\}\) in row \(i\). On a seam, define its particle bit \(q_i\) to be \(b_i\) if the site immediately to its left is black, and \(1-b_i\) otherwise. At the initial boundary we use the checkerboard signs of a virtual column \(0\) and a single occupied incoming edge at the deleted corner. This edge simply declares that corner already matched; it is not an edge of the rectangle. The initial particle set is therefore \[\alpha=\{3,5,\ldots,2p+1\},\] and, because \(m\) is odd, the empty final boundary has particle set \[\beta=\{2,4,\ldots,2p\}.\] Both sets have size \(p\).

Consider a column whose black rows are odd. At a black row the incoming and outgoing particle bits are \(1-b_i^{\rm in}\) and \(b_i^{\rm out}\); at a white row they are \(b_i^{\rm in}\) and \(1-b_i^{\rm out}\). An incoming or outgoing horizontal dimer leaves the particle unchanged. A vertical dimer instead moves one particle from its black endpoint to its white endpoint. Thus particles stay or hop to an adjacent row, with hops allowed only from black to white rows. Disjoint such moves determine the dimers in the column uniquely: the rows unused by horizontal dimers are paired by the hops.

Let \(A\) be the \(n\times n\) matrix whose rows index outputs and columns index inputs: \[ A_{ij}=\mathbf 1_{\{i=j\}} +\mathbf 1_{\{j\ \mathrm{odd},\ |i-j|=1\}}. \tag{2}\] In the sector with \(j\) particles, the transfer is \(\Lambda^j A\), written in the increasing-index wedge basis. This is the nonintersecting-path determinant viewpoint of Lindström and Gessel–Viennot [16, 9]; in this one-column setting its positivity can be checked directly. To check its signs, an allowed bijection between source and destination rows cannot reverse their order: such a reversal would require opposite hops across the same adjacent pair, whereas only black-to-white hops are allowed. A hop also cannot pass a staying particle. Hence every surviving determinant term has positive sign and is exactly one legal collection of disjoint moves. The transfers in consecutive columns alternate between \(A\) and \(A^*\). For two independent matchings we take their tensor product.

Write \(T^{(j)}_{b,a}\) for the transfer in sector \(j\) from the seam after column \(a\) to the seam after column \(b\), so it contains \(b-a\) column steps. The number of single matchings is \[Z=\langle e_\beta,T^{(p)}_{m,0}e_\alpha\rangle, \qquad Z_{\rm dd}=Z^2\] for the two-replica partition function. All these matrix entries count matchings with positive weights.

Singular modes and the cost of changing particle number

The alternating transfers have compatible singular bases. The open-boundary sine modes belong to the transfer-matrix analysis developed by Lieb [15] and Rasmussen–Ruelle [18]. We give the derivation with the alternating bases and odd-width zero mode needed by the cap calculation. This gives both the large-sector penalty used below and the summability needed for the short-interval estimate in the next section.

The boundary caps

A spectral estimate for the middle of the rectangle is useful only if the boundary factors have the correct normalization. Let \[L_a=T^{(p)}_{a,0}e_\alpha, \qquad R_b=(T^{(p)}_{m,b})^*e_\beta\] be the ordinary single-replica cap vectors. The double-replica vectors are \(L_a\otimes L_a\) and \(R_b\otimes R_b\).

Cap diagrams and modified endpoint signs

We now turn the spectral penalty into a geometric estimate. Superpose the two matchings inside one cap, retaining the multiplicities of its edges but forgetting which matching supplies each single edge. After doubled edges are suppressed, each component is either an interior closed loop or a simple arch joining two single seam edges. The arches are disjoint. An ordinary interior loop has two alternating matching assignments; a doubled edge has only one.

At a single seam edge, the difference of the two particle bits is \(+1\) or \(-1\). It equals the signed crossing in the direction of the cycle obtained by directing first-matching edges from black to white and second-matching edges from white to black. Each arch has opposite particle-difference signs at its two ends. Either choice of those opposite signs determines its alternating matching assignment uniquely. These descriptions also hold for the cap at the missing corner: the virtual incoming edge occurs in both replicas and produces no open single strand.

A tear is the following formal modification of an arch. Instead of requiring opposite endpoint signs, prescribe both signs to be \(+\), or both to be \(-\), with weight one for that prescription. The uncolored arch and all its edge multiplicities remain unchanged. This defines a vector of seam states even though the modified assignment need not be a pair of matchings inside the cap. A positive tear increases the first particle number by one and decreases the second by one; a negative tear has the reverse effect. Geometrically it can be regarded as a source or a sink on the arch.

Here is the coefficient rule for the resulting formal vector. Fix an uncolored cap diagram \(D\) and its prescribed tears, and let \(c(D)\) be its number of internal single loops. At a seam site with multiplicity zero or two, the two occupancy bits are forced, and hence so are the particle bits. At the single seam sites, choose endpoint signs subject to the opposite-sign rule on every unmarked arch and the prescribed equal signs on every torn arch. Let \(\mathcal A(D)\) be the set of particle-state pairs \((Q_1,Q_2)\) obtained in this way. The diagram contributes \[2^{c(D)}\sum_{(Q_1,Q_2)\in\mathcal A(D)} e_{Q_1}\otimes e_{Q_2}.\] Each permitted endpoint assignment is counted once. In particular, a prescribed tear replaces the two ordinary arch assignments by one new assignment; it is not applied separately to each old coloring. Summing these contributions over diagrams, marks, and their specified weights defines the marked vector.

Proof of traversal tightness

Marked caps and the analytic bound

If either deep region is absent, there is nothing to prove. Otherwise choose two cap seams in the interior of the rectangle, \(x=a\) and \(x=b\), with \[c-\rho_-<a<c<b<c+\rho_+.\] They remain a positive macroscopic distance from the middle seam and from the longitudinal ends of the rectangle. In the left cap an arch is eligible if it reaches \(x\le c-\rho_-\); in the right cap it is eligible if it reaches \(x\ge c+\rho_+\). Here an arch is a connected nondoubled path in the cap with its two endpoints on the cap seam. Let \(E_M\) be the event that each cap has at most \(M\) eligible arches, where \(M\ge1\). Choose in each cap a fixed strictly interior line between the deep line and the cap seam. Every eligible arch makes two traversals between this shallower line and the cap seam, even when the deep line itself is a longitudinal side of the rectangle. Proposition 4 therefore gives \[ \lim_{M\to\infty}\limsup_{\delta\downarrow0}\mathbb P(E_M^c)=0. \tag{18}\] The same event \(E_M\) will be used for all intervals of the partition.

For now assign each eligible arch \(A\) a weight \(\lambda(A)\in[0,1]\) determined only by its own uncolored cap diagram and the choice of \(A\). We first compute with these weights and then choose them to make the signed contraction nonnegative. Their dependence on one cap alone allows the two marked cap vectors to be formed separately.

In each cap select one eligible arch and tear it as in Lemma 7, with weight \(\lambda(A)\). Use two plus signs at the left cap and two minus signs at the right cap. These are two sources: their apparently different signs are due to the opposite boundary normals of the two caps. Sum over the selections, and set the marked cap vector to zero when its cap has more than \(M\) eligible arches. Denote the resulting left and right vectors by \(V_L\) and \(V_R\). In the particle convention of the preceding section they lie in the two-replica sectors \((p+1,p-1)\) and \((p-1,p+1)\), respectively, where \(n=2p+1\) is the number of transverse sites. In cap-vector subscripts, the seams are indexed by their column numbers, as in Section 2. Lemma 7 gives \[\|V_L\|\le M\|L_a\otimes L_a\|, \qquad \|V_R\|\le M\|R_b\otimes R_b\|.\]

At the middle seam let \(S(I)\subseteq\{1,\ldots,n\}\) be the indices of sites in \(I\). The insertion \(\mathcal O_I\) sums, over \(i<j\) in \(S(I)\), the operation which removes the pair \(i,j\) from the first exterior factor and creates it in the second. It vanishes unless both sites have first-replica occupancy one and second-replica occupancy zero. The resulting pair of occupancies is reversed at both sites. Thus \[\mathcal O_I: \Lambda^{p+1}\mathbb R^n\otimes\Lambda^{p-1}\mathbb R^n \longrightarrow \Lambda^{p-1}\mathbb R^n\otimes\Lambda^{p+1}\mathbb R^n.\] Its increasing-wedge sign in the site basis is \[ (-1)^{\#\{\text{single occupations strictly between }i\text{ and }j\}}. \tag{19}\] Indeed the two exterior factors contribute their usual deletion and insertion signs; doubly occupied intervening indices contribute twice, and the other nonzero contributions are precisely the single occupations. The operation preserves total seam-edge multiplicities site by site, also on rows where the particle convention complements edge occupancy. Figure 1 shows the two cap sources and the two middle-seam sinks in a same-loop contribution.

The analytic estimate supplies a factor \(|S(I)|^2\) for every permitted choice of weights. To use it for a probability, we must choose the weights so that the full signed contraction is nonnegative and detects the event we want to exclude. We now compute its contributions before making that choice.

The pointwise color calculation

Fix an ordinary uncolored double-dimer configuration \(\omega\). Each nondoubled loop has two assignments of its alternating edges to the two matchings; doubled edges have only one. For each selected pair of cap arches and each sink pair \(i<j\) in \(S(I)\), count the modified color assignments satisfying the two source rules and the two insertion rules. Multiply by (19), divide by the ordinary number of assignments, and multiply by the two cap weights \(\lambda\). Sum over the choices, and define the result to be \(X^\lambda_{M,I}(\omega)\); set it to zero on \(E_M^c\). Lemma 7 gives the exact identity \[ \mathbb EX^\lambda_{M,I} =\frac{\langle V_R,\mathcal T_+\mathcal O_I\mathcal T_-V_L\rangle} {Z_{\rm dd}}. \tag{23}\]

To compute this color ratio, cut the uncolored configuration at the middle seam \(x=c\). A marked loop not reaching that seam has no possible sink and contributes zero. Otherwise the selected left cap arch lies in a unique arch of the same loop in \(\{x<c\}\). Direct its two branches away from the source. Ordinary strands propagate these directions to the middle seam, where both branches leave the left half-plane. The induced endpoint signs are therefore \(++\). On the right the same argument gives \(--\), because both branches leave \(\{x>c\}\). An unmarked middle-seam arch has one entrance and one exit, hence opposite endpoint signs.

Several choices of a remote eligible arch can lie on the same middle-seam arch. We keep them as separate marked choices, each with its own weight. Moving a source along that arch changes no intersection with \(I\). Thus the color calculation below applies to each remote choice and preserves its multiplicity when we sum.

Rank the singly occupied edges along the whole middle seam, and give rank \(r\) the alternating sign \(\sigma_r=(-1)^r\). Formula (19) becomes \[ -\sigma_i\sigma_j \tag{24}\] when \(i,j\) now denote the two ranks. On any one simple loop these rank signs agree with oriented intersection signs up to a common sign. To see why other loops do not affect this assertion, take two seam hits of the chosen loop. They have the same inside/outside relation to every other disjoint loop. Each such loop therefore contributes an even number of hits between them. After these even contributions are removed, the assertion is the usual alternation of entering and leaving a Jordan domain along a line.

For a precise parity identity, order the endpoints \(A,B\) of \(I\) in the increasing seam direction, and let \(d(g)\) be the number of loops strictly enclosing \(g\). Start rank numbering at the first hit of the whole seam, so that the region before all hits is exterior to every loop. Then \[ t_g=-(-1)^{d(g)} \bigl(\mathbf 1_{\{B\in\operatorname{Int}(g)\}} -\mathbf 1_{\{A\in\operatorname{Int}(g)\}}\bigr). \tag{25}\] Indeed at a hit entering \(g\), the nesting parity just before the hit is \(d(g)\), and at a hit leaving \(g\) it is \(d(g)+1\). Since \(\sigma_r=(-1)^r\), summing these signed changes telescopes to (25), regardless of how often \(g\) hits the seam.

The loops separating the endpoints of \(I\) form at most two nested chains, one around either endpoint, with their common ancestors omitted. Order each chain from its outermost loop inward. The signs \(t_g\) alternate along each chain. If both chains are nonempty, the two outermost signs are opposite. Successive members of either chain have ancestor depths differing by one. The two outermost members, when both exist, border the same common-ancestor region and have equal depths, but the endpoint differences in (25) have opposite signs. Common ancestors change both depth parities equally; other nonseparating loops do not affect the identity. Endpoints outside the rectangle are permitted in this description.

Nested arches and positivity

Only loops separating the endpoints of \(I\) contribute to the different-loop sum, since the other loops have \(t_g=0\). For such a loop \(g\), let \[U_g=\sum_{\substack{A\text{ eligible left arch}\\A\subset g}}\lambda(A), \qquad V_g=\sum_{\substack{A\text{ eligible right arch}\\A\subset g}}\lambda(A).\] These totals still depend on the weights to be chosen; empty sums are zero. The total contribution from marks on distinct loops is, by Lemma 10, \[ -\frac14\sum_{g\ne g'}U_gV_{g'}t_gt_{g'}. \tag{26}\] Individual summands can be negative: two loops whose positions in a chain differ by two have the same sign \(t_g\). We choose the weights so that the total mark weights decrease by a factor of at least two along either nesting chain. The resulting alternating sums will control the complete expression (26).

The weights must still be determined within each cap. For an eligible arch \(A\), let \(b(A)\) count the eligible arches in that cap whose endpoints strictly span those of \(A\) along its seam. Choose \[ \lambda(A)=w(A):=(2M)^{-b(A)}. \tag{27}\] On \(E_M\) every such weight belongs to \([q_M,1]\), where \[ q_M=(2M)^{-(M-1)}. \tag{28}\] From now on \(U_g,V_g\) use this choice, and write \(X_{M,I}=X^w_{M,I}\). Every nonzero \(U_g\) or \(V_g\) is at least \(q_M\), because it contains at least one eligible-arch weight. To connect the endpoint nesting chains to these cap weights, we show that every eligible arch of an inner separating loop is spanned by an eligible arch of the outer loop. That additional spanning arch supplies a factor \(1/(2M)\); the cutoff of \(M\) arches will then give the required factor \(1/2\) for the total weights.

We can now finish the detection argument. Lemma 10 makes every same-loop term of \(X_{M,I}\) nonnegative. Lemmas 11 and 12 make its complete different-loop term nonnegative. Thus \[ X_{M,I}\ge0. \tag{29}\]

If condition (i) of Proposition 8 occurs, each deep endpoint of the tested arc lies in an eligible arch of its cap. Select those two arches. Moving the endpoints to the marks inside these arches changes no intersection with \(I\), since the arches stay strictly on their respective sides of the middle seam. If the tested arc has intersection number \(a\) and the whole loop has number \(e\), the two mark-to-mark sums have product \(a(a-e)\). Here \(a\ne0,e\) and \(e\in\{-1,0,1\}\), so this product is a positive integer. The weighted same-loop term is at least \(q_M^2/2\).

Suppose instead that condition (ii) occurs and no tested arc gives condition (i). Each of its two nonzero arc sums must then equal the corresponding whole-loop sum. The two loops separate the endpoints of \(I\) and each has an eligible arch in each cap. In particular there is a distinct pair with \(U_gV_{g'}>0\). Their full different-loop sum is at least \(q_M^2/16\). We have proved the pointwise bound \[ X_{M,I}\ge\frac{q_M^2}{16}\, \mathbf 1_{E_M\cap\{\textnormal{condition (i) or (ii) holds on }I\}}. \tag{30}\]

Summing the short intervals

A uniform comparison with finite rectangles

Write \(\mathcal R_{\delta,R}\) for a Temperleyan rectangle whose coarse bottom side is at height \(\delta\), whose other sides are within \(O(\delta)\) of \(x=-R\), \(x=R\), and \(y=R\), and whose lower left coarse corner is deleted. Rounding the sides to the coarse grid gives odd numbers of fine-grid vertices in both directions. The precise rounding will be immaterial. Agreement of matching states on a set means agreement on every edge having at least one endpoint in that set.

Here is the finite graph to which we apply the Temperley correspondence [13]. Let \(G\) be the coarse rectangular grid of mesh \(2\delta\), including its lower left corner \(r\), and let \(G^*\) be its plane dual, with exterior vertex \(f_\infty\). The fine-grid vertices are the vertices of \(G\), its edge midpoints, and its bounded face centers. A midpoint is joined to the endpoints of its coarse edge and to its incident bounded face centers. Deleting \(r\) gives exactly \(\mathcal R_{\delta,R}\); the exterior face-node is omitted from this matching graph.

A spanning tree \(T\) of \(G\) is directed toward \(r\). Its complementary dual tree \(T^*\) is directed toward \(f_\infty\). Each remaining primal vertex is matched to the midpoint of its outgoing edge in \(T\). Each bounded face center is matched to the midpoint of the primal edge crossed by its outgoing edge in \(T^*\). The corner \(r\) is incident with the exterior face, so the correspondence is a bijection. All edge weights are one: uniform matchings correspond to uniform primal trees and, by complementation, to uniform dual trees. In \(G^*\) we retain a separate edge to \(f_\infty\) for every primal boundary edge, including parallel edges.

Wilson’s algorithm [21, 17] samples this finite dual tree by loop-erasing walks stopped on the tree already drawn, starting with \(f_\infty\). Each walk chooses uniformly among the four edge options at a bounded face. A proposed step crossing a primal boundary edge is absorbed at \(f_\infty\), and we retain that boundary edge as its physical contact. Theorems 13 and 16 of [17] permit any fixed order of starting sites. We will confine the dual paths needed near \(K\) and then recover the primal directions from the cycles they determine. The following walk estimate supplies the uniform attachment probability used in that confinement.

The two geometric consequences in the half-plane

We next state exactly what is preserved when the connections outside a window change. A directed loop always has the orientation fixed by the matching labels: first-matching edges go from black to white and second-matching edges from white to black. For a directed path portion \(\alpha\) and a transverse segment \(I\), retain the notation \(\iota_I(\alpha)\) for its signed intersection sum. The path endpoints are kept off the test line, and the segment endpoints off the lattice graph. A fixed line between lattice columns or rows may be used as a seam; it need not bisect the crossed edges.

From local traversals to compactness in the disk

We now use Corollary 15 to control parametrizations, not merely the number of loops. The use of crossing control and multiscale time changes to obtain compactness follows the strategy of Aizenman–Burchard [1]. We use the direct partition construction below, which requires only our fixed-scale traversal tightness. For a collection of parametrized loops in \(\overline{\mathbb D}\), let \(N_\eta\) be the largest total number of parameter-disjoint portions whose two endpoints have Euclidean distance greater than \(\eta\). Parameter interiors are disjoint along each individual circle; common endpoints are allowed. Disjointness is automatic between different loops. This number is invariant under reparametrization and reversal.

The distinction between this proposition and the desired conclusion is essential. Traversal bounds retain every macroscopic path, but they do not by themselves exclude collapsed interiors or retracing. Proposition 16, together with the fixed-puncture law, will rule out those possibilities and identify every surviving curve in Section 5.

A joint limit and its puncture signatures

Fix any sequence \(\delta_n\downarrow0\). By Proposition 18, pass to a subsequence with convergent ordered, parametrized loop collections. On a suitable probability space their representatives satisfy \[ F\circ\gamma_{n,j}\longrightarrow F\circ\gamma_j \quad\text{uniformly on }\mathbb T \tag{38}\] for every retained nonconstant entry \(j\). The convergence also retains every loop above each positive diameter, with multiplicity. One may pad a finite list by constants and then discard constant limiting entries. In particular, for each \(a>0\) all loops of diameter at least \(a\) lie in a finite initial portion of the lists, eventually. Indeed, choose an index whose limiting diameter is less than \(a/2\); its approximating diameter is eventually less than \(a\), and diameter ordering gives the same bound for every later entry. We may enlarge this joint realization by further tight coordinates and take further subsequences below.

We choose the test lines before choosing the punctures. For a continuous real function on an interval, the set of its local maximum and minimum values is countable. Indeed, each such value is the maximum or minimum on some compact interval with rational endpoints inside a corresponding extremum neighborhood. This argument includes nonstrict extrema and flat intervals. Apply it to the coordinates of all \(\gamma_j\) in countably many finite parameter charts. Fubini’s theorem gives deterministic countable dense families of vertical lines and positive-height horizontal lines such that, almost surely, none of their levels is a local extremal value of any \(\gamma_j\). We also avoid every lattice coordinate in the chosen mesh sequence. Consequently, at each visit of a limiting path to a test line, every parameter neighborhood contains points on both strict sides of that line.

On every test line fix successively finer uniform grids, with mesh tending to zero, and bounded testing ranges increasing to cover its finite part. Choose grid origins so that their endpoints avoid the graphs for every \(n\). For vertical lines include the intervals meeting height zero; endpoints at nonpositive height have inside indicator zero for every discrete loop. Let \[Q=\{q_1,q_2,\ldots\}\subset\mathbb H\] contain all interior grid endpoints and be dense in \(\mathbb H\), with all added points chosen off the countable union of the lattice graphs. All these choices are deterministic. For each fixed \(q\in Q\) and all sufficiently small meshes, \(q\) lies in a unique bounded lattice face. Replacing it by that face center preserves every loop membership, and the centers converge to \(q\), as required in Theorem 2. By Proposition 3, almost surely no point of \(Q\) lies on a canonical \(\mathrm{CLE}_4\) trace.

For each discrete loop define \[a_{n,j}(q)=\mathbf 1\{q\text{ lies inside }\gamma_{n,j}\},\qquad q\in Q.\] Set all these indicators to zero for a padded constant entry. Adjoin these binary coordinates and the finite-puncture multiset records to the joint subsequence. The binary product is compact; the finite-puncture coordinates are tight by Theorem 2. The ambient space is the Polish product \[C(\mathbb T,\overline{\mathbb D})^{\mathbb N} \times\{0,1\}^{\mathbb N\times\mathbb N} \times\prod_{m\ge1}\mathcal R_m,\] where \(\mathcal R_m\) is the countable discrete space of finite reduced multiset records for \(q_1,\ldots,q_m\). Choose a compact restriction in each coordinate with a summable allocation of the error probability. Their product is compact, and the union bound proves joint tightness without any independence assumption. The Prokhorov and Skorokhod theorems [5] therefore supply a further subsequence and a realization on one probability space such that \[ a_{n,j}(q)=a_j(q)\quad\text{eventually, for each fixed }j,q. \tag{39}\] The limiting path marginal is unchanged by this further extraction and realization, so the probability-one generic-line property chosen above still holds. We call \(\{q\in Q:a_j(q)=1\}\) the signature of the entry \(j\). This definition does not assert that the signature is already realized by the winding of \(\gamma_j\).

Fixed tests followed by a slowly refining grid

We next combine signature stabilization with the geometric estimate. The order of limits matters: no puncture convergence theorem with colliding or mesh-dependent punctures will be used.

Enumerate the test lines and their fixed bounded testing ranges, bounded windows with rational coordinate bounds, and positive rational side buffers. For each finite initial batch, Proposition 16 says that the probability of any opposite-sign violation on its grid intervals tends to zero when the grid mesh tends to zero after the lattice-mesh upper limit. Choose a sufficiently fine grid level for the first batch, then a lattice-mesh threshold making this failure probability small. At that fixed grid level, there are only finitely many endpoint indicators for any fixed finite prefix of the loop list. Equation (39) lets us decrease the lattice-mesh threshold so that failure of their simultaneous stabilization is also small. Repeat with increasing batches and loop prefixes, using a summable sequence of error probabilities.

Passing to the resulting subsequence and applying Borel–Cantelli gives grid levels \(r_n\to\infty\) with the following almost-sure property. For every fixed finite batch, eventually its level-\(r_n\) intervals have no local opposite-sign violation, and all indicators used for each fixed finite prefix of loops agree with their stabilized signatures. The grids themselves were fixed before any of these lattice-mesh thresholds were chosen. We shall use this procedure again after identifying the traces, adding further countably many coordinates which are then known to stabilize.

Here and below, a directed subpath means a restriction to a parameter interval; it need not be injective in the limit. For a directed bounded path \(\alpha\) and an oriented line interval \(I=[u,v]\), with \(u,v\) off its image and the path endpoints off the line, denote its oriented intersection index by \(\iota_I(\alpha)\), with the convention of Proposition 16. Orient \(I\) consistently with that convention. For polygonal paths this is the signed crossing sum. In general, close \(\alpha\) by a path avoiding \(I\) and set \[\iota_I(\alpha)=\operatorname{wind}(\alpha\text{ with its closing path},u) -\operatorname{wind}(\alpha\text{ with its closing path},v).\] The value does not depend on the closing path, since a closed path avoiding \(I\) has equal winding about its endpoints. This definition is additive under concatenation and stable under uniform perturbations avoiding \(u,v\), with endpoints staying off the test line. For a Jordan loop it is, up to the loop’s orientation, the difference of the two inside indicators. In particular, it belongs to \(\{0,1,-1\}\).

The following elementary partition observation permits use of Proposition 16, despite potentially infinitely many line hits.

Thus all pieces relevant to a fixed bounded test range satisfy some fixed positive side buffers. On sufficiently close discrete approximations they satisfy the same conditions. Realized partitions do not require an uncountable extension of the probability estimate: that estimate already quantifies over all subpaths within the fixed window and buffers. The countable families of windows and rational buffers cover every partition just constructed.

Excluding invisible paths and identifying the traces

Fix a nonconstant limiting entry \(\gamma_j\). For a bounded testing range on a selected line, apply Lemma 20. On every interval of the slowly refining grids, all nonzero indices of the resulting discrete portions have the same sign, by Proposition 16. Their sum is the whole-loop index. Consequently, \[ \iota_I(\alpha_n)\ne0 \quad\Longrightarrow\quad a_{n,j}(u)\ne a_{n,j}(v) \qquad(I=[u,v]) \tag{40}\] for every retained portion \(\alpha_n\) of this partition. Endpoints outside \(\mathbb H\) are understood to have indicator zero.

There is a useful local consequence. Suppose a subpath of \(\gamma_j\) stays in a small finite ball and crosses a selected line from one strict side to the other. Retain that subpath in the partition. Its discrete approximation has net signed crossing \(+1\) or \(-1\) with the entire line. Since the approximation stays in a slightly enlarged ball, that crossing is the sum of its indices on grid intervals in the enlarged ball. At least one of those indices is nonzero. Equation (40) then forces an endpoint-indicator switch on one such interval. For sufficiently fine grids all relevant intervals lie in any fixed larger ball. By the simultaneous stabilization already arranged, the same switch occurs in the limiting signature.

This has identified the sets traced by all macroscopic loops and their multiplicities. It has not yet identified their classes in the curve metric: a degree-one traversal can move backward over a substantial arc before continuing forward. The final geometric estimate excludes precisely this behavior.

The exclusion of backtracking

We first make explicit a local fact about Jordan curves. No smoothness or transverse derivative is assumed.

All fixed grid endpoints now avoid the limiting traces: interior endpoints belong to \(Q\), and the traces themselves lie in \(\mathbb H\). Fix a finite grid, a finite prefix of retained nonconstant entries, and finitely many directed parameter intervals with rational endpoints. For each such interval whose two limiting path endpoints are strictly off its test line, uniform convergence preserves its index on every interval of the grid for all sufficiently large \(n\). A parameter interval with a limiting endpoint on the line imposes no test. There are only finitely many tests, and each included pair has a positive realized distance from its line. Their simultaneous failure probability therefore tends to zero by bounded convergence.

Repeat the earlier diagonal selection with these additional index equalities. At stage \(k\), choose a finer deterministic grid for the first \(k\) geometric tests, then keep that grid fixed while choosing \(n_k\) for the first \(k\) retained entries and rational parameter intervals. The geometric violations, membership mismatches, and new index mismatches can together be given probability less than \(2^{-k}\). Borel–Cantelli yields their eventual exclusion for every fixed test, entry, and rational interval whose limiting endpoints are off the line. The indices can be added only at this stage: trace identification has supplied the avoidance of grid endpoints needed for their stability. This second extraction preserves all the earlier almost-sure conclusions and again fixes each finite grid before taking the lattice mesh sufficiently small.

The required collection space and the full limit

Every nonconstant limiting entry is now one canonical \(\mathrm{CLE}_4\) loop as a curve, and every canonical loop occurs once. Proposition 18 controls the positive-diameter tails, so each closed-disk limit has exactly the nested \(\mathrm{CLE}_4\) law. Since the starting mesh sequence was arbitrary, the subsequence criterion gives full convergence in this ambient space.

The remaining task is to obtain compact containment in the interior collection space. The following lemma separates this topological step from the identification of the curves. Write \(X\) for the closed-disk collection space with metric \(\rho\), and \(S\subset X\) for its subspace of interior collections.

For our laws, only tightness on compact positive mesh intervals remains to be checked. On any fixed lattice the possible finite cycles form a countable set, and each cycle can occur at most once. At each positive diameter cutoff, choose finitely many such cycles to capture every loop above that cutoff with arbitrarily high probability. Make the errors summable over decreasing cutoffs. The collections of distinct fixed-lattice cycles satisfying all these restrictions form a compact subset of \(S\): diagonalize membership of the possible cycles, and at any fixed cutoff match the finitely many retained cycles exactly. The unmatched tails have vanishing diameter. Thus the fixed lattice law is tight in \(S\).

Moreover, \(\mathcal L_\delta\) has exactly the law of \(\delta\mathcal L_1\). The maps \[T_\delta=F\circ(z\mapsto\delta z)\circ F^{-1}\] are jointly continuous on \([a,b]\times\overline\mathbb D\) for every \(0<a<b<\infty\). They induce jointly continuous maps on collections as well: their common modulus of continuity controls the distances of matched loops and the diameters of unmatched loops, while a change in \(\delta\) changes \(T_\delta\) uniformly. Images of a compact restriction for \(F(\mathcal L_1)\) under these maps are compact subsets of \(S\). The laws with \(\delta\in[a,b]\) are therefore uniformly tight in \(S\).