Proof. Force all cells meeting a central square of radius \(h\) in \(Q_0\) first black and then white, and denote the difference of crossing probabilities by \(D_\varepsilon(h)\). Proposition 4.5 of [10] gives, for each fixed small \(h>0\), a common limit of this quantity in the Voronoi and triangular models. The upper arm bound and [10], “A square can transmit a pivotal change,” compare \(D_\varepsilon(h)\) with the four-arm probability from radius \(h\) to radius \(1\), with constants uniform in small \(h\), after taking the mesh limit. The corresponding triangular four-arm probabilities have exponent \(\lambda=5/4\) by [14].

Fix \(\eta>0\). For an arbitrarily large but fixed ratio \(L=1/h\), these comparisons and qualitative convergence imply, for all sufficiently large \(r\) depending on \(L\), \[c_\eta L^{-\lambda-\eta/3} \le A(r,Lr)\le C_\eta L^{-\lambda+\eta/3}.\] Choose \(L\) so large that both these constants and the constants in (2) are absorbed by an additional factor \(L^{\eta/3}\). Only after fixing this \(L\) choose a starting radius above the convergence threshold. Iterating over radii \(r,Lr,L^2r, \ldots\) gives four-arm bounds with exponents between \(\lambda-\eta\) and \(\lambda+\eta\). The bounded initial range and the final incomplete ratio change only the multiplicative constants, by (1) and (2). Equation (4) then yields \(C_\eta^{-1}R^{-\lambda-\eta}\le f(R)\le C_\eta R^{-\lambda+\eta}\). Letting \(\eta\downarrow0\) proves the lemma. ◻

Proof. Walls in a triangulation neither branch nor cross. If fewer than four inner ends are connected to outer ends, the wall pairing visible from the outer cycle is independent of the inner cap: there is no choice of pairing on zero or two surviving ends. Thus the left side of (6) is zero. If all four are transmitted, their cyclic order is preserved, with any possible winding retained. The intervening monochromatic bands carry black connectivity to black connectivity and white connectivity to white connectivity. Hence \(H_{\mathsf b}=1\) and \(H_{\mathsf w}=0\). This is also the usual Hex duality proof after each run is contracted to a vertex. Composing the planar pairings proves the last assertion; any closed walls created in the composition have weight one. ◻

Proof. Every compact subset of a lobe persists by Jordan convergence. The lobe joins the back across every compact subinterval of its open seam. Conversely, every point of the limiting boundary is approached by the approximating boundaries. Contacts between different lobes do not open passages: the individual Jordan pieces are attached along their specified seams, not through a point of contact. These facts give kernel convergence with the indicated accesses. The persisting hole and separating back exclude degeneration of the modulus. The conformal kernel theorem therefore gives convergence of normalized annular maps on interiors and convergence of their moduli.

It remains to control the boundary parameters. Fix a nonconstant open subarc of one of the finitely many Jordan boundary pieces and a point in the persisting back. Jordan-map convergence in a lobe, or in a disk patch along an unchanged back boundary, gives a positive lower bound for the harmonic measure of a smaller subarc as seen from a point in that patch. A Harnack chain through persisting interiors gives the same kind of lower bound from the fixed back point. In round annular coordinates the harmonic-measure densities from its preimage are bounded above uniformly: the moduli and the preimage stay in compact nondegenerate ranges. Consequently no fixed nonconstant subarc of the boundary traversal can be squeezed into an angular interval of length tending to zero.

The Jordan convergence of the finitely many pieces gives convergence of the full boundary traversals in curve order. Partition those traversals into finitely many small image arcs, and apply the preceding lower bound to intervening nonconstant subarcs. A failure of equicontinuity in angular parameters would traverse one such fixed subarc in a vanishing angular interval, a contradiction. This proves equicontinuity of the boundary maps. Interior convergence then identifies every uniformly convergent subsequence, giving convergence on the closure. Small crosscuts at the finitely many joins give the same conclusion there. There is no constant boundary interval in the limiting conformal map, by uniqueness for a holomorphic function continuous on a boundary arc. The boundary order and converging marked accesses thus also identify the limiting switches. ◻

Proof. First the mesh in annular coordinates tends to zero. Otherwise a connected cell or triangle portion of vanishing physical diameter would have preimages with a subsequential nontrivial continuum limit on which \(\phi\) is constant. Such a continuum cannot meet the interior, where \(\phi\) is one-to-one, and cannot lie on the boundary, whose finitely many simple arcs contain no constant interval. Here graph portions, rather than whole cells crossing the domain boundary, are used.

Remove small coordinate neighborhoods of the switches and of the finitely many joins of simple bank arcs. On each remaining compact bank interval, use disk patches whose boundary consists of that bank and cuts strictly inside the annulus. Their physical boundaries are Jordan and their graph approximations converge in curve order. Their uniformizing coordinates agree, with fixed-scale margins, with the annular coordinates; reflection in the limiting round bank controls distortion on smaller patches. Thus the local boundary-arm estimates of [10] apply on these patches even when another prime-end bank has the same physical image.

Cover these intervals by \(O(s^{-1})\) coordinate disks of radius \(s\). Implement a motion much smaller than \(s\) by successively replacing bank intervals, with tapered ends. At graph level choose short transverse joins in disjoint narrow strips from the old cycle to its replacement. Substitute the intervals in cyclic order, retaining the old cycle off the interval currently being changed and erasing the joins once the whole cycle is replaced. The graph approximation in curve order ensures that these are simple cuts. Each join has the same constant color as its run. Moves containing switches or bank joins are instead included in the exceptional neighborhoods. Two nearby nonnested contours can be compared through a common slightly inset contour.

Consider one ordinary move on a black bank. If a capped connection test changes, choose a winning black connection in one test and a winning dual white connection in the other. Contract the prescribed runs and use the same cap in both tests. Purely wired connections do not change, so a free segment of a winning path must meet the support of the move. The test neighborhood avoids every other run and mark. The white path has two free branches leaving that neighborhood. At least one black branch must also leave freely: a black branch already reaching the same prescribed run outside the moved interval could be joined there without the local change. Taking simple paths after contraction gives disjoint white branches, and the opposite-colored black branch is disjoint from them in the common fair region. This gives three free arms in a slightly shifted half-patch, with fixed-factor margins. The same argument applies with the colors interchanged. Forced sites on intermediate cuts are not counted as free arms.

For any fixed larger chart scale, the mesh-first bound for a single disk is \(O(s^{1+\eta})\). The union over \(O(s^{-1})\) disks therefore costs \(O(s^\eta)\). The number of exceptional neighborhoods is fixed. A change there requires a fair arm approaching from positive parameter distance: connections already provided by the same run or by the cap do not change. This arm has positive physical diameter, uniformly after fixing its outer parameter scale, by the no-continuum-fiber observation above. The ordinary physical one-arm bound consequently makes the exceptional cost tend to zero, including at a contact between different accesses. One may move each exceptional neighborhood as a single piece, so this cost is not multiplied by the number of ordinary intervals.

First take the mesh limit, then make the ordinary motions small compared with \(s\), then let \(s\) tend to zero, and finally shrink the exceptional neighborhoods. This proves the assertion. The same proof with moving marks, or with a moving round radius, proves the stated continuity. ◻

Proof. On every strict Jordan disk patch the joint law is the critical triangular percolation law by [10]. This law is conformally invariant; see [2] and the quad-crossing formulation in [6]. Scalar Cardy sandwiches identify smooth images of polygon tests and nearby graph crossing conventions with the same limit.

Choose a finite-length smooth cut opening the annulus. Any finite collection of tests compactly contained off the cut fits in a strict Jordan disk, so its joint law agrees in the physical domain and in the conformal chart. To recover tests meeting the cut, use the Schramm–Smirnov gluing theorem [12], only in strict Jordan disk patches. For the critical triangular crossing law, the crossing of a regular quad in such a patch is measurable, modulo null sets, with respect to countably many tests avoiding the finite smooth cut in that patch. Its hypotheses hold here: independence in the discrete model, RSW, one-arm decay, and a four-arm exponent strictly greater than one. Regular continuity tests may be used throughout, by Cardy sandwiches.

Each required quad meeting the annular cut has a slightly larger Jordan disk neighborhood inside the annulus. The recovering measurable rule in that neighborhood is the same in the chart and in its image, since their whole disk laws already agree. All recovering tests belong to the common countable off-cut family. Hence the recovered crossings have the same joint law as well. This proves the assertion. White crossings are included by Hex duality and continuity. No gluing assertion along a rough slit is used. ◻

Proof. By Lemma 10, it suffices first to consider smooth inset annuli with spare space. We show that ordinary quad crossings determine the limiting value of each capped black connection test. Sample fair colors throughout a slightly larger annular neighborhood in that spare space, overriding them on prescribed runs only when evaluating a capped test. Ordinary tube tests use the fair colors. Compare two candidate sequences, one of which may be in round coordinates. Couple their countable ordinary crossing vectors through the common limiting law of Lemma 11. Along further subsequences include the cap indicators in the coupling; the curve tightness proved below also allows the chosen winning paths to be included. The two limiting collections are coupled to have the same ordinary crossing vector.

Fix a small favorable perturbation of the black test in the second sequence. Move black banks into the annular fluid and extend their runs slightly past the old switches; retreat white banks from the old fluid. Use positive slacks and taper the motions within the old white intervals, putting each new switch on the retreated part. Thus every old black attachment lies behind a new black bank, the new white banks are positively separated from the old fluid, and any removed fluid is in separate collars of the expanded black runs. All these choices take place in the spare space.

On a positive configuration in the first sequence, contract each prescribed run and attach the cap edge when it is black. Take an outermost simple winning path between the two outer black terminals. For example attach an external terminal edge and take the appropriate boundary of the union of cycles containing it. Contracting connected boundary runs and attaching the cap preserves the planar embedding. The resulting path has no bypass on its chosen exterior side. Since it visits each contracted run at most once, it decomposes into a bounded number of free black segments joining different runs, together with wired or cap joins. Crop their endpoints by a vanishing graph distance if they end adjacent to a wire.

We recall why these free segments admit simple limiting curves. Many oscillations across an annulus supply many disjoint free arms; BK and one-arm decay bound their probabilities. Successive-excursion compactness therefore gives tightness in the curve metric. A bulk self-contact with two positive excursions would give four black traversals. In two gaps on the no-bypass side, intermediate transverse cuts and Hex duality give two additional white traversals: a black transverse crossing would instead be a bypass. The six arms are disjoint after fixed-factor crops, also for backtracking pieces. The mesh-first six-arm exponent greater than two excludes such contacts by a covering argument. At a smooth boundary bank, a repeated visit to the same point gives at least three free traversals, including when one of the visits is an endpoint; the boundary exponent greater than one excludes these repeated visits. A single tangential bank contact need not be excluded: the favorable perturbation accommodates it. At the finitely many switches one free arm suffices. These are precisely the local no-bypass and free-segment arguments of [10]. Contracted runs do not change the bulk argument, whose working annulus avoids them. Constant pauses may be removed.

In a subsequential coupled representation, fix the favorable displacement and then crop each limiting free segment a small positive amount at its ends, keeping the new endpoints strictly behind the expanded black banks. A cropped endpoint may still lie on an old bank; the ambient tube may straddle that bank within the spare space. This positive crop is distinct from the vanishing graph crop used to remove incidences to a wire. Surround the cropped simple segment by a thin Jordan tube, with end caps in those endpoint neighborhoods. The approximating free graph segments supply ordinary fair-color crossings of this ambient tube. The tube can be approximated with positive margins by a quad from the countable ordinary test family. Its crossing occurs in the first sequence, and hence in the second for all sufficiently large indices in the coupling. The witnesses may be chosen from the countable family after taking the subsequential limit; coordinatewise convergence still applies to each of the finitely many chosen witnesses.

Clip these transferred crossings when they hit the new black banks. Excursions outside the new fluid can only lie in the collars now wired black. They can be shorted through the corresponding run, and collars of different black banks are separated. This operation also handles arbitrarily many backtracks: at graph level every passage between a removed collar and the fluid meets its boundary cycle. The transferred tubes, together with the unchanged cap or wire joins, therefore give a winning black connection in the favorably perturbed second test.

Apply this comparison to subsequences realizing the upper and lower limits of the unperturbed cap probabilities. The upper limit of the first is bounded by the lower limit of the second plus its boundary-motion error. Remove the favorable perturbations using Lemma 10 and reverse the roles of the sequences. Their upper and lower limits coincide. This proves convergence and conformal invariance of each cap test on inset annuli. Taking the inset size to zero proves the same assertions under Definition 8. The cap identity gives the transmission limit. Allowing the marks and round radii to move in the same comparison proves continuity in \((p,x,y)\).

Finally consider random remaining domains under the stated coupling. For almost every convergent domain realization, the deterministic statement applies to its fresh product-color law; continuity identifies the same limit for \(T^\pm\) at the actual conformal data. All probabilities are bounded by one, so bounded convergence gives convergence in mean absolute value. This conclusion does not assert a uniform conditional estimate over arbitrary atypical explored boundaries. ◻

Proof. If two consecutive switches coalesce, four-interface transmission requires a fair arm from their intervening shrinking run to the other boundary. This arm cannot use another prescribed run: it lies between the two corresponding transmitted walls. A mesh-first one-arm bound, with a fixed outer scale and then a shrinking inner scale, proves vanishing. The bound does not depend on the positions of the other switches and therefore also covers simultaneous collisions. Conformal invariance gives simultaneous rotational invariance, and fair colors give global color symmetry. Moving the first switch to the next switch reverses the alternating run labels on that component. By the defining convention, this interchanges the two inner prescriptions. ◻

Proof. We first derive a local equation with just one switch active. Write the annulus as the cylinder \(0<\operatorname{Im}z<p\), with real period \(2\pi\), and take the active switch on its lower boundary. Parametrize its exploration by the decrease of the modulus. In the annulus Loewner normalization with zero regular constant at the driving pole, the angular kernel is \[H_p(u)=\lim_{N\to\infty}\sum_{k=-N}^N \cot\frac{u-2ikp}{2}.\] The pole has residue \(2\), and the imaginary part on the other boundary is \(-1\). The locality theorem for annulus \(\mathrm{SLE}_6\) [17] identifies the local driving angle with Brownian motion of variance parameter \(6\). Only locality before leaving a small simply connected neighborhood of the active switch is used here.

For completeness, the absence of an additional driving drift depends on the normalization. In a chordal chart the pole part is \(2/(z-U_t)\). If \(F_t\) is a changing local conformal coordinate, comparison of pole residues gives the new clock \(F_t'(U_t)^2\,dt\). Comparison of the regular terms, both normalized to have zero constant, gives \(\partial_tF_t(U_t)=-3F_t''(U_t)\). This cancels the Itô correction \(\kappa F_t''(U_t)/2\) precisely for \(\kappa=6\). No condition on a remote target in the annulus enters this calculation.

Other switches on the active boundary have velocity \(H_p(u)\), at angular difference \(u\) from the active one. Switches on the opposite boundary have angular velocity \(\operatorname{Re}H_p(u+ip)\). Pairing the positive and negative terms of the series gives, uniformly with any fixed number of derivatives, \[ H_p(u)-\cot(u/2)=O(e^{-2p}), \qquad \operatorname{Re}H_p(u+ip)=O(e^{-p}). \tag{9}\] The remainders are smooth periodic functions of \(u\), including at \(u=0\). Thus, when \(x_i\) is the active switch, its equation is \[ \begin{aligned} 0={}&-\partial_pT+3\partial_{x_ix_i}T +\sum_{j\ne i}H_p(x_j-x_i)\,\partial_{x_j}T\\ &+\sum_{j=1}^4\operatorname{Re}H_p(y_j-x_i+ip)\, \partial_{y_j}T. \end{aligned} \tag{10}\] For an active switch \(y_i\), the equation is obtained by exchanging \(x\) and \(y\). Indeed the cylinder change \(z\mapsto-z+ip\) exchanges the boundaries and reverses both angle coordinates; the angular kernels are odd, so transforming back preserves the displayed signs. Averaging these eight equations gives (8). The series estimates give the asserted bounds on \(B_p\). The averaged expression treats the labels within each tuple symmetrically; therefore it is permutation equivariant. At a collision its two corresponding components agree, proving tangency. The other symmetries follow in the same way from the kernels.

We justify the Markov identity underlying these equations before assuming differentiability. On a round annulus choose an arbitrarily small crosscut around the active switch, separated from all other switches and from the other boundary component. Approximate the smooth bank by chord-free graph boundaries and complete the neighborhood to a Jordan patch, putting its second color switch remotely. Up to the pre-query stop at the crosscut, the exploration in this patch is exactly the exploration in the annulus: no color beyond the crosscut is read. The stopped-exploration results of [10] give convergence of the path together with its remaining domains. Crosscuts may be chosen as the semicircle crosscuts in the stopped-target construction; the probability of an endpoint stop vanishes when endpoint neighborhoods shrink.

The unaltered part, obtained by cutting off the fixed crosscut neighborhood, is a Jordan annular back with unchanged boundary traversal. For the remaining face containing this back, the stopped-target-face and remaining-domain statements of [10] give finitely many Jordan side pieces attached to the back along open subintervals of that crosscut. Components already disconnected from this face are discarded; no assertion of finiteness is being made about all components swallowed by the exploration. We check that its limit is still doubly connected. In the closed crosscut neighborhood, remove the side-lobe interiors and their open attaching intervals. The remaining closed set is connected to the original outer complement. Indeed join any of its points to the old outer bank by a path in that neighborhood whose interior avoids the crosscut. If this path enters a lobe, replace its portion from first entry to last exit by the corresponding subarc of the lobe boundary off its open attachment. The entry and exit lie on this retained bank, not on the crosscut. This bank meets the crosscut only at its own endpoints and enters no other lobe interior, since the Jordan interiors are disjoint. Thus each replacement avoids all lobe interiors and open attachments; finitely many replacements give a path in the complement. The unchanged inner hole is separated from this connected outer complement by the persisting annular back. These are precisely the two nondegenerate complementary components. Lemma 9 gives convergence of annular uniformizations on their closures, with the prime-end traversals and marks in order. The hole and a separating open bulk persist throughout this local exploration, so the moduli remain nondegenerate. Thus the residual annuli satisfy the hypotheses of Proposition 12. The stopping edge becomes the replacement switch, the other marks are unchanged, and the colors in the remaining face are fresh. Transmission topology and bounded convergence therefore give the local mean-value identity for \(T^\pm\) at each of these hull stops.

This identity first implies the active equations in the viscosity sense. Work modulo simultaneous rotation and let a smooth test function touch \(T\) from above or below. On a sufficiently small crosscut stop the modulus and image marks remain in the touching neighborhood. Indeed the hull capacity tends to zero, and annular maps normalized at a remote mark tend to the identity on the other compact bank intervals, by reflection. The image of the tip is squeezed between neighboring unaffected bank points. Applying Itô’s formula to the touching function contradicts a strict inequality with the wrong sign in the active equation, because the stop takes positive time. The same touching function can be used for all eight active equations, so their average also holds in viscosity form. On every compact subset of the open chamber the averaged equation is a smooth, uniformly parabolic linear equation, also after the rotation quotient. Interior parabolic regularity then gives smoothness and the classical equation. Simultaneous rotation invariance removes the quotient. ◻

Proof. We first justify using the diffusion all the way to the chamber boundary. The drift in one tuple is \(\frac14(\sum_{j\ne i}\cot((w_i-w_j)/2))_{i=1}^4\). On the lifted convex chamber, use \(-\log m\), shifted to be nonnegative, as a Lyapunov function. If \(\alpha=4/3\) is the density exponent, the cotangent identity shows that its generator has a singular contribution proportional to \((\alpha-\alpha^2)\sum_{i<j}\csc^2((w_i-w_j)/2)\). This coefficient is negative, so the generator is bounded above. More explicitly, the one-tuple contribution is \[D(-\log J^{4/3})= \frac{10}{3}-\frac1{24}\sum_{i\ne j} \csc^2\frac{w_i-w_j}{2}.\] Stopping on inner exhaustions proves that no collision occurs in finite time. Interior existence and pathwise uniqueness therefore give a global diffusion. A bounded added drift, such as the time-dependent \(B_p\), preserves noncollision by Girsanov’s theorem on each bounded time interval. The divergence form of \(D\) gives symmetry and invariance of \(\mu\), first on interior exhaustions and then by noncollision.

The Feynman–Kac representation for (13) uses \(D+B\cdot\nabla\) and potential \(V\), with coefficients read at decreasing moduli from the evaluation modulus down to the initial modulus. Its initial data at \(p_0\) obey \(|W(p_0)|\le C/h\). To justify this representation despite the unbounded transformed data, first apply the killed representation to \(T\) on inner chamber exhaustions. Lemma 13 says that the lateral values of \(T\) tend uniformly to zero on each compact modulus interval. Their contribution therefore disappears. The \(h\)-transform on the exhaustion gives the stated representation; pass to the limit using noncollision. It suffices to do this for the nonnegative \(T^\pm\) and then use linearity.

Here are the uniform smoothing estimates. The negative logarithm of \(m\) is convex on the lifted chamber, so the drift of \(D\) is monotone dissipative. Couple two diffusions by the same Brownian motion and add to the second a drift toward the first of magnitude their initial distance divided by the allotted time, until they meet. Dissipativity forces meeting by that time. Girsanov’s theorem removes the extra drift with all fixed moments bounded in terms of the time and the initial distance. Localization inside the chamber justifies the coupling, and the preceding change-of-measure argument excludes collisions for the controlled diffusion. Hölder’s inequality gives the dimension-free form \[|P_tF(z)|^q\le C(q,t,\operatorname{dist}(z,z'))P_t(|F|^q)(z').\] One can always choose lifts of \(z,z'\) at bounded distance. Integration in \(z'\) against the invariant probability \(\mu\) proves the asserted \(L^q\)-to-supremum estimate.

A bounded drift perturbation and a bounded potential on a unit time interval preserve initial smoothing from \(L^4(\mu)\). Indeed Cauchy–Schwarz applied to the Girsanov density bounds the perturbed expectation of \(|f|\) by \(C(P_1|f|^2)^{1/2}\); the unperturbed \(L^2\)-to-supremum bound for \(|f|^2\) then gives \(C\|f\|_{L^4(\mu)}\). There is exactly this integrability here: \(h^{-1}\in L^4(\mu)\), since \(h^{-4}m=1\). After the first unit interval \(W\) is bounded. Its bound persists for all subsequent times because \(\int_{p_0}^\infty\|V_p\|_\infty\,dp<\infty\).

The same coupling proves a Lipschitz estimate for bounded data on a unit interval. In the presence of \(B\), a control of size at most a fixed constant times the initial distance forces meeting; before meeting the distance stays bounded by the same multiple of the initial distance. The second moment estimate for the Girsanov exponential bounds its total-variation cost by that distance. The two Feynman–Kac weights differ by at most the same order, since \(V\) has a uniform Lipschitz bound. Thus the unit-time Lipschitz norm is bounded by a constant times the initial supremum norm. Apply this on the last unit interval to obtain (14). ◻

Proof. For a mean-zero vector \(v\in\mathbb R^4\) the Hessian of the negative logarithm of the one-tuple density satisfies \[\nabla^2(-\log J^{4/3})[v,v] =\frac13\sum_{i<j}\csc^2\frac{r_i-r_j}{2}(v_i-v_j)^2 \ge\frac43|v|^2.\] The Brascamp–Lieb inequality [1, 7], applied on the convex alcove, therefore gives \[\operatorname{Var}(F)\le\frac34\int|\nabla_rF|^2\,d\mu_r.\] The Dirichlet form has factor \(3/8\), yielding the gap \(1/2\). One may first use inner convex exhaustions of the alcove and then pass to the limit.

For the two tuples, simultaneous rotation invariance allows only center frequencies \((n,-n)\), whose energy cost is \(3n^2/16\). A cyclic relabeling sends \(\theta\) to \(\theta+\pi/2\) and acts on \(\mathcal A\) by a measure-preserving affine isometry. If a function changes sign under this relabeling, its relative-angle mean must vanish unless \(e^{in\pi/2}=-1\), that is, unless \(n\equiv2\pmod4\). The same assertion holds separately for the other tuple. Therefore frequencies not congruent to \(2\) modulo \(4\) pay at least \(1/2+1/2=1\) in relative energy. The remaining frequencies have \(|n|\ge2\) and pay at least \(3/4\) in center energy. At \((n,-n)=(0,0)\) the two relative means vanish, giving the gap \(1\).

These inequalities give \(L^2\) semigroup decay through the symmetric Dirichlet form \[\mathcal E(F,F)=\frac38\int|\nabla F|^2\,d\mu.\] There is no hidden absorbing boundary: the constant function is in the form domain. Cutoffs approaching the collision faces have vanishing energy because the density vanishes with power \(4/3>1\). ◻

Proof. For any transformed solution put \(F_p=B_p\cdot\nabla W(p)+V_pW(p)\). Lemma 15 implies \(\|F_p\|_\infty\le Ce^{-p}\) once \(p\) is sufficiently large. Variation of constants gives \[ W(p)=P_{p-p_1}W(p_1)+\int_{p_1}^pP_{p-s}F_s\,ds. \tag{18}\] This identity also follows directly by Itô’s formula with \(D\), first stopping on inner exhaustions: on the interval in question the solution and the forcing are bounded, and collisions are not hit.

For \(W_-\) the forcing respects simultaneous rotation invariance and the two cyclic sign changes. Apply the \(3/4\) gap to (18), and then apply the unit-time \(L^2\)-to-supremum smoothing to all semigroup times at least one. For shorter times use the supremum bound on \(F_s\). The convolution with \(e^{-s}\) is \(O(e^{-3p/4})\). After projecting by \(\Pi\), which commutes with \(D\), the gap is \(1\) and the convolution is \(O((1+p)e^{-p})\). This proves (15). It does not require \(B_p\) to commute with independent rotations: the projected forcing is estimated after projection.

For either \(W^+\) or \(W^-\) separately, \(D\) has a positive gap off constants. Indeed the center gap is \(3/32\) and the relative gap is at least \(1/2\). The same argument shows exponential convergence of \(W-\int W\,d\mu\) to zero. Also \(\partial_p\int W\,d\mu=\int F_p\,d\mu=O(e^{-p})\), so its mean converges exponentially. The first estimate of (15) makes the two limiting constants equal; call their common value \(K\). Multiplication by \(e^{-\lambda p}h\) proves (16).

The unit-interval Lipschitz estimate in Lemma 15, applied to the homogeneous perturbed evolution for \(W_-\), improves its Lipschitz bound to \(Ce^{-3p/4}\). To obtain time regularity, the same diffusion has expected angle displacement at most \(C\sqrt\ell\) in time \(\ell\le1\), uniformly in its initial point. Here is a bound that remains valid near a collision. Apply Itô’s formula to squared distance from the initial lift. Each pair in the scalar product with the singular drift contributes a constant multiple of \[(v-v_0)\cot(v/2),\qquad v,v_0\in(0,2\pi),\] which is bounded above uniformly: use \(v-v_0\le v\) for \(v\le\pi\) and \(v-v_0\ge v-2\pi\) for \(v\ge\pi\). The bounded additional drift and the Brownian quadratic variation then give \(\mathbb E|Z_\ell-Z_0|^2\le C\ell\), by localization and Gronwall’s inequality. The Feynman–Kac representation over time \(\ell\), the spatial Lipschitz bound, and the bounded potential give the time estimate in (17). Finally \(J\) is a bounded Lipschitz nonnegative function on the closed chamber, and taking a cube root is \(1/3\)-Hölder. The assertion for \(h\) follows by taking products. ◻

Proof. For a fixed realization, pushing \(v_{\Sigma_r}\) to any intermediate state set contracts total variation. When that set has two states, its zero-mass image has the form \(\alpha(\delta_B-\delta_W)\), with \(|\alpha|\le\|v\|_r\). Pushing the two-state difference through the other end gives a signed measure \(q\) with \(\|q\|_{\mathrm{TV}}\le2\). Hence the pointwise norm of the replacement error is bounded by \(2\|v\|_rF\). The same argument, summing absolute masses before a two-state reduction, handles discarded events. No regularity of the dependence of \(\alpha\) or \(q\) on the seam data is needed.

On the auxiliary localization successes, \(F\) is a function of marked Poisson variables independent of both original seams and has the stated full-plane comparison law. Integrate the pointwise bound conditional on the output seam. Auxiliary failures contribute the final \(o(1)\) term, also uniformly, because their supports are off the original seams. The seam success indicators themselves are not deleted. ◻

Proof. First consider two tips approaching the same point of the barrier. Use a graph disk patch with its base on the barrier and a half-annulus on the exploration side. Extend each stopped wall through its last triangle to the base, only for this geometric argument; no barrier color is queried. The two wall fingers give two proper traversals of the half-annulus, with fixed margins at its two semicircular ends. Following the sites on their banks gives three disjoint free arms in the sectors on the two exterior sides and between the fingers. These sites are on the exploration side, not on the barrier. Short spokes through successive wall triangles keep each bank in its sector. The base through the tips prevents an exterior bank from escaping around a finger’s end. Cropping the last mesh steps gives the same count when two stopping triangles share a barrier vertex.

This is the free-bank-shadow count for two traversals in the boundary argument of [10]. The local boundary three-arm exponent is strictly greater than one. Covering the barrier by \(O(s^{-1})\) patches of radius \(s\) therefore excludes colliding tips as \(s\downarrow0\), mesh first. The BK inequality applied to the free shadows also makes the number of fixed-scale traversals tight.

For the frontier assertion, fill the starting side of the starting cycle with a disk, using the sphere for the outside experiment. Attach the queried triangles along the explored bands and add all edges between revealed vertices. The resulting graph is two-connected: attaching a triangle along an existing edge preserves two-connectivity, as does adding an edge between existing vertices. Thus the boundary of its face containing the unqueried barrier is a simple cycle. The barrier and its paths towards the unexplored middle remain in that face. In particular each stopping triangle is incident with this same face on its unread side.

There is a useful invariant: every revealed vertex has a revealed path of its own color to the starting cycle. It holds initially, and a newly queried triangle vertex has the color of one endpoint of the incoming heterochromatic edge, extending such a path. Thus an added heterochromatic edge between revealed vertices cannot belong to an unstarted wall. Such a wall would either close in the annulus or return to the barrier without visiting the starting cycle; in either case it would separate one bank from that cycle, contradicting its revealed same-color path. Every heterochromatic revealed-vertex edge therefore belongs to the traced wall system.

To identify the colors along the frontier, extend the stopped walls to the barrier as above. These proper arcs, together with the walls that paired back to the starting cycle, divide the explored collar into regions with their consistent bank colors. Every revealed vertex is either a starting-cycle vertex or was queried in a wall triangle. Its color agrees with the relevant bank. An added edge cannot cross one of these wall arcs, and the preceding invariant excludes any untraced switch there. Previously crossed switch edges have triangles on both sides in the revealed bands. The only switches exposed to the remaining face are consequently the surviving stopping edges.

With zero or two switches there is just one noncrossing cap pairing. Every input state gives that same pairing, so a zero-mass difference vanishes. With six or more surviving wall fingers, their bank shadows give six disjoint site arms in successive sectors between the starting cycle and the barrier. Fixed-factor crops remove the two ends, where different shadows could otherwise meet. The arbitrary-color six-arm bound is applicable, so no particular numbering or color order of the six arms is required. ◻

Proof. The BK and successive-excursion argument gives tightness of the four frontier arcs. Indeed repeated annular traversals provide disjoint free arms; all sites used, including those on the auxiliary starting cycle, have their original fair colors. This is applied first at fixed radii and then at successively smaller excursion scales. The contact arguments below are the face-boundary arguments in [10]; we give the details relevant to the present frontier.

Every frontier vertex off the starting cycle belongs to a revealed wall band. On its revealed side it has an opposite-colored neighbor with a bank path back to the starting cycle. Consequently a visit of a frontier arc to the stopping circle other than at its designated endpoint, with a positive excursion on both sides of the visit, gives three free arms in a half-patch: two along the arc and one along the opposite bank. The same argument excludes an intermediate return to an endpoint. The boundary three-arm exponent and a covering of the circle exclude all such visits. A nonconstant interval contained in the circle is excluded by the one-arm bound at a countable dense set of circle points. Constant pauses can be removed from the parametrization. Distinctness of the four tips follows from Lemma 22.

Consider next a self-return of one monochromatic frontier arc at a point away from the starting and stopping circles. Separate the two visits by a fixed positive excursion scale. In a smaller annulus around the proposed contact, the arc supplies four proper traversals. Its opposite bank paths lie on the revealed side of the frontier cycle. They may all be joined through the starting cycle when making the planar separation argument. This joining set need not be monochromatic at the starting cycle: the working annulus misses that cycle, and within the working annulus every part of the joining set is genuinely of the opposite color.

At least two complementary sectors have a frontier traversal on their boundary whose revealed side faces the sector. Each such sector contains an opposite-colored longitudinal crossing. To verify this assertion, choose transverse cuts at intermediate scales with margins, leaving both ends of that frontier traversal beyond the cuts. If the joined neighbor set did not cross the sector longitudinally, planar duality would provide a transverse crosscut avoiding it. Start this crosscut from that frontier traversal and stop at its first subsequent hit of the frontier cycle. In the disk on the revealed side, the two traversal ends and the two crosscut ends alternate along the frontier. The crosscut therefore separates their neighbor spokes, contradicting the joined neighbor set, which lies on the revealed side and avoids the crosscut.

The argument remains valid when the traversals backtrack. Choose the transverse cuts along intermediate graph cycles, or apply Hex duality with such a cycle temporarily wired in the sector. Two disjoint cuts bound a quadrilateral whose side arcs belong to the chosen traversals and whose interior avoids those whole proper traversals, not merely the selected side subarcs. Thus the crosscut’s first subsequent frontier hit lies outside the whole traversal from which it started, so the alternating boundary order used above persists under backtracking. The crosscut is in the quadrilateral’s interior, whereas the end spokes are beyond the cuts. They thus avoid the crosscut even if other portions of the frontier enter the sector. Cropping the margins leaves two disjoint opposite-colored arms in different sectors, in addition to the four arms along the monochromatic arc. This gives six disjoint bulk arms. The bulk exponent strictly greater than two excludes these contacts by a covering argument. At a self-contact on the starting circle, the four arc traversals alone suffice: the arbitrary-color four-arm exponent is strictly greater than one, and the possible contact locations lie on a fixed circle.

All these exclusions are made for distinct visits with fixed positive excursion scales before those scales decrease. Countably many scales and finite ball coverings make the assertions simultaneous. They bound joint probabilities with the four-tip event; no division by the probability of a rare history is involved.

We now identify the remaining domain; see Figure 3. Cut along the barrier, using the four short incidences from the tip edges through their terminal triangles. Between consecutive tips, the frontier arc and its corresponding barrier interval bound a lobe polygon. The four lobes are attached over open intervals to the untouched region on the middle side of the barrier. Each lobe converges in order to a Jordan lobe, because its frontier arc is simple and meets the barrier only at its endpoints. Different lobe boundaries may touch in physical projection; their interiors remain disjoint, and their distinct accesses are retained. With two well-separated barriers there is a fixed open annular region between the ends.

We verify that adjoining the lobes still gives an annulus, even when their boundaries touch. On either side of this open annular region, start with the closed disk bounded by the corresponding stopping circle, using the sphere for the outer side. The complementary set on that side is obtained by removing the lobe interiors and their open attachment intervals. Each lobe’s free frontier arc remains in this complement: it cannot enter another lobe’s interior, since their Jordan interiors are disjoint, and it meets the stopping circle only at its own endpoints, never on another open attachment. Connect a point of the remaining closed set to the preserved starting-side region by a path in the closed disk, with its interior off the stopping circle. Whenever the path enters a lobe, replace the portion from its first entry to its last exit by the intervening subarc of that lobe’s free frontier. The entry and exit lie on this free arc. The replacement avoids every lobe interior and every open attachment, so it cannot undo previous replacements. There are only four lobes; after finitely many replacements the path is wholly in the complementary set. Thus each side has a connected complement. The two such nondegenerate continua are separated by the open annular region and exhaust the complement of the residual domain. The domain is therefore doubly connected, with its two boundary components physically apart.

Lemma 9 now gives uniform convergence of the closed-annulus parametrizations with their prime-end order. The boundary traversal consists of finitely many individually simple arcs. These are exactly the hypotheses required by Proposition 12.

Finally extract, from any sequence of scales, a further sequence on which the localized full-plane geometries are good in the sense of [10]. Conditional on those geometries, couple convergent subsequences of the frontier histories just described. All unqueried colors in the residual face remain independent and fair. Apply Proposition 12 path by path to the converging graph annuli. The continuum function on the actual approximating conformal data also converges, by its continuity. Bounded convergence proves (33) along the further sequence and hence along the original sequence. Unused colors on the removed sides of the frontiers may be revealed or integrated in their own end variables; they do not affect this argument. ◻

Proof. Before the first reduction at level \(j\), there are at least six tips at the preceding level. Lemma 22 then requires six arms from the starting region to that level. The mesh-first six-arm bound gives \(C2^{-(2+\tau)j}\), with a changed constant for the first few levels. The same statement holds at the outer end, uniformly in its original seam: the necessary six-arm event is read in the independent region, while attachment variables and success indicators have total conditional mass at most one. Multiplying the two separate mass bounds proves (35). For used pairs the physical end experiments also have disjoint buffered supports, so their joint law is the product used in the operator contribution. Arbitrary input weights are bounded pointwise by their total variation, as in Lemma 21; the number of switches on the input seam does not enter the estimate.

To bound the remainder, run the inner exploration to its longest allowed level. If it never reduces, it has paid a six-arm probability over \(n-O(1)\) dyadic levels. Otherwise let \(j\) be its first reduction level. If the reduction has fewer than four tips, its contribution is zero. In the remaining case run the outer experiment as far as (34) allows, possibly through no additional level. If it does not reduce in time, the two experiments together have paid six-arm probabilities over at least \(n-O(1)\) levels. Their buffered supports remain disjoint. Summing the resulting bounds over the at most \(O((n+1)^2)\) possible terminal levels proves (36). The finitely many auxiliary localization failures and near-coincidences of tips cost \(o(1)\) for fixed \(L\), uniformly by Lemma 21. ◻

Proof. Choose a ratio \(L\) from Proposition 20 so large that the constant per block in (31) is absorbed by a smaller power saving. Choose \(s_*>r_0(L)\) sufficiently large. For given \(s\), use radii \[r_j=L^j r_0,\qquad 0\le j\le N, \qquad r_N=s,\quad s_*\le r_0<Ls_*.\] All screens use the base process. At a successful seam, the inserted point cannot reach or alter its cycle, by localization. The two inside state laws consequently have the same state space and each has mass one. Their difference has zero mass, norm at most two, and is even and isotropic.

If every seam is successful, iterating (31) bounds the message by \(C(s/r_0)^{-2-\nu}\) for some \(0<\nu<1\). Since \(r_0\) stays in a fixed compact range of positive scales, this gives (49) for that contribution.

For the remaining contribution, group configurations by the largest failed radius \(v=r_j<s\). The exact partition, with the final success kept in every term, is \[\boldsymbol1_{G_{r_N}} =\prod_{i=0}^{N}\boldsymbol1_{G_{r_i}} +\sum_{j=0}^{N-1}\boldsymbol1_{G_{r_j}^c} \prod_{i=j+1}^{N}\boldsymbol1_{G_{r_i}}.\] Form a fresh difference of inside state laws at the next cut \(Lv=r_{j+1}\), multiplying both laws by the same base-process indicator \(\boldsymbol1_{G_v^c}\). Earlier screen outcomes are left unrestricted. This weighted difference still has zero mass for each successful seam at \(Lv\): for each common base configuration the two state laws have equal total mass. It is also even and isotropic. The collar at \(v\) is disjoint from that at \(Lv\), so conditional on the latter seam its norm is at most \[ 2\mathbb P(G_v^c\mid\Sigma_{Lv}) \le C e^{-cv^2}. \tag{50}\] Success at \(Lv\) ensures that both configurations use the same cut there, independently of what happened at smaller radii.

Transfer this reset message through the successful cuts above \(Lv\). Those success restrictions ensure that \(v\) is indeed the largest failed radius; no restriction on earlier failures was needed. The sum of these contributions is bounded by \[C\sum_{v\in\{r_0,\ldots,r_{N-1}\}} e^{-cv^2}\left(\frac{s}{Lv}\right)^{-2-\nu} \le C s^{-2-\nu}.\] The final inequality follows by summing the exponentially decaying weights times \(v^{2+\nu}\) over the geometric sequence. This proves the claim. Outside data beyond the final screened collar have no effect on the inside law or on the argument, which proves the stated stability. ◻

Proof. For one block write \(M=M_1+E\) as in (30). Since \(M_1\) has rank at most one, \(M_1\otimes M_1\) vanishes on antisymmetric tensors. On that subspace, \[\|M\otimes M\| \le 2\|M_1\|\,\|E\|+\|E\|^2 \le C L^{-2\lambda-\delta}.\] Choose \(L\) large enough to absorb \(C\) into a smaller exponent saving \(\nu'>0\), simultaneously with its choice in Corollary 25. Iterating the estimate and using \(\|u\wedge v\|_\pi\le2\|u\|_r\|v\|_r\) gives (51). The argument holds first for finite tensor sums and then for their projective-norm completion. Ordinary Markov pushes contract total variation, so fixed-factor omitted end intervals affect only the constant after rounding to the ratio-\(L\) grid. ◻

Proof. Let \(\eta_t\) have intensity \(t\) and fair colors. Euclidean scaling gives \[ G_R(t):=\mathbb EH_R(\eta_t)=f(\sqrt t\,R). \tag{54}\] We justify differentiation of the full-plane expectation before using this identity. All the estimates in this paragraph are uniform for \(t\) in a compact subinterval of \((0,\infty)\).

For \(n\ge n_0(R)\), let \(H_{R,n}\) be the same test computed from the points in \(B(0,2^n)\), together with the central insertion. A locality screen in a collar well inside \(B(0,2^n)\) determines every cell meeting \(RQ_0\), including its intersections with the boundary. Consequently \(H_{R,n}=H_R\) outside an event of probability at most \(C_R\exp(-c_R4^n)\). The same screen gives \(H_{R,n+1}=H_{R,n}\). Finite-volume Poisson differentiation follows by differentiating the absolutely convergent Poisson series; see also [8]. Equivalently, for a bounded function \(F\) of the process in a bounded set \(D\), \[ \frac{d}{dt}\mathbb EF(\eta_t\cap D) =\mathbb E\left[F(\eta_t\cap D) \left(\frac{N_t(D)}t-|D|\right)\right]. \tag{55}\] Here \(N_t(D)\) is the number of Poisson points in \(D\). Apply this to \(F=H_{R,n+1}-H_{R,n}\) on \(D=B(0,2^{n+1})\). The score in parentheses has second moment \(|D|/t\). Cauchy–Schwarz therefore bounds the derivative by \(C_R2^n\exp(-c_R4^n)\). The differences of the expectations and of their derivatives are uniformly summable. Hence \(G_R\) is \(C^1\) and its derivative is the limit of the finite-volume derivatives.

The add-one version of finite-volume differentiation is \[\frac{d}{dt}\mathbb EH_{R,n}(\eta_t) =\int_{B(0,2^n)} \mathbb E\bigl[H_{R,n}(\eta_t\cup\{y^{\rm f}\}) -H_{R,n}(\eta_t)\bigr]\,dy.\] For fixed \(y\) the integrand converges to its full-plane counterpart. For \(|y|\ge C R+C\), a screen surrounding \(RQ_0\) and the origin, at distance comparable to \(|y|\) and strictly short of \(y\), prevents the insertion from changing any cell used by the test. Its failure probability is at most \(C_R e^{-c_R|y|^2}\). The screen can be taken inside \(B(0,2^n)\) whenever \(y\in B(0,2^n)\) and \(n\) is sufficiently large; thus this is also a dominating bound for the finite-volume integrands. On a fixed bounded set the bound \(2\) suffices. Dominated convergence proves absolute integrability and identifies \(G_R'(1)\) with the integral in (53). Equation (54) proves the asserted derivative formula and, locally at each \(R\), the \(C^1\) assertion for \(f\). ◻

Proof. Use the exponent \(0<\nu<1\) of Corollary 25 and the one-arm exponent \(\beta>0\) from the critical inputs. All screens and arm probes below have fixed-factor buffers. Independent probes mean probes determined by disjoint sets of Poisson coordinates, not unrestricted Voronoi events in disjoint planar regions.

We first explain how arm requirements can be multiplied by a signed message estimate. The integrand in (52) is a mixed difference: one difference changes the central color, and the other inserts the point at \(y\). If this mixed difference is nonzero, changing the central color changes a crossing for at least one setting of the insertion, and changing the insertion changes a crossing for at least one setting of the central color. A screened probe separated from both changes reads the same base colors in all four configurations. Thus the necessary-arm implications for a local change apply to these common colors; see [10]. They apply even when the two crossing changes occur in different configurations.

The conditional version is equally important. Every formal noncrossing pairing can be drawn as disjoint wall arcs in a topological disk. Coloring its complementary regions alternately realizes the prescribed boundary runs and the pairing. Thus two formal cap states give two planar fillings agreeing on every exterior probe; the deterministic necessary-arm argument does not require Poisson geometry inside the replaced disk. Fix a successful cut about \(y\), its seam, and all exterior variables. The exterior test is now a function of the inside connection state. If a necessary central or enclosing four-arm event does not occur, the central black-minus-white test is zero for every such state. If a necessary side or corner arm event does not occur, each crossing test with the central color fixed is constant across inside states. These are the same planar necessary-arm implications, applied to two possible cap states on the cut. The insertion message has zero mass, so it annihilates the latter constants. Its conditional answer is consequently bounded by its total variation norm times the necessary probe indicators. Those indicators are measurable outside the cut. Disjoint buffered supports then allow their probabilities to be multiplied.

We use screened arm probes with failures allowed, whose probabilities are still bounded by a constant times their intended arm bounds; see the convention in Section 2 and [10]. Indeed, group failures by the largest failed radius \(v\) and retain the arm test starting a fixed factor beyond \(v\), independently of the failure. For a four-arm probe starting at \(r\), the relevant estimate is \[e^{-cv^2}A(C_0v,R) \le C A(r,R)(v/r)^g e^{-cv^2} \qquad(C_0v<R),\] where \(g\) is the polynomial exponent in (1). For a side or corner probe the corresponding endpoint losses are bounded by \(C(v/d)^2\) or \(C(v/u)^\beta\), respectively. If a failed screen crosses the junction between side and corner probes, the lost factors are therefore only polynomial. If the retained test has no remaining radial range, the same comparison follows from the polynomial lower bound, or directly from the displayed power factor for a side or corner probe. Summing over geometric \(v\) preserves the indicated arm cost as long as the retained probes and the failure event have disjoint buffered supports. A failure large enough to reach the central probe requires a separate argument below: that probe is then dropped, not treated as independent. An unweighted exponentially small error alone would not suffice when \(R\) is arbitrarily large.

Proof. Take concentric cuts about the origin, starting at radius \(10m\) and ending at a radius comparable with a small fixed fraction of \(R\), with consecutive radii in ratio \(L\). Round the final radius down to the last such scale. Its ratio to \(R\) is bounded below by a positive constant depending only on the already fixed \(L\). All cut collars lie strictly inside both rectangles and outside both inserted points. If necessary, enlarge the fixed lower bound on \(R/m\); the remaining bounded ratios follow by increasing \(C\), since all four scalar quantities in the determinant are bounded.

First restrict every test to success of every cut screen, with the screens evaluated on the common base process. The probability discarded is bounded by \[C\sum_{j\ge0}\exp\bigl(-cm^2L^{2j}\bigr) \le Ce^{-c'm^2}.\] At the first cut let \(a\) be the signed connection-state message obtained by changing the central color. Let \(b_y\) be the difference of this message with and without the extra fair point at \(y\). Both have zero mass conditional on each seam, and their total variation norms are bounded by \(2\) and \(4\), respectively. They depend on neither \(R\) nor \(R'\): on screen success their construction uses only the common interior Poisson variables and the first seam. In particular this description does not require disjoint neighborhoods of \(0\) and \(y\).

Let \(T\) be the composed transfer through the intervening cuts. The exterior crossing tests in \(RQ_0\) and \(R'Q_0\) define bounded linear functionals \(\ell_R\) and \(\ell_{R'}\) at the final cut. The four restricted expectations are \[\ell_R(Ta),\quad \ell_{R'}(Ta),\quad \ell_R(Tb_y),\quad \ell_{R'}(Tb_y).\] Their determinant is the evaluation of \(T^{\otimes2}\) on \(b_y\otimes a-a\otimes b_y\) by \(\ell_R\otimes\ell_{R'}\). Corollary 26 therefore bounds it by \(C(m/R)^{2\lambda+\nu'}\). Restoring the discarded events changes the determinant by at most \(Ce^{-cm^2}\), because each scalar factor is bounded. This proves (61). ◻

Proof. Set \(g(R)=Rf'(R)/f(R)\), which is defined and continuous for large \(R\) by Lemma 27 and (4). We first prove a uniform estimate on each interval \([R,2R]\). Fix \[\zeta=\frac{\nu'}{4(2+2\lambda+\nu')},\qquad \eta=\frac{\nu'}8,\qquad m=R^\zeta.\] The exponent estimate (5) implies \(f(s)\ge s^{-\lambda-\eta}\) for all sufficiently large \(s\). For \(R\le R'\le2R\), divide (61) by \(f(R)f(R')\) and integrate over \(|y|\le m\). The result is \[\begin{align*} \left|\int_{|y|\le m}\frac{K_R(y)}{f(R)}\,dy -\int_{|y|\le m}\frac{K_{R'}(y)}{f(R')}\,dy\right| &\le C R^{2\lambda+2\eta}m^2 \bigl[(m/R)^{2\lambda+\nu'}+e^{-cm^2}\bigr]\\ &\le C R^{-\nu'/2}. \tag{63}\end{align*}\] Indeed the power in the first term is exactly \[-\nu'+2\eta+\zeta(2+2\lambda+\nu')=-\nu'/2,\] and the exponential term is smaller. For large \(R\), the chosen \(m\) satisfies all the scale restrictions of Lemmas 28 and 29. Lemma 28 bounds each complementary normalized integral by \(C R^{-\zeta\xi}\). Since \(g(R)=2\int K_R(y)/f(R)\,dy\) by (53), we obtain \[ \sup_{R\le R'\le2R}|g(R')-g(R)|\le CR^{-\alpha}, \qquad \alpha=\min\{\nu'/2,\zeta\xi\}>0. \tag{64}\]

The values \(g(2^nR_0)\) form a Cauchy sequence, because the successive errors in (64) are summable. The same estimate between dyadic points shows that \(g(R)\) has a finite limit \(\gamma\) as \(R\to\infty\), and summing along \(R,2R,4R,\ldots\) gives \[|g(R)-\gamma|\le C\sum_{j\ge0}(2^jR)^{-\alpha} \le C'R^{-\alpha}.\] It remains to identify this limit. Integration yields \[\log f(R)-\log f(R_0)=\int_{R_0}^R g(s)\,\frac{ds}s.\] Dividing by \(\log R\) shows that \(\log f(R)/\log R\to\gamma\). By (5) the same limit is \(-\lambda\). Thus \(\gamma=-\lambda\), proving (62). ◻

Proof of Theorem 2. By Proposition 30, \[\frac{d}{dR}\log\bigl(R^\lambda f(R)\bigr)=O(R^{-1-\alpha}).\] The right side is integrable at infinity. Since \(f(R)>0\) for large \(R\), the logarithm has a finite real limit. Consequently \[ R^{5/4}f(R)\longrightarrow a\in(0,\infty). \tag{65}\] Positivity and finiteness follow from convergence of the logarithm, rather than merely boundedness of the normalized probabilities. In fact the argument also gives \(R^{5/4}f(R)=a(1+O(R^{-\alpha}))\).

Recall that \(S_\varepsilon=f(1/\varepsilon)\), \(m_\varepsilon=\varepsilon^{-2}\), and \(q_\varepsilon=(\mathbb EN_\varepsilon(Q_0))^{-1}\). The calibration limits (3) are \[m_\varepsilon q_\varepsilon S_\varepsilon\longrightarrow c\in(0,\infty), \qquad q_\varepsilon\mathbb EN_\varepsilon\longrightarrow B\in(0,\infty).\] Equation (65) gives \(S_\varepsilon\sim a\varepsilon^{5/4}\), hence \(q_\varepsilon\sim(c/a)\varepsilon^{3/4}\). It follows that \[\varepsilon^{3/4}\mathbb EN_\varepsilon\longrightarrow \frac{Ba}{c} =:c_V\in(0,\infty).\] Here \(N_\varepsilon\) counts color-pivotal sites for the original unwired unit-square event. The Poisson insertion was used only to differentiate \(f\); it has not changed the pivotal count in this last formula. Taking reciprocals proves the equivalent assertion for the near-critical scale. ◻