Lemma 49 ([Separation under a harmonic landing]). For every >0 there is c()>0 such that, for all sufficiently large R , (Q(Φ_3R )∧Q(Θ_2R )>c() Φ_4R ,Θ_R,Z^σ)>1- . The estimate is uniform in admissible records for which the conditioning …

Write F _ r,s =σ(Φ_r,Θ_s) , and write q_ o ,q_ i for the two numerical quality formulas above, before the following common live test. An exact pair is called open if both pairs of strand tips exist, neither observed interface has closed or prematurely merged, and the observed walk has made no forbidden contact with any observed strand or with the original exterior; the prescribed terminal contact at zero is the sole exception. The raw qualities are zero if these conditions fail. A pair is called live when (Z^σ F_ r,s )>0 ; its two qualities are the raw qualities when live and are set to zero otherwise. A live pair is open. We shall update only one of the two records at a time. For the outer record put ρ=r and e=-1 , and for the inner record put ρ=s and e=1 . If the selected record has quality q>0 , put L=qρ, ρ_1=(1+10eq)ρ. Let q_1 denote its quality after this update, including the live-pair test, with the other record held fixed. We first prove, uniformly for 3.3R<r≤4R, R≤s<1.7R, 0<L<R/200, L≤ρ_ other q_ other , where ρ_ other and q_ other are the radius and live quality of the unchanged record, that (q_1≥2q F_ r,s )≥c_0>0. Here q=L/ρ<1/200 , so the cap is inactive. The geometric, barrier, and walk construction will be proved in the wider layout buffer 2.8R<r≤4R, R≤s<1.84R, 0<q≤1/100, for any jointly admissible open pair, using its selected raw quality q , again denoted by q there. Only the positive-support step will use the smaller- physical-separation premise in (G3). For the low-quality updates used later we also prove the following joint form. Put F= q_1<2q , and, on F∩q_1>0 , make one further update of the same record to ρ_2=(1+10e q_1)ρ_1 , leaving the other record fixed. Put q_2=0 on q_1=0 . If q<1/500 , r>3.25R , and s<1.61R , then (F,q_2=0 F_ r,s ) ≥c_1(F F_ r,s ). Both subsequent radii in (G4) remain in the wider layout buffer just displayed. The estimates condition on the stopped records, not on Z^σ . The three local centerlines. The selected two strand tips and walk tip have pairwise distance at least L . Each strand tip has distance at least L from the entire old walk, and the walk tip has distance at least L from the entire old strand. All those histories are on the old side of the rounded circle. Their deviation from the Euclidean circle is bounded by a lattice constant c_ . Below, fixed errors O(c_) include the short access from an actual tip to a circular polar coordinate. Such accesses can be chosen in the unobserved side: for an outer tip use the interior of an incident selected hexagon to reach the Euclidean circle; for an inner tip use the unselected-hexagon construction given below. Cut the cyclic order of the three access angles at a largest gap and lift them to a_1<a_2<a_3<a_1+2π . Write g_i=a_ i+1 -a_i for i=1,2 and g_3=2π-a_3+a_1≥2π/3 . Define δ_i=(5q-g_i)_+, (Δ_1,Δ_2,Δ_3) =12(-δ_1-δ_2, δ_1-δ_2, δ_1+δ_2). Use a fixed tie rule in choosing the cut. Then |Δ_i|≤5q . During interpolation by a parameter t∈[0,1] , the first two gaps are g_i+tδ_i ; the cut gap is g_3-t(δ_1+delta_2)≥2π/3-10q>1.99 . Thus the cyclic order is preserved, no already larger gap is reduced except the largest one, and all three final circular gaps are at least 5q . For the moment suppose L≥L_0 , where the fixed threshold L_0 will absorb all lattice errors. From the short circular access, retain a straight radial stem of length L/10 . Spread the angles linearly to a_i+Δ_i by radial progress 5L , and keep these angles fixed for the final radial progress 5L , ending at radius ρ_1 . Continue the held rays to total radial progress 11L for the barriers and to total progress 13L only for a topological completion. The spread width is 4.9L+O(c_) . During spreading the radii stay between .95ρ and 1.05ρ ; using the conservative factor 1.1 , after increasing L_0 the ratio of arclength speed to radial speed is at most v_* =√1+(5.5/4.89)^2<1.51. The total length of a barrier, including its held extension and the endpoint construction below, is at most 11L+5.5L+O(L/2000)+O(c_)<17L . Here are the required uniform separation estimates. For L_0 large, the circular access gaps are at least .98q . Even through total progress 13L , the common radius is at least .87ρ-O(c_) . The corresponding centerline points have distance at least 2(.87ρ-O(c_))(.49q)>.84L . If two points have radial difference at least L/4 , their distance has that lower bound. Otherwise compare at a common radius and use (G3b), losing less than 1.51L/4 ; the remainder exceeds L/4 . The same argument applies to a stem, a spread piece, or a held extension. For a short access piece, use its bounded length and the following old-side estimate. If H is an opposite old history and a centerline point has radial progress u , then d(z,H)-v_*u-O(c_), u-O(c_) ≥L1+v_*-O(c_)>.38L. This also proves the distance from each complete local centerline to the entire opposite old history. Consequently any two complete centerlines are separated by L/4 , after one further increase of L_0 . Let t_i be the three target points at radius ρ_1 . Every piece of a different centerline before the held part is radially at least 5L-O(c_) from t_i . The distance from t_i to any ray whose circular angular distance is at least 5q is at least ρ_1(5q) : for an angle at least π/2 the lower bound is ρ_1 , and otherwise it is the perpendicular distance to the infinite ray. This covers the entire held piece and its extension. Since ρ_1≤1.1ρ and q≤.01 , it follows that d(t_i,the entire other centerline) ≥4.4qρ_1. This bound includes the bounded access errors. It is the final held length 5L , rather than only the final angular gap, which controls the earlier spreading portions. Four barriers with fixed parameters. Put w=L/2000 , ℓ=w/10 , and d=w/(200A) , where A is a sufficiently large fixed constant. Smooth the two centerline corners in O(w) neighborhoods, preserving the initial radial stem, radial monotonicity, and all preceding estimates with a small fixed loss. The two sides of a regular radius- w tube about each strand centerline give its two side lanes. Such a tube is a strip: the stem and held pieces are radial, the middle piece is a graph over radius with slope bounded by (G3b), and the two corners can be rounded with curvature at most 1/(10w) . The strips for distinct centerlines are disjoint. For an outer update, complete only the two outer observed arms by these two centerlines and a simple connector at radial progress 13L . This is a prospective oriented chord K_ o ∪J . For an inner update, use the inner observed strand and the two outward centerlines, joining at radial progress 13L , to obtain a prospective oriented simple loop. Let K_0 be the selected old field system, and denote the two ideal sign sides of K_0∪J by Ω_+ and Ω_- . They are Jordan disks, with the unbounded side of an inner loop viewed on the sphere. For an actual attachment in side s , delete from Ω_s the closed lattice triangles meeting K_0 , take the component incident to the connected free collar of J , and take its prime-end completion. The deleted triangles attach to the K_0 boundary, so this component has connected complement and its resolved completion is a disk. Its non- J boundary incidences away from tips and marks are the pure sign- s bank. This is the attachment side; a raw pinched component of D(K) is not being called a Jordan disk. The connector is only a topological separator; no rectangle, shift, or barrier follows it. It stays at least 2L-O(w) beyond the barrier endpoints. The other stopped field record is disjoint from the completion because (1-13q)r≥.87(2.8R)>1.84R, (1+13q)s≤1.13(1.84R)<2.8R. No compatibility of the sampled inner orientation is assumed here. We spell out the old-end attachment; it also handles a shadowing return of the other old arm. At an actual tip take the circle of radius 4w . The two side lanes meet it at transverse offsets ±w and radial depth √15w , up to O(c_) . Follow the circle from the chosen lane away from the radial stem. The long arc between these two lane points stays at distance at least w-O(c_) from the local J ; the rest of J is farther by the stem length and the centerline separation. The two lane points are on different sides of K_ o ∪J , or of the inner loop. Thus this arc, which avoids J , must hit the old field boundary. At its starting point the full-boundary clearance exceeds 3w : the old record is on the old side of the rounded circle, and the curvature error is O(w^2/ρ)+O(c_)=o(w) . Stop at its first point a of full-boundary distance ℓ , and append a globally shortest segment from a to b∈∂D(K) . The retained circular arc has clearance at least ℓ . The segment cannot cross J , since its length is ℓ<w-O(c_) , and it cannot cross a different fixed boundary before its endpoint. It is therefore in the chosen actual side. Its endpoint lies on an old string, not on the remote record or exterior, and d(b,J)≥.89w-O(c_), d(b,every observed tip)≥3.8w-O(c_). For later caging, identify this actual endpoint with the ideal bank as follows. In the incident deleted triangle selected by its nearest prefix and meeting K_0 , join b to the first K_0 -hit. The link has diameter at most one and stays in the signed ideal side by (G3f). It is only an identification through forbidden material, inaccessible to a new dual continuation away from an active strand tip. Coincident links are retained as separate incidences in their boundary order. This link is not a free path and is not used to prove the pocket. The nearest segment has exact clearance d(b+tu,∂D(K))=t for 0≤t≤ℓ . This attachment has the pure pocket required by Proposition 46 ([Uniform barriers for straight-attachment layouts]). Take the radius- d circular crosscut in the resolved attachment side which the nearest segment meets. The circle misses J by (G3f), so its two boundary incidences are on the pure bank. The retained circular arc has distance at least ℓ>d from b . From its lane endpoint a short transverse segment reaches the radial stem at depth about 4w ; this segment also has full-boundary clearance greater than d . It follows that the tail and a collar of J are on the same side of the crosscut, while the initial nearest prefix is on the other side. The crosscut theorem in the Jordan side makes that latter component a pocket bounded by the crosscut and the single pure boundary subarc which excludes J . The nearest prefix crosses every smaller circle once. The remote record is on the J -side: in the outer case it lies in the connected region below radius r-L/20 , and in the inner case in the region above radius s+L/20 ; these regions reach the stem and miss the radius- d crosscut. Removing the remote record cannot alter the pocket. This proof does not assert that the pure subarc or the pocket has small diameter. After deleting only the nearest prefix of length 3d/5 , its remainder has clearance at least 3d/5 ; the retained circular arc and lanes have clearance at least ℓ , and the free endpoint is remote from all pins. Thus (UB1) holds with a=1/2 , including the bounded lattice error. A point within d/5 of the boundary lies on the nearest prefix at radius less than d/2 , proving the ordinary near-boundary barrier condition as well. The endpoint testing window is B(b,O(d)) . Every component of an old simple strand cut by its generous testing square meets the rim, since all strand tips and the remote record are outside that square by (G3f). The incident component also reaches the rim. This proves local thickness, even with arbitrarily many returns. All tested pure pins and the traced pin chain are string pins: these windows are near radius at most 4R , whereas the original exterior is outside 12R . A remote pure subarc may traverse exterior, but no exterior pin enters the quantitative attachment estimate; its constant is the interior-string constant of Proposition 46 ([Uniform barriers for straight-attachment layouts]). Use the four side barriers only through radial progress 11L . Their radial span exceeds d , and their diameters are less than 17L . Take common barrier scale R=32L . Choose the fixed mesh and interpolation neighborhoods in the proposition so that every full endpoint module has radius at most w/100 and every tube enlargement has width at most w/100 . The full domains C_± and all supports P_± lie in their corresponding Ω_± at distance at least .8w from J , by the lane offset, (G3f), and the two w/100 losses. A free path between opposite sides cannot traverse fixed K , hence must cross J . This proves (UB2). For the ordinary internal distance, an access-labelled admissible path has its interior in U . Under the preceding forbidden-material bank identifications it starts on opposite ideal sides of the continuous separator K_0∪J ; it cannot meet K_0 , and therefore crosses J∩U . Its diameter is at least .8w>2d . Thus the ordinary opposite-barrier separation also holds, with the fixed choices θ= d32L = 112\,800\,000A , m=4, a=1/2, κ= .8w32L = 180\,000 . The observed component near an endpoint supplies the required anchor. There are a bounded number of smooth pieces and turns, and all rectangle counts in the proposition depend only on (G3g), not on q . The probability of avoiding all four barriers is therefore at least a fixed p_ b >0 under either actual inner orientation. What the barriers force. Add only for topology a transverse gate from each side barrier's free endpoint to the corresponding centerline at radial progress 11L . The two gates for tip i end at the same point g_i∈J , beyond the stopping circle. In the ideal Ω_s , adjoin the forbidden- material endpoint link; the side barrier plus this link and gate is a proper crosscut A_ i,s from the pure bank B_s to J . The two same-sign crosscuts are disjoint. Parametrize B_s and J from tip 1 to tip 2. Their J -endpoints have this order, so their B_s -endpoints have the same order: otherwise the four endpoints alternate and the crosscuts must meet. This conclusion permits both attachments to be on shadowing parts of the old strand. Glue the two Jordan sides along J , retaining distinct incidences on the two fixed banks. The union Y_i=A_ i,+ ∪A_ i,- is a proper crosscut in this split-bank domain. The preceding nonalternation gives disjoint tip-1 and tip-2 end components and a central component. Extra fixed observations can only remove points. In the relative boundary of an end component the two virtual J portions cancel; its only exits are actual barriers, fixed boundary, and its beyond-stop gates. A fresh interface prefix which avoids the barriers and is stopped at the new circle therefore stays in its end component. The two prefixes cannot merge there. For an outer update the completed chord must connect the two old tips; for an inner update the complete component either joins the two continuations or reaches the original exterior, which is beyond the new circle. Thus neither case can lose a target tip before the stopping circle. On the new circle all cap and turning pieces are behind, and each held lane crosses in one bounded lattice block. The end component meets that circle precisely in the short interval between its two lane blocks. Each new strand tip is consequently within O(w+c_) of its target. No metric confinement of the rest of that prefix in a narrow tube is asserted. Put ν=L/1000 . Require the fresh walk to follow the width- ν tube about its centerline through a terminal extension of length 2ν , including a full entrance ball of radius L/100 about its starting vertex and the radius- ν terminal ball. The terminal ball extends at least ν beyond the new circle. The entire corridor meets that circle only within ν+O(c_) of the target: the entrance ball is 10L away, the spreading pieces are at least 5L away, and the remaining tube follows the held ray. Stop the successful walk at its first crossing of the new rounded circle in this window. The entrance ball is not clipped at the old circle and may meet the old walk. By (G3c) it misses the old field, and centerline separation makes the whole tube and ball miss the side barriers and their gates, including the hexagon error. They lie in B_ 5R , so they also miss the original exterior outside B_ 12R . Let X be the ideal ambient disk or sphere slit only along the selected old dual system K_0 , with its bank accesses retained and the short forbidden endpoint links included in Y_1∪Y_2 . Let W be the union of the closed hexagons of every old and fresh visited walk vertex, not just the polygonal walk edges. Every cell interior avoids K_0 , which lies on hexagon edges. Consecutive visited cells share an edge not crossed by K_0 : for the old record this is the open no-prior-contact condition, including the reversed first step from zero, since otherwise its forward predecessor would already be adjacent to the interface; for the fresh part it is the tube clearance. Thus W K_0 has a connected lift to X , with the prescribed access along the old contacts of the origin cell. Those boundary contacts are not asserted to be in the open surface. The old walk misses the crosscuts because away from the O(w) tip caps they lie on the new side of the old circle, while the old quality protects the caps. Together with the tube separation, this makes W disjoint from Y_1∪Y_2 , including their links and gates, after the hexagon error. The terminal cell meets the central circle interval. Hence the connected lift of W K_0 lies in the central component of X(Y_1∪Y_2) , even if it crosses virtual J . A fresh dual prefix stays in an end component, so it cannot meet W off K_0 . It cannot newly enter K_0 away from an active strand tip; the pre-existing origin contact remains allowed. Put a ball about the deterministic walk target of radius 2qρ_1+ν+O(c_) , including the full landing-window and hexagon errors. Equations (G3c)--(G3d), the 5L radial hold, and the O(w) gates show that this ball misses both paired crosscuts Y_i . It also misses the old field by the 10L radial gap. It misses the remote stopped record by (G3e), including this ball radius. The ball is connected and contains the central target; its lift to the split-bank domain is wholly central even if it crosses virtual J . Thus the actual walk tip is at distance at least 2qρ_1 from the entire newly observed field. Conversely each new field tip is within O(w+c_) of its target; (G3d) bounds its distance to the entire new walk tube, and the radial gap bounds its distance to the entire old walk. Both exceed 2qρ_1 . Finally the two field targets have distance at least 2ρ_1(5q/2) , which still exceeds 2qρ_1 after the landing errors. Hence every term in the new quality numerator is at least 2qρ_1 , and no forbidden contact or premature merger has occurred. The walk probability. For x,y≠0 , put G_0(x,y)=∑_ n≥0 _x(S_n=y, n<T_0). This killed Green function is finite and symmetric. If the exact outer and reversed inner walk records end at a and b , the unexposed middle path has law B_ a,b (m)= 6^ -|m| G_0(a,b) , m:a⟶b, m∩0=∅. The fixed prefix and suffix contribute a common factor to the walk weight; summing the remaining weights gives G_0(a,b) . All revisits of either recorded history or circle are permitted. Reversal is exactly B_ b,a , including all loops, and the field record does not change this law before conditioning on Z^σ . Let E be the successful entire-corridor first-crossing event specified above, stopped at τ . Summing the still unexposed remainder in (GW1) gives for an outer extension B_ a,b (E) = _a[1_EG_0(S_τ,b)] G_0(a,b) . The full entrance ball and width- ν tube have length O(L) and are covered by a fixed number of overlapping balls of radius comparable to L . Their entire corridor and a fixed relative enlargement containing the doubled Harnack balls stay between radii .89r and 1.01r for an outer update, or between .99s and 1.12s for an inner update. The remote-pole gaps are bounded below by .89(2.8R)-1.84R>.65R, 2.8R-1.12(1.84R)>.73R, and the enlargement is also a fixed positive distance from zero. Since L≤.04R , these are uniform relative Harnack margins. Ordinary walk traverses the ball chain with a fixed positive probability. Positive harmonicity of G_0(·,b) on its doubled balls compares its value on the landing window with G_0(a,b) , so (GW2) is at least a fixed p_ w >0 . The full entrance ball permits the necessary old-circle recrossings. After any exact outer prefix ending at a' , the remaining law is B_ a',b ; reversing it gives the inner estimate with G_0(·,a') . The current stopped layers remain strictly nested, and the same displayed pole margins hold for every current joining vertex in the buffer. Thus this also applies after an earlier one-record update, without an independence assertion. Field and walk are independent given their respective stopped records and the actual field orientation. The bound p_ b p_ w survives averaging over that orientation, and gives an open raw output of quality at least (2q)∧1/100 with this uniform probability for L≥L_0 . This part has not used the live test. Bounded lattice scales. We use a simultaneous form of bounded insertion. Given a fixed number m of fixed-size patches near observed tips and compatible sign prescriptions on a bounded number of their free vertices, their joint probability under the original exact-strand conditioning has a positive uniform lower bound. Fix first the shell failure parameter 1/(4m) . For each fixed patch size choose the corresponding long-slack enlargement from the proof of Lemma 41 ([Bounded local insertion]), with this parameter and its required deficit. Merge clusters whenever their already chosen enlarged boxes, with one neighbor layer, intersect, recomputing the enclosing box with the same parameter after a merge. There are at most m-1 merges, and overlap bounds each new diameter by a deterministic function of the previous ones. Thus the final boxes have uniformly bounded sizes and no edges between their interiors; there is no later enlargement. The arbitrary animal parameter in Corollary 11 ([Sparse contacts with signs]) makes each shell failure less than 1/(4m) . With probability at least 3/4 all shells are good. Construct the prescribed feasible filling separately inside every box, keeping all shells fixed. On their disjoint interiors perform one joint contraction toward these fillings. Its Jacobian is t^ ∑N_i , with t>0 and ∑N_i bounded by the patch parameters. Integrating proves the joint bound. No first fence pattern is used as a new conditioning for a second invocation. For L≤20 use the deterministic triangular/hexagonal routing in . It uses only the local rounded-circle geometry, the full-history quality inequalities, and positive completion support. Its longest boundary arc and extremal left/right dual continuations leave a bounded primal route avoiding all adjacent hexagons, with doubled quality and preserved completion support. The same construction with first exit in place of first entry gives the inner route; the fixed ratio r/s>1.52 keeps the opposite record out of its bounded patches. The simultaneous insertion argument just proved replaces only the field probability input. A prescribed primal route of at most M steps has, under either Green transform in (GW2), probability at least 6^ -2M : for the current remote pole b , harmonicity gives G_0(v_ j+1 ,b)≥G_0(v_j,b)/6 at each step. Thus this branch has a uniform positive probability without drawing sublattice-width barriers. For 20<L<L_0 we give a direct finite construction. Put h_ hex =1/√3 . For an outer strand tip, join it through an incident selected hexagon to a point of the Euclidean radius- r circle; this costs at most 2h_ hex and avoids the old dual strand. For an inner tip choose an incident unselected hexagon with center v , join the tip to v through its interior, and choose a lattice unit vector e_v with e_v·v/|v|≥√3/2 . An unselected hexagon has |v|≥s+1/2 , since it contains the radius- 1/2 disk about its center, and boundary incidence gives |v|≤s+3h_ hex . For every integer k≥1 , |v+ke_v|≥|v|+k√3/2>s+h_ hex , so these successive hexagons are unselected. Their center segments remain in the unselected union. Follow them until the radius- s+7/4 circle is first met. This occurs after center-path length at most (7/4-1/2)/(√3/2)<1.444 ; the whole tip access has length less than 2.022 and avoids the inner strand. Use the radial circular access for the walk, with its bounded first-entry or first-exit error. The outer access points have pair distance at least L-4/√3>.88L . Use (G3a) with its signed shifts bounded by 4.12q , an initial radial prefix to radius r-8 , spreading to r-5L , and the final held piece to r-10L . The spread width is at least 4.6L . Comparing spread parameters at threshold 1/12 gives either radial separation 4.6L/12>.383L , or angular gap at least (.88-4.12/12)q>.536q , whose chord exceeds .50L . The same comparison handles a prefix against a spread or held piece. The short access pieces have length at most 2/√3 and stay within a bounded vanishing radial sagitta of the old circle. Against another prefix, either their radial gap is at least 7.7 , or that prefix is within 2/√3+7.7+o(1) of its own tip; the original separation then leaves more than 20-4/√3-7.7-o(1)>9.9 . Against a spread piece the radial gap is at least 8-o(1)>7.66 , and held pieces are farther. Thus the outer complete centerlines have mutual distance >7.66 . Their speed ratio is at most √1+(4.12/4.6)^2<1.35 , so the outer analogue of (G3c), with old histories outside radius r-1 , is d(z,H)-2/√3-1.35u, u-1>.36L>7.2. For the inner access circle the pair distance is at least L-4.044>.797L ; since s≥2000 , its angular gaps are at least .796q . The centered shifts in (G3a) are at most 4.204q . Take the straight prefix to radius s+10 , spread to s+5L , and hold to s+10L . The spread width is at least 4.5L . At parameter threshold 1/11 , the two alternatives give radial separation 4.5L/11>.409L or angular gap (.796-4.204/11)q>.413q , whose chord exceeds .412L . A short access has radius at most s+7/4 , whereas a spread piece has radius at least s+10 , giving gap 8.25 . Against another prefix, either the radial gap is at least 7.8 , or that prefix is within 2.022+7.8 of its own tip; the original tip separation then leaves more than 20-2.022-(2.022+7.8)>8.15 . Hence all inner complete centerlines have mutual distance greater than 7.8 . The old inner strand is inside radius s+2/√3 and the whole old reversed walk inside radius s+1+1/√3 . The speed ratio after the access is less than 1.41 . The counterpart of (G3c) is therefore greater than 7.5 for L≥20 ; explicitly, for radial depth u≥7/4 , it is at least L-2.022-1.41(u-7/4), u-1-1/√3>7.5. The short access itself has distance at least L-2.022 from the opposite old history. Use finite side lanes of width w_b=3 and first full-boundary stop distance ℓ_b=4/3 . Near a tip take the appropriate one-sided outer boundary of the radius-three neighborhood of its short access and straight prefix, truncated at outer depth six or inner depth eight. The short access has diameter less than three, so its radius-three disks have a common interior core; adjoining the monotone radial prefix gives a local strip with an end cap. Its side frontier has distance three from the local J . The future spread starts two radial units farther. Its tangential slope is at most .982 , so even in the direction of the offset point its distance there is at least _ t≥0 √(2+t)^2+(3-.982t)_+^2>3.5 , up to a vanishing polar-curvature error; earlier points of the end collar are farther. Other centerlines are farther than 7.66 . Thus this local frontier has distance three from the full J , and it joins the side boundary of the full radius-three tube. Its deep point has full deleted-boundary clearance greater than 6-1-o(1)>4 in the outer case and 8-2/√3-1-o(1)>5.8 in the inner case. The old strand must meet this frontier near the tip. For the outer case take its first point at distance ten from the tip. A J -point of inward depth at least four is at least four from the old strand; a shallower one is within 4+2/√3<5.16 of its own tip, giving distance greater than 10-5.16>3 from that old point, for either local J . For the inner case a J -point of outward depth at least five is at least 5-2/√3>3 from the old strand; a shallower one is within 2.022+(5-7/4)<5.273 of its own tip, again leaving distance greater than 10-5.273>3 . These old points exist because the relevant old strand continues a macroscopic distance to the exterior or to the incident-edge neighborhood. Its arc starts on J and exits the radius-three neighborhood. It cannot exit through the deep cross-section, which is strictly on the new side of the old radial bound. In each actual Jordan side the local strip frontier from the deep lane must therefore end on the old boundary; an earlier shadowing return only ends it sooner. Stop at the first clearance ℓ_b and nearest-attach. Every point of this open finite barrier then has distance at least 3-4/3=5/3>2/√3 from J . Continue along the held lane to a free endpoint beyond the new circle and add the same virtual gate as above. To identify the actual endpoint b∈∂D(K) with its bank of the central chord or loop, choose the incident deleted triangle which meets K , and join b inside that triangle to its first K -hit. The triangle has diameter one, less than d(b,J)≥5/3 , so this link stays on the same bank. It is only an identification inside forbidden material, not a free path or a sign prescription. Coincident links are resolved in their boundary order. An actual dual continuation cannot use such a link or enter its endpoint away from an observed tip. Perturb these finitely long barriers off dual vertices. Prescribe sign s at every free primal vertex whose closed hexagon meets an open sign- s barrier. An already fixed hexagon meeting it has sign s : its interior lies on that bank of the prospective chord or loop, and by (G3k) it cannot also meet J . Remote fixed observations miss these bounded lanes. No free hexagon meets opposite-sign barriers, for a path through that hexagon between the two sides would have to cross J , requiring diameter at least 5/3 , greater than the hexagon diameter. These are compatible prescriptions on at most M(L_0) vertices in at most two bounded patches near the old tips. The simultaneous insertion argument gives them a joint positive probability. At a transverse crossing of an open barrier, both hexagons adjacent to the crossed dual edge meet that barrier and have its strict sign. A zero interface cannot cross it, nor pass through its fixed endpoint away from a tip. The paired-crosscut caging proof above therefore applies literally. For the walk, the cells successively met by its centerline give a primal nearest-neighbor route in its full radius-one tube; at a cell-vertex tie insert an incident neighboring cell. Include the full radius-one entrance ball and extend the route two units beyond the target circle. In the inner case the final crossed cell center has radius at least s+10L+2-1/√3>s+10L+1/√3 , so it is unselected and the route has exited. Stop at first exit; use first entry in the outer case. The route has at most M(L_0) steps and controls the whole new walk history, including revisits. The fixed radial buffer keeps every route vertex away from zero and stops it before the remote joining vertex. Equations (G3i)--(G3j) and the centerline separations leave fixed lattice room between this tube and the finite sign fences. The new field tips lie within five lattice units of their targets. The targets are at radial distance 5L from all earlier pieces and at distance at least 4.8qρ_1 from every other held ray; subtracting the target, lane, and hexagon errors (at most eight units) still leaves more than 2qρ_1 for L>20 . Old histories are separated by the 10L-O(1) radial gap. The expanded target-ball and central- component argument controls the entire fresh field, as before. The finite route probability is at least 6^ -2M(L_0) by (GW2). Thus the direct construction gives an open raw output of quality at least (2q)∧1/100 with a uniform positive probability on the remaining bounded range, before the live test. The same deterministic construction works with any fixed upper bound in place of L_0 ; only the finite cell count and its probability constant change. Positive support after direct steering. The branch L≤20 already includes the completion support in its cited discrete routing conclusion. Suppose now that L>20 in (G3), and fix any exact output pair in the successful open raw event just constructed. The unchanged physical separation is at least L , by the hypothesis of (G3), and the updated one is at least 2(1-10q)L>1.9L . Thus both raw physical separations exceed twenty. The input live pair has a compatible cyclic matching of its two strand accesses and its walk access, obtained from any completion witnessing Z^σ . The target order and the landing error bounds show that direct steering preserves this matching. It is a geometric matching; the inner orientation sampled for the raw event is not retained. Fix a sufficiently large absolute H , to be specified below. From the fixed output pair prospectively update any raw physical separation ℓ<H , leaving the other record fixed, until both are at least H . Use the direct finite sign-fence construction above with its fixed upper range enlarged to L_0,H . All its deterministic inequalities hold for L>20 ; only its finite cell count changes with this upper range. This construction and its finite walk route use only joint admissibility and openness, not the live test. In particular they give, with positive conditional probability, an open output with raw quality at least twice the input raw quality whenever the cap is inactive. The fresh walk remains inside B_ 5R , so it creates no contact with the original exterior. After each such event forget its auxiliary sign prescriptions and orientation and retain an admissible exact output pair. There are countably many exact path outputs, so one has positive conditional probability. This is a prospective positive-support construction, not a uniform lower bound. For R≥1000H , every selected raw quality in this continuation is at most H/R≤10^ -3 . Each successful update therefore multiplies its physical separation by at least 1.8 . For either record the sum of its own starting physical separations before reaching H is less than 1.8H/(1.8-1)=2.25H ; its total radial change is less than 22.5H . The original (G3) output has r>3.135R and s<1.785R . Hence the whole prospective continuation remains in the wider 2.8R,1.84R layout buffer. It ends after finitely many updates, and retains openness, the cyclic matching, and the absence of every forbidden strand or exterior contact. At this final raw pair use only the deterministic centerlines and capped side curves from the proof of Lemma 48 ([Hookup probability]), not its quantitative probability assertion. Take a=H/(4R) . Both normalized raw qualities are at least a ; the rounded-circle errors are absorbed by choosing H large. The deterministic estimates there give centerline separation at least cH and lane width at least w_H=c'H , for fixed absolute c,c'>0 . Fix H so large that w_H exceeds one hundred times every absolute cell error below. The compatible cyclic matching chooses the orientation of the inner strand and the side s_0 containing both walk accesses. Write Γ=K∪J for the resulting simple ideal chord, with its two sign sides Ω_± . Here J consists of the two free connectors in the unobserved annulus, with the radial end stems just specified in that proof. Use the sign- + and sign- - side curves along just one of these connectors. At an endpoint concatenate the open curve with a link through the selected same-side sector of its incident deleted triangle, ending in the interior of a pure strand-adjacent hexagon. If necessary the link uses at most two old hexagons of that same sign. It does not include the actual K endpoint: at a tiling vertex use the specified access and end in a pure edge-side cell. Perturb the concatenated curve within its side so that it crosses tiling edges transversely and misses tiling vertices. The list of crossed hexagons is then an edge-connected primal chain from the outer sign bank to the inner sign bank. Every old sign site on this chain has the indicated sign, because its hexagon interior lies in Ω_± . No hexagon belongs to both chains: its interior avoids K , and a path inside it between the two sides would have to cross J , contradicting the distance of each open side curve and endpoint link from J , which is greater than the hexagon diameter. Join the two walk accesses by the third deterministic centerline, including its actual walk-tip access pieces, and list the hexagons it crosses after a generic perturbation. Their centers give a finite primal route between the exact walk-tip vertices; its full neighbor layer stays in the free third corridor by the choice of H . Concatenate it with the outer record and the reversal of the inner record to obtain a candidate path Π ending at zero. For every nonzero vertex of Π , request sign s_0 on that vertex and all its lattice neighbors. These requests are compatible with all old signs. Indeed the old walked hexagons cannot cross J : their histories are on the old sides of the stopping circles, and the only end neighborhoods of J are separated from those histories by the physical gap H , including one neighbor layer. They cannot cross K by openness. Their accesses and the middle route are in Ω_ s_0 , so all their nonzero hexagons lie on that side. If a neighboring old sign hexagon were on the opposite side, the common primal edge would meet Γ . A K edge there is exactly a forbidden prior adjacency, and a J crossing is excluded by the preceding clearance. An exterior neighbor on the opposite sign arc would likewise make the common primal edge cross K or J , contrary to observed no-contact or the same clearance; a same-side Dirichlet neighbor is retained unchanged. The same side and hexagon-diameter argument excludes an overlap with the opposite sign chain. Requests of sign s_0 cause no conflict with the other chain. The origin may occur in a neighbor layer and then has the compatible sign s_0 ; only the star centered at the origin is omitted. Choose 0<b<1/4 . Give every requested free sign site, including all old string sites, the value +b or -b , give all other free sites value zero, and retain the original Dirichlet pins. No site has received opposite requests. The retained old signs include the literal tail signs through each crossing triangle and first edge on the other side of its rounded stopping layer; these preserve the specified first entrance or exit and the exact in-disk component. Each free--free edge has height difference at most 2b<1 , and each free--pin edge has difference at most b+1/2<1 ; the pin--pin constraints were already feasible. A relative-open neighborhood therefore satisfies all old exact strand signs, the two chains, the nonzero walk stars, and the hard Lipschitz constraints. It has positive volume in this finite domain, and in particular proves positive probability for the chosen inner orientation. Along a simple zero strand the consecutive same-side hexagons are equal or edge-adjacent. Each of the two forced chains consequently joins the corresponding sign next to e_σ^* , through the inner bank and outer bank, to its corresponding exterior sign arc. The zero component through e_σ^* cannot be a closed cycle, since a cycle would separate one of these two sign clusters from the exterior. Interior zero vertices have degree two and the only boundary sign changes are the two marks, so this component is the boundary chord. At every nonzero vertex of Π , all neighbors have its one strict sign; no zero edge is incident to its hexagon. Thus Π meets the chord first at its allowed endpoint zero. It has no earlier exterior contact by the same raw openness and corridor clearance. The finite middle walk route has positive conditional Green-bridge probability. The strict field event and this route therefore give Z^σ positive conditional probability at the terminal H -pair. Each of the finitely many raw success transitions leading there had positive conditional probability, with its auxiliary orientation and fences forgotten at the retained exact record. Applying the tower property successively along this finite continuation gives positive hookup probability at the original fixed direct output pair. Every direct successful output in (G3) is live. Combining this with the uniform raw steering bound and the already supported L≤20 branch proves (G3). The one-record collapse estimate. For the outer update the proof of (G4) follows the stopped first-approach decomposition in ; we record the exact conditioning because it is needed below. Reveal the new two field arms first, but not the whole intervening walk. Write x,y for their tips, z for the eventual new walk tip, and S^ for the newly exposed walk portion. The failure event F is covered, including ties, by the following events: M_0, = q_1=0 ,; M_1, = d(S^, x,y )<(2qρ_1)∧|x-y| ,; M_2, =F∩ρ_1q_1=d(z,new whole strand history),; M_3, =F∩0<ρ_1q_1=|x-y|. Events involving missing tips are empty, that case being in M_0 . The cap is inactive because 2q<1/100 . The old opposite histories cannot make the new quality insufficient: their radial gap is 10L , whereas 2qρ_1≤2.2L . At the bounded lattice scales this is understood with the exact first-entry/first-exit version of the cited discrete construction. For M_1 , stop S^ at its first visit within a=(2qρ_1)∧|x-y| of either new strand tip. Condition only on the new field arms and this stopped walk prefix, and call the approached tip x . The other tip is at least a away. The preceding annular width is 10qρ , whereas a≤2qρ_1 ; for either direction their ratio is at least 5/(1+10q)>250/51 . At diverging scales the rounded-circle error therefore still leaves width at least 4.8a . The approached strand crosses the annulus about x from radius a/4 to radius 2a . Starting at this first approach, stop a local trial at a field hit, entry to either radius- a/4 tip ball, or exit from B(x,2a) . The source's planar stopped-walk estimate gives a fixed probability of the field-hit alternative. Its fixed relative enlargement avoids the remote joining vertex and zero in the present buffer, so Harnack and (GW2) give the same assertion for the Green bridge. The trial need not finish before the first crossing of the ρ_1 -circle. If that crossing occurs first at z with d(z,H)<a/4 , where H is the whole new strand history, interrupt the trial there. Before that time the walk has stayed outside the two radius- a/4 tip balls, and before its first approach it stayed at distance at least a from both tips. Since |x-y|≥a and the cap is larger than a/4 , either the pair is already dead or ρ_1q_1=d(z,H) . The direct M_2 hitting estimate at this complete first record, with scale d(z,H) , then kills hookup during the next update with a fixed probability. If instead the crossing has d(z,H)≥a/4 , every quality term at that crossing is at least a/4 , unless hookup is already dead. Its next radial displacement is therefore at least 10(a/4)=2.5a . A field hit in the local trial, which stays in B(x,2a) , occurs before that next circle; the strict 0.5a margin absorbs the rounded-circle error at diverging scales. A trial hit before the first crossing also kills hookup. Thus the union of a local hit before interruption and an interruption contains the un-interrupted local-hit event, so it has a fixed lower probability. On the interruption event, condition on the record at interruption, not on whether the trial would later hit; the direct estimate then supplies a fixed conditional death probability. Conditioning there and integrating proves (q_2=0,M_1 F_ r,s ) ≥c(M_1 F_ r,s ). This is not asserted after fixing the later escaping part of the failed walk. On M_2 the same direct hitting estimate starts at the new walk tip at scale ρ_1q_1 , and may be used after that full new record is fixed. Its next radial displacement is ten times this scale. At bounded scales the exact discrete first-approach and finite routing argument of the source, with bounded insertion, gives the same fixed lower bounds. On M_3 , one single-sign circular-clearance barrier around the midpoint of [x,y] forces the strands to join before the next circle, whose radial gap is 10|x-y| . Its nearest endpoints, pure pockets, and local thickness are those proved above for the circular construction, and the remote record is outside its annulus. Proposition 46 ([Uniform barriers for straight-attachment layouts]) applies at diverging scales and bounded insertion at bounded scales. M_0 is immediate. Thus (G4b) holds for all four M_i with one c>0 . Summing the possibly overlapping events gives (G4) with c_1=c/4 . For an inner update use the same decomposition for the reversed first-exit walk and the outward-growing strand. The entire old histories now lie inside the old rounded circle, so the radial-gap calculation reverses without changing its constants. Reverse the whole Green bridge as in (GW1)--(GW2); its local first-approach law has the remote outer joining vertex as pole. The same first-approach hitting argument and the reflected single-sign circular barrier apply. The quantitative buffers ensure these local regions miss the fixed outer record. This proves the joint form (G4) for either selected record, without claiming a hazard bound conditional on a complete failed intermediate record. We now iterate the one-record estimates at the smaller physical separation. Write ℓ_ o =r q_ o , ℓ_ i =s q_ i , p= ℓ_ o ∧ℓ_ i R , β= 32 , where both live qualities are zero at death. Fix a_0=β^ -N ≤10^ -3 , to be chosen below, and put B = (r,s):7R/2≤r≤4R, R≤s≤3R/2 . Starting from (r,s)=(4R,R) , stop at death, when p≥a_0 , or when the radii leave B . Stopping is checked only at round boundaries; let τ be the first terminal boundary. At a live round boundary choose a record with smaller physical separation ℓ=pR , using a deterministic tie rule. Its quality is q=ℓ/ρ≤p<a_0 , so the cap in (G3) is inactive. Attempt its (G3) update, leaving the other record fixed. On success its new physical separation satisfies ℓ_1=ρ_1q_1≥2(1+10eq)ℓ≥1.8ℓ. On failure q_1<2q , flag the round and immediately execute the same record's (G4) update. If q_1=0 , end dead without another radial update. Otherwise use q_1 to set the collapse radius and then end the round. After a first success, let U≥ℓ be the unchanged physical separation. If U≥βℓ , end the round. If U<βℓ , it is now the smaller separation by (G5a); attempt its (G3) update. Its quality is at most U/R<βp<.0015 , so its cap is also inactive and (G4) applies on failure. Flag and immediately collapse that same record on failure, again making no further update if its failed quality is zero. End the round after this second attempt or its collapse. A flag-free round has p_ k+1 ≥βp_k . A failed growth and its following collapse are one transition for the round filtration: the intervening full record is not added before (G4) is used. The final nested records determine that restriction, the chosen radii, and the flag, without observing any still unexposed bridge. Let F_k be the exact pair at the start of round k , let J_k indicate its flag, and let A_k mean that the state is live. Decrease c_1 , if necessary, so 0<c_1<1 , and put ϱ=1-c_1 . The joint form of (G4) says that for either failed attempt F , (F, survival after the immediate collapse F_ pre ) ≤ϱ(F F_ pre ). For the second failure apply this after the first success and integrate its exact record. The two failure branches are disjoint, so (J_k=1,A_ k+1 F_k) ≤ϱ(J_k=1 F_k), (J_k=0 F_k)≥c_0^2. The second bound uses at most two (G3) successes. No hazard bound is asserted after specifying the complete failed intermediate record. For later division, consider any reachable nondead exact pair with radii in B , including a pair where p≥a_0 has just been attained. In a separate prospective continuation update the smaller physical separation until both are at least p_*R , where p_*=1/200 . Provisionally stop also at the first exit from the (G3) input buffer r>3.3R , s<1.7R . At every prospective update before that exit, the selected quality is less than p_*R/ρ≤p_*=.005 . Thus the cap is inactive and each success multiplies that record's physical separation by at least 1.8>β . Between its own updates its radius and intrinsic quality stay fixed, since the all-success pair remains live. For either record every prefix of its own starting-separation sequence therefore has sum less than ββ-1 p_*R=.015R. An elementary radius changes by exactly ten times that starting physical separation. Hence every prospective prefix has r_ f >3.35R, s_ f <1.65R. This contradicts the proposed first exit from the (G3) input buffer, so the continuation reaches its quality target within that buffer. At the end both normalized qualities are at least p_*/4 , the ratio lies in (9/8,5) , and 6s_ f <9.9R<12R . Lemma 48 ([Hookup probability]) applies. There are at most 20,_β(p_*/p) successful updates. With α=2_β(c_0^ -1 ) , this proves (Z^σ F) ≥c(p∧p_*)^α R. The prospective continuation is used only for this lower bound and may leave B ; (G5c) supplies its larger buffer. Set all flags after τ to zero. Freeze the state after a nondeath stop, and keep its live indicator zero after death. Let N_k=∑_ j<k J_j . By (G5b), W_k=1_ A_k ϱ^ -N_k is a nonnegative supermartingale. For I_n=(β^ -n ,β^ 1-n ] , put M_n=∑_ k<τ 1_ A_k, p_k∈I_n , and let T_ n,m be the m -th counted start. Between two visits to a band there is a completed flagged round: without a flag each transition multiplies p by at least β , so it cannot return. The flag may occur in another band. Optional stopping at bounded times, followed by monotone convergence of the hitting indicators, gives on S=T_ n,1 <∞ (T_ n,m <∞ F_S)≤ϱ^ m-1 . Here F_S is the stopped sigma-field on the first-visit event. At a live start in B , the ratio r/s lies in [7/3,4] and s∈[R,3R/2] , so the upper bound in Lemma 48 ([Hookup probability]) is C/ R . Apply it at T_ n,m , use (G5d), and divide by (G5) at the first visit, where β^ -n <p<a_0<p_* . Integrating that first-visit record yields (M_n≥mΦ_ 4R ,Θ_R,Z^σ) ≤Cβ^ αn ϱ^ m-1 . Choose A>αβ/(-ϱ) . For some ξ<1 , (G6) gives (M_n> AnΦ_ 4R ,Θ_R,Z^σ) ≤Cξ^n . A failed physical separation satisfies ℓ_1=ρ_1q_1<2(1+10eq)ℓ≤2.2ℓ . The sums of elementary physical inputs in the three possible round types are consequently less than 3.2ℓ , 2.5ℓ , and ℓ+1.5ℓ+2.2(1.5ℓ)=5.8ℓ , respectively. Thus the sum of outer loss and inner gain in one round is less than 58pR . For p<a_0≤10^ -3 , a complete round begun in B has intermediate radii r>3.442R>3.25R, s<1.558R<1.61R, even when its endpoint triggers stopping. These are the strong (G4) buffers; the selected normalized qualities before their growth attempts were already shown to be less than 1/500 , as (G4) requires. Its failed output need not remain the smaller physical separation. Low-quality rounds have n≥N+1 . Outside the exceptional events, ∑_ k<τ p_k ≤(A+1)∑_ n≥N+1 nβ^ 1-n =3(A+1)(N+3)β^ -N . Choose N so large that the resulting radial movement is less than R/2 and C∑_ n≥N+1 ξ^n</3 . The radii cannot then leave B while p<a_0 . An infinite live low-quality run would contain infinitely many flags, since after a last flag repeated multiplication by β reaches a_0 . The supermartingale shows that survival through infinitely many flags has probability zero. Conditional on Z^σ , therefore, p≥a_0 is attained before buffer exit with probability at least 1-/3 . On that attainment event condition on the exact transcript H . Since its radii lie in B , q_ o ∧q_ i ≥a_0/4, p∧p_*≥a_0. Apply the one-record late-collapse argument to the outer record down to 3R , keeping the attained inner record fixed; apply its outward version to the inner record up to 2R , keeping the attained outer record fixed. These are two estimates under the same H , not estimates conditioned on the other final record. The argument in applies with these fixed buffers. A pair of close strand tips is screened at dyadic circles about its midpoint by the single-sign circular-clearance barriers proved above. Each required breach costs a fixed factor, giving C_ a_0 t^ ζ_ s for separation tR . For a walk--strand approach, expose the walk only to its first tR -approach and use Beurling up to the next fixed intermediate circle, giving C_ a_0 t^ ζ_ w . Leave the remaining bridge unexposed in both cases. The augmented upper bound in (G1) then gives the joint-with-hookup bound C_ a_0 (t^ ζ_ s +t^ ζ_ w )/ R . No (G3) call is made in this late step. The field-radius pairs for the augmented (G1) factors are (3R,s_H) and (r_H,2R) , with respective ratios in [2,3] and [7/4,2] . Their clearance requirements are 6s_H≤9R and 6(2R)=12R ; the intermediate walk prefix is covered by the augmented upper bound. Divide by (G5), which at the attained pair is at least c a_0^α/ R . Choose 0<t<.03 so small that, under the same H,Z^σ , the union bound for the two late-collapse events is less than 2/3 . The bad events here mean that a defining physical separation is at most tR . Either final normalized quality at most t/3 implies its corresponding bad event. Taking c()=t/3 , together with attainment, proves (G2).