Proof. Park’s two-point observable supplies a massive-holomorphic \(f\) and its square integral \(h\), with the asserted arc values, bounds and interior regularity; see [6]. Equation (6) gives the displayed scalar equation with precisely the coefficient \(4m\). The source states uniqueness in its observable and square-integral class. We verify that the bounded scalar class used here is also unique.

If \(h_1,h_2\) are two bounded classical solutions and \(q=h_1-h_2\), define \[\beta=4m\frac{\nabla h_1+\nabla h_2} {|\nabla h_1|+|\nabla h_2|},\] with value zero when both gradients vanish. The identity \((|p|-|p'|)(|p|+|p'|)=(p+p')\cdot(p-p')\) shows that \[\Delta q+\beta\cdot\nabla q=0,\qquad |\beta|\le4m.\] Extend \(\beta\) boundedly and measurably outside \(O\) and use the diffusion \(\mathrm dX=\sqrt2\,\mathrm dB+\beta(X)\mathrm dt\), killed on exit from \(O\). Bounded-drift Girsanov constructs this diffusion and makes its law equivalent to Brownian law on every finite time interval. Its exit time is almost surely finite. Indeed, in a fixed time block a Brownian coordinate increment larger than the diameter of \(O\) plus the maximal drift displacement forces exit. This event has a positive probability uniform over the starting point and the past; iteration gives a geometric bound on the number of such blocks.

Brownian exit from a simply connected domain has a prime-end limit, and its harmonic measure gives mass zero to each specified prime end. Finite-horizon equivalence and then a countable union over horizons transfer these assertions to \(X\). In particular its exit prime end is almost surely different from \(c,d\). Itô’s formula on compact interior exhaustions makes \(q(X)\) a bounded stopped martingale. At exit its limit is zero by the prescribed boundary limits. Bounded convergence and optional stopping give \(q=0\).

For continuity, take any converging sequence of bounded marked domains. Approximate each domain by a rhombic domain sufficiently finely that its discrete square integral and reconstructed squared fermion approximate \(h\) and \(Dh\) on the first finitely many compact sets. The mesh can also be chosen so that this diagonal sequence converges to the limiting marked domain. Park’s convergence theorem applies to that sequence. Uniqueness just proved identifies every subsequential limit with the scalar solution in the limiting domain. Interior estimates for the massive equation give convergence of first derivatives on smaller compact sets. This proves the stated continuity and does not impose a factorization condition on the scalar solution class. ◻

Proof. Pair the medial half-edges locally at each revealed rhombus. Pairings already fixed can be removed from the product defining the law. A closed medial loop contributes its fixed factor \(\sqrt2\); the remaining pairings give the same loop law in the continuing component. In the cluster formulation this is precisely the spatial domain Markov property with cluster weight \(2\). The two boundary colors meet at the current and terminal corners.

Clipping along an open primal diagonal leaves a primal wired edge on the retained side; clipping along a dual-open diagonal has the dual interpretation. These are the half-rhombus boundary conventions in the observable problem of Proposition 5. If a bank contacts itself, its different occurrences are read in boundary order. Components behind the contact have no continuing change point and may be removed. Thus no new interior weight or extra boundary-change point is introduced. This proves the assertion even when the physical polygon has pinches: the simply connected polygon is the one with its incident sectors kept separate. Equivalently one may draw a thin polygonal collar within the corresponding half-rhombi. A fixed number of neighboring cell layers contains the entire change.

Finally, the ratio of the two local edge weights stays in a compact subset of \((0,\infty)\), by the angle bound and the convergence of the abstract angles to the geometric angles. Opening an edge changes the number of FK clusters by either zero or one. The two conditional odds therefore differ from the weight ratio by at most the fixed cluster factor \(2\). Both outcomes have a common positive lower bound. ◻

Proof. We first describe the comparison, since simply summing over components would not prove the assertion. Work in the graph of color \(c\), keeping boundary-sector copies until the comparison is made. Let \(H\) be the union of all its portions in the regions under consideration. An actual \(c\)-path cannot exit \(H\) through a noncircular bank: that bank is wired in the other color. Consequently every exterior connection that can influence \(H\) enters through its radial-end vertices. Conditioning on all edges off the shell may impose any partition of these vertices. Dominate it by the single partition wiring all radial ends together, including the inner and outer ends.

Identify copies that project to the same ambient lattice vertex. For \(q=2\) this is an increase in boundary identification, hence it increases open-edge connectivity. Embed the resulting graph into the ambient lattice annulus. Edges missing because of a bank or a deleted sector may be viewed as conditioned \(c\)-closed. Release those conditions and add the remaining ambient vertices and edges. Random-cluster monotonicity couples these operations so that every open edge of \(H\) remains open in the larger graph. Each counted arm projects to an actual arm of the ambient graph. The construction couples the entire union at once and therefore pays no multiplicity factor.

This also explains why winding around the annulus causes no additional problem. A winding region changes the partition induced at the radial ends, but that partition was already dominated by the maximal one. An artificial wire is used only in the measure and is not counted as an edge in the arm event. Thus an opposite-color circuit in the interior of a buffered subannulus blocks every counted arm, even if the boundary condition artificially connects the two radial ends.

In a subannulus of fixed ratio, cover a circuit by a fixed number of overlapping rectangles. Proposition 5 bounds below the conditional probability of each opposite-color crossing under the worst boundary condition. FKG glues those crossings into a circuit, giving a probability at least \(c_*>0\), uniformly under conditioning off that subannulus. Use a family of disjoint buffered subannuli of fixed ratio between scales \(K_0(r\vee\delta)\) and \(R/K_0\). Exposing them successively gives an upper bound \((1-c_*)^N\), with \(N\ge c\log(R/(r\vee\delta))-C\). This is (14). The same exposure argument proves the product claim for disjoint annuli. Restriction to a rectangle inside any shell induces a boundary partition that is dominated by its fully wired partition, so arbitrary off-shell conditioning is harmless. ◻

Proof. At a stopping time, complete the currently exposed tile if the drawing has been stopped in its interior. This reveals only a bounded number of local turns. Use the graph domain of Lemma 6. Consider an annulus whose inner circle meets its boundary, and a component of its intersection with the domain that does not separate the two current marks.

Such a component cannot have accessible banks of both colors. Indeed, an interior path joining the two banks, together with short terminal pieces in their incident sectors, would be a crosscut joining the two Dobrushin arcs. The two change points alternate with the ends of that crosscut on the prime-end circle, so it separates them. Removing the whole annular component would then also separate them, a contradiction. An interface crossing in a component with one bank color has an actual arm of the other color beside its strand. The neighboring diagonals join in boundary order along the strand; trimming a fixed number of cells at each circular end removes any terminal ambiguity. Components with no accessible bank can be assigned either color, since the strand has both adjacent cluster arms.

Apply Lemma 7 separately to the two possible missing colors. For radii larger than a fixed multiple of the mesh this gives \(K(r/R)^\alpha\) for the probability of any unforced crossing, with no dependence on the number of components. For \(R\) exceeding a fixed multiple of \(L\) a crossing is impossible; thus all rectangle scales used by the comparison are uniformly bounded.

It remains to give the drawing convention at submesh scales. In each rhombus use simple polygonal tracks through its side midpoints for its two possible pairings. Choose the tracks and the boundary half-cell tracks with a clearance \(c_\eta\delta\) between nonincident pieces. The initial and terminal tracks leave their boundary corners transversely. Compactness of the angle interval gives uniform nondegeneracy of these finitely many types of local templates, including their lengths, turning angles, and transverse clearances. This is a perturbation inside a bounded number of cells.

If \(r\ll\delta\), an inner circle meeting the current boundary can be visited by the future track only within \(O(r)\), measured along the incident template, of the current or terminal mark. Away from consecutive pieces this follows from clearance. Along consecutive pieces it follows from their uniformly nondegenerate angles. In that local picture both incident banks occur. A subannulus with inner radius larger than a fixed multiple of \(r\), outer radius smaller than \(c_\eta\delta\), and still inside the original annulus separates the mark from the distant continuation. Its component is therefore forced. Consequently, if \(R/\delta\) is bounded and \(R/r\) is sufficiently large, there is no unforced crossing at all. If \(R/\delta\) is larger than a prescribed large constant, use instead Lemma 7 between scale \(K\delta\) and \(R/K\). Choosing that constant first makes its upper bound smaller than the constant required in G2. These two cases cover all submesh inner radii. The argument is conditional on the entire revealed past, as required. ◻

Proof. We give a length–area proof; in fact only the lower conformal-radius bound is needed. First let \(V\) be a finite polygon, put \(c=c_0\), and let \(E\subset V\) be connected with diameter \(d<c/64\). Koebe’s theorem gives \(B(z_0,c/4)\subset V\). If \(E\) meets \(B(z_0,c/8)\), Cauchy’s estimate for the bounded analytic function \(\chi\), on a slightly larger interior disc, gives \(\operatorname{diam}\chi(E)\le16d/c\).

Otherwise choose \(x\in E\), put \(r=2d\) and \(R=c/16\). Then \(E\subset B(x,r)\), whereas \(z_0\notin B(x,R)\). For a regular radius \(s\in(r,R)\), let \[L(s)=\int_{V\cap\partial B(x,s)}|\chi'(z)|\,|\mathrm dz|\] be the total image length of all circular components. Cauchy–Schwarz and polar integration give \[\int_r^R L(s)^2\frac{\mathrm ds}{s} \le 2\pi\int_{V\cap\{r<|z-x|<R\}}|\chi'(z)|^2\,\mathrm dA(z) \le2\pi^2.\] There is therefore a regular \(s\) for which \[L(s)\le \ell:= \left(\frac{2\pi^2}{\log(R/r)}\right)^{1/2}.\] The source circle is either wholly inside \(V\), or cuts it into finitely many disjoint crosscuts. Their images have total length at most \(\ell\). When \(\ell<1/4\), each image crosscut lies within distance \(\ell\) of one of its circle endpoints. The shorter boundary-circle arc joining its endpoints lies in that same ball. The enclosed Jordan region has diameter at most \(2\ell\) and does not contain \(0\); it is exactly the noncentral shadow of that crosscut. A full image circle of length \(\ell\) has bounded interior of diameter at most \(\ell/2\).

The connected set \(\chi(E)\) lies in a component separated from \(0\) by this finite collection of image crosscuts. Such a component is contained in the noncentral shadow of one crosscut: adding disjoint crosscuts one at a time gives a tree of components, and the first edge on the route from the central component identifies a separating shadow. Thus \(\operatorname{diam}\chi(E)\le2\ell\). This proves a modulus of order \((\log(c/(32d)))^{-1/2}\), independent of physical area, diameter, or the number of boundary pieces.

For a general simply connected \(V\), first connect any two points of \(E\) by a compact path in an arbitrarily small open neighborhood of \(E\) contained in \(V\). Such a neighborhood can be chosen connected, as a union of small discs centered on the connected set \(E\). Its physical diameter is at most \(d+o(1)\). Exhaust \(V\) by finite Jordan polygons containing this path and an interior reference disc. Their conformal radii eventually exceed \(c/2\), and their normalized maps converge on compact sets. The polygon estimate passes to the two chosen points. Taking their supremum proves the assertion for \(E\).

Accessible endpoints are handled by truncating their actual approaching arcs and then taking prime-end limits. For a boundary-following connected lift in one of the finite polygon domains used here, write \(f=\chi^{-1}\) and approximate its parameter image \(C\subset\overline\mathbb D\) by \(tC\), \(t\uparrow1\). The continuous physical boundary extension of \(f\) makes \(f(tC)\) an interior approximation with the same limiting physical diameter. This extension need not be injective at a slit, which is why the lift, not an unlabelled physical boundary set, was specified. A retained cell sector and a fixed number of neighboring sectors have physical diameter \(O(\delta)\); the last assertion follows. ◻

Proof. Apply [3] to Proposition 9. Their modified simple medial drawing agrees with the separated drawing used above. The changes of drawing have vanishing disc diameter by Lemma 10, and so do not change the limiting curve law.

The disc maps used in that theorem fix the two endpoints, whereas our normalization fixes an interior point and angle. Marked Carathéodory convergence makes the intervening disc automorphisms converge to a nondegenerate automorphism. Thus the two normalizations give the same tightness and convergence statements. Choose the half-plane maps \(\phi_\delta\) taking the marks to \(0,\infty\) so that \(\phi_\delta\to\phi\) on compact subsets; the remaining positive dilation is fixed by the interior normalization. The joint half-plane conclusion then has precisely the capacity convention \(\partial_tg_t=2/(g_t-W_t)\) used here. ◻

Proof. Work first in the normalized initial disc. The kernel of the geometric slit domains on the nonswallowed side is identified by joint Loewner convergence. To compare a graph domain with its geometric slit, take a compact interior path and a compact neighborhood of it in \(O_t\). They have positive distance from the limiting past. For all large \(n\), neither the discrete past nor its cell collar meets this neighborhood. Join the path to a fixed interior point near the terminal mark. A sufficiently small terminal neighborhood is untouched by the past through the given finite time, because the target is reached only at the end. The remaining connection to the terminal boundary occurrence can therefore be made in that unchanged neighborhood. The compact path belongs to the active graph component.

This proves that every protected compact subset of \(O_t\) is retained. Conversely, no kernel from such a basepoint can cross the limiting slit. A component joined only by a neck collapsing to that slit is not part of its kernel. Thus removal of discarded components and the cell collar changes do not alter the active kernel.

Kernel convergence alone does not identify the moving prime end. Choose times \(s>t\), arbitrarily close to \(t\), for which \(\gamma(s)\) is interior to \(O_t\). Such times exist: a whole interval of trace confined to the old boundary and hull would add no hull and hence no capacity, contrary to strict capacity growth. For fixed \(s\), a compact corridor from \(\gamma(s)\) to an untouched target neighborhood is retained in the graph domain by the preceding paragraph. The corresponding discrete continuation from its moving corner to a point converging to \(\gamma(s)\) is an actual connected chain in that domain, up to its incident boundary sectors. Include all forced steps before the next decision; they remain in the same chain.

By uniform curve convergence, the diameter of this chain in the initial disc can be made arbitrarily small by first taking \(s-t\) small and then taking \(n\) large. Normalize the current domains at a fixed protected bulk point. Their conformal radii there are uniformly nondegenerate. Lemma 10 therefore makes the current-chart diameter of the chain small. Its interior endpoint converges by kernel convergence, whereas its other end is the current corner occurrence. As \(s\downarrow t\), the image of the interior endpoint tends to \(W_t\), by the trace property. The moving corner consequently tends to the same prime end. There is no use of marked convergence in applying Lemma 10; it is a uniform statement for connected interior sets. In particular, near a self-contact the actual continuation chain selects the correct incident sector.

Finally, returning to physical coordinates uses the compact convergence of the initial uniformizations. The proof can be localized at any protected bulk basepoint in \(O_t\), so it does not require the original basepoint to remain unswallowed. ◻

Proof. At a discrete restart, the future graph problem is Dobrushin by Lemma 6. Proposition 9, conformal transport, and the equivalence of G2 and G3 give the same conditional power constants in its current chart. For the first event, use a half-annulus centered at \(x\), of inner radius \(d\) and a fixed small outer radius avoiding \(0\). It is unforced for the marks \(0,\infty\), and its inner circle meets the real boundary. Entry into the ball requires its crossing. A ball later swallowed without being visited does not create an entry event.

For the second event, write \(r_0=\operatorname{crad}_{O_0}(w)\). Koebe’s theorem puts the old boundary at distance at least \(r_0/4\). A decrease of the active conformal radius below \(\varepsilon r_0\) therefore requires the new trace to approach \(w\) within \(O(\varepsilon r_0)\). The annulus from that scale to \(r_0/8\) lies in the old domain. The interface has an actual cluster arm across it. The blocking-circuit proof of Lemma 7, now in a bulk annulus, gives the bound. Only physical radii bounded by the original diameter are used. The infimum stops at swallowing; assigning radius zero after swallowing would give a different and false assertion.

We explain the limiting conditional statement. First stop before a protected point has small chart height, before chart coordinates or the driver leave fixed bounds, and before a fixed capacity horizon. Joint convergence and Lemma 12 make the current charts converge on all compact sets used in the estimate. Apply the discrete conditional inequality, multiplied by any bounded nonnegative continuous function of the stopped driver history. Pass to the limit with slightly enlarged balls and relaxed thresholds; then decrease those enlargements. A monotone-class argument gives the conditional inequality for the limiting history.

On such a protected interval, a clock with rate \(|g_t(\phi(w))-W_t|^{-2}\) is continuous and strictly increasing. Its inverse stopping times are obtained by convergence of these clocks. First-crossing times of angular levels can either be taken at generic levels or approximated on clock-time grids. Continuity, followed by exhaustion of the protected intervals, gives all stopping times used here. This procedure concerns only the driver filtration. It does not presume that conformally mapping a nonsimple future trace through an old swallowed pocket is a valid operation. ◻

Proof. First consider deterministic sequences of active graph domains along a joint curve–driver convergence as above. Lemma 12 and Proposition 5 identify the locally uniform limit of the squares of their fermions as \(D h_t\). The same reasoning for every sequence of times \(t_n\to t\) gives convergence locally along the evolution.

We also need continuity of the limiting square under this domain variation. It follows directly from the deterministic observable theorem: approximate each marked continuum domain by an admissible lattice domain and choose the meshes diagonally. Interior compactness and uniqueness of the bounded two-point solution identify every subsequential limit. Equivalently, bounded solutions of the displayed elliptic equation are compact on interior discs and their prime-end boundary values are controlled by the same two-point boundary estimates. Thus the limiting square is continuous in the marked kernel topology, locally on protected compacts. The assertion applies uniformly along a compact family of convergent curve–driver paths: otherwise a contradicting sequence of times and paths would give a deterministic violating sequence of domains.

It remains to lift convergence of squares to signed martingales. We give the argument rather than presuming a continuous choice of sign. Work at finitely many test edges near fixed protected bulk points, stopped before they lose a larger protection buffer, and before the observables exceed the deterministic interior bound. At a genuine exploration decision there are two possible next pairings. Each next domain and its observable are determined by the revealed past and the hypothetical pairing. The two conditional probabilities are at least \(p_0>0\), by Lemma 6.

Fix a tolerance \(\varepsilon>0\). Stop immediately before the first decision at which either hypothetical next square changes by more than \(\varepsilon\), or either hypothetical next domain loses the smaller protection buffer, while the prevalue remains in the larger one. This is a predictable decision index: no future continuation is used to test the two hypothetical domains. Select one bad alternative measurably if there are two. Conditional on reaching this index, the probability of selecting a bad alternative is at least \(p_0\). Consequently \[ p_0\mathbb P\{\text{a bad-alternative stop occurs}\} \le \mathbb P\{\text{an observed bad step occurs}\}. \tag{15}\] The right side tends to zero by the preceding square convergence, the protected-domain convergence, and the fact that individual cell steps have vanishing capacity increment on a protected finite capacity interval. There is no sum over the number of decisions.

Suppose now that the current value is \(f\), with \(|f|\ge a>0\), and both next squares are within \(\varepsilon\) of \(f^2\). For \(\varepsilon\ll a^2\), each next value lies within \(K\varepsilon/a\) of either \(f\) or \(-f\). If one alternative were near \(-f\), the real projection of the conditional average onto the direction \(f\) would be at most \[|f|-2p_0|f|+K\varepsilon/a.\] For \(\varepsilon\ll p_0a^2\), this contradicts the martingale identity. Thus neither alternative changes sheets where the square is bounded away from zero.

This statement gives tightness through zeros as well. On a short time interval, uniform continuity of the limiting square implies one of two possibilities. Either all its amplitudes are small, in which case every square root is small, or its values stay in a small disc not containing zero, in which case the signed field stays on one continuous root branch and has a small oscillation. Rapidly moving zeros cannot produce a fixed-size oscillation: such an oscillation would require either a nonzero sheet switch or a visible oscillation of the square. Deterministic forced steps add no martingale decision and are handled by the same square convergence. Combining this argument with (15) proves continuity tightness at each protected test point.

Take countably many test points and rational protected neighborhoods, together with a countable collection of stopping levels. Extract jointly with the curve and driver. Spatial compactness of the fermions up to one common sign, furnished by Proposition 5, identifies the limits on overlapping neighborhoods and gives the whole field. The spatial equation passes to the limit. Its interior elliptic estimates give all spatial derivatives on smaller compact sets. In particular the field is locally continuous in space and time, with the asserted square.

Finally, before any tighter protected stop, the discrete fields are bounded martingales: their conditional-expectation interpretation is valid at every still active test edge and their terminal phase is unchanged. Multiply each martingale identity by a bounded continuous function of finitely many earlier curve–field evaluations and pass to the joint limit. The uniform bound gives uniform integrability. A monotone-class argument gives the conditional expectation identity in the joint natural filtration. Stopping at generic levels of continuous protection quantities, or first using a tighter buffer and then exhausting, removes any discontinuity of the stopping operation. This proves precisely the stated local martingale assertion on the live domain. It gives no assertion about a zero extension through swallowing, and none will be used. ◻

Proof. The imaginary part of the Loewner equation gives \(\partial_tY_t=-2Y_t/r_t^2\). This differentiates \(g_t(\phi(w))\), not \(W_t\): the driver is real, so its subtraction does not affect the imaginary part. If \(H_t=g_t\circ\phi\), then \[\operatorname{crad}_{O_t}(w)=\frac{2Y_t}{|H_t'(w)|},\qquad \partial_t\log|H_t'(w)|=-2\operatorname{Re}\frac1{Z_t^2}.\] It follows that \[\partial_t\log\operatorname{crad}_{O_t}(w) =-\frac2{r_t^2}+2\operatorname{Re}\frac1{Z_t^2} =-\frac{4Y_t^2}{r_t^4}.\] The clock has a strictly positive continuous rate on every compact live interval. Changing variables proves all the assertions. ◻

Proof. For the first assertion, decrease \(d_*\) so that \(|x|\) stays, for example, between \(3/4\) and \(1\). A half-annulus centered at \(x\), with a fixed outer radius less than \(1/4\) and inner radius \(d\), misses the tip \(0\) and the target \(\infty\) and is unforced. A visit to its inner ball entails an unforced crossing. Conditional G3 and its uniform conformal transport therefore give the asserted bound. A subsequent swallowing of the ball without visiting it is not a crossing and causes no difficulty for this statement.

For the second assertion, Koebe’s conformal-radius comparison gives \[\operatorname{dist}(w,\partial O_\tau)\ge R/4.\] If \(w\) is still alive and its conformal radius drops below \(\varepsilon R\), its distance to the current boundary is less than \(\varepsilon R\). For sufficiently small \(\varepsilon\), this new nearby boundary cannot be a piece of the old boundary, which is at distance at least \(R/4\). The future interface must therefore approach within \(\varepsilon R\) of \(w\). Such an approach supplies an actual FK arm through the annulus with inner radius comparable to \(\varepsilon R\) and outer radius, say, \(R/8\). This annulus is initially in the protected bulk. In each of a sequence of disjoint buffered shells, rectangle crossings and FK positive association give a blocking circuit with a probability bounded below, uniformly under every conditioning outside that shell. Their product gives \(K\varepsilon^a\). The physical outer radius is uniformly bounded by the assumed diameter bound. Larger values of \(\varepsilon\) are absorbed by increasing \(K\).

Here these statements are inherited by the subsequential law through the localized conditional convergence of Corollary 13. We detail the role of the stopping times, since no stochastic differential equation for \(W\) is available. First restrict to a finite capacity interval on which the test point has positive chart height, its relevant coordinates stay bounded, and its physical access and conformal radius are protected. Driver convergence and the Loewner ODE then imply convergence of its coordinates, derivatives, and point clocks. The inverse clock is continuous on such an interval because its rate is positive and continuous. In the discrete conditional future, finish the current tile when necessary and use the uniform crossing bounds; the tile error disappears under the same protections. One passes these bounds by multiplying them by bounded continuous functions of the stopped driver history, slightly enlarging visiting balls and relaxing conformal-radius thresholds before taking the limit. The conditional inequalities consequently hold in the natural driver filtration.

For a bounded live stopping time, approximation from above by stopping times on a deterministic time grid gives the same conclusion. Current charts and coordinates are continuous under the protections; relaxed thresholds remove a possible equality at the limiting time. First angular-level times can equivalently be approximated on clock grids, with a slightly relaxed level, and then obtained by continuity. Finally exhaust the finite horizons and the protections. For the physical annulus estimate, take the mesh to zero at each fixed positive protected radius before removing that protection. Thus the annuli used in the limiting estimate are always above the lattice cutoff during its proof. This establishes the two estimates, including all their live clock restarts, without assuming a continuum domain Markov property or a semimartingale driver. Only the natural past is conditioned on; no enlargement revealing future information is used. ◻

Proof. Shrink the ball radius by a fixed factor whenever needed below; this only changes the constants. Because the trace misses the original ball, its connected upper half-ball is disjoint from the trace at every time. All points of the smaller upper half-ball are in the same component of the trace complement. They are therefore disconnected from infinity simultaneously, if disconnection occurs. The real boundary germ at \(x\) will have this same swallowing time, as we verify below. This assertion uses avoidance of a whole ball, not merely avoidance of the point \(x\).

Before this common lifetime the map extends by Schwarz reflection to a univalent map on a smaller full ball centered at \(x\). Its derivative \(l_t'(x)\) is positive. The driver is outside this reflected image: if \(W_t=l_t(\xi)\) for a point in the ball, symmetry and univalence force \(\xi\) to be real. The local inverse would then give \[\gamma(t)=\lim_{y\downarrow0}l_t^{-1}(W_t+\mathrm iy)=\xi,\] contrary to trace avoidance.

Fix any \(t_0\) before the common bulk lifetime. Since \(l_{t_0}(x)\ne W_{t_0}\), the complex Loewner ODE has a flow \(\Phi_{t,t_0}\), holomorphic near \(l_{t_0}(x)\), for \(t\) in a time neighborhood of \(t_0\) within this lifetime (one-sided at \(0\)). On a sufficiently small fixed disc \(D\) about \(x\), the composition \(\Phi_{t,t_0}\circ l_{t_0}\) is defined and holomorphic. Forward and backward ODE uniqueness identify it with \(l_t\) on \(D\cap\mathbb H\); the identity theorem extends this equality to \(D\). Hence \(t\mapsto l_t(x)\) satisfies \(\partial_t l_t(x)=2/(l_t(x)-W_t)\) before the common bulk lifetime.

Thus the real Loewner solution at \(x\) cannot meet the driver before the common bulk lifetime. Conversely, if that real solution survived a finite bulk disconnection time, its distance from the continuous driver would be positive on a slightly longer compact time interval. Continuous dependence for the complex Loewner ODE would then make all sufficiently close points \(x+\mathrm iy\) survive that longer interval. This contradicts the simultaneous disconnection of all points in the half-ball. The real and bulk lifetimes therefore agree; the argument also covers an infinite lifetime by exhaustion.

Koebe’s theorem and interior distortion yield, with \(r_x(t)=|l_t(x)-W_t|\), \[ r_x(t)\ge c d\,l_t'(x),\qquad \left|\frac{l_t''(x)}{l_t'(x)}\right|\le\frac Kd, \qquad Y_t\asymp X_0l_t'(x),\qquad r_t\asymp r_x(t). \tag{23}\] For the last comparison, distortion first gives \(|l_t(z_0)-l_t(x)|\le KX_0l_t'(x)\); choosing \(c_0\) sufficiently small and using the first inequality makes this a small fraction of \(r_x(t)\).

Differentiate the deterministic Loewner equation in its spatial variable. On the real reflected interval it gives \[ \partial_t\left(\frac{l_t''}{l_t'}\right)(x) =\frac{4l_t'(x)}{(l_t(x)-W_t)^3}. \tag{24}\] The real denominator cannot change sign before swallowing. Hence the right-hand side has a fixed sign. The primitive starts from zero and is bounded in absolute value by \(K/d\) through (23). It follows that \[ \int\frac{l_t'(x)}{r_x(t)^3}\,\mathrm dt\le\frac Kd \tag{25}\] up to any strictly earlier time and then up to the lifetime by monotone convergence. This argument differentiates a map with respect to space and capacity; it never differentiates the continuous driver.

The distortion inequalities give \(X_t/X_0\asymp l_t'(x)/r_x(t)\le K/d\). Since \(\mathrm ds=\mathrm dt/r_t^2\) and \(r_t\asymp r_x(t)\), \[\int\frac{X_s}{X_0}\,\mathrm ds \le K\int\frac{l_t'(x)}{r_x(t)^3}\,\mathrm dt\le\frac Kd.\] Taking the product of this bound and the supremum bound proves (22). Disconnection introduces no additional term: the integrals stop at the common lifetime, and the killed angular process is zero beyond its clock lifetime. ◻

Proof. Translate and dilate the current chart to make the starting radial coordinate equal to one. The corresponding capacity scaling leaves \(\mathrm dt/r_t^2\), and hence the point clock, unchanged. Time zero and the notation \(O_0\) in this proof refer to the chosen restart.

For a large \(u\) take \(d=A/u\), with a fixed \(A\) larger than both the constant in (21) and \(1/c_0\). If \(X_0\le c_0d\), the reflected-ball bound makes the supremum at most \(u\) unless the trace visits \(B(x,d)\). Lemma 18 bounds that exception by \(Kd^a\). If \(X_0>c_0d\), the deterministic inequality \(X\le1\) already gives \(X/X_0<u\). Increasing \(K\) handles the bounded range of \(u\) for which \(d\) is not yet sufficiently small. This proves (26).

For tightness, fix a sufficiently small \(d>0\). If \(X_0\le c_0d\), then \(I\le K/d^2\) outside an event of conditional probability at most \(Kd^a\), by Lemmas 18 and 19. If \(X_0>c_0d\), the conformal-radius identity instead gives \[I=\frac1{4X_0^2} \log\frac{\operatorname{crad}_{O_0}(w)} {\inf_{t<T_w}\operatorname{crad}_{O_t}(w)}.\] This is an equality of extended nonnegative quantities, with the infimum taken on the live interval. By (20), \[\mathbb P\{I>L\mid\text{past}\} \le K\exp(-4a c_0^2d^2L) \qquad\text{if }X_0>c_0d.\] For \(L\ge K/d^2\) the two cases therefore give a uniform bound by \(Kd^a+K\exp(-4a c_0^2d^2L)\). Given an error probability, first choose \(d\) small and then \(L\) large. This proves the asserted tightness. ◻

Proof. Choose \(p<1/4\) so small that \(4p<a\), with \(a\) from (26). Integrating that tail gives \[ \mathbb E\left[\left(\sup_{s\ge0}\frac{X_s}{X_0}\right)^{2p} \,\middle|\,\text{past}\right]\le K_p. \tag{29}\] We next find a deterministic clock time \(H\) at which the conditional \(p\)-moment contracts strictly.

Fix a small error tolerance \(\alpha>0\), to be chosen below. Choose \(\rho\in(0,1/2)\) so small that \(K(2\rho)^a\le\alpha\). By occupation tightness choose \(L\) with \(\mathbb P\{I>L\mid\text{past}\}\le\alpha\), uniformly over the restart. Take \(H>L/\rho^2\). If the angular process is alive throughout \([0,H]\) and never reaches \(\rho X_0\), then \(I\ge\rho^2H>L\). Thus with conditional probability at least \(1-\alpha\) it either dies or reaches \(\rho X_0\) by \(H\). At the first live hitting time, the restarted supremum estimate gives \[\mathbb P\left\{\sup_{\text{after hit}}X_s>X_0/2 \,\middle|\,\text{past at hit}\right\} \le K(2\rho)^a\le\alpha.\] The killed process never rebounds. Consequently \[ \mathbb P\{X_H/X_0>1/2\mid\text{past}\}\le2\alpha. \tag{30}\] Using (29) and conditional Cauchy–Schwarz on the exceptional event, \[\mathbb E[(X_H/X_0)^p\mid\text{past}] \le 2^{-p}+K_p^{1/2}(2\alpha)^{1/2}.\] Choose \(\alpha\) sufficiently small to make the right-hand side a constant \(\lambda<1\). All choices were uniform in the starting angle and past.

Apply this contraction successively at clock times \(H,2H,\ldots\) while alive; after death the conditional contribution is zero. This gives \(\mathbb E[X_{nH}^p\mid\text{initial past}]\le\lambda^nX_0^p\). For \(s=nH+u\), \(0\le u<H\), the restarted supremum moment bounds the last partial interval by a fixed factor. With \(b=-H^{-1}\log\lambda>0\), enlarging \(K\) proves (28). Only this \(p\)-moment is asserted to contract; the higher moment needed later comes from (29). ◻

Proof. On a surviving clock interval, Lemma 17 gives \[ r_s=\frac{Y_0e^{-2s}}{X_s},\qquad t(s)=Y_0^2\int_0^s\frac{e^{-4v}}{X_v^2}\,\mathrm dv. \tag{32}\] Since \(X_v\le1\), the second integral is at least a fixed constant times \(\min(1,s)\). Therefore \[ \frac{r_s}{\sqrt{t(s)}} \le\frac{K}{X_s\sqrt{\min(1,s)}}. \tag{33}\] For \(0<\epsilon\le p\), the inequality \(X_s^{2p-\epsilon}\le X_s^p\) and (28) imply \[ \mathbb E\left[X_s^{2p} \left(\frac{r_s}{\sqrt{t(s)}}\right)^\epsilon; s<\sigma_*\right] \le K\min(1,s)^{-\epsilon/2}X_0^p e^{-bs}. \tag{34}\] Its integral is uniformly finite when \(\epsilon<2\).

For the other ratio put \(q_s=\min_{0\le v\le s}X_v\) on \(\{s<\sigma_*\}\). This minimum is strictly positive on each surviving compact clock interval. Equation (32) gives \[ \frac{\sqrt{t(s)}}{r_s} \le\sqrt{s}\,e^{2s}\frac{X_s}{q_s}. \tag{35}\] Partition the surviving event into the disjoint bins \[B_k(s)=\{s<\sigma_*,\quad 2^{-k-1}X_0<q_s\le2^{-k}X_0\},\qquad k=0,1,\ldots.\] For \(k\ge1\), on \(B_k(s)\) the process has reached \(2^{-k}X_0\) while alive by clock time \(s\). Restarting the \(2p\)-moment supremum estimate there gives \[ \mathbb E[X_s^{2p};B_k(s)]\le K X_0^{2p}2^{-2pk}. \tag{36}\] For \(k=0\) the same bound follows by restarting at time zero. Independently, \(X_s^{2p}\le X_s^p\) and the small-power contraction give \[ \mathbb E[X_s^{2p};B_k(s)]\le K X_0^p e^{-bs}. \tag{37}\] Taking the geometric mean of these two upper bounds yields \[ \mathbb E[X_s^{2p};B_k(s)] \le K X_0^{3p/2}2^{-pk}e^{-bs/2}. \tag{38}\] No exponential contraction of the \(2p\)-moment has been used.

On \(B_k(s)\) we have \(q_s^{-\epsilon}\le2^{(k+1)\epsilon}X_0^{-\epsilon}\). Combining this with (35), \(X_s^{2p+\epsilon}\le X_s^{2p}\), and (38), we obtain the full bin sum \[\begin{align*} &\mathbb E\left[X_s^{2p} \left(\frac{\sqrt{t(s)}}{r_s}\right)^\epsilon; s<\sigma_*\right] \\ &\qquad\le K s^{\epsilon/2}e^{2\epsilon s}X_0^{-\epsilon} \sum_{k\ge0}2^{k\epsilon}\mathbb E[X_s^{2p};B_k(s)] \\ &\qquad\le K s^{\epsilon/2}e^{-(b/2-2\epsilon)s} X_0^{3p/2-\epsilon} \sum_{k\ge0}2^{-(p-\epsilon)k}. \tag{39}\end{align*}\] Choose once and for all \[ 0<\epsilon<\min\{p,b/4,2\}. \tag{40}\] The geometric series converges, \(X_0^{3p/2-\epsilon}\le1\), and the remaining expression is integrable at both zero and infinity. Together with (34), this proves (31). All estimates are uniform as \(X_0\) tends to zero. Killed paths contribute nothing to either integral. ◻

Proof of Proposition 16. Fix \(t>0\), and divide \(\mathbb H\) into the dyadic shells \[A_j(t)=\{v\in\mathbb H:2^j\sqrt t\le|v|<2^{j+1}\sqrt t\}, \qquad j\in\mathbb Z.\] Write \(\vartheta=\arg v\) and \(L_j(t)=\int_{A_j(t)}\mu_t(v)|v|^{-2}\,\mathrm dA(v)\). Cauchy–Schwarz with the two factors \(\mu_t\sin^p\vartheta/|v|\) and \(\sin^{-p}\vartheta/|v|\) gives \[\begin{align*} L_j(t)^2 &\le \left(\int_{A_j(t)} \frac{\mu_t(v)^2\sin^{2p}\vartheta}{|v|^2}\,\mathrm dA(v)\right) \left(\int_{A_j(t)} \frac{\sin^{-2p}\vartheta}{|v|^2}\,\mathrm dA(v)\right)\\ &\le K_p\int_{A_j(t)} \frac{\mu_t(v)^2\sin^{2p}\vartheta}{|v|^2}\,\mathrm dA(v). \tag{41}\end{align*}\] Indeed the second factor is exactly \[(\log2)\int_0^\pi(\sin\vartheta)^{-2p}\,\mathrm d\vartheta<\infty,\] because \(2p<1\). This is where the small angular power is retained.

For the \(\epsilon\) chosen in (40), weighted Cauchy–Schwarz over a finite set of shell indices gives \[\left(\sum_j L_j(t)\right)^2 \le\left(\sum_{j\in\mathbb Z}2^{-\epsilon|j|}\right) \sum_j2^{\epsilon|j|}L_j(t)^2.\] On \(A_j(t)\), the weight \(2^{\epsilon|j|}\) is bounded by a constant times \((|v|/\sqrt t)^\epsilon+(\sqrt t/|v|)^\epsilon\). Increasing the finite sets of indices and using monotone convergence therefore gives the extended-integral inequality \[ J_t^2\le K\int_{\mathbb H} \frac{\mu_t(v)^2\sin^{2p}(\arg v)}{|v|^2} \left[ \left(\frac{|v|}{\sqrt t}\right)^\epsilon +\left(\frac{\sqrt t}{|v|}\right)^\epsilon \right]\mathrm dA(v). \tag{42}\]

Under \(w=\psi_t(v)\) the area formula is \[\mu_t(v)^2\,\mathrm dA(v)=m^2\,\mathrm dA(w).\] The angular coordinate and radial coordinate in (42) become \(X_t(w)\) and \(r_t(w)\) from (18). Thus Tonelli’s theorem gives \[\begin{align*} \mathbb E\int_0^\infty J_t^2\,\mathrm dt &\le K m^2\int_\Omega \mathbb E\int_0^{T_w} X_t(w)^{2p} \left[ \left(\frac{r_t(w)}{\sqrt t}\right)^\epsilon +\left(\frac{\sqrt t}{r_t(w)}\right)^\epsilon \right]\frac{\mathrm dt}{r_t(w)^2}\,\mathrm dA(w)\\ &\le K m^2\operatorname{area}(\Omega), \tag{43}\end{align*}\] where the last inequality is precisely Lemma 22 for each fixed physical point, followed by its point-clock change of variables. The constants are independent of \(w\). There is no need to choose a common exceptional event for all physical points: Tonelli applies to the nonnegative jointly measurable integrands and the uniform expected bound for each fixed point. The single time \(t=0\) has no effect on the integral. This proves (17) and the almost-everywhere finiteness assertion. ◻

Proof. We first justify the gradient estimate for a bounded solution of \(\Delta h=-4m|\nabla h|\) in a bounded domain. The following elementary interior argument keeps its constants uniform as the domain is slit. On a disc \(B(x,R)\) compactly contained in the domain, let \[N=\sup_{y\in B(x,R)}(R-|y-x|)|\nabla h(y)|.\] One can first replace the disc by a slightly smaller disc, so that \(N<\infty\). Choose \(y\) attaining at least \(N/2\), and put \(d=R-|y-x|\). On \(B(y,d/2)\) the gradient is at most \(2N/d\). Decompose \(h\) there into its harmonic extension of the boundary data and its Green potential. The derivative of the Poisson kernel at the centre, and the integrability of the derivative of the disc Green kernel, give \[|\nabla h(y)|\leq \frac{K}{d}\|h\|_\infty+ Kd\,\|\Delta h\|_{L^\infty(B(y,d/2))} \leq \frac{K}{d}\|h\|_\infty+KmN.\] Multiplication by \(d\) absorbs the last term when \(mR\) is below a fixed constant. Taking \(R\) comparable to \(\min\{\operatorname{dist}(x,\partial O_t),m^{-1}\}\), and using \(0\leq h_t\leq1\), proves \[|\nabla h_t(x)|\leq K\bigl(\operatorname{dist}(x,\partial O_t)^{-1}+m\bigr).\] The distance is bounded above by the physical diameter. Koebe’s one-quarter estimate compares distance with conformal radius and gives (49). Interior regularity needed in this argument also follows directly from the constant-coefficient equation for \(M\); indeed \(\Delta M=4m^2M\).

For the representation, fix \(t\) and write \(k=h_t-u_t\). Exhaust \(O_t\) by images of concentric discs of radii \(r\uparrow1\) under an interior normalized Riemann map. Green’s formula in each exhaustion domain is \[k(z)=H_r(z)+4m\int_{O_t^{(r)}} G_{O_t^{(r)}}(z,w)|M_t(w)|^2\,\mathrm dA(w),\] where \(H_r\) is the harmonic extension of \(k\) from the exhaustion boundary. In disc coordinates the functions \(k\) are uniformly bounded and tend to zero radially at every boundary point except the two marks. The Poisson formula and dominated convergence imply \(H_r(z)\to0\). The Green kernels increase to \(G_t\); monotone convergence therefore proves (48), as well as nonnegativity of \(k\). Since \(0\leq h_t,u_t\leq1\), this also proves (50). This reasoning uses prime-end boundary values and requires no physical boundary regularity. ◻

Proof. Choose a smooth nondecreasing function \(\chi:[0,\infty)\to[0,1]\) which is zero on \([0,1]\) and one on \([2,\infty)\). Write \(Y_t(w)=\operatorname{Im}\Phi_t(w)\), and insert the factor \(\chi(Y_t(w)/\lambda)\) in \(Q_t\). First also insert time-independent compact physical cutoffs \(0\leq\rho_N\leq1\) increasing to one. Denote the resulting potential by \(Q_t^{N,\lambda}(z)\).

For fixed \(N,\lambda\) the product formula is legitimate entirely before lifetimes. Here is one explicit way to implement this assertion. For each \(w\) in the compact support, stop its martingale when \(Y_t(w)\leq\lambda\), and freeze it thereafter. On the set where the cutoff is nonzero this stopped martingale equals \(M_t(w)\). Moreover, on a bounded time horizon, \[|\Phi_t'(w)|\leq |\phi'(w)|\exp(2T/\lambda^2) \quad\text{as long as }Y_t(w)\geq\lambda.\] Together with (49), this bounds these stopped martingales uniformly over the compact support. Their pointwise expected brackets are therefore uniformly bounded. The kernel is bounded above by \(G_\Omega(z,w)\); its only interior singularity is logarithmic and is integrable in \(w\). These bounds justify stochastic Fubini for the compact integral. Equivalently, approximate its spatial integrals by simple functions and use the covariance Cauchy–Schwarz inequality. The singularity can also be tested first against a smooth compact function of \(z\).

Both time-dependent factors multiplying \(|M|^2\) decrease. Formula (54) gives one nonnegative loss. The other is \[-\mathrm d\chi(Y_t/\lambda) =\frac{2Y_t}{\lambda|Z_t|^2}\chi'(Y_t/\lambda)\,\mathrm dt.\] Take \(\lambda\) smaller than a fixed fraction of the protected height of \(z\). On \(\lambda\leq Y_t(w)\leq2\lambda\) the explicit half-plane kernel gives \(G_t(z,w)\leq K_zY_t(w)\). Including the area Jacobian from (51), the absolute cutoff loss is at most \[\begin{align*} &K_z\int_{\lambda\leq\operatorname{Im}v\leq2\lambda} \frac{(\operatorname{Im}v)^2}{\lambda|v|^2} \frac{\mu_t(v)}m|F_t(v)|^2\,\mathrm dA(v)\,\mathrm dt \\ &\hspace{25mm}\leq K_z\int_{\operatorname{Im}v\leq2\lambda} \frac{\mu_t(v)}{|v|^2}\,\mathrm dA(v)\,\mathrm dt. \tag{57}\end{align*}\] Constants may depend on the fixed mass and the protection of \(z\), but not on \(N\). By (44), \(\mathbb E\int_0^T J_t\,\mathrm dt<\infty\). Dominated convergence shows that the expected total loss in (57) tends to zero as \(\lambda\downarrow0\). The ordinary kernel loss is bounded by \(K_zJ_t\,\mathrm dt\), by (52).

Apply the product formula to \(Q^{N,\lambda}\) and take expectations. Its bracket contribution is nonnegative, and its initial and terminal potentials lie between zero and \(1/(4m)\). Consequently \[\mathbb E\int_0^T\int \rho_N(w)\chi(Y_t(w)/\lambda)G_t(z,w) \,\mathrm d[M(w),\overline{M(w)}]_t\,\mathrm dA(w) \leq K_z\left(1+\mathbb E\int_0^T J_t\,\mathrm dt\right).\] If an additional martingale localization is used in deriving this inequality, positivity allows its removal by Fatou’s lemma. Letting \(N\uparrow\infty\) and \(\lambda\downarrow0\) monotonically proves (55).

For completeness, the covariance inequality controlling the martingale integrals is \[ \begin{aligned} &\left|\int G_t(z,w)\,2\operatorname{Re} (\overline{M_t(w)}A_t(w))\,\mathrm dA(w)\right|^2\\ &\qquad\leq 4\left(\int G_t(z,w)|M_t(w)|^2\,\mathrm dA(w)\right) \left(\int G_t(z,w)|A_t(w)|^2\,\mathrm dA(w)\right). \end{aligned} \tag{58}\] Here \(A\) can be any Hilbert realization of a finite family of covariance densities; approximation gives the same inequality for spatial integrals. For the difference of two cutoffs one inserts the absolute cutoff difference in both factors. The first factor is uniformly bounded by (50), and the expected integral of the second tends to zero by (55). Thus the martingale terms converge in \(L^2\), uniformly in probability on the stopped time interval. The positive bracket terms converge in expected total variation. The kernel-loss terms converge in \(L^1\), and the cutoff loss disappears by (57). At each fixed time the potentials converge by their nonnegative integral representation. Passing to the limit in the product identity proves (56) and its continuous version. In particular the cutoff does not leave a measure concentrated on swallowing times. The positive bracket limits are continuous as well: along a subsequence their total variation tails converge uniformly to zero, and the approximating bracket integrals are continuous.

The construction applies to every protected point in a countable dense set. Spatially testing before applying the same argument gives the distributional statement. Local elliptic estimates, made explicit in Lemma 28, identify these versions with point evaluations wherever those are used below. ◻

Proof. Take \(z,z'\) joined by a fixed path inside a protected physical disc. The identity \[h_t(z)-h_t(z')=\operatorname{Im}\int_{z'}^z M_t(\zeta)^2\,\mathrm d\zeta\] shows that this difference is a continuous semimartingale. Indeed the field is uniformly bounded after protection, the expected brackets are uniformly bounded on the compact path, and covariance Cauchy–Schwarz justifies the one-dimensional spatial stochastic integral in its product formula. Subtracting (56) at the two points proves that \(u_t(z)-u_t(z')\) is a semimartingale.

For complex \(a,a'\in\mathbb H\) and real \(w\), define \[H(a,a',w)=\frac1\pi\bigl(\arg(a-w)-\arg(a'-w)\bigr).\] Its derivative in the real variable is \[\partial_wH(a,a',w)=\frac1\pi\bigl(P(a-w)-P(a'-w)\bigr).\] Whenever \(P_t(z)\ne P_t(z')\), the implicit-function theorem therefore recovers \(W_t\) smoothly from \(u_t(z)-u_t(z')\), \(\Phi_t(z)\), and \(\Phi_t(z')\) in a neighbourhood of the current state. The last two processes have finite variation by the Loewner equation. This proves semimartingality on that neighbourhood without applying stochastic calculus to \(W\) in advance.

We give a stopping localization that covers every finite time, so that repeated swallowings cannot leave an exceptional accumulation time. Enumerate rational physical discs with closure in \(\Omega\), rational positive chart-protection levels, and finite collections of rational pairs in each disc. Stop before chart height on the closed disc falls below its specified level, before the coordinates and \(W\) exceed specified bounds, or at a specified finite horizon. All these quantities are continuous before loss of protection. Also stop when the maximum of the absolute Poisson-kernel differences over the finite collection falls below a specified positive threshold. If the initial inequalities fail, set the stopping time equal to zero.

For each such choice, the allowed coordinate states form a compact set on which at least one of the finitely many derivatives \(\partial_wH\) is bounded away from zero. Cover this compact set by a finite collection of implicit-function neighbourhoods, each with a smaller neighbourhood whose closure is contained in the first. Select a valid smaller neighbourhood measurably, and change charts only on leaving the corresponding larger neighbourhood. The fixed positive separation and continuity of the finitely many coordinate processes prevent an accumulation of chart changes on a protected finite interval. For clarity, let \(S_k\) be the successive chart-change times, capped at this protection stop \(\sigma\) and kept equal to \(\sigma\) once it is reached. Each finite concatenation gives the semimartingale \(W^{S_k}\); the chart selected at \(S_k\) is \(\mathcal F_{S_k}\)-measurable, so its stopped increment is a finite sum of predictable stopped integrals of the corresponding smooth implicit function. Define \[R_k=\begin{cases}S_k,&S_k<\sigma,\\ \infty,&S_k=\sigma. \end{cases}\] These are increasing stopping times, and the nonaccumulation of chart changes gives \(R_k\uparrow\infty\). Since \((W^\sigma)^{R_k}=W^{S_k}\), they are ordinary semimartingale localizers for \(W^\sigma\). Thus \(W\) stopped at every enumerated protection time is a semimartingale.

To see that these times cover all horizons, fix a sample path and \(T<\infty\). The active domain \(O_T\) is nonempty; choose a rational closed disc contained in it. This disc survives on \([0,T]\), and compactness provides positive chart-height and finite coordinate bounds there. The function \(P_t\) is not constant on this disc: \[|\nabla P_t(z)|=\frac{|\Phi_t'(z)|}{|Z_t(z)|^2}>0.\] At each time some rational pair has different Poisson-kernel values. Continuity and compactness of \([0,T]\) select finitely many pairs whose maximum difference stays above a positive threshold. Some localization in the enumeration therefore lasts beyond \(T\).

Finally, finite maxima of semimartingale stopping localizations are again such localizations, using \(W^{\sigma\vee\tau}=W^\sigma+W^\tau-W^{\sigma\wedge\tau}\). The successive finite maxima tend to infinity almost surely, by the previous paragraph. This is a global semimartingale localization sequence. The normalized continuous local-martingale and continuous finite-variation decompositions agree on overlaps, because a continuous finite-variation local martingale is constant. They therefore glue to the canonical decomposition \[ W=W^{\mathrm M}+V, \tag{59}\] where \(W^{\mathrm M}\) is a continuous local martingale and \(V\) has locally finite variation. At this stage \(V\) and \([W^{\mathrm M}]\) may have singular parts. ◻

Proof. We describe the construction on a countable exhaustion; this also specifies the simultaneous exceptional set. Let \(B\) range over rational physical discs with closure in \(\Omega\). Stop the entire field on \(B\) at one common stopping time, before the minimum chart height on its closure falls below a positive rational level and before a fixed time or coordinate bound is exceeded. Include slightly larger discs whenever interior evaluations on \(B\) will be used. The resulting stopped field is uniformly bounded, by (49) and the Loewner derivative estimate in the proof of Lemma 26. Freezing the whole field at the same time preserves its equation in the spatial variable.

Let \(\mathcal H_B\) be the real Hilbert space of square-integrable complex functions on \(B\) which solve \(\overline\partial f+\mathrm im\overline f=0\) distributionally, with inner product \(\operatorname{Re}\int_B f\overline g\,\mathrm dA\). It is a closed subspace of real \(L^2(B;\mathbb C)\) and hence is separable. Interior estimates for \(\Delta f=4m^2f\) show that evaluation and every spatial derivative on a smaller disc are bounded linear functionals on \(\mathcal H_B\). Their Riesz representatives depend smoothly on the evaluation point: the Taylor remainder estimates follow from the same interior bounds one derivative further.

The stopped field is a continuous square-integrable \(\mathcal H_B\)-valued martingale. To verify this assertion, integrate its bounded scalar martingale identities against a countable dense set of \(L^2\) tests; approximation gives all Hilbert-space conditional expectation identities. Joint local continuity gives Hilbert continuity. If \((e_j)\) is a real orthonormal basis, its trace bracket is \[\sum_j[\langle M,e_j\rangle]_t, \qquad \mathbb E\sum_j[\langle M,e_j\rangle]_T =\mathbb E\|M_T-M_0\|_{\mathcal H_B}^2<\infty.\] Enumerate all discs and protections. Multiplying their centred stopped martingales by sufficiently small positive deterministic weights makes their direct sum a continuous square-integrable martingale in a separable real Hilbert space. This is possible because each stopped field has a deterministic norm bound. For the driver use the normalized martingale \[N_t^0=\int_0^t\frac{\mathrm dW_s^{\mathrm M}}{1+[W^{\mathrm M}]_s}, \qquad [N^0]_t=\frac{[W^{\mathrm M}]_t}{1+[W^{\mathrm M}]_t}\leq1.\] It is square-integrable and can be included as one real coordinate. The joint covariance of this direct sum retains all cross-brackets, including those between different discs and this normalized driver. The covariance vector of the original driver is recovered by multiplying that coordinate’s vector by \(1+[W^{\mathrm M}]_t\).

A positive trace-class covariance measure has an operator density with respect to its trace measure. This can be constructed directly: take Radon–Nikodym derivatives of its matrix entries in a countable orthonormal basis, impose positivity on the countable collection of rational finite linear combinations, and complete the resulting positive form. Its finite trace makes it a bounded positive trace-class operator at almost every clock time. Its positive square root applied to the Riesz representatives of evaluation gives the vectors in (60). Real and imaginary evaluations supply the complexified vector \(A_{\mathrm{phys}}\). Smoothness of the Riesz representatives proves smoothness in \(w\). On overlapping protected discs the evaluation martingales coincide before either stop; the bracket of their difference is zero there. Impose this equality first at rational spatial points and then use smoothness. The local realizations therefore agree on their overlaps on one common full set.

Add \(\mathrm dt\), \(|\mathrm dV|\), and \(\mathrm d[W^{\mathrm M}]\) to the trace clock. Also add the countably many positive Green-integrated bracket measures from (55), using rational protected points or nonnegative compact tests in \(z\). Their cumulative processes are finite on their stopped intervals. There is no issue in making this sum locally finite: for any continuous increasing process \(H\), the measure \((1+H_t)^{-2}\mathrm dH_t\) has total mass at most one and has the same null sets as \(\mathrm dH\); use these measures with summable positive weights. Every required measure is therefore absolutely continuous with respect to \(\mathrm dL\). Matrix Radon–Nikodym derivatives with respect to this clock give the same construction, now including singular parts of \(\mathrm dV\) or \(\mathrm d[W^{\mathrm M}]\).

Choose the countable smooth test families dense in the \(C^\infty\) topology for each fixed compact support in a countable protected exhaustion, and include a nonzero nonnegative bump in each protected disc. For each such nonnegative test, nonnegative Fubini identifies the density of its Green-integrated bracket measure with the corresponding integral of \(G_t|A_{\mathrm{phys}}|^2\). Fix a time in the resulting common full set and a surviving bump \(\eta\), and put \[K_\eta(w)=\int\eta(z)G_t(z,w)\,\mathrm dA(z).\] Then \(\int K_\eta|A_{\mathrm{phys}}|^2\,\mathrm dA<\infty\). For \(z\) in any compact subset of the live domain, the explicit half-plane kernel bounds \(G_t(z,w)\) and its first \(z\)-derivatives by a constant times \(K_\eta(w)\) outside a slightly larger compact subset. Locally, their logarithmic and inverse-distance singularities are integrable against the smooth spatial fields. This proves (61) at every live interior point on the same full set. Weighted Cauchy–Schwarz gives the corresponding tail control for the mixed Green integrals; (58) and (50) identify their compact-truncation limits with the covariance vectors of the noncompact martingale integrals. The potential expressions define continuous spatial distributions, so density of the chosen tests extends the identities and their distributional derivatives to all compact tests on this same full set. An arbitrary current conformal chart, and the boundary-approaching balls chosen later, are deterministic spatial choices after a time in this full set has been fixed. ◻

Proof. Differentiating (45) gives \[ DG(z,v)=\frac{\operatorname{Im}v} {\pi(z-\overline v)(z-v)}. \tag{72}\] For \(z\in B_s\) and \(v\notin N_s\) this implies \[ |DG(z,v)|\leq K\frac{\operatorname{Im}v}{(s+|v|)^2}. \tag{73}\] For \(v\in N_s\) the additional singular bound is \[ |DG(z,v)|\leq \frac K{|z-v|}. \tag{74}\] The first bound follows by comparing both denominator factors to \(s+|v|\); when \(|v|\) is comparable to \(s\), the exclusion of \(N_s\) keeps \(|z-v|\) at least \(s/4\). The second follows because \(|z-\overline v|\geq\operatorname{Im}z+\operatorname{Im}v\). For each fixed \(v\in\mathbb H\), uniformly as \(z\in B_s\) tends to zero, \[ DG(z,v)\longrightarrow \frac{P(v)}\pi. \tag{75}\]

We record two consequences of analyticity of \(\psi'\). Its subharmonic mean inequalities on discs of radius comparable to \(s\), contained in \(B(\mathrm is,3s/4)\), give \[\begin{align*} \sup_{N_s}\mu &\leq\frac K s \left(\int_{B(\mathrm is,3s/4)}\mu^2\,\mathrm dA\right)^{1/2} =o(s^{-1}), \tag{76}\\ \sup_{N_s}\mu &\leq\frac K{s^2}\int_{B(\mathrm is,3s/4)}\mu\,\mathrm dA \leq K\int_{B(\mathrm is,3s/4)}\frac\mu{|v|^2}\,\mathrm dA=o(1) \quad\text{if }J<\infty. \tag{77}\end{align*}\] The vanishing follows from absolute continuity of the corresponding area integral. Define the finite measure \[ \mathrm d\rho(v)=\frac{\operatorname{Im}v}{1+|v|^2} \mu(v)|A(v)|^2\,\mathrm dA(v). \tag{78}\] On \(N_s\), for small \(s\), its density is comparable to \(s\mu|A|^2\). In particular \[ \int_{N_s}s\mu|A|^2\,\mathrm dA=o(1). \tag{79}\]

We now estimate each potential separately.

The potential in (63). Off \(N_s\), (73) and \(|F|^2\leq K/y\) bound it by \[K\int_{\mathbb H}\frac\mu{(s+|v|)^2}\,\mathrm dA=o(s^{-1}).\] To see the little-oh rather than merely a big-oh, split at a fixed small radius \(\varepsilon\). On its inner side the \(L^2\) norm of \((s+|v|)^{-2}\) is \(O(s^{-1})\), multiplied by an arbitrarily small \(L^2\) norm of \(\mu\). On the outer side the integral has a bound independent of \(s\), by \(\mu\in L^2\). First let \(s\) tend to zero and then let \(\varepsilon\) tend to zero. This also controls infinity. On \(N_s\), (74), \(|F|^2\leq K/s\), and \(\int_{N_s}|z-v|^{-1}\mathrm dA(v)\leq Ks\) give the uniform bound \(K\sup_{N_s}\mu=o(s^{-1})\). This proves (67).

If \(J<\infty\), the far integrand is dominated by \(K\mu/|v|^2\). Equations (75) and dominated convergence give its limit \[\frac4\pi\int_{\mathbb H}P(v)\mu(v)|F(v)|^2\,\mathrm dA(v)=C.\] The near part tends uniformly to zero by (77). The portion of the limiting integral removed with \(N_s\) tends to zero by the same \(J\) majorant. This proves (70).

The mixed potential in (64). Covariance Cauchy–Schwarz against the finite measure (78) bounds the far part uniformly in \(z\in B_s\) by \[ K\rho(\mathbb H)^{1/2} \left(\int_{\mathbb H}\frac{(1+|v|^2)\mu(v)}{(s+|v|)^4} \,\mathrm dA(v)\right)^{1/2}. \tag{80}\] Here the square of the kernel contributes \(y^2\), and the weights \(|F|^2\leq K/y\) and \(\mathrm d\rho\) remove both factors of \(y\). The integral inside parentheses is \(o(s^{-3})\): near zero this follows from the \(O(s^{-3})\) \(L^2\) norm of \((s+|v|)^{-4}\) and the preceding small-radius truncation argument; away from zero its kernel is square-integrable, including its \(|v|^{-2}\) tail. Thus (80) is \(o(s^{-3/2})\). If \(J<\infty\), multiply that integral by \(s^2\) and write it against \(\mu(v)|v|^{-2}\mathrm dA(v)\). The remaining factor \[\frac{s^2|v|^2(1+|v|^2)}{(s+|v|)^4}\] is bounded for \(0<s\leq1\) and tends to zero for each fixed \(v\ne0\). Dominated convergence improves (80) to \(o(s^{-1})\).

For the near part put \(q(v)=\mu(v)\operatorname{Re}(\overline{F(v)}A(v))1_{N_s}(v)\). Hilbert norms give \[ \|q\|_{L^2(\mathbb H)}^2 \leq \frac{K\sup_{N_s}\mu}{s^2} \int_{N_s}s\mu|A|^2\,\mathrm dA. \tag{81}\] The truncated kernel \(1_{\{|v|<2s\}}/|v|\) has \(L^1\) norm \(O(s)\). Young’s inequality, followed by the factor \(O(s^{-1})\) converting an \(L^2\) norm on \(B_s\) to its normalized-area norm, therefore gives \[ \begin{aligned} \|\text{mixed near part}\|_{L^2_{\rm av}(B_s)} &\leq \frac{K(\sup_{N_s}\mu)^{1/2}}s \left(\int_{N_s}s\mu|A|^2\,\mathrm dA\right)^{1/2}\\ &=o\bigl((\sup_{N_s}\mu)^{1/2}/s\bigr). \end{aligned} \tag{82}\] Hilbert-valued Young’s inequality follows from the scalar convolution inequality for the norm, so no finite-dimensional restriction is involved. Equations (76) and (77) yield precisely (68) and (71). This step uses weighted integral control of \(A\), not a pointwise boundary bound.

The bracket potential in (65). Its far part is bounded by \[K\int_{\mathbb H}\frac{1+|v|^2}{(s+|v|)^2}\,\mathrm d\rho(v)=o(s^{-2}).\] Indeed multiplication of the integrand by \(s^2\) makes it bounded uniformly for \(s\leq1\) and tending to zero pointwise. Finite \(\rho\) then gives dominated convergence. For the near part, first integrate the singular kernel in \(z\): \[\begin{align*} &\frac1{|B_s|}\int_{B_s}\int_{N_s} \frac{K\mu(v)|A(v)|^2}{|z-v|}\,\mathrm dA(v)\,\mathrm dA(z)\\ &\hspace{20mm}\leq\frac K s\int_{N_s}\mu|A|^2\,\mathrm dA =\frac K{s^2}\int_{N_s}s\mu|A|^2\,\mathrm dA=o(s^{-2}). \end{align*}\] This proves (69) using only (62), and completes the proof. ◻

Proof of Proposition 24. Work at a time in the simultaneous full set of Lemma 28. The uniform leading expansion (67) implies \(|F(z)^2|\asymp s^{-1}\) on \(B_s\) for all sufficiently small \(s\). Consequently division by \(F^2\) is legitimate there. Bilinearly squaring (68) and dividing by \(4F^2\) gives \[ A(z)\cdot A(z)=\frac{|U|^2}{4\pi z^3}+o(s^{-3}) \quad\text{in }L^1_{\rm av}(B_s). \tag{83}\] For example, if \(R=2FA-U/(\pi z^2)\), the cross-error is controlled by \(\|U/(\pi z^2)\|_{L^2_{\rm av}}\|R\|_{L^2_{\rm av}}\) and the quadratic error by \(\|R\|_{L^2_{\rm av}}^2\); multiplication by \(1/F^2=O(s)\) gives the displayed order. This justifies the use of averaged errors in a bilinear, rather than Hermitian, square.

Compare the cubic coefficients in (65), using (69). This yields \[ \frac{|U|^2}{4}=|U|^2-4\tau, \qquad |U|^2=\frac{16}{3}\tau. \tag{84}\] Coefficient comparison in these averaged statements can be read without pointwise subsequences: set \(z=s\zeta\), multiply by \(s^3\), and test on the fixed disc \(B(\mathrm i,1/4)\) against a smooth test with nonzero pairing with \(\zeta^{-3}\). The lower-order and little-oh terms vanish.

Consider first the singular part of the clock, where \(\tau=0\). Equation (84) gives \(U=0\). Now the same weak mixed estimate gives the stronger conclusion \[A\cdot A=\frac{R\cdot R}{4F^2}=o(s^{-2}) \quad\text{in }L^1_{\rm av}(B_s).\] In (65) only \(\nu/(\pi z^2)\) can remain at order \(s^{-2}\), by (69). Rescaling and testing at that order proves \[ \nu=0\quad\text{on }\{\tau=0\}. \tag{85}\] In particular this conclusion has not used \(J<\infty\) on a set singular to capacity time.

On \(\{\tau>0\}\), (44) ensures \(J<\infty\) at \(\mathrm dL\)-almost every time: a capacity-null set has zero \(\tau\mathrm dL\) measure, hence zero \(\mathrm dL\) measure on \(\{\tau>0\}\). We can therefore use (70) and (71). Uniformly on \(B_s\), \[\frac1{F^2}=\pi z-\pi^2Cz^2+o(s^2).\] The refined mixed remainder is \(o(s^{-1})\) in normalized \(L^2\); its cross term, after division by \(F^2\), is \(o(s^{-2})\) in normalized \(L^1\). We obtain \[ A\cdot A=\frac{|U|^2}{4\pi z^3} -\frac{|U|^2C}{4z^2}+o(s^{-2}). \tag{86}\] Comparing the quadratic coefficients in (65), after cancelling the cubic ones, gives \[\frac\nu\pi-2C\tau=-\frac{|U|^2C}{4}, \qquad \nu=\frac{2\pi}{3}C\tau.\] Together with (85) this identity holds on the whole clock. Hence, in physical capacity time, \[\mathrm d[W^{\mathrm M}]_t=\frac{16}{3}\mathrm dt, \qquad \mathrm dV_t=\frac{2\pi}{3}C_t\,\mathrm dt.\] The continuous martingale characterization of Brownian motion shows that \(B=\sqrt{3/16}\,W^{\mathrm M}\) is Brownian, with the harmless initial constant chosen to make \(W_0=0\). Finally (52) and (44) give \[\mathbb E\int_0^T C_t^2\,\mathrm dt\leq K\mathbb E\int_0^T J_t^2\,\mathrm dt<\infty.\] This proves (46). The stochastic proof used the joint filtration only for the fermion martingales and their calculus. The energy input itself concerns the marginal driver law, so its use here does not require conditional crossing estimates in a larger, possibly anticipative filtration. ◻

Proof. For a continuous driver and a fixed time, the Loewner ordinary differential equation determines the maps on all unswallowed compact sets. The inverse maps depend continuously on the driver on such protected sets. This also gives the marked kernel topology of \(O_t\): in the centered chart the two marks are always \(0,\infty\).

Proposition 2 supplies a unique \(h_t\) and kernel-continuous interior values and gradients. To spell out the measurability consequence, exhaust \(\mathbb H\) by deterministic compact sets \(K_j\) avoiding \(0\) and the real line. The integrals \[C_t^{(j)}=\frac4\pi\int_{K_j}P(v)m|\psi_t'(v)| |\nabla(h_t\circ\psi_t)(v)|\,\mathrm dA(v)\] are Borel functions of the stopped driver. One can first use compact sets on which the maps and their derivatives satisfy fixed protection bounds and then exhaust these bounds. The continuity assertion in Proposition 2 applies because all physical active domains lie in the fixed bounded domain \(\Omega\). Taking an increasing exhaustion \(K_j\) makes \(C_t=\lim_j C_t^{(j)}\) an extended nonnegative Borel function.

All these constructions at time \(t\) use only the path up to \(t\). More explicitly they are Borel functions of \((t,x^t)\), where \(x^t(s)=x(s\wedge t)\) and the stopped-path map is continuous on each finite time interval. They are therefore progressively measurable. Set the extension to zero on the measurable set where the extended integral is infinite or the prescription is undefined. The admissibility conditions retain the requirement that the actual integral be finite almost everywhere, so this convention adds no solutions by substituting a value on a positive-measure set of undefined coefficients. ◻

Proof. Fix a finite horizon \(T\). On canonical path space define extended energy and its first level by \[A_t(x)=\int_0^t b(s,x)^2\mathrm ds\in[0,\infty],\qquad \sigma_n(x)=\inf\{t\ge0:A_t(x)\ge n\}\wedge T.\] Use the usual right-continuous augmentation for stopping times. Even for extended energy, \(\sigma_n\) is a stopping time: the strict sublevel events \(\{\sigma_n<t\}\) can be expressed using rational times less than \(t\), and right continuity gives the weak sublevel events.

Let \(X\) be any candidate solution and put \(\tau_n=\sigma_n(X)\). By Equation (88), \(A_t(X)\) is finite and continuous on \([0,T]\), and \(A_{\tau_n}(X)\le n\). The stochastic exponential \[ Z_n=\exp\left\{-\frac1{\sqrt\kappa} \int_0^{\tau_n} b(s,X)\mathrm dB_s -\frac1{2\kappa}A_{\tau_n}(X)\right\} \tag{89}\] is a true martingale density by the bounded-energy Novikov criterion; for this criterion and the following Girsanov change of measure, see [4]. It is strictly positive. Under the equivalent measure \(Q_n\) with density \(Z_n\), the process \[\widetilde B_t=B_t+\frac1{\sqrt\kappa} \int_0^{t\wedge\tau_n}b(s,X)\mathrm ds, \qquad 0\le t\le T,\] is Brownian motion. Set \(Y_t=\sqrt\kappa\widetilde B_t\). It has the full Wiener law \(\mathsf W_\kappa\) on \([0,T]\) and agrees with \(X\) through \(\tau_n\).

Nonanticipation implies \(A_t(Y)=A_t(X)\) for \(t\le\tau_n\). If \(\tau_n<T\), that energy is strictly below \(n\) earlier and equals \(n\) at \(\tau_n\). Consequently \[ \sigma_n(Y)=\tau_n. \tag{90}\] If \(\tau_n=T\), the paths agree on the whole horizon and the same identity holds. Possible nonintegrability of \(b(s,Y)^2\) immediately after \(\tau_n\) cannot change this first level: it has already been attained. This observation also proves, under the fixed full Wiener law, that \(A_{\sigma_n}(Y)\le n\) almost surely. A hypothetical Wiener path whose energy jumps from below \(n\) to infinity before this first attained prefix has Wiener measure zero whenever a finite-energy solution exists.

The inverse of Equation (89), written under \(Q_n\), is the canonical Wiener functional \[ R_n(y)=\exp\left\{\frac1\kappa \int_0^{\sigma_n(y)} b(s,y)\mathrm dy_s -\frac1{2\kappa}A_{\sigma_n(y)}(y)\right\}. \tag{91}\] The stopped energy bound just established defines this stochastic integral in \(L^2(\mathsf W_\kappa)\), and hence as a measurable functional of the canonical path, up to a single Wiener-null set. One may use predictable simple-process approximations to choose that version once for all candidate solutions. In particular, both the stopped path and its inverse density are fixed by the same Wiener law. For every bounded Borel functional \(F\), \[ \mathbb EF(X^{\tau_n}) =\int F(y^{\sigma_n(y)})R_n(y)\, \mathsf W_\kappa(\mathrm dy). \tag{92}\] The right side does not depend on the candidate solution. Thus all candidate stopped laws coincide.

Under each candidate, \(A_T(X)<\infty\) almost surely; hence \(\tau_n=T\) for all sufficiently large \(n\) on each sample path. Bounded convergence in Equation (92) proves equality of the full laws on \([0,T]\). No assertion that \(\sigma_n\to T\) under unweighted Wiener measure is needed. Since \(T\) was arbitrary, the laws on the full continuous path space coincide. ◻

Proof. Consider first any such approximating sequence. The marked basepoint normalization and common diameter bound place all its domains in a fixed physical ball. Proposition 11 gives tightness jointly for curves and drivers; every subsequential limit has the full continuous Loewner trace and terminal completion required in Definition 3. Proposition 24, together with Proposition 16, identifies its driver as a weak solution of Equation (11) with Equation (12).

Apply Lemma 31 to the extension in Lemma 30, with \(\kappa=16/3\) and \(b=(2\pi/3)C\). Every admissible continuum solution has the same driving law. A continuous driving path determines its Loewner hulls. Whenever it has a trace, that trace at each finite time is the boundary limit \[\lim_{y\downarrow0}\phi^{-1} \bigl(g_t^{-1}(W_t+\mathrm iy)\bigr),\] read in disc coordinates if a physical boundary value is ambiguous. Thus the full capacity trace, its orientation and its continuous terminal completion are determined by the driver. Inserting an extra curve in a swallowed component would violate the trace requirement and does not create another admissible curve law.

Existence requires only one bounded approximation. Such approximations exist, for example on square rhombic meshes: exhaust \(\Omega\) by simply connected polygonal interiors, approximate a chosen interior access sequence for each marked prime end, and then refine the mesh diagonally. The polygonal approximation can be kept in a fixed neighborhood of \(\Omega\); compact kernel convergence and convergence of the two prime-end coordinates are preserved by the diagonal choice. With any nomes having the required asymptotic ratio, compactness supplies a subsequential limit and hence an admissible continuum law.

This use of a subsequence is an existence proof for the already specified continuum equation; the definition and uniqueness do not depend on its choice. For every bounded admissible approximation, all tight subsequences have that unique law. The subsequence criterion gives full-sequence convergence. Section [sec:localization] removes the common diameter restriction. ◻

Proof. Choose as many consecutive fixed dyadic rings about \(\xi\) as needed, all with radii below \(a\). For a radius \(s\) in their range, the circular crosscut contains the inward radial point \((1-s)\xi\). All these radial points belong to a fixed compact subset of \(\mathbb D\), because the smallest radius is positive. Compact convergence of \(f_n\) puts their physical images in a fixed neighborhood of the bounded limit domain. In particular they are separated from \(\partial B(0,S-2)\) by a positive gap \(g\), uniformly for all large \(n\).

Let \(L_n(s)\) be the length of only that part of \(f_n(\partial B(\xi,s)\cap\mathbb D)\) lying in \(B(0,S-2)\). Polar Cauchy–Schwarz and univalence imply \[ \int L_n(s)^2\,\frac{\mathrm ds}{s} \le 2\pi \int_{f_n^{-1}(B(0,S-2))}|f_n'(z)|^2\,\mathrm dA(z) \le 2\pi^2(S-2)^2 . \tag{93}\] The right side bounds only the area of a fixed physical ball. It makes no assumption on \(\operatorname{area}(\Omega_n)\). By taking enough fixed dyadic rings, a regular radius can be selected for which \(L_n(s)<\min(\rho,g)\). If its full preimage arc left the physical ball, the segment from the anchored radial point to its first exit would already have length at least \(g\) inside the ball, contradicting this inequality. The whole arc therefore stays in the ball and has length at most \(\rho\). Radii can be chosen away from the finitely many contour degeneracies of the polygon.

For a fixed prescribed parameter accuracy, apply this construction to a finite collection of circle points whose small caps cover \(\partial\mathbb D\). Images of remote arcs lie uniformly close to \(\partial\mathbb D\): every fixed compact subset of \(\mathbb D\) has bounded physical preimage contained in \(B(0,S-1)\) for all large \(n\). They avoid the bounding crosscuts of the chosen caps, since those have physical preimages in \(B(0,S-2)\). Connectedness therefore traps each remote image in one of the small caps. This proves the uniform diameter assertion. If the remote arc is a crosscut, join its endpoints by the short boundary-circle interval. Its noncentral shadow is contained in the same cap.

For finitely many distinct prescribed points, first choose disjoint outer parameter neighborhoods and then perform the selection within each. Every radius range remains fixed at this stage, although its lower endpoint can be very small. All mesh limits are taken after these ranges have been fixed. ◻

Proof. First note that filling preserves a small attachment property. The central component contains \((1-\varepsilon_n)\mathbb D\). A point in a discarded component lies in the remaining boundary collar. Follow its outward radial segment to its first encounter with an obstruction or the circle. Before that encounter the segment stays in the same discarded component and has length at most \(\varepsilon_n\). Append the obstruction’s small attachment when necessary. Every point of the filled complement thus has a connected route of diameter at most \(2\varepsilon_n\) to the old outer complement.

We give a quantitative crosscut proof of (94). Suppress \(n\) and suppose \[(1-d)\mathbb D\subset V\subset\mathbb D,\qquad \operatorname{diam}A_x\le e\] for a connected complement attachment \(A_x\) from each complement point \(x\) to the old exterior. The normalized map \(F:\mathbb D\to V\) extends continuously to the physical boundary, since \(V\) has a finite, locally connected piecewise analytic boundary, with occurrences retained at slits. Put \(a=F'(0)\). Schwarz’s lemma, also applied to \(F^{-1}\) on \((1-d)\mathbb D\), gives \(1-d\le a\le1\). Schwarz–Pick applied to \(F(z)/z\) yields \[ |F(z)-z|\le\frac{2d}{1-|z|}. \tag{95}\]

Fix \(\zeta\in\partial\mathbb D\) and \(0<h<1\). Length–area on the circular crosscuts \(\partial B(\zeta,t)\cap\mathbb D\), \(h<t<\sqrt h\), gives one whose image \(\sigma\) has length at most \[L_h=\frac{2\pi}{\sqrt{\log(1/h)}}.\] It contains \(F((1-t)\zeta)\), whose distance from \(\zeta\) is at most \(\sqrt h+2d/h\) by (95). Let \(p,q\) be the endpoint occurrences of \(\sigma\). Their physical locations have distance at most \(L_h\). Join them in the complement by their \(e\)-attachments and the shorter arc of \(\partial\mathbb D\) between the attachment endpoints. This gives a connected complement set within distance \(L_h+3e\) of \(p\).

The complement is locally path connected in the present finite-arc setting. For any \(\epsilon>0\), the joining continuum can therefore be replaced by a simple complement arc within its \(\epsilon\)-neighborhood. Together with \(\sigma\) it forms a Jordan curve. If the physical endpoints coincide, \(\sigma\) itself is the corresponding closed curve; the same separation argument applies to its two boundary occurrences. When \[2L_h+\sqrt h+2d/h+3e+\epsilon<1,\] this curve is in a ball missing \(0\). Its bounded side contains the noncentral image cap. It follows that \[|F(\zeta)-\zeta| \le2L_h+\sqrt h+2d/h+3e+\epsilon .\] Let \(\epsilon\downarrow0\). The bound is uniform in \(\zeta\), and the maximum principle gives the same bound on \(\overline\mathbb D\). Choose \(h\) small first and then \(d,e\) small. This proves (94). The argument uses the number of obstructions nowhere.

Finally, if \(w=F_n(\zeta)\), then \(|F_n^{-1}(w)-w|=|\zeta-F_n(\zeta)|\). This proves the inverse statement with prime-end occurrences, without making an inverse single-valued at an unlabelled physical slit point. ◻

Proof. For all large \(n\), none of the indicated circles is wholly contained in \(\Omega_n\). Otherwise simple connectivity would put its full interior in \(\Omega_n\), contradicting kernel convergence to the much smaller bounded domain. At a generic circle, its components inside a finite polygon are open arcs with endpoints on boundary occurrences. Those incident to \(U_{n,j}\) are its visible gates.

Cut the domain at all these circular arcs. The adjacency graph of the resulting components, with one edge for each crosscut, is a finite tree: successively adding disjoint proper crosscuts splits one simply connected component into two and preserves connectedness and absence of cycles. Root this tree at \(U_{n,j}\). The visible gates are exactly its root-adjacent edges, and their noncentral shadows are the corresponding disjoint branches. Every point outside the circle belongs to a nonroot component, so the unique first edge of its root path is its mandatory gate. Crosscut separation makes every domain path to that point cross this gate. The tree uses boundary occurrences, so the argument also applies at repeated physical vertices.

A connected later gate lies outside every earlier circle and hence in one unique earlier shadow. For two disjoint crosscuts the root-oriented shadows are either disjoint or nested, by the same tree argument. These ancestors are therefore consistent across levels. A shadow may contain further components inside the earlier circle; this does not affect the root path or the mandatory-gate property.

The total length of gates on a circle is at most \(2\pi(S+1)\), proving the count for large gates. A small gate fits in a ball of radius \(K2^{-j}\) centered at any one of its points. Successive level gaps are comparable to \((j+2)^{-2}\). Choose \(c_0\) sufficiently small that the outer balls lie in disjoint radial bands, and omit finitely many initial levels so that their inner radii are smaller than the outer radii.

The center-to-far-point route must cross each mandatory gate. Its appended core portion has radius at most \(S\), whereas all gates have larger radius. Thus each crossing is on the original portal-to-far-point path. Its two endpoints lie outside every indicated outer ball, so visiting the small gate requires an annular crossing. At lattice scale the path and gate may be moved within incident cells. Since \(2^{-j}\ge\delta_n\) for \(j\le N_n\), these changes are absorbed by the fixed constant in the inner radius. The smallest radial gap between outer bands is at least \(c(N_n+2)^{-2}\). Since \(\delta_n(N_n+2)^2\to0\), any fixed number of buffer cells preserves their disjointness uniformly through the moving final level \(N_n\). ◻

Proof. The marked parameter endpoints converge to two distinct circle points. Around each choose a small circular cap of angular size at most \(a\). These two caps are disjoint. Choose their bounding circular crosscuts using Lemma 34, so that their physical preimages lie in \(B(0,S-2)\). Call them endpoint fences. Their parameter radii are in fixed intervals and can be chosen to avoid every point of \(H\) on the fences themselves. A point of \(H\) coinciding with an endpoint is inside its fence. Other points of \(H\) can be kept away by reducing the endpoint neighborhood first.

Join a middle point of each fence to \(0\) by fixed compact paths in the parameter disc. Their physical preimages stay in \(B(0,S-2)\) for all large \(n\), by compact convergence. The paths, together with the fences, form two protected access chains. Now choose the still smaller disjoint caps about the points of \(H\), applying Lemma 34 within fixed radius intervals. Their physical cuts have diameter at most \(\rho\). They avoid the protected paths and fences, since those are compact and their boundary endpoints are away from \(H\). When an \(H\)-point is an endpoint limit, its cap is placed entirely behind that endpoint’s fence. For all large \(n\), the finitely many large gates lie inside their designated \(H\)-caps.

Replace each protected fence and access path by the chain of cells it crosses, including terminal boundary half-cells. A generic small perturbation avoids vertices, so the chain is connected through sides. Its disc distance from the original path tends to zero by Lemma 10. Explore from an endpoint until just before taking the turn in the first cell of its protected chain, even if that turn would be forced. No tile of that chain has been clipped.

The stopping event is unavoidable for a path to the opposite endpoint, since that path must cross the fence. Before the stopping event, the explored head stays on the endpoint side of the fence, up to its neighboring cells. Thus its parameter diameter is at most \(Ka+o(1)\). No discarded component can contain the whole protected chain: its connected access has not been clipped, and the continuing path must still reach the fence. In particular the center-to-front-corner access remains on the continuing side. The two explorations are in disjoint endpoint neighborhoods and may be carried out in either order.

Lemma 6 now gives an ordinary Dobrushin problem with the two front corners as its change points. The procedure has stopped on the first local arrival; it has not conditioned the unknown connecting path to hit an interior target. All head changes are within the two small parameter caps and are attached to the old boundary there. The active-side convention fills only components behind these obstructions. Their attachments and fillings satisfy Lemma 35. ◻

Proof. Declare all opposite-color vertices incident to cells of the cap reaching \(|z|\ge S+3\) to be seeds, and reveal the exterior cells. The seed region is topologically attached to the original cap bank. Indeed, from any point of its exterior component, an outward radial route reaches its first exit from the finite polygon while staying outside the cutoff; it cannot encounter the portal, which lies strictly further in. Include the cells of such routes in the exterior region when recording its boundary. This is a topological attachment of an exposed set. The seeds need not belong to one open opposite-color cluster.

Search every opposite-color cluster meeting this region, revealing incident edges until its full inner frontier is known. Stop with failure if the search or its necessary clipping cells reach the portal buffer. On success, every edge at the inner frontier is closed for the searched color, hence is an open diagonal of color \(c\), or is an original \(c\)-bank edge. The union of the seed region and the searched clusters is attached to the bank. Take the component containing the portal on its unsearched side, and fill the holes discarded on the other side. Its new boundary is therefore a collection of \(c\)-paths attached to that same bank, not a collection of interior wired islands.

Every tested edge is either in the removed seed-and-cluster region or is one of the known frontier edges. None is strictly in the retained unqueried component. Conditioning on the full record, the FK spatial Markov property imposes exactly its original Dobrushin wires, extended by the new \(c\)-frontier. The retained interface cannot cross that frontier into the discarded side. The event of success is already determined by the record, so it imposes no extra condition on retained edges.

For a failure witness, choose a reached vertex in the portal buffer and trace its searched-color cluster back to its first exterior seed. That seed is at radius at least \(S+3-K\delta_n\). The path between it and the last buffered cells consists of actual open edges; no artificial connection within the whole seed region is needed. If a last incident edge triggers failure before being queried, the known path ends only \(K\delta_n\) short of the portal, which gives the stated endpoint allowance.

All constructions can be performed on incident sectors before identifying repeated boundary vertices. Frontiers of a multiply armed cluster or of a fragmented portal are treated on their accessible side. Filling holes removes them from the retained domain and does not query inside it. Apply the construction to caps in a deterministic order. On success in an earlier cap only a wired frontier outside the remaining core has been added, so the same argument restarts in every later cap. ◻

Proof. We compare all possible failure arms of one portal type and one bank color together. Inside an annulus separated from the portals and from the two ends of an arm, their accessible side banks are free for the intruding color. Lemma 7 applies to this union, including all portal fragments and boundary-sector occurrences. The comparison wires radial ends only; it counts actual open arms and permits arbitrary off-shell conditioning. It therefore gives product bounds in disjoint buffered annuli without a factor for the number of fragments. Earlier successful searches merely add favorable single-color frontiers and do not change this argument.

If failure ends at a portal on an \(H\)-cut, the actual intrusion arm from Lemma 39 joins a ball containing that entire cut to a fixed outer radius, say \(1/2\). The source is outside \(S+3-O(\delta_n)\), whereas the cut lies inside \(S-2\). Applying Lemma 7 to all fragments of that cut and summing over the finitely many \(H\)-cuts gives the first term in (97).

Now suppose failure ends at a circle-\(S\) portal. Join its endpoint to \(0\) through the central core, which excludes the initial \(H\)-caps and stays inside radius \(S\), up to the fixed cell buffer. Select a far point on the arm at radius \(S+2\). This center-to-far-point route has the mandatory initial gate sequence of Lemma 37.

If any gate at a level \(j\le J\) is large, it lies inside its selected parameter \(H\)-cap. The arm therefore visits that cap, while its core extension stays on the inner side. It crosses the corresponding \(H\)-cut. This gives an arm through an annulus about the physically tiny cut, again up to a bounded number of cells. Circle-\(S\) portals are outside these annuli because the \(H\)-cut lies inside \(S-2\). The union comparison for the circle-portal type therefore gives the same first term.

Otherwise let \(k>J\) be the first level at which the gate is large. There are at most \(K2^k\) choices for it, and each fixes all its ancestors. The earlier ancestors are small. Figure 2 illustrates their surrounding annuli. For \(j_*\le j<k\), the same arm crosses an annulus with inner scale \(K2^{-j}\) and outer scale \(c_0(j+2)^{-2}\). These annuli lie in disjoint radial bands and miss the circle-\(S\) portals. Both endpoints of the arm are outside the bands. Applying the conditional product bound gives \[K2^k\prod_{j_*\le j<k} K\bigl(2^{-j}(j+2)^2\bigr)^\alpha .\] The two possible bank colors cost only another fixed factor.

Finally, if all gates through \(N_n\) are small, specify the connected initial-domain sector of an incident cell at the first crossing of radius \(S+2\). That sector fixes the ancestor chain, since it is connected and lies outside all the tested circles. The angle bound gives cell areas at least \(c_\eta\delta_n^2\) and bounded local degree. A bounded annular band therefore has at most \(K\delta_n^{-2}\) possible terminal cells and sectors. The product through \(N_n\) gives the last term.

For completeness, the logarithm of the product through \(k\) is \[-\frac{\alpha\log2}{2}k^2+O(k\log(k+2)).\] It dominates both the factor \(2^k\) and, when \(k=N_n\), the factor \(\delta_n^{-2}=\exp(O(N_n))\). The last term vanishes, and the tail sum can be made arbitrarily small by increasing \(J\). At every fixed stage the number \(|H|\) is finite, so \(\rho\) can then make the first term small. Cell buffers and generic contour perturbations are absorbed once the stage is fixed and the mesh is sufficiently small. All comparison annuli have uniformly bounded physical radii. ◻

Proof. Let the prescribed parameter accuracy and failure probability decrease to zero in stages. At one stage, choose \(J\) so that the tail in (97) is smaller than its failure tolerance. Pass to the subsequence determining the finite set \(H\). Choose endpoint fences with the prescribed small parameter size and protect their access chains. Next choose the \(H\)-caps with still smaller parameter size and physical cut diameter \(\rho\) making the first term of (97) small. Only then take the mesh small enough for all deterministic contour and cell comparisons at that stage. Iterate and take a diagonal subsequence.

On success, Lemma 39 has removed all parts beyond \(S+3+K\delta_n\). The central core and the two front corners remain, and every new boundary is a wired extension of one original bank. This proves the uniform diameter assertion and the conditional-law assertion. The failure probability tends to zero by Lemma 40.

It remains to verify coordinate control uniformly in the explored outcomes. In initial disc coordinates, the removed heads and their filled components are small attached obstructions by Lemma 38. Thus their removal changes normalized coordinates by \(o(1)\) along the diagonal stages, by Lemma 35.

The portals in the resulting domain also have uniformly vanishing parameter diameter. For the \(H\)-portals this follows from their chosen angular sizes. For circle-\(S\) portals it follows from Lemma 34, since the entire circle is outside \(B(0,S-1)\). Their noncentral shadows are consequently small. Every subsequent searched or discarded portion lies behind one of these portals. Its points attach to the old boundary within that small shadow. Any remaining boundary-cell modifications have vanishing parameter diameter by Lemma 10 and attach through their incident sectors. Filling discarded components preserves the small attachment property. A second application of Lemma 35 therefore gives uniform \(o(1)\) closeness between the final and initial normalized maps.

In particular all fixed kernel compacts are eventually unmodified; the successive cuts lie in parameter neighborhoods shrinking to the circle. The two new marks are within the endpoint fence neighborhoods, whose diameters tend to zero. This proves marked Carathéodory convergence of every deterministic sequence of successful outputs. The same reasoning is uniform over outcomes, because only the deterministic stage tolerances, cell sizes, and parameter shadows were used.

Finally the original curve on success is the first head, followed by the retained interface, followed by the reverse of the second head. Its middle part is exactly the ordinary conditional interface already described. Uniform coordinate closeness and vanishing head diameters show that this concatenation and the retained curve have distance \(o(1)\) in the oriented uniform metric modulo increasing reparametrization. One may allocate a vanishing parameter interval to each head, or use continuous nondecreasing reparametrizations and approximate them by strictly increasing ones. This proves the last assertion. ◻

Proof. Apply Proposition 41 to any subsequence. Conditional on a successful record, the retained law is one of the bounded laws of Proposition 32. Their convergence is uniform over the possible successful outputs in the required sequential sense. Indeed, if a bounded continuous test function of curves had an error bounded below along some sequence of outputs, choosing one such output at each mesh would contradict that proposition, since every chosen domain sequence has the same marked limit and a common diameter bound.

Integrate this convergence against the distribution of the successful records. The failure probability vanishes, and the original and retained curves have vanishing distance on success. Thus this further subsequence converges to the same continuum law. Every subsequence admits such a further subsequence, which proves full-sequence convergence. In particular the limiting law is independent of the lattices, domain approximations, and nome sequence. ◻