Proof. For vertices \(p_i^+\) and \(p_j^-\) on opposite arms, optimality and the triangle inequality give \[T(z,p_i^+)+T(z,p_j^-) =T(p_i^+,p_j^-) \leq T(w,p_i^+)+T(w,p_j^-).\] Subtract the right-hand terms and let \(i,j\to\infty\). This proves (6). Additivity and asymptotic linearity imply that its left-hand side is \[(\rho_{p^+}+\rho_{p^-})(w-z)+o(|w|) \qquad (|w|\to\infty).\] Putting \(w=z+ne_k\) and \(w=z-ne_k\), dividing by \(n\), and letting \(n\to\infty\), for \(k=1,2\), proves that the sum of the functionals vanishes. The assertion about moving the base point follows from tail invariance in 3.

The exclusion of a deterministic countable set is 5. ◻

Proof. Let \(K=\{x:m(x)\leq1\}\), and let \(K^*=\{\rho:\rho(x)\leq m(x)\text{ for all }x\}\) be the dual unit ball. For \(\rho\in\partial K^*\) the exposed face \[F_\rho=\{x\in K:\rho(x)=1\}\] is a point or a line segment. Its radial image is \(A(\rho)\). If this image contains two directions, it is a nondegenerate closed circular arc. The relative interiors of these arcs are disjoint for distinct \(\rho\): a direction in such an interior corresponds to the relative interior of a nondegenerate face of \(K\), whose supporting line, and hence its normalized functional, is unique. Each nondegenerate arc contains a rational angle. Thus only countably many functionals have more than one contact direction.

There are also only countably many members of \(C\) that are isolated on at least one circular side. Indeed, they are endpoints of components of \(\partial K^*\setminus C\), with the finite-set cases included. The open components are disjoint and each contains a point of a fixed countable dense subset of the circle.

Consider a remaining \(\rho\in C\), so that \(A(\rho)=\{d\}\) and \(C\) accumulates at \(\rho\) from both circular sides. A lattice rotation makes \(d_2>0\). The contact correspondence is upper semicontinuous: if \(\rho_n\to\rho\) and \(d_n\in A(\rho_n)\) with \(d_n\to d'\), then \(m(d')=\rho(d')\). Compactness of the unit circle therefore supplies an open boundary neighborhood of \(\rho\) on which every contact has vertical coordinate at least some \(c>0\).

Choose once and for all a countable dense subset \(D_0\) of \(C\). In the neighborhood just obtained, choose one point of \(D_0\) on either side of \(\rho\). The compact boundary arc joining them through \(\rho\) can be taken inside that neighborhood. There are only countably many pairs from \(D_0\) and finitely many choices of lattice coordinates, so all such arcs form a countable family. For \(D\) take the union of \(D_0\), the two countable exceptional sets described above, and their negatives.

For completeness, suppose two chart points \(\rho,\lambda\) had the same horizontal coordinate and \(\lambda(e_2)>\rho(e_2)\). Choosing any \(d\in A(\rho)\), for which \(d_2>0\), would give \(\lambda(d)>\rho(d)=m(d)\), contrary to \(\lambda\in K^*\). The same argument shows that each chart point is the top endpoint of the vertical section of \(K^*\) at its horizontal coordinate. The continuous horizontal coordinate is thus injective on the arc and strictly monotone along it. Orient \(J\) so that it increases. On the opposite lower arc, the horizontal coordinate again increases from left to right; negation reverses the order of the corresponding labels.

Finally, 9 excludes \(D\) simultaneously and identifies the opposite end. The contact restriction implies that rays of the chosen upper label tend to height \(+\infty\), and those of the negative label tend to height \(-\infty\). This gives the claimed orientation. ◻

Proof. If two rays meet infinitely often, choose a shared vertex beyond their finite initial portions. The geodesic segments from that vertex to every later common vertex agree by finite uniqueness. The later common vertices escape to infinity, so the two tails agree. Tail invariance of the Busemann function then gives equality of their labels. Thus distinct labels have only finitely many common vertices.

Suppose the tails of \(p\) and \(q\) are disjoint. Choose an integer height \(h\) above both starting sites, above all intersections, and above finite prefixes preceding the disjoint tails. After their respective last visits to row \(h\), both rays stay strictly above that row. For sufficiently large \(N\), their first visits to row \(N\) occur after those last visits, since the prefixes ending at the last visits are finite. The intervening arcs are disjoint strip crossings. Their order on row \(N\) therefore agrees with the fixed order of the last visits to row \(h\). This proves stabilization.

Assume this stable order is \(p\) to the left of \(q\). For a positive integer \(n\), set \(u=(-n,0)\) and \(v=(n,0)\). Let \(\gamma_j\) be the concatenation of the finite geodesic from \(v\) to \(p_j\) and the tail of \(p\) after \(p_j\). This concatenation is used as a walk; it need not be a ray or be simple. Define \[\delta_j=T(v,p_j)-B_p(v,p_j)\geq0.\] Since \(B_p(v,p_j)=B_p(v,p_0)+T(p_0,p_j)\), the definition of \(B_p\) implies \(\delta_j\to0\) for this fixed \(v\). Every vertex \(w\) of \(\gamma_j\) satisfies \[ 0\leq T(v,w)-B_p(v,w)\leq\delta_j. \tag{9}\] On the initial geodesic this follows by decomposing its cost and the Busemann increment at \(w\), with the remaining deficit nonnegative. On the tail it follows from the triangle inequality and exact calibration of \(p[p_j,w]\). Define \(\xi_j\) in the same way by joining \(u\) to \(q_j\) and then following \(q\), and write its error as \(\delta'_j\to0\).

We next control all row-zero visits of these walks, including visits very far from the origin. The horizontal directions are outside both contact sets, so \[\gamma= \min_{\lambda\in\{\rho_p,\rho_q\},\,\sigma\in\{-1,1\}} \bigl(m(\sigma e_1)-\lambda(\sigma e_1)\bigr)>0.\] Write \(a_p(x)=B_p(0,x)-\rho_p(x)\), and define \(a_q\) similarly. Given \(\varepsilon>0\), the uniform metric estimate and these two rooted asymptotics give a finite \(K_\varepsilon\) such that, for all lattice sites \(x,y\), \[\begin{align*} |T(x,y)-m(y-x)|&\leq\varepsilon(|x|+|y|)+K_\varepsilon,\\ |a_p(x)|+|a_q(x)|&\leq\varepsilon|x|+K_\varepsilon. \end{align*}\] For \(w=(t,0)\) with \(d=|w-v|\geq n/2\), we have \(|v|+|w|\leq2n+d\leq5d\). Additivity therefore gives the explicit bound \[ T(v,w)-B_p(v,w) \geq (\gamma-10\varepsilon)d-3K_\varepsilon. \tag{10}\] The identical estimate holds with \(u,q\) in place of \(v,p\). Choose \(\varepsilon<\gamma/40\) first, and then \(n\) so large that \(3K_\varepsilon\leq\gamma n/8\). The right-hand side of (10) is at least \(\gamma d/2\geq\gamma n/4\). For this fixed \(n\), choose \(j\) so large that both approximate-calibration errors are smaller than \(\gamma n/4\). It follows that every row-zero visit of \(\gamma_j\) has horizontal coordinate in \((n/2,3n/2)\), whereas every such visit of \(\xi_j\) lies in \((-3n/2,-n/2)\).

Both walks eventually follow their exact upward tails. Their last row-zero visits thus exist and have the order \(\xi_j,\gamma_j\). Choose a row \(N\) above the finite joining paths, the discarded original prefixes \(p[p_0,p_j]\) and \(q[q_0,q_j]\), and the prefixes ending at these last row-zero visits, and high enough for the stable order of \(p,q\). The first visits of the two walks to this row occur on their exact tails and have the opposite order \(\gamma_j,\xi_j\). Their portions from the last row-zero visits to these first row-\(N\) visits are strip crossings. The planar fact gives a common vertex \(w_j\).

At that vertex, domination by the metric and (9) give \[\begin{align*} B_p(u,v) &=B_p(u,w_j)-B_p(v,w_j)\\ &\leq T(u,w_j)-T(v,w_j)+\delta_j\\ &\leq B_q(u,w_j)-B_q(v,w_j)+\delta_j+\delta'_j\\ &=B_q(u,v)+\delta_j+\delta'_j. \end{align*}\] Let \(j\to\infty\) with \(n\) fixed. We obtain \(B_p(u,v)\leq B_q(u,v)\). Finally, the rooted asymptotics at \(u\) and \(v\) give \[B_p(u,v)=2n\rho_p(e_1)+o(n),\qquad B_q(u,v)=2n\rho_q(e_1)+o(n).\] Division by \(2n\) proves (8). If the labels are distinct, injectivity of \(r\) makes the order strict. Reflection in the horizontal axis proves the lower-ray version; this reflection preserves \(e_1\), so it also preserves the horizontal parameter used in that version. ◻

Proof. Let \(\Lambda=\{\rho_\ell,\rho_r,-\rho_\ell,-\rho_r\}\). For each of these four deterministic functionals, 4(iii) provides a unique ray \(g^\lambda_z\) from every vertex \(z\) and a common Busemann function \(B_\lambda\). Let \(\sigma_\lambda=1\) for the upper endpoints and \(\sigma_\lambda=-1\) for the lower endpoints. Define \[ H(z)=1+\max_{\lambda\in\Lambda} \sup_{w\in g^\lambda_z} \bigl(s|w_1-z_1|-\sigma_\lambda(w_2-z_2)\bigr)_+. \tag{13}\] Each supremum is finite. Indeed, the limit directions of this endpoint ray lie strictly inside its cone of slope \(2s\), and translating the origin to its fixed starting point does not change its limit directions. Outside a finite prefix the displayed positive part is therefore zero.

We prove (12) first for these endpoint rays. Put \(a_\lambda(x)=B_\lambda(0,x)-\lambda(x)=o(|x|)\). If \(w\in g^\lambda_z\), calibration gives \[ T(z,w)=B_\lambda(z,w) =\lambda(w-z)+a_\lambda(w)-a_\lambda(z). \tag{14}\] Fix \(L\) and \(\eta>0\). Whenever \(|z|\leq Ln\) and \(d=w-z\) satisfies \(|d|\geq\eta n\), we have \[|z|+|w|\leq 2Ln+|d| \leq(1+2L/\eta)|d|.\] The uniform metric approximation, the four rooted asymptotics, and (14) thus imply \[ \sup_{ \substack{\lambda\in\Lambda,\ |z|\leq Ln,\ w\in g^\lambda_z\\ |w-z|\geq\eta n}} \frac{m(w-z)-\lambda(w-z)}{|w-z|}\longrightarrow0. \tag{15}\] To make the uniformity explicit, bound each rooted error by \(\varepsilon|x|+K_\varepsilon\) and the metric error by \(\varepsilon(|z|+|w|)+K_\varepsilon\). The quotient in (15) is then at most \(2\varepsilon(1+2L/\eta)+3K_\varepsilon/(\eta n)\); take \(n\to\infty\) and then \(\varepsilon\downarrow0\).

For each \(\lambda\in\Lambda\), the continuous function \(m(d)-\lambda(d)\) has a strictly positive minimum on \[\{d\in S^1:\ \sigma_\lambda d_2\leq s|d_1|\},\] because all its zeros lie strictly inside the cone of slope \(2s\). Taking the minimum over the four functionals preserves positivity. It follows from (15) that, for all sufficiently large \(n\), every endpoint-ray displacement with \(|d|\geq\eta n\) lies in the unshifted cone of slope \(s\). For the remaining displacements, \[\bigl(s|d_1|-\sigma_\lambda d_2\bigr)_+ \leq(s+1)|d|<(s+1)\eta n.\] Consequently the maximum in (12) is at most \(1+(s+1)\eta n\) eventually. Letting \(\eta\downarrow0\) proves that equation for the \(H(z)\) in (13).

It remains to trap every intermediate ray with exactly this offset. Fix \(z\) and an upper ray \(p\) with \(r(\rho_\ell)<r(\rho_p)<r(\rho_r)\). Write \(a=g^{\rho_\ell}_z\), \(b=g^{\rho_r}_z\), and \(Q=Q^+(z,H(z))\). The bounding rays lie in \(Q\). Finite uniqueness implies that they have a common finite prefix ending at a vertex \(v\), and then disjoint tails. Indeed, a reunion after their first split would give two finite geodesics from \(z\) to the reunion vertex. Their common prefix is finite because their labels differ.

Suppose, for a contradiction, that \(p\) visits a vertex \(w\notin Q\). Choose an integer \(N\) above the heights of the entire common prefix of \(a,b\), above the finite prefix of \(p\) ending at \(w\), and large enough for the three pairwise orders in 11 to have stabilized. Let \(a_N,b_N,p_N\) denote their first arrivals on row \(N\). They are strictly ordered as \[(a_N)_1<(p_N)_1<(b_N)_1.\] Each first-arrival path lies strictly below that row until its endpoint. The two arcs \(a[v,a_N]\) and \(b[v,b_N]\), together with the horizontal segment \([a_N,b_N]\), form a simple closed polygonal curve \(\mathcal J\). Let \(\mathcal D\) be its bounded Jordan domain. The curve lies in \(Q\), including its top segment by convexity. Since a bounded Jordan domain is contained in the convex hull of its boundary, \(\mathcal D\subset Q\).

The common prefix \(a[z,v]=b[z,v]\) need not lie in \(\mathcal D\); viewed from the Jordan curve it may form a slit on either side. The following first-arrival argument treats that possibility explicitly; see 1. The last edge into \(p_N\) is vertical from below. Since \(p_N\) is in the relative interior of the top segment, the part of that edge sufficiently close to \(p_N\) lies inside \(\mathcal D\). Starting at \(w\notin Q\), the ray therefore passes from outside \(\mathcal D\) to its inside before its first row-\(N\) arrival. It cannot enter through the top segment, since that would already be a visit to row \(N\). Hence, after \(w\), it meets one of the two bounding arcs at a lattice vertex \(u\), possibly \(u=v\). Both its prefix \(p[z,u]\) and the corresponding bounding prefix are finite geodesics. Uniqueness makes them identical. The bounding prefix is contained in \(Q\), whereas \(p[z,u]\) contains \(w\notin Q\), a contradiction. This argument also rules out departures made before the bounding rays split.

Thus every intermediate upper ray is in \(Q\). Rays with an endpoint label are the unique endpoint rays. Reflection proves the same statement for lower rays. The reasoning was deterministic on the single event already specified and applied to an arbitrary ray, so it establishes the simultaneous all-ray assertion. ◻

Proof. For upper rays put \(h=\max(x_2,y_2)\). Both rays have a last visit to row \(h\), because they tend upwards. Call the two last vertices \(\widehat x,\widehat y\). The cone bounds give \[|\widehat x_1-x_1|\leq\frac{h-x_2+H_x}{s},\qquad |\widehat y_1-y_1|\leq\frac{h-y_2+H_y}{s}.\] Since \((h-x_2)+(h-y_2)=|x_2-y_2|\), (16) implies \(\widehat x_1<\widehat y_1\). If the rays did not intersect, their portions from these last visits to their first visits to a sufficiently high row would be disjoint strip crossings. Their order there would remain \(p,q\), contradicting 11 and the strict reverse order of their labels. For lower rays use \(h=\min(x_2,y_2)\) and the last visits before departure towards height \(-\infty\). The same estimates and the lower form of 11 apply. ◻

Proof. Put \(d=y_1-k_1\). If a vertex \(w\) belongs to the ball of radius \(R\) about \(k\), then \[|w_2-y_2|\le R, \qquad |w_1-y_1|\ge d-R.\] Membership in either comparator cone would therefore require \(R\ge s(d-R)-H_c\). Thus \[ d>R+\frac{R+H_c}{s} \tag{41}\] makes both base comparator rays avoid the entire ball. Their edge weights are unchanged in \(T^k\). Since every other edge weight only increases, each comparator remains a geodesic ray in \(T^k\). 7 preserves its label.

Increase the separation further, if necessary, to apply 13 with offsets \(H_p,H_c\). The upper marked ray is to the left at its starting site but has label strictly east of \(\alpha\). The same ordering holds for the lower rays, since \[r(-\beta)=-r(\beta)<-r(\rho)=r(-\rho).\] The upper comparator therefore meets \(p^+\), and the lower comparator meets \(p^-\). Call intersection vertices \(w^+\) and \(w^-\), respectively. By (41), each intersection is outside the ball. Consequently it occurs after the whole prescribed initial gain portion of the corresponding marked arm. This remains true if an arm leaves and later reenters the ball, since every vertex of its prescribed initial portion lies in the ball.

Let \(g^+\) and \(g^-\) be the increases on those portions, so \(g^++g^-\ge b\). Writing \(\operatorname{cost}_T\) for the cost of a specified finite path in the base weights, domination by passage time and then by a Busemann increment gives \[\begin{split} \operatorname{cost}_{T^k}(p^+[k,w^+]) &\ge g^++\operatorname{cost}_T(p^+[k,w^+]) \ge g^++B_\alpha(k,w^+),\\ \operatorname{cost}_{T^k}(p^-[k,w^-]) &\ge g^-+\operatorname{cost}_T(p^-[k,w^-]) \ge g^-+B_{-\beta}(k,w^-). \end{split}\] The marked segment from \(w^-\) through \(k\) to \(w^+\) is minimizing in \(T^k\). A competing finite walk follows the lower comparator backwards from \(w^-\) to \(y\), then follows the upper comparator to \(w^+\). The cost of this walk in \(T^k\) equals its base cost, namely \[B_{-\beta}(y,w^-)+B_\alpha(y,w^+).\] No disjointness between the two comparator portions is required: a finite walk is an admissible competitor and its cost counts every traversal. Comparison with the marked segment yields \[b+B_\alpha(k,w^+)+B_{-\beta}(k,w^-) \le B_\alpha(y,w^+)+B_{-\beta}(y,w^-).\] Additivity cancels the increments ending at the intersection vertices and gives exactly (39). Both the avoidance bound and the separation in 13 have the asserted dependence on the offsets and \(R\). ◻

Proof. Set \(w=r(\beta)-r(\alpha)>0\) and, for \(j\ge0\), write \[A_j=B_\alpha(0,k_j)+B_{-\beta}(0,k_j) =-wjl+e_j.\] The sign in this formula uses \(\alpha(e_1)+(-\beta)(e_1)=r(\alpha)-r(\beta)=-w\). The two fixed-label asymptotics give \(e_j=o(jl)\) as \(j\to\infty\). Define \[E_N=\max_{0\le j\le N}|e_j|.\] Then \(E_N/(Nl)\to0\) almost surely. To see the needed uniformity, for each \(\delta>0\) choose \(J\) so that \(|e_j|\le\delta jl\) when \(j\ge J\). The earlier errors have a finite maximum \(C_J\), and \(E_N\le\max\{C_J,\delta Nl\}\). Let \(N\to\infty\) and then \(\delta\downarrow0\). This argument controls random first and last marks without replacing them by deterministic endpoints.

Let \(m_N\) be the number of marks in \(\{k_0,\ldots,k_N\}\). If \(m_N\ge2\), list their indices as \(i_1<\cdots<i_{m_N}\). Summing the comparison over consecutive marks and using additivity gives \[\begin{split} b(m_N-1) &\le A_{i_1}-A_{i_{m_N}}\\ &=wl(i_{m_N}-i_1)+e_{i_1}-e_{i_{m_N}} \le wNl+2E_N. \end{split}\] If \(m_N=0\) or \(1\), the same upper bound applies to \(b(m_N-1)_+\). Therefore, in all cases, \[ bm_N\le b+wNl+2E_N, \qquad \limsup_{N\to\infty}\frac{m_N}{Nl}\le\frac wb \quad\text{almost surely}. \tag{43}\] The expectation step uses only the deterministic bounds \[0\le\frac{m_N}{Nl}\le\frac{N+1}{Nl}\le\frac2l, \qquad N\ge1.\] Fatou’s lemma applied to the nonnegative variables \(2/l-m_N/(Nl)\) gives the bounded reverse Fatou inequality \[\limsup_{N\to\infty}\mathbb E\frac{m_N}{Nl} \le\mathbb E\left[\limsup_{N\to\infty}\frac{m_N}{Nl}\right] \le\frac wb.\] Equal marginal probabilities imply \(\mathbb Em_N=(N+1)q_{\alpha,\beta}\) by linearity of expectation. Hence \(q_{\alpha,\beta}/l\le w/b\), as asserted. In particular, this proof takes no expectation of either a Busemann error or a Busemann increment. ◻

Proof. Let \(a\) be the bottom of the support of \(G\). It is finite, and \(G(\{a\})=0\). As \(b\downarrow a\) with \(b>a\), the probability \(G([0,b))\) tends to zero and is positive. Choose \(b_0>a\) so that this probability is less than \(1/64\), and choose \(a_0\in(a,b_0)\). The definition of \(a\) gives \(p_a>0\), proving (44).

The edges of a simple path are distinct. For a specified path of length \(j\), independence and a union bound over subsets of its edges bound the probability that at least \(j/2\) of them are unusable by \[2^j q^{j/2}.\] This also bounds the probability of fewer than \(j/2\) usable edges. There are at most \(4^j\) nearest-neighbor paths of length \(j\) from a given vertex, so the union bound over simple paths is at most \(\theta^j\). Summing over \(j\ge n_0\) proves (45); the further union bound over the \((6n+1)^2\) starting vertices proves (46). Finally, the events in (45) decrease as \(n_0\to\infty\), and their probabilities tend to zero. Countability gives the last assertion simultaneously at all vertices. These estimates use no moment of an individual edge weight. ◻

Proof. Fix a unit vector \(v\) and let \(e_k=\{k,k+v\}\) for sites \(k\) on a horizontal grid whose spacing will be chosen below. Use one iid base environment \(t\) and independent samples \(\widehat t_{e_k}\) with law \(G\). The individual environment \(t^k\) equals \(t\) off \(e_k\) and equals \(\widehat t_{e_k}\) on \(e_k\). Each individual environment has precisely the iid law. We take the common full-measure event for the base and all these countably many individual environments, including the fixed-label exclusions at every endpoint in \(D_I\) and \(-D_I\).

For a bin \((\alpha,\beta)_I\), let \(A_e^{\alpha,\beta}(H)\) denote the event in an iid environment that \(e\) is usable, \(\mathcal R_e(H)\) holds, and a bigeodesic whose upper label is in this bin uses \(e\). Mark \(k\) in this bin when \[ t_{e_k}\le a_0 \quad\text{and}\quad A_{e_k}^{\alpha,\beta}(H)\text{ holds in }t^k. \tag{49}\] The original variable \(t_{e_k}\) is independent of the entire individual environment \(t^k\): that environment consists of the independent sample \(\widehat t_{e_k}\) and the other base coordinates. Consequently the common marginal marking probability is exactly \[ q_{\alpha,\beta} =p_a\,\mathbb P\bigl(A_e^{\alpha,\beta}(H)\bigr), \tag{50}\] where \(e\) is any fixed translate of \(\{0,v\}\). The regularity test is measurable in the individual environment, so it does not alter this independence calculation. Dependence between marks at different sites is immaterial.

On a marking event, the one changed edge is increased by at least \(b_0-a_0\). Every witnessing bigeodesic passes through \(k\), and the edge \(e_k\) is the first edge of one of its arms based there. Use that edge as the gain portion of that arm and use the empty portion of the other arm. Thus the modification and both gain portions fit in a ball of radius one, and the total gain is at least \[b=b_0-a_0>0.\] The regularity event in \(t^k\) confines both marked arms with offset \(H\).

We must also establish confinement of the base comparator rays at every marked site; base regularity was not assumed in (49). Let \(q\) be any base ray from \(k\) with label in \(I\) or \(-I\). A simple path from \(k\) can traverse \(e_k\) only as its first edge. Indeed, a later traversal in either direction would revisit \(k\). If \(q\) avoids \(e_k\), all its weights are unchanged and it remains a ray in \(t^k\) by monotonicity. Its label is unchanged by 7, so \(\mathcal R_{e_k}(H)\) in \(t^k\) puts it in the appropriate cone at \(k\).

If \(q\) uses \(e_k\) first, its tail from \(k+v\) avoids \(e_k\) and remains a ray in \(t^k\), with the same label. That tail lies in the corresponding cone at \(k+v\) with offset \(H\). For either choice of sign, the inequality \[\pm(w_2-k_2-v_2)\ge s|w_1-k_1-v_1|-H\] implies \[\pm(w_2-k_2) \ge s|w_1-k_1|-\bigl(H+|v_2|+s|v_1|\bigr).\] Including the initial vertex \(k\), and then the initial edge by convexity, confines the whole base ray with offset at most \(H+1+s\). This proves the needed base comparator control at both fixed labels \(\alpha\) and \(-\beta\), uniformly over all bins. It explains why regularity was required at both endpoints of \(e_k\).

Choose an integer grid spacing \(l=l(H)\) large enough for 19 with \(R=1\), \(H_p=H\), and \(H_c=H+1+s\). It may be chosen so that \[ 2\le l(H)\le C_0(H+1) \tag{51}\] for a chart-dependent constant \(C_0\), uniformly in the unit vector \(v\). The lower bound makes the edges \(e_k\) at distinct grid sites distinct. Any two different marked sites are now sufficiently separated. Applying 20 and (50) to each bin yields \[\mathbb P\bigl(A_e^{\alpha,\beta}(H)\bigr) \le\frac{l(H)}{(b_0-a_0)p_a} \bigl(r(\beta)-r(\alpha)\bigr).\] For the finite partition \(\Pi_j\), let \(K_j(e,H)\) count its occupied bins among the labels counted by \(N_e(H)\). Linearity of expectation and (38) give \[ \mathbb EK_j(e,H) \le\frac{l(H)W_I}{(b_0-a_0)p_a}. \tag{52}\]

On the event excluding \(D_I\) as bigeodesic labels, refinement never loses a counted label, so \(K_j(e,H)\) is nondecreasing. Every finite collection of distinct counted labels is eventually in distinct bins. Thus \[K_j(e,H)\uparrow N_e(H),\] also when the latter is infinite: apply the separation assertion to arbitrarily large finite collections. Monotone convergence in (52), followed by (51), proves (48). Translation invariance and the finite number of edge directions give one constant for all \(e\). ◻

Proof. The counting argument groups forks by label and extracts disjoint forward branches within each group. Positive fork density would force a cubic expected number of usable edge–label pairs in large boxes; uniform confinement and 22 give an arbitrarily small cubic upper bound.

It suffices first to exclude upper forks in one chart. Work on the simultaneous event of 18. Every upper fork label is exceptional by the first test in 15, since two different rays of that label start at its split vertex. Consequently all lower rays with the opposite label coalesce, simultaneously for every label arising in this way. In particular, a finite collection of bigeodesics with one such upper label has a common backwards tail. A vertex sufficiently far along that common tail is a common past root for the entire finite collection.

Let \(F_n\) be the number of vertices in \([-n,n]^2\cap\mathbb Z^2\) that are upper forks for this chart, and let \[p_{\mathrm{fork}}=\mathbb P(0\text{ is an upper fork for this chart}).\] The chart and the fork definition are translation invariant, so \[ 0\le F_n\le(2n+1)^2, \qquad \mathbb EF_n=(2n+1)^2p_{\mathrm{fork}}. \tag{53}\] Fix \(\varepsilon>0\) and put \(H_n=\lceil\varepsilon n\rceil\). Let \(G_n\) be the intersection of the event \(U_n\) in 21 and the event that \(\mathcal R_e(H_n)\) holds for every edge with both endpoints in \([-2n,2n]^2\). Uniform all-ray confinement in 12 gives the latter event eventually almost surely: all its endpoint sites lie in a box of radius a fixed multiple of \(n\), and their required offsets are \(o(n)\). Therefore, for this fixed \(\varepsilon\), \[ \mathbb P(G_n^c)\longrightarrow0. \tag{54}\] No rate uniform in \(\varepsilon\) is asserted or needed.

We claim the following deterministic counting inequality on \(G_n\), for \(n\ge2\): \[ \frac n4 F_n \le\sum_{\substack{e:\text{both endpoints in }[-2n,2n]^2}} N_e(H_n). \tag{55}\] Choose a witnessing pair at each of the finitely many fork sites counted by \(F_n\), and group the choices according to their exact upper label. These finite choices are only used to prove a pathwise inequality; no measurable enumeration of labels is required.

Consider a group containing \(q\ge1\) distinct fork sites. There are at most \(2q\) witnessing bigeodesics in this group. Their lower rays coalesce by 18. Pairwise coalescence of a finite collection gives a common tail by going beyond all their finitely many merging locations. Choose a vertex \(o\) on this tail, earlier than every selected split, and orient each witness forwards from \(o\) toward its upper end.

The union of these forward paths is a rooted tree. To justify this directly, the initial path from \(o\) to any vertex \(z\) on a witness is a finite geodesic. If two witnesses visit \(z\), finite uniqueness makes their entire paths from \(o\) to \(z\) equal. They therefore cannot diverge and later meet, and common edges cannot carry conflicting forward orientations. Each visited vertex other than \(o\) has one parent, and the resulting graph has no cycle.

Let \(\mathcal T\) be the finite subtree obtained by taking the union of the root paths to the two selected children immediately after every selected split. Each of the \(q\) split vertices has at least two children in \(\mathcal T\). For any finite rooted tree with \(L\) leaves, \[ L-1=\sum_{u:\,d^+_{\mathcal T}(u)>0} \bigl(d^+_{\mathcal T}(u)-1\bigr), \tag{56}\] where \(d^+_{\mathcal T}(u)\) is its number of children. Indeed, the sum of the outdegrees is the number of edges, one less than the number of vertices; subtracting the number of nonleaves gives (56). Thus \(L\ge q+1\).

Every leaf of \(\mathcal T\) is an endpoint of one of the root-to-child paths defining \(\mathcal T\). Otherwise a defining path containing that leaf would continue to another vertex of \(\mathcal T\), contradicting its being a leaf. In particular, every leaf is a selected child of a fork site, and hence lies in \([-n-1,n+1]^2\). The root paths are allowed to leave and reenter both boxes; only their terminal children determine this location of the leaves. Selected split vertices may also be ancestors of one another, but each is still a distinct vertex with at least two children and contributes at least one to (56).

For each leaf choose a witnessing bigeodesic containing its defining root-to-child path, and continue forwards along that witness. These continuations are pairwise vertex disjoint. In fact, if two met at \(w\), uniqueness of the finite geodesic from \(o\) to \(w\) would put both leaves on a single root path. They would be comparable in the ancestor order, contrary to being distinct leaves of \(\mathcal T\). The same reasoning rules out an intersection between one continuation and the root path to another leaf: its own leaf would then be an ancestor of the other one. An intersection with an earlier point of its own root path contradicts simplicity. These conclusions hold throughout the plane, including parts of \(\mathcal T\) outside either box.

Each leaf has infinity-norm distance at least \(n-1\) from the boundary of \([-2n,2n]^2\). Its continuation eventually reaches this boundary, since an upper ray in the chart has height tending to \(+\infty\). Stop the continuation at its first boundary visit. The resulting path is simple, all its edges have both endpoints in the outer box, and its length is at least \(n-1\ge n/2\). Its starting point is in \([-3n,3n]^2\), so \(U_n\) supplies at least half its edges as usable. It therefore supplies at least \(n/4\) usable edges inside the outer box. Every such edge lies on the witnessing bigeodesic and satisfies \(\mathcal R_e(H_n)\) on \(G_n\). It contributes the group’s label to \(N_e(H_n)\).

For this group the leaf continuations are disjoint, so no edge is credited twice with that label. Groups with different labels credit different pairs \((e,\rho)\), even if they use the same geometric edge. Thus the count is of distinct edge–label pairs. Summing over the groups credits at least \[\frac n4\sum_{\text{groups}}L \ge\frac n4\sum_{\text{groups}}q =\frac n4 F_n\] pairs, proving (55).

We now take expectations, retaining the scale of the bad-event loss. By (53), \[\mathbb E[F_n\mathbf 1_{G_n}] \ge (2n+1)^2 \bigl(p_{\mathrm{fork}}-\mathbb P(G_n^c)\bigr).\] The loss after multiplication by \(n/4\) is at most \[ \frac n4(2n+1)^2\mathbb P(G_n^c)=o(n^3) \qquad\text{for fixed }\varepsilon. \tag{57}\] There are \(2(4n)(4n+1)\le40n^2\) edges with both endpoints in the outer box. From (55) and 22, \[ \frac n4(2n+1)^2 \bigl(p_{\mathrm{fork}}-\mathbb P(G_n^c)\bigr) \le 40C n^2\bigl(\lceil\varepsilon n\rceil+1\bigr). \tag{58}\] Divide by \(n^3\) and let \(n\to\infty\) for this fixed \(\varepsilon\). Equations (54) and (58) yield \(p_{\mathrm{fork}}\le40C\varepsilon\). Sending \(\varepsilon\downarrow0\) gives \(p_{\mathrm{fork}}=0\). Countability of vertices excludes every upper fork in this chart almost surely. Reversing upper and lower gives the same conclusion for lower forks, and the countable union over charts proves the proposition. ◻

Proof. Take the common full-measure event of [prop:opposite-coalescence,prop:no-forks] and the structural properties. Let the two bigeodesics have upper label \(\rho\). If their upper Busemann functions differ, \(\rho\) passes the second exceptional label test of 15. The simultaneous conclusion of 18 makes their lower rays coalesce. Choose a vertex \(o\) sufficiently far along their common lower tail. The forward portions from \(o\) are upper geodesic rays. They cannot have the same tail, since tail invariance in 3 would make their upper Busemann functions equal. They must therefore have a first split. Finite uniqueness prevents any later meeting, so this split is an upper fork with a common past, contrary to 23.

If the lower Busemann functions differ, the lower label \(-\rho\) is exceptional by the same second test. Apply 18 with upper and lower reversed to coalesce the upper rays. Reversing the preceding argument then produces a forbidden lower fork. This proves both equalities in (59).

All these deductions are pathwise on the stated common event. They do not take an uncountable union of fixed-label null events, nor assert coalescence of every one-sided family at random labels. Absolute continuity transfers the event to each individual changed environment; countable intersection gives the final assertion. ◻

Proof of 1. Suppose bigeodesics exist with positive probability. Each bigeodesic contains a lattice vertex. Countability of the vertices and the charts in 10, together with the exclusion of their deterministic exceptional labels, yields a chart in which a bigeodesic through some fixed vertex has positive probability. Translate that vertex to zero and use the lattice coordinates of the chart. By 12, its arms are contained in \(Q^\pm(0,H)\) for some finite \(H\) almost surely. Taking the countable union over integer \(H\) gives a deterministic \(H_0\geq1\) for which the event \(\mathcal B(0)\) in 7 has probability \(p>0\). Removing the countably many deterministic partition endpoints does not change this probability, by the fixed-label backward-finiteness assertion in 4.

Apply 28 for each positive integer \(M\), with the fixed spacing \(l\) from (75). We next impose two filters on its marks. The first supplies the base comparator rays required for 19; the second supplies their required separation within each bin.

For a horizontal site \(k\), let \(\mathcal R_K(k)\) be the event that, in the unchanged base environment, every ray from \(k\) with label in \(I\) lies in \(Q^+(k,K)\) and every ray from \(k\) with label in \(-I\) lies in \(Q^-(k,K)\). By 12, \[\mu(\mathcal R_K(k)^c)\longrightarrow0\qquad(K\longrightarrow\infty).\] Choose one deterministic \(K\geq1\), independent of \(M\), such that this failure probability is at most \(p/8\). It is the same at all horizontal sites by translation invariance. Retain a mark when \[\widetilde J_M(k)=J_M(k)\mathbf1_{\mathcal R_K(k)}=1.\] Unconditional subtraction, without any independence assertion, gives \[ q_M:=\mathbb P(\widetilde J_M(k)=1) \geq\mathbb P(J_M(k)=1)-\mu(\mathcal R_K(k)^c) \geq p/8. \tag{80}\] This filter is imposed after selecting the paths. Its complement has an existential ray witness, so \(\mathcal R_K(k)\) is a universally measurable environment event by 8. It remains a separate filter; it is not placed inside an existential path projection. In particular no new alternation of path quantifiers is used to define the selected marks. The translated rule continues to give equal marginals along the grid.

The near radius \(R_0\), spacing \(l\), and comparator offset \(K\) are fixed independently of \(M\). For each \(M\) we choose \(R(M)\) and the comparison separation below, refine the partition, and only then increase \(M\). Fix \(M\) for the moment. Its radius \(R(M)\) is a fixed finite number. The selected arms have offset \(H_0\), the base comparator rays have offset \(K\), the changed region has radius at most \(D_{R(M)}\) from (74), and the prescribed gain portion has radius at most \(R(M)\). The comparison in 19 therefore holds at a finite deterministic horizontal separation \(S(M)\). The same \(S(M)\) works for every bin of every partition \(\Pi_j\): the cone bounds control all the relevant comparator labels in the chart, and the geometric bounds on the changed region and gain portion are independent of the bin. Increase \(S(M)\) if necessary to be nonnegative. Its possible growth with \(M\) has no effect on the already fixed grid spacing \(l\).

For the finite partition \(\Pi_j\), retain a site with \(\widetilde J_M(k)=1\) only if there is no other site \(y\) with \(\widetilde J_M(y)=1\) and \[0<|y-k|\leq S(M)\] whose selected label lies in the same bin as \(\rho_k\). Here \(k,y\) lie on the horizontal grid, so the displayed distance is their horizontal separation. Denote this final indicator by \(J_{M,j}(k)\) and its common marginal probability by \(q_{M,j}\). The test concerns at most \(2\lfloor S(M)/l\rfloor\) other sites and is measurable from the previously selected labels and marks. At a fixed \(j\), two surviving marks in the same bin have separation strictly greater than \(S(M)\), so 19 applies to consecutive surviving marks of that bin with gain \(b=M\).

For completeness, the disappearance of this last filter is a pointwise finite-set argument. On the common full-measure event of 7.3, fix an outcome with \(\widetilde J_M(k)=1\). By 30, every marked site \(y\ne k\), and in particular every one of the finitely many retained neighbors within distance \(S(M)\), has \(\rho_y\ne\rho_k\). The nested partitions eventually put each such pair in different bins. The maximum of their finitely many separating indices is finite, so \(k\) survives the last filter for every subsequent partition. If \(\widetilde J_M(k)=0\), all of its final indicators are zero. We have therefore proved \[ J_{M,j}(k)\longrightarrow\widetilde J_M(k) \quad\text{almost surely as }j\longrightarrow\infty, \qquad q_{M,j}\longrightarrow q_M, \tag{81}\] where the probability convergence follows by bounded convergence. No rate of separation of random labels is required.

For a bin \((\alpha,\beta)_I\) of \(\Pi_j\), write \(q_{M,j}^{\alpha,\beta}\) for the probability that a given site survives and its selected label lies in that bin. 20, with \(b=M\), gives \[q_{M,j}^{\alpha,\beta} \leq\frac lM\bigl(r(\beta)-r(\alpha)\bigr).\] Every selected label avoids all partition endpoints, so each surviving mark belongs to exactly one bin. Summing over the finitely many bins and using their total width \(W_I\) yields \[ q_{M,j}\leq\frac{lW_I}{M}. \tag{82}\] The label-budget bound uses only equal marginal marking probabilities and the pathwise comparison. In particular the dependencies introduced by the shared raises and both filters are permitted.

Now send \(j\) to infinity for this fixed \(M\). Equations (80), (81), and (82) imply \[\frac p8\leq q_M\leq\frac{lW_I}{M}.\] All of \(p,l,W_I\) were fixed before \(M\) was chosen. Taking any integer \(M>8lW_I/p\) is a contradiction.

The reduction to a chart involved countably many vertices and charts. Within the construction the excluded deterministic labels, integer radii, sites, and union metrics form countable collections; equality of the Busemann pairs and distinctness of marked labels hold pathwise for their random labels. Thus the contradiction excludes the existence event itself and proves almost-sure absence of all bigeodesics, with no prescribed deterministic direction. ◻

Proof. Choose a vertex \(a\) after the last exit of the target ray from \(H\). The segment from \(a\) to each later \(v_j\) minimizes for both metrics, so \(T_H(a,v_j)=T(a,v_j)\). Each of the sequences \[T_H(x,v_j)-T(a,v_j),\qquad T(x,v_j)-T(a,v_j)\] is decreasing by the triangle inequality along this calibrated tail. They are bounded in absolute value by \(T_H(x,a)\) and \(T(x,a)\), respectively. These bounds are finite because the half-plane lattice is connected and all edge weights are finite. Their limits therefore exist and are finite. Their difference is nonnegative, since \(T_H\geq T\), and defines \(D_{\mathcal G}(x)\).

Any two rays of the family have a common tail, so changing the target ray leaves the limit unchanged. A translation by \(kz\) preserves \(H\) and replaces the target ray by another translated ray of the same family. This proves covariance, hence stationarity. No choice of the auxiliary anchor \(a\) enters the definition of \(D_{\mathcal G}\). Measurability follows from the measurable ray choice and the countable limits defining the distances. Subtracting the two distance limits at \(x\) and \(y\) proves (83) and the existence of \(B_{\mathcal G}^H\). ◻

Proof. Set \(X_k=\min\{\Delta(k,k+1),1\}\). Then \[0\leq A_n:=\frac1n\sum_{k=0}^{n-1}X_k \leq\frac{\Delta(0,n)}n,\qquad A_n\leq1.\] Thus \(A_n\to0\) in probability and \(\mathbb EA_n\leq\varepsilon+\mathbb P(A_n>\varepsilon)\to\varepsilon\) for every \(\varepsilon>0\). Stationarity gives \(\mathbb EA_n=\mathbb EX_0\) for all \(n\), so \(X_0=0\) almost surely. Stationarity, countability and additivity prove the assertion for every pair. Neither integrability of \(\Delta\) nor ergodicity is needed. ◻

Proof. Let the two families be \(\mathcal G,\mathcal G'\), with common functional \(\rho\). By [2], their contact sector is contained in the interior of one of the eight half-planes \(H\). Both families eventually enter every translate of \(H\).

We first recall the geometric part of [2], which does not use its subsequent expectation assertion. There is positive probability that the ray from the origin stays in \(H\): on an eventual-entrance ray, a vertex minimizing the normal coordinate starts a tail contained in its translated half-plane. Finite uniqueness and coalescence identify this suffix with the selected ray from that vertex. Countability of the vertices and covariance give positive probability at a fixed vertex. Boundary translations of an iid environment are ergodic, also for diagonal boundaries. Hence there are such starting vertices in both directions along the boundary.

Choose two of these roots on opposite sides of the origin. The two contained rays coalesce, and their portions up to their first common vertex form a finite simple connecting arc. If the origin lies on this arc, its selected suffix is already contained in \(H\). Otherwise, choose consecutive boundary contacts of the arc on opposite sides of the origin. The intervening subarc, together with the boundary segment, encloses a bounded region adjacent to the origin. Choose a vertex \(w\) on the common tail of the two bounding rays and the target ray, outside this region and beyond the connecting arc. Every restricted minimizer from the origin to a target beyond \(w\) must meet the separating subarc. From that meeting point, the bounding-ray suffix to the target lies in \(H\) and is a full-plane minimizer. Restricted uniqueness forces this suffix, hence passage through \(w\). The prefix to \(w\) is then the fixed restricted minimizer to \(w\). This proves stabilization and coalescence with the full-plane family, including when the bounding rays revisit the boundary. Applying the construction at every boundary root gives \(\mathcal G_H,\mathcal G'_H\); their common Busemann functions on the boundary are exactly the restricted limits in 31.

Suppose that the families differ with positive probability. Because each restricted family coalesces, the event that the two families coalesce is invariant under boundary translations. If they coalesce, finite uniqueness makes their rays from any common root identical. Boundary ergodicity therefore makes distinctness an almost-sure event. The same invariance and ergodicity make their asymptotic planar order deterministic, as in [2]. Orient the boundary vector \(z\) so that the finite crossing comparison in equation (12) of that source gives \[ \Delta(k,l):=B_{\mathcal G}^H(kz,lz) -B_{\mathcal G'}^H(kz,lz)\geq0, \qquad k<l. \tag{84}\] For clarity, the comparison chooses a crossing of \(\mathcal G_H(kz)\) with \(\mathcal G'_H(lz)\) and common-tail vertices \(x\) of the primed rays and \(y\) of the unprimed rays, both beyond the crossing. Swapping the two pieces at the crossing gives walks \(\pi_x\) from \(kz\) to \(x\) and \(\pi_y\) from \(lz\) to \(y\), and \[\Delta(k,l) =\bigl(T_H(\pi_y)-T_H(lz,y)\bigr) +\bigl(T_H(\pi_x)-T_H(kz,x)\bigr).\] Both summands are nonnegative finite excess costs. This identity is purely geometric and does not use expectations.

Put \(Z(x)=D_{\mathcal G}(x)-D_{\mathcal G'}(x)\). Equation (83) gives \[\Delta(0,n)=B_{\mathcal G}(0,nz)-B_{\mathcal G'}(0,nz) +Z(0)-Z(nz).\] The first difference divided by \(n\) tends to zero almost surely, by the full-plane linearity and the equality of the two functionals. For each \(\varepsilon>0\), finiteness and stationarity give \[\mathbb P(|Z(nz)|>n\varepsilon)=\mathbb P(|Z(0)|>n\varepsilon)\longrightarrow0.\] Consequently \(\Delta(0,n)/n\to0\) in probability. The process \(\Delta\) is additive and stationary under boundary shifts, so 32 and (84) imply \(\Delta(k,l)=0\) for every pair, almost surely.

The final uniqueness argument is the finite one in [2]. Distinct same-root restricted rays have a finite common initial portion. Choose a deterministic bound on its length that holds with positive probability; boundary ergodicity gives two such roots farther apart than twice this bound. In the crossing comparison just described, at least one swapped walk then differs from the corresponding minimizing path. Indeed, if both swapped walks were their unique geodesics, their crossing vertex would lie in the common initial portion from each root. If these portions have at most \(m\) edges, the lattice distance between the roots would be at most \(2m\), contrary to their choice. Since both nonnegative excesses sum to zero, a distinct swapped walk is also minimizing, contrary to uniqueness. Thus the restricted families agree. Their coalescence with the full-plane families and finite uniqueness imply \(\mathcal G=\mathcal G'\). ◻