Proof. Uniform ellipticity implies the strict ellipticity assumed by the companion’s directional zero–one law. It therefore upgrades \(P_0(A_\ell)>0\) to \(P_0(A_\ell)=1\). Since \(A_\ell=A_{\ell/\|\ell\|}\), Theorem 1 gives the stated velocity and positive projection.

Fix this vector \(v\) and a direction \(u\in S^{d-1}\). If \(v\cdot u>0\), the vector law of large numbers gives \(X_n\cdot u/n\to v\cdot u>0\) almost surely, hence \(P_0(A_u)=1\). Conversely, if \(P_0(A_u)>0\), the preceding argument applied to \(u\) gives a deterministic vector \(w\) with \(X_n/n\to w\) almost surely and \(w\cdot u>0\). The two probability-one convergence events intersect, and uniqueness of a vector limit gives \(w=v\). Thus \(v\cdot u>0\). These two implications prove the equality of direction sets and the assertion on the closed opposite hemisphere. ◻

Proof. The global height minimum is attained almost surely. Thus for some deterministic \(n,x\) the event that \(X_n=x\) and that all later heights are at least \(h(x)\) has positive probability. The quenched Markov property at \(n\) gives \[E\big[P_{0,\omega}(X_n=x)P_{x,\omega}(D_h)\big]>0.\] Since the first factor is at most one, translation invariance gives \(p_h>0\).

At a first strict record, every earlier departure row has smaller height. The fresh-row rule therefore gives conditional raw probability \(p_h\) that the record is true. Test the first positive-time strict record. If an actual drop later detects its failure, test the next strict record above the whole past observed by that failure time. All candidates and failure detections are path stopping rules. Transience supplies another candidate after every finite failure, and each candidate has success probability \(p_h\). Thus a true record is eventually found almost surely.

For the iid assertion, enumerate the possible first words. A word from \(0\) to \(z\) has positive length, every preterminal height in \([0,h(z))\), and every intermediate strict record invalidated by a drop before its end. Together with a suffix staying at or above \(h(z)\), these conditions say exactly that this is the first word under \(D_h\). Indeed a later suffix above \(h(z)\) cannot invalidate any earlier lower record. The prefix weight and the suffix factor \(p_h\) use disjoint departure rows. Dividing by the initial conditioning probability \(p_h\) leaves the raw prefix weight as the word probability and an independent translated suffix of the original conditioned law. Countable enumeration and iteration give the iid words and infinitely many true boundaries. For the first raw true record the same argument applies, without the restriction that its prefix have nonnegative height. This argument does not apply an annealed Markov property at a future-dependent cut.

Under \(D_h\), a fixed positive ordinary record index \(i\) is true with probability equal to the raw probability of reaching that record before dropping below zero: the fresh suffix contributes \(p_h\), which cancels the conditioning. This probability is at least \(p_h\), since all record indices are reached on \(D_h\). The true indices are a renewal process with iid increments \(S_*\). Its positive-index count through \(m\), divided by \(m\), converges almost surely and in expectation to \(1/\mathbb E S_*\). This remains true with limit zero when the mean is infinite, by the strong law on truncations; the ratios are bounded by one. The preceding lower bound on each indicator therefore yields \(\mathbb E S_*\le p_h^{-1}\). Each ordinary record increases height by at most one, so \(L\le S_*\). This argument uses record indices, not a lattice structure for the values of \(h\). ◻

Proof. Suppose instead that \(\mathbb E R=\infty\), and put \(I(s)=\mathbb E\min(R,s)\). This function is increasing, concave, unbounded, and satisfies \(I(s)=o(s)\), since \(R\) is finite almost surely. Write \(H_k=\sum_{j=1}^kL_j\), with \(H_0=0\).

A family of likely finite templates. The width strong law gives constants \(c>0\), \(C\), and \(C_0\) such that \[ck-C_0\le H_k\le Ck+C_0\qquad(k\ge0)\] holds with probability greater than \(.95\). Fix an integer \(C_1\) with \(C_1c>C+4\), then \(b>1/c\), and then \(\eta>0\) so small that \(4C_1\eta<.1\). For large \(a\), choose the least positive integer \(N\) with \(I(aN)\ge\eta a\). Then \(N\to\infty\), and minimality and concavity give \[\eta a\le I(aN)<2\eta a\] for all sufficiently large \(a\).

With probability greater than \(.8\), the first \(C_1N\) words satisfy the height bounds above and \[\sum_{j=1}^kR_j\le ak\qquad(1\le k\le C_1N). \tag{2}\] Here is the maximal estimate behind this assertion. Truncate each radius at \(aC_1N\). Its expectation is at most \(C_1I(aN)<2C_1\eta a\). The union bound for a truncation among the first \(C_1N\) radii is therefore at most \(2C_1\eta\). For a stationary nonnegative sequence \((Z_j)\), \[\Pr\!\left(\max_{1\le k\le n}\frac1k\sum_{j=1}^kZ_j>a\right) \le \frac{\mathbb E Z_1}{a}.\] To see this, mark the starts among \(1,\ldots,M\) admitting a witness interval of length at most \(n\) with average greater than \(a\). Select such intervals greedily from left to right. They are disjoint and cover all marked starts, so their total sum is greater than \(a\) times the number of marked starts. The selected intervals lie in \(1,\ldots,M+n\). Take expectations, divide by \(M\), and let \(M\to\infty\) to remove the boundary contribution. Applied to the truncated radii, this costs at most another \(2C_1\eta\). Together with the height event this proves the asserted \(.8\) bound. This is the nonnegative-sequence form of Hopf’s maximal ergodic inequality; see Garsia (Garsia 1965).

The polynomial lower bound. Fix a large \(B>1\). Choose \(i_0\) so that \[\sum_{i\ge i_0}\Pr(L>i)<\frac{\eta}{4B},\qquad C(i-1)+C_0+i\le(C+2)i\quad(i\ge i_0).\] Choose \(a\) sufficiently large that \(N\ge i_0\) and \(I(Bai_0)<\eta a/4\). Tail integration, followed by removal of the width exceptions, gives \[\frac{\eta}{2B}\le \lambda:=\sum_{i=i_0}^N\Pr(R>Bai,\ L\le i) \le2\eta.\] For the lower bound, the unrestricted sum is at least \([I(Ba(N+1))-I(Bai_0)]/(Ba)\ge3\eta/(4B)\); for the upper bound it is at most \(I(BaN)/(Ba)\le I(aN)/a\).

Count the indices \(i\) at which this large-word event occurs and the preceding words satisfy the height and radius bounds through \(i-1\). Independence gives mean at least \(.8\lambda\). Dropping the preceding-word restrictions bounds the second moment by \(\lambda+\lambda^2\). The count is therefore positive with probability at least \(c_2/B\), where \(c_2>0\) does not depend on \(B\) or \(a\). At some point of that word, \[\|X_n\|>(B-1)ai, \qquad 0\le h(X_j),\quad M_n:=\max_{j\le n}h(X_j)\le(C+2)i.\] The height bound holds throughout this prefix because every word ends at a strict record and has no earlier height above its endpoint. One of the \(2d\) signed coordinate vectors \(u\) thus satisfies \[\frac{Y(X_n)}{a(1+M_n)}>c_3B, \qquad Y(x)=u\cdot x+ba h(x), \tag{3}\] for a fixed \(c_3>0\) and sufficiently large fixed \(B\). Consequently, under the raw law the union of these events over \(u\) has probability at least \(p_hc_2/B\), before the path leaves \(\{h\ge0,\ M\le(C+2)N\}\).

A fresh continuation caps the ratio. Fix \(u\) and put \(Q_n=Y(X_n)/[a(1+M_n)]\). Let \(\zeta\) be the exit time just described. We claim that constants \(\epsilon_0>0\) and \(K<\infty\), independent of \(B,a\) and the threshold \(t>0\), have the following property: at every finite prefix ending at the first crossing \(\sigma<\zeta\) of \(Q>t\), a continuation of conditional raw probability at least \(\epsilon_0\) exits the strip before \(Q\) exceeds \(t+K\).

At the crossing, \[Y(X_j)\le ta(1+M_j)\le ta(1+M_\sigma)<Y(X_\sigma) \quad(j<\sigma).\] Thus the arrival is a strict \(Y\)-record. A step changes \(Y\) by at most \(ba+1\), so \(Q_\sigma\le t+b+1\) when \(a\ge1\). Choose an integer \(M_0'>C_0\) with \(M_0'b>1+bC_0\). First prescribe \(M_0'\) steps of height \(+1\). Each increases \(Y\) by at least \(ba-1>0\). At their endpoint impose a translated \(D_h\)-suffix whose first \(C_1N\) words satisfy the template above. In its word \(k+1\), every site has \(Y\)-increment from the launch at least \[M_0'(ba-1)+ba(ck-C_0)-a(k+1)>0.\] This holds uniformly in \(0\le k<C_1N\) for large \(a\): the coefficient of \(k\) is \(a(bc-1)>0\), and the constant term is positive by the choice of \(M_0'\). These words therefore avoid every old departure row. Their terminal true cut has height at least \[h(X_\sigma)+M_0'+cC_1N-C_0>(C+2)N.\] The entire remaining suffix stays above this cut, and so also avoids all old departures. The template stays above its own starting height, which is strictly above every script departure. Thus the whole infinite continuation uses no old departure row: first the \(Y\) barrier provides separation, and then the terminal height does. The fresh-row factorization gives probability at least \[\epsilon_0=.8\,\kappa^{M_0'}p_h>0.\] Figure 1 records these two separation mechanisms.

It remains to bound the ratio on this continuation. The positive part of the launch coordinate \(u\cdot X_\sigma\), divided by any later denominator, is at most \(t+b+1\), since \(h(X_\sigma)\ge0\) and the denominator is nondecreasing. The later height term contributes at most \(b\). The script adds at most \(M_0'\) to the ratio. In word \(k+1\), the additional coordinate contribution is at most \[\frac{M_0'/a+k+1}{A_0+ck} \le\max\left\{\frac{M_0'+1}{A_0},\frac1c\right\}, \qquad A_0=1+M_0'-C_0>1.\] The numerator uses the bound on the sum of radii through word \(k+1\); the denominator uses the height of that word’s starting boundary. Thus any fixed \[K>2b+1+\max\left\{M_0',\frac{M_0'+1}{A_0},\frac1c\right\}\] proves the claim. Exit occurs by the last prescribed template endpoint.

The exponential upper bound. Let \(\sigma_j\) be the first crossing before \(\zeta\) of \(t_j=1+jK\). The continuation just constructed prevents \(\sigma_{j+1}<\zeta\). Summing its bound over the finite prefixes at \(\sigma_j\) yields \[P_0(\sigma_{j+1}<\zeta) \le(1-\epsilon_0)P_0(\sigma_j<\zeta).\] This conditions only on the observed crossing prefix; it does not reveal whether an infinite template has failed. Hence the event in [eq:3] has probability at most \(C'e^{-c'B}\) for this \(u\). Taking the union over signed coordinates contradicts the lower bound \(p_hc_2/B\) when \(B\) is large. All estimates use the permitted order: fix the geometry constants, then \(B\), then \(i_0\), then take \(a\) large. This proves [eq:1]. ◻

Proof. Word endpoints obey the vector strong law, with mean displacement \(w\) satisfying \(h(w)=\mathbb E L>0\). Proposition 4 also gives \(R_k/k\to0\) almost surely: for each \(\varepsilon>0\), the probabilities \(\Pr(R_k>\varepsilon k)\) are summable. Thus interpolation inside each word gives the same limiting direction \(r=w/\|w\|\), without any duration moment. The completed-word count tends to infinity. Lemma 3 transfers the conclusion through the finite initial raw prefix.

Some signed coordinate of \(r\) is positive. Relabel it as the first coordinate, preserving the lattice, uniform ellipticity, and the iid row law. Then \((X_n)_1\to+\infty\) almost surely. The proofs of Lemma 3 and Proposition 4 apply again with this coordinate height, using only its transience. We retain the same notation for the transformed walks and laws. ◻

Proof. Uniform ellipticity excludes confinement forever in a fixed finite-height strip, even quenched: blocks of sufficiently many consecutive upward steps have a fixed positive chance. Hence \(a_H(x)\downarrow q(x)\). Also \(q(je_1)\) uses only layers at or above \(j\), so the event that \(q(je_1)>0\) infinitely often is a tail event of the independent layers. For every \(m\), the union of these events over \(j\ge m\) has probability at least \(P(q(0)>0)>0\), by stationarity and \(Eq(0)=p>0\). The tail zero–one law makes their limsup almost sure. A finite upward script connects zero to one such site and gives \(q(0)>0\). Translation and countability give positivity at every site simultaneously.

At layer \(H\), let \(\mu_H=K_H(0,\cdot)/a_H\) be the endpoint probability distribution. It is \(\mathcal F_H\)-measurable, and the quenched Markov property, restricted to a suffix never below layer \(H\), gives \[q(0)\ \ge\ a_H\int q(x)\,\mu_H(dx).\] Each \(q(x)\) on that layer has the original marginal law independently of the lower rows. If \(q(0)/a_H<s\), then more than half the \(\mu_H\)-mass has \(q(x)\le2s\). Conditional expectation followed by Markov’s inequality therefore gives \[P\big(q(0)/a_H<s\mid\mathcal F_H\big) \le2P\big(q(0)\le2s\big).\] The endpoint values \(q(x)\) need not be independent of one another. Partitioning by a finite stopping layer gives the same conditional bound there.

Choose \(0<\delta<\kappa\) sufficiently small that \(\rho:=2P(q(0)\le2\delta/\kappa)<1\), and set \[H_j=\inf\{H\ge0:a_H<\delta^j\},\qquad j\ge1.\] These are stopping layers. One upward step after a successful crossing gives \(a_{H+1}\ge\kappa a_H\), so on \(H_j<\infty\), \(a_{H_j}\ge\kappa\delta^j\). The event \(H_{j+1}<\infty\) implies \(q(0)<\delta^{j+1}\), and hence \(q(0)/a_{H_j}<\delta/\kappa\). The preceding conditional bound yields \[P(H_{j+1}<\infty\mid\mathcal F_{H_j})\le\rho \quad\text{on }\{H_j<\infty\}.\] Since \(a_H\downarrow q(0)\), the event \(H_j<\infty\) is exactly \(\{q(0)<\delta^j\}\). Iteration gives its probability at most \(\rho^{j-1}\). Summing over \(\delta^{j+1}\le q(0)<\delta^j\), and choosing \(\lambda>0\) with \(\rho\delta^{-\lambda}<1\), proves [eq:4]. ◻

Proof. All conclusions below using [eq:5] are conditional on that estimate.

Spatial confinement before exit. The raw path before leaving the height interval \([0,N]\) stays in \(\|\underline X\|_\infty\le CN^2\), except with probability at most \(Ce^{-cN}\). To prove this, fix a signed lateral coordinate \(u\). By Corollary 5, a sufficiently large fixed \(b>0\) makes \(Y(x)=u\cdot x+bx_1\) transient in its positive direction. Thus the event of staying forever at \(Y\ge0\) has positive raw probability, by the minimum argument of Lemma 3.

For a sufficiently large fixed \(C_2\), the path through its first relative height hit \(N+1\) has norm at most \(C_2N\) with probability tending to one. Indeed the ray gives \(\|X_n\|\le(C_2/2)(X_n)_1\) eventually, and the earlier path has a finite maximum norm. The intersection of this event with the preceding no-backtracking event therefore has probability bounded below uniformly for large \(N\).

At a first crossing of \(Y>jC_3N\), before strip exit, the walk is at a strict \(Y\)-record. Impose that favorable translated suffix using the fresh rows at or above the record. If \(C_3\) is sufficiently large, the suffix precludes crossing \((j+1)C_3N\) before strip exit: its \(Y\)-displacement through relative height \(N+1\) is at most \((1+b)C_2N\), the launch overshoot is bounded, and absolute strip exit occurs no later than this relative height hit. The finite-prefix threshold recursion used in Proposition 4, now with spacing \(C_3N\), gives exponential decay through \(N\) thresholds. Since \(u\cdot X\le Y\) before exit, the finite union over signed lateral coordinates proves the confinement assertion.

Super-polynomial width tails.

Now \(\Delta u_N:=u_N-u_{N+1}\ge0\) is the annealed probability of reaching \(N\) before the drop but dropping before \(N+1\). In environments good at all sites of the strip and box (where \(a_{N+1}(x)\ge(N+1)^{-\beta}\)), the portion of this event inside the box has quenched probability at most \[(1-(N+1)^{-\beta})^{N+1}.\] Indeed, after reaching \(N\), at first descending-level visits to each of \(N,N-1,\ldots,0\) still in the box, there is at least the indicated chance to hit top \(N+1\) before the next descent by one. Condition [eq:5], translation, a polynomial union bound, and the box claim thus give super-polynomial decay of \(\Delta u_N\).

The true levels under \(P_0^+\) form the renewal process of iid widths \(L_{\rm s}\), with renewal masses exactly \(u_n\): impose the fresh \(D\)-suffix at the first reach of \(n\) before drop, and cancel \(p\). If \(G_j=P_0^+(L_{\rm s}>j)\), then \(\sum_{j=0}^n G_j u_{n-j}=1\) by the last renewal up to \(n\), so \[G_{n+1}=\sum_{j=0}^n G_j\Delta u_{n-j}.\] Put \(f_k=\Delta u_{k-1}\) for \(k\ge1\). This nonnegative kernel has total mass \(\theta=1-p<1\) and all polynomial moments. The recurrence, with \(G_0=1\), gives \(G=\sum_{r\ge0}f^{*r}\), including unit mass at zero for \(r=0\). If \(\theta>0\), normalize \(f\) by \(\theta\). For every positive integer \(m\), the \(m\)-th moment of its \(r\)-fold convolution is at most \(r^m\) times its \(m\)-th moment, by \((z_1+\cdots+z_r)^m\le r^{m-1}\sum_i z_i^m\). Summing with the geometric weights \(\theta^r\) proves \(\sum_n n^mG_n<\infty\). The case \(\theta=0\) is immediate. In particular the width tail \(G_n\) decays super-polynomially.

Time integrability and velocity. The raw exit time \(T_N\wedge T_{-1}\) exceeds \(N^A\) with super-polynomially small probability, for a sufficiently large fixed \(A\). To see this, use confinement and [eq:5] on a slightly enlarged box and strip, requiring \(a_N(x+e_1)\ge N^{-\beta}\) for every relevant site with \(0\le x_1<N\). A polynomial union bound controls the exceptional environments.

From any visit to such a site \(x\), the quenched probability of no further visit before exit is at least \(\kappa a_N(x+e_1)\): take one upward step and then advance \(N\) before dropping below the new layer. This event forces original strip exit before revisiting \(x\), even though the auxiliary target may be higher than \(N\). The quenched Markov property consequently bounds each site’s visit count by a geometric tail with success probability at least \(\kappa N^{-\beta}\). There are at most \(CN^{2d-1}\) sites in the confined box. If the exit time exceeds \(N^A\), one site has at least \(N^A/(CN^{2d-1})\) visits. For \(A>2d-1+\beta\), the union of these geometric-tail events is super-polynomially small, as are the bad-box and bad-environment events.

On \(D\), if \(L_{\rm s}\le N\), the first word ends at its first arrival to height \(L_{\rm s}\), and hence \(\tau_{\rm s}\le T_N\). Thus \[P_0^+(\tau_{\rm s}>N^A) \le P_0^+(L_{\rm s}>N) +p^{-1}P_0(T_N\wedge T_{-1}>N^A).\] Both terms have super-polynomial decay, proving \(\mathbb E\tau_{\rm s}<\infty\). The iid-word strong laws now give the velocity as mean displacement divided by mean duration; its first coordinate is \(\mathbb E L_{\rm s}/\mathbb E\tau_{\rm s}>0\). The integrable spatial radius controls the inter-endpoint deviations, so this limit holds at every path time. The finite raw initial prefix does not change it. Finally the velocity is a positive multiple of the ray in Corollary 5, which has positive projection on the original \(\ell\). Undoing the signed coordinate permutation proves the stated conclusion. As an almost-sure deterministic limit, this velocity is unique for the law \(\nu\). ◻

Proof. Work first with the raw pair laws. Test a new integer level above both explored path pasts, pausing each walk at its first arrival there. All departure rows revealed by these prefixes lie strictly below the candidate level. The conditional probability that both arrivals are true is \(p^2\) in independent environments and \(p_2\) at the arrival sites in a shared environment. Both probabilities have a uniform positive lower bound.

If the candidate fails, resume the two walks, for example one step of each per round, until one drops below the candidate. This failure is detected in finite time. Then test a level strictly above both explored pasts. Each walk tends to positive infinite height almost surely; this also holds for the shared raw pair by its marginals. Thus, after every detected failure, another candidate can be reached. The uniform success bound proves the existence of a common true level above any prescribed height. Conditioning on the two initial no-drop events preserves this almost-sure conclusion.

For the continuation law, specify a pair of finite words from \(x,y\) to their first hits \(x',y'\) of a common terminal level \(L>0\). Call this word pair admissible when both prefixes avoid the initial drop and every intermediate positive level has a witnessed failure of common truth: at least one prefix has dropped below that level after its first hit. These are exactly the required prefix conditions. Indeed, once truth is imposed at \(L\), any failure of truth at an earlier level must already have occurred in the specified prefixes.

Let \(A_{x,y}^{\rm sh}(w_1,w_2)\) be the raw annealed prefix weight of an admissible word pair. Its quenched weight is the product of the two path weights in the shared environment, including any repeated use of a row. Every departure row lies below the terminal layer. The suffixes constrained by truth use only rows at or above that layer, so site-row independence gives the conditional word probability \[P_{x,y}^{++}\bigl((w_1,w_2)\text{ is the first common word}\bigr) = A_{x,y}^{\rm sh}(w_1,w_2) \frac{p_2(x',y')}{p_2(x,y)}.\] More generally, for any event \(B\) of the suffix pair, its joint probability with this word pair is \[A_{x,y}^{\rm sh}(w_1,w_2) \frac{p_2(x',y')}{p_2(x,y)} P_{x',y'}^{++}(B).\] Here \(B\) is read with the new sites as starting sites. In particular, the factor \(p_2(x',y')/p_2(x,y)\) is part of the shared word law; it does not cancel in general.

In independent environments the corresponding terminal and initial conditioning probabilities are both \(p^2\), so their ratio is one. Translation of each walk separately then makes the relative suffix law \(P_0^+\otimes P_0^+\), independent of the word just traversed. Enumerate the countably many admissible finite word pairs and iterate these factorizations to obtain the stated continuation rules. Summing over a finite stopping index gives the same rules there. The information at such an index does not include any unseen upper environment. ◻

Proof. The probability that a deterministic level \(h\) is true under the single conditioned law is \(u_h\), by the exact-truth identity in Section 2. Independence therefore makes the renewal mass for common words equal to \(u_h^2\ge p^2\). The renewal-count strong law, including its infinite-mean version obtained by truncation, implies \(\bar L\le p^{-2}\). The lower bound follows from \(L\ge1\).

For each of the two paths, let \(N_i\) be the number of its single words used before the first positive common boundary. Since single widths are positive integers, \(N_i\le L\). The event \(N_i\ge j\) is determined by that path’s first \(j-1\) words and the independent boundary sequence of the other path. It is consequently independent of the radius of its \(j\)-th word. The finite mean single-word radius from Section 2 gives \[\mathbb E\sum_{j=1}^{N_i}R_{{\rm s},j}^{(i)} =\mathbb E N_i\,\mathbb E R_{\rm s} \le \bar L\,\mathbb E R_{\rm s}<\infty.\] The absolute value of \(S\) is bounded by the sum of these two total radii. Finally, uniform ellipticity permits two short paths with different second-coordinate displacements to level one. Imposing fresh truth there gives positive probability of an exact common cut with nonzero \(S\).

We record the renewal estimate that supplies local tightness. For either the single renewal sequence or the independent common sequence, let \(L'\) be its word width and \(A\) any almost surely finite size of the word, such as its maximal displacement. If \(A_{\rm word}\) is the size of the word based at the last boundary at or below a deterministic level \(h\), then \[\Pr(A_{\rm word}>s) \le \sum_{j=0}^h\Pr(L'>j,\ A>s) \le \mathbb E[L'\mathbf1_{\{A>s\}}].\] To obtain the first inequality, sum over the possible preceding boundary levels \(h-j\), use independence of the next word, and bound each renewal mass by one. Since \(\mathbb E L'<\infty\), the final quantity tends to zero as \(s\to\infty\). The same reasoning with \(A=L'\) gives tightness of the covering width. Thus first-hit positions at a deterministic layer differ from their preceding boundary positions by tight amounts. A fixed interval of integer levels meets only the words covering those levels, so it also has tight total displacement, uniformly in its base. Shared marginals inherit this statement from their uniform domination by the single conditioned law.

It remains to prove [eq:6]. Test the \(m\) levels \[h=H,H+B,\ldots,H+(m-1)B.\] Call failure at \(h\) visible if, after first hitting \(h\), some walk drops below it before its first hit of \(h+B\). In the raw law, the probability that all \(m\) tests have visible failure is at most \((1-\underline p_2)^m\). Indeed, at each pair of first-hit arrivals to \(h\), all previous visible-failure tests are already determined, while both fresh truths at \(h\) would preclude the next failure. Passing to the conditioned pair law costs at most \(\underline p_2^{-1}\).

If \(h\) is not common true but has no visible failure, some walk drops below \(h\) after its first hit of \(h+B\). That walk has no single true boundary between \(h\) and \(h+B\), inclusive; its word covering \(h\) therefore has width greater than \(B\). The covering-word estimate and marginal domination bound this alternative by \[C\mathbb E[L_{\rm s}\mathbf1_{\{L_{\rm s}>B\}}].\] A union bound over the \(m\) tests now gives \[P_{x,y}^{++}\bigl(\sigma(H)>H+(m-1)B\bigr) \le \underline p_2^{-1}(1-\underline p_2)^m +Cm\mathbb E[L_{\rm s}\mathbf1_{\{L_{\rm s}>B\}}].\] Choose \(m\) large and then \(B\) large. This proves uniform tightness of the height overshoot. The fixed-interval displacement bound then proves the other assertions in [eq:6]. Finally, the first common boundary at or above \(H\) is a stopping index of the common-word chain, so its suffix rule follows from Lemma 8. ◻

Proof. The ratio \(m(r)/r^2=\mathbb E[(S^2/r^2)\wedge1]\) is continuous and nonincreasing. There is a bounded interval of strictly positive values of \(|S|\) having positive probability; its contribution is strictly decreasing once \(r\) exceeds that interval. This proves the monotonicity and eventual strict monotonicity of \(n\). Monotonicity of \(m\) proves the first inequality in [eq:7], and \(S^2\wedge r^2\le r|S|\) proves the second, hence \(n(r)\to\infty\).

For [eq:8], replace \(S_j\) by \(S_j\mathbf1_{\{|S_j|\le z\}}\). These truncated variables have mean zero by symmetry. The probability of any replacement among the first \(H\) variables is at most \(H\Pr(|S|>z)\). Off this event the original partial sums agree with the truncated sums, whose expected squared maximum is at most \(4H\mathbb E[S^2\mathbf1_{\{|S|\le z\}}]\) by the square maximal inequality. On the replacement event, the capped squared maximum is at most \(z^2\). Adding these bounds gives the first inequality in [eq:8]; Markov’s inequality for the capped square gives the second. ◻

Proof. In a raw first walk, select the first level \(h\le H\), if any, such that \(T_h<T_{-1}\) and \(|Y_h-b_h|>z\). Let \(A\) be the event that such a level exists. Conditional on this prefix and its selected level, an independent raw second walk reaches \(h\) before dropping, with its displacement on the opposite side of the median \(b_h\), with probability at least \(p/2\). Here the conditioned no-drop law only lower bounds a prefix probability; it is not imposed as a separate test on that second walk.

At the two first-hit endpoints, impose both fresh truths. Their probability factor is \(p^2\). The resulting event implies the two initial no-drop events and has \(h\) as an exact common true level, with \(|\Delta_h|>z\). Dividing its raw probability by \(p^2\), its probability under the independent conditioned pair law is at least \(P_0(A)p/2\). At a common cut of height at most \(H\), the difference is a partial sum of at most \(H\) common-word increments, so \[\Pr(M_H^S>z)\ge \frac p2P_0(A) \ge \frac{p^2}{2}P_0^+(W_H>z).\] This proves [eq:9]. All of these adaptive prefix transfers can also be obtained by summing the corresponding finite stopped-word identities separately over the possible levels \(h\). ◻

Proof. By [eq:7]–[eq:9], choose a fixed large \(C_0\), then \(t\) small enough that \(C_0\sqrt t\le1\), to obtain \[P_0^+(W_{2H}>C_0\sqrt t\,r)<p/4.\] For \(i+j\le2H\), the event that \(i\) is exactly true under \(P_0^+\) has probability \(u_i\ge p\), and its suffix has the fresh conditioned law. There is therefore positive probability on this event that \(Y_i\), \(Y_{i+j}\), and \(Y_{i+j}-Y_i\) are all within \(C_0\sqrt t\,r\) of \(b_i\), \(b_{i+j}\), and \(b_j\), respectively. For the first two requirements use the preceding maximal bound, and for the third use the same bound on the fresh suffix. Consequently, \[b_{i+j}=b_i+b_j+O(\sqrt t\,r) \qquad(i+j\le2H),\] uniformly in the indices.

Set \(d_j=b_j-jb_H/H\) for \(0\le j<H\). Apply the last relation to \(j+j\), and, when \(2j\ge H\), also to \(H+(2j-H)\). This gives \[2d_j=d_{2j\bmod H}+O(\sqrt t\,r)\] with a uniform error. Taking the maximum absolute value over \(0\le j<H\) bounds that maximum by half itself plus \(C\sqrt t\,r\). Since the error at \(j=H\) is zero, this proves [eq:10]. ◻

Proof. On the joint environment-and-path space, \[\frac{d\widetilde P_0}{dP_0^+}=\frac{p}{q(0)}.\] This density is integrable but need not be bounded. For any prescribed error, truncate it and then approximate the truncation by a bounded nonnegative environment function \(g\) depending on finitely many rows. Thus \(g\) approximates \(p/q(0)\) in \(L^1(P_0^+)\), and \(E_0^+g\) tends to one as the approximation error tends to zero. We will remove the finite initial segment that can be affected by \(g\).

First check the deterministic centering error caused by such a removal. If \(\limsup P_0^+(W_H>z/10)\ge0.3\), Equation [eq:11] is immediate with a sufficiently large fixed \(C\). Otherwise these probabilities are eventually less than \(0.3\). For every fixed integer \(s_0\), local tightness then gives \[|b_{h+s}-b_h|\le0.3z \qquad(0\le s\le s_0,\ h+s\le H)\] for all sufficiently large indices in the sequence. Indeed, each of \(|Y_h-b_h|\le z/10\) and \(|Y_{h+s}-b_{h+s}|\le z/10\) has probability greater than \(0.7\). By Lemma 9, the probability of \(|Y_{h+s}-Y_h|\le z/10\) tends to one uniformly over these indices. The three events have a nonempty intersection, which proves the claimed deterministic inequality by the triangle inequality.

Let \(s\) be the first positive single true boundary strictly above every row used by \(g\). Its index in the single-word chain is a finite stopping index. For a sufficiently accurate approximation, \(E_0^+g>0\). Under the probability measure with density \(g/(E_0^+g)\) relative to \(P_0^+\), the translated suffix at \(s\) has law \(P_0^+\), independently of the prefix. To verify this, apply exact-word factorization separately to each finite prefix through the selected boundary. Both its path weight and \(g\) use only rows below that boundary; all rows for the conditioned suffix are fresh. This remains true for rows used by \(g\) that the prefix has not visited. The separation is by height, not by which rows the walk happened to reveal.

For fixed \(g\), first choose a deterministic \(s_0\) so large that \(s\le s_0\) except for an arbitrarily small \(g\)-weighted error. As \(z\to\infty\), we may further restrict to \[\max_{j\le s}|Y_j|\le z/10, \qquad \max_{j\le s_0}|b_j|\le z/10,\] with a further error tending to zero. On these restrictions no offense \(|Y_j-b_j|>z\) can occur before \(s\). After \(s\), write the relative suffix displacement as \(Y'_u=Y_{s+u}-Y_s\). The centering estimate above gives, for \(s<j\le H\), \[|Y_j-b_j| \le |Y_s|+|Y'_{j-s}-b_{j-s}|+|b_{j-s}-b_j| \le0.4z+|Y'_{j-s}-b_{j-s}|.\] Thus \(W_H>z\) requires the fresh suffix to have its own \(W_H>z/10\). If \(H\le s\), no offense remains at all.

The \(g\)-weighted probability is consequently bounded in the limsup by \(E_0^+g\) times \(\limsup P_0^+(W_H>z/10)\), plus the restriction errors. Return to \(\widetilde P_0\) using the \(L^1\) density approximation, and send all approximation and restriction errors to zero. Enlarging the same fixed \(C\) to include the earlier trivial case proves Equation [eq:11].

Finally, in a fixed environment the event \(D\) is a subevent of \(\{T_H<T_{-1}\}\). On first-hit prefixes, changing between these two conditional laws costs total variation at most \(1-q(0)/a_H\). Therefore the distance between \(\widetilde P_0\) on those prefixes and the \(P\)-average of \(K_H/a_H\) is at most \[E(1-q(0)/a_H)\longrightarrow0,\] since \(a_H\downarrow q(0)>0\) almost surely. This proves the last assertion. ◻

Proof. Fix \(v_0>0\), set \(w_0=v_0/8\), and write \(\theta=b_H/H\). The proof uses one estimate for loss of mass in a narrower slope tube, then restarts that estimate after its first failure. All stopping tests will use finite-height kernels.

Proof. Fix \(s>0\), put \(H=\lfloor sn(r)\rfloor\), and let \(k=\lfloor H/\bar L\rfloor\). First we justify the approximation \[\Delta_H-\sum_{i=1}^kS_i=o_{\Pr}(r).\] By Lemma 9, the displacement from the preceding common boundary to the first-hit position at \(H\) is tight. The renewal strong law puts the number of completed common words within \(o_{\Pr}(n(r))\) of \(k\). No independence between this count and the increments is required: on the event that its distance from \(k\) is at most \(an(r)\), bound the sum error by the maximal sums in the deterministic forward and backward windows of that length around \(k\). For every fixed \(\rho>0\), [eq:7]–[eq:8] bound the probability of such an error exceeding \(\rho r\) by \(O_\rho(a)+o(1)\). Let \(a\downarrow0\), and include the tight decoration error, to obtain the approximation.

For \(s=1\), set \[V=\sum_{i=1}^k(S_i^2\wedge r^2).\] Its mean is comparable to \(r^2\), since \(k\sim n(r)/\bar L\), and \[\mathbb E V^2 \le kr^2m(r)+k^2m(r)^2=O(r^4).\] The second-moment inequality therefore makes \(V\) exceed a fixed positive multiple of \(r^2\) with uniformly positive probability. Conditional on the magnitudes \((|S_i|)\), the nonzero signs are independent fair signs. The sum \(\sum_{i=1}^kS_i\) has conditional variance \(Q=\sum_{i=1}^kS_i^2\ge V\) and conditional fourth moment at most \(3Q^2\). A further second-moment inequality gives a uniformly positive conditional probability of a displacement of order \(\sqrt Q\), hence of order \(r\) on the preceding event. The approximation, with a smaller displacement constant, proves the first line of [eq:13].

For the second line, take \(s=t\) and set \(p_r=\Pr(|S|>lr)\). The assumptions and the definition of \(m\) give \[\epsilon\le n(r)p_r\le l^{-2}.\] Among \(k\sim tn(r)/\bar L\) summands, the probability that exactly one has magnitude greater than \(lr\) is therefore at least \(c_{l,\epsilon}t\) for each sufficiently small fixed \(t>0\) and large \(r\). Conditional on the index of that exceptional summand, the others are independent symmetric variables conditioned to have magnitude at most \(lr\). Their sum has variance at most \[\frac{k\,\mathbb E[S^2\mathbf1_{\{|S|\le lr\}}]}{1-p_r} \le Ctr^2.\] For \(t\) sufficiently small, Chebyshev’s inequality gives probability at least \(1/2\) that their sum has magnitude at most \(lr/2\). On this event they cannot cancel more than half the exceptional summand. Thus the deterministic sum has magnitude greater than \(lr/2\) with probability at least a constant times \(t\). Keep \(t\) fixed while \(r\to\infty\); the \(o_{\Pr}(r)\) approximation then proves the claimed bound at threshold \(lr/4\).

Finally, if two iid endpoint displacements differ by more than \(a\), at least one differs from any given deterministic center by more than \(a/2\). The probability of this single-walk absolute deviation is at least half the two-walk separation probability. One of its two signs has at least half that mass. This proves the final assertion after adjusting constants. ◻

Proof. Write \((L_i,S_i)\) for the successive independent common-word widths and difference increments, and put \[C_m=\sum_{i=1}^mL_i,\qquad A_m=\sum_{i=1}^mS_i, \qquad N(h)=\max\{m:C_m\le h\}.\] The difference at the preceding common cut is \(\widehat\Delta_h=A_{N(h)}\). As with the first-hit process, we interpolate its values linearly on the height grid \(h/n(r)\).

The limit at common cuts. First consider \(A_m/r\), with \(n(r)\) summands per unit time. The Gaussian-sequence assumption permits a symmetric truncation \[S_i^{(r)}=S_i\mathbf1_{\{|S_i|\le\eta_r r\}}, \qquad \eta_r\downarrow0,\] for which \[n(r)\Pr(|S|>\eta_r r)\longrightarrow0, \qquad \frac{\mathbb E[(S_i^{(r)})^2]}{m(r)}\longrightarrow1.\] Indeed, for every fixed \(0<\eta<1\), \[0\le m(r)-\mathbb E[S^2\mathbf1_{\{|S|\le\eta r\}}] \le r^2\Pr(|S|>\eta r)=o(m(r));\] a sufficiently slow diagonal choice of \(\eta_r\) gives both requirements. The probability that any summand is removed on a fixed time interval tends to zero.

The variables \(S_i^{(r)}/r\) have mean zero, variance \((1+o(1))/n(r)\), and fourth moment at most \(\eta_r^2\) times that variance. Taylor expansion of their characteristic functions therefore gives the standard Brownian finite-dimensional laws. For tightness, the fourth-moment computation for independent centered sums, followed by the fourth-moment maximal inequality, bounds the expected maximal fourth power on a block of \(\lfloor\delta n(r)\rfloor\) summands by \[O(\delta^2+\delta\eta_r^2).\] A union bound over the blocks in a fixed time interval, followed by \(\delta\downarrow0\), proves tightness. Thus \(A_m/r\) has the standard Brownian functional limit.

The width strong law gives \[\sup_{0\le h\le Tn(r)} \left|\frac{N(h)}{n(r)}-\frac{h}{\bar L n(r)}\right| \longrightarrow0 \quad\text{in probability}.\] Applying this time change to the tight sum process, whose limits are continuous, proves the Brownian limit for \(\widehat\Delta_h/r\), with variance parameter \(1/\bar L\). The counts and the sums need not be independent. The maximal normalized jump tends to zero as well, so the stated interpolation convention has the same limit.

Passing from cuts to every first hit. Lemma 9 makes \(\Delta_h-\widehat\Delta_h\) tight uniformly at deterministic levels \(h\). This already gives the required finite-dimensional limits. To prove path tightness, fix \(\delta>0\) and partition the height range into consecutive blocks of \(\lfloor\delta n(r)\rfloor\) levels. There are only finitely many block starts for fixed \(T,\delta\).

Suppose that within one of these blocks, with start \(a\), \[|\Delta_h-\Delta_a|>\rho r\] for some level \(h\) in the block. Under the raw independent pair law, select the first such offending level among prefixes for which both walks have avoided the initial drop. The selection is determined by the paired prefixes through that level. The raw probability of finding an offense is at least \(p^2\) times its probability under the independent conditioned pair law. Imposing both fresh truths at the selected level multiplies each selected prefix weight by \(p^2\), leaves the offense unchanged, and implies both initial no-drop events. After division by the initial normalizer \(p^2\), both probabilities below are under the conditioned pair law, and \[\Pr(\text{an offense}) \le p^{-2}\Pr(\text{an offense at a common cut}).\] The same statement holds for the union of the blocks by selecting its first offense.

At a common cut \(h\), the offending difference satisfies \[|\Delta_h-\Delta_a| \le |\widehat\Delta_h-\widehat\Delta_a| +|\Delta_a-\widehat\Delta_a|.\] The maximum of the second term over the finitely many deterministic block starts is \(o_{\Pr}(r)\). The first term is controlled by the modulus of the common-cut process, which has a continuous Brownian limit. Consequently, for each \(\rho>0\), the limsup probability of an offense tends to zero as \(\delta\downarrow0\). Adjacent-block bounds and interpolation give tightness of \(\Delta_h/r\), proving the first assertion.

Recovering a single path. Let \(\mu_r\) be the law in continuous path space of the interpolated single-path process \(Y_h/r\), on a fixed compact time interval. Its iid differences are tight by the first assertion. We first show that \(\mu_r\) is tight after suitable deterministic path translations. Choose compact sets \(K_j\) capturing the difference laws with errors \(\varepsilon_j^2\), where \(\varepsilon_j\downarrow0\) and \(\sum_j\varepsilon_j<1\). If a prospective center \(f\) is sampled from \(\mu_r\), then \[\int\mu_r(f+K_j)\,\mu_r(df)\ge1-\varepsilon_j^2.\] Markov’s inequality shows that the centers for which \(\mu_r(f+K_j)<1-\varepsilon_j\) have \(\mu_r\)-probability at most \(\varepsilon_j\). Some center path \(f_r\) therefore works for every \(j\), and the translated laws are tight. We may choose \(f_r\) with the same interpolation as the original paths.

Along a subsequence on which these translated laws converge, let \(G\) be the continuous limiting process. Its iid two-copy difference has the Brownian law just proved. Cramér’s normal decomposition principle (Cramér 1936), in the iid-difference form verified below, applied to every scalar linear combination of finitely many coordinates of \(G\), shows that \(G\) has Gaussian finite-dimensional laws. Its covariance is half that of the difference process, namely \[\operatorname{Cov}(G(s),G(t))=\sigma_*^2\min(s,t).\] Only a deterministic mean function remains to be identified.

For completeness, suppose that \(Z-Z'\) is normal for iid real variables \(Z,Z'\). Choose \(C_0\) with \(\Pr(|Z'|\le C_0)>0\). Intersecting either tail of \(Z\) with this event bounds that tail by a constant times a Gaussian tail. Hence \(f(z)=\mathbb E e^{zZ}\) is entire and obeys \(|f(z)|\le\exp(C(1+|z|^2))\). The identity \(f(z)f(-z)=\mathbb E e^{z(Z-Z')}\), first on the real axis and then everywhere, shows that \(f\) has no zeros. Its entire logarithm \(g\), chosen with \(g(0)=0\), satisfies \(\operatorname{Re}g(z)\le C(1+|z|^2)\). On every circle the mean of \(\operatorname{Re}g\) is zero, so its absolute integral mean is also \(O(1+|z|^2)\). The Fourier coefficient formula on circles of arbitrarily large radius then makes every Taylor coefficient of \(g\) of order greater than two vanish. Thus \(g\) is quadratic, real on the real axis, with \(g''(0)=\operatorname{Var}Z\ge0\). It follows that \(Z\) is normal, allowing a point mass.

Write \(m_G(t)=\mathbb E G(t)\). Sample continuity and convergence in law of Gaussian marginals at convergent times imply that \(m_G\) is continuous. Each marginal has the unique median \(m_G(t)\), including at time zero. At a grid time \(h/n(r)\), the number \[\frac{b_h}{r}-f_r(h/n(r))\] is a median of the translated marginal. Along every sequence of grid times converging to \(t\), weak path convergence and uniqueness of the limiting median make these medians converge to \(m_G(t)\). Compactness of the time interval makes the convergence uniform over grid times; interpolation gives uniform convergence on the whole interval. Use a slightly longer interval when interpolating at the endpoint. Subtracting the interpolated medians from the original process therefore removes precisely the limiting mean function. Every subsequential limit is Brownian motion of variance parameter \(\sigma_*^2\), proving the single-path assertion. ◻

Proof. We first prove the additive relation. It suffices to consider arbitrary sequences of indices for which \(k/n(r)\to s\) and \(j/n(r)\to t\), and to pass further so that \(k\) is fixed or tends to infinity. Let \(\mathcal C_k\) be exact single truth at level \(k\). Under \(P_0^+\), this event has probability \(u_k\ge p\). Conditional on \(\mathcal C_k\), the prefix has the raw law conditioned on \(T_k<T_{-1}\), and its relative suffix is an independent fresh \(P_0^+\) path.

If \(k\to\infty\), the total variation distance between this prefix law and the \(P_0^+\) prefix law is at most \(1-p/u_k\to0\). If \(k\) is fixed, its displacement and median are fixed random and deterministic quantities, respectively, and their normalized difference tends to zero. Lemma 17 therefore gives, under the conditional law, \[\frac{Y_{k+j}-b_k-b_j}{r} \ \Longrightarrow\ N(0,\sigma_*^2(s+t)).\] The same lemma gives this normal limit under the unconditional \(P_0^+\) law when the centering is \(b_{k+j}\).

Set \(d_r=(b_k+b_j-b_{k+j})/r\). The unconditional distribution centered by \(b_{k+j}\) dominates \(p\) times the conditional distribution with that same centering. Tightness first forces \(d_r\) to be bounded. Along a further subsequence with \(d_r\to d\), the limiting domination would read \[N(0,v)\ \ge\ pN(d,v),\qquad v=\sigma_*^2(s+t),\] as measures. For \(v>0\), a nonzero shift makes the ratio of the shifted normal density to the unshifted one unbounded. For \(v=0\), the two point masses have different supports unless \(d=0\). Thus \(d=0\) in every case. Compactness of the ranges of \(k/n(r)\) and \(j/n(r)\) proves the uniform relation [eq:15].

To turn approximate additivity into a linear approximation, put \(N=N_r\), \(\theta=b_N/N\), and \(d_j=b_j-j\theta\) for \(0\le j<N\). Apply [eq:15] to \(j+j\), and also to \(N+(2j-N)\) when \(2j\ge N\). Uniformly over these indices, \[2d_j=d_{2j\bmod N}+o(r).\] The maximum of \(|d_j|\) is therefore at most half itself plus \(o(r)\), so it is \(o(r)\). For \(h\le Tn(r)\), write \(h=qN+j\), where \(q\) remains bounded, and apply [eq:15] a bounded number of times. This proves \(\max_{h\le Tn(r)}|b_h-h\theta_r|=o(r)\). Finally, at \(h=1\) this gives \(|b_1-\theta_r|=o(r)\); since \(b_1\) is fixed, \(\theta_r=o(r)\). ◻

Proof. Relative to the single conditioned environment-and-path law from \(x\), the first marginal has density \[\frac{p\,q(y)}{p_2(x,y)}.\] For a fixed integer \(J\), replace \(q(y)\) by \(a_J(y)\), then normalize. The resulting probability law differs from the original one in total variation by at most \(C(u_J-p)\), uniformly in \(x,y\). Indeed \(a_J\ge q\), and the added mass before normalization is \[\frac{E[q(x)(a_J(y)-q(y))]}{p_2(x,y)} \le\frac{u_J-p}{\underline p_2}.\] All these normalized laws have density bounded by a common constant relative to the single conditioned law.

The weight \(a_J(y)\) uses only rows between the common base and the layer \(J\) above it, excluding the latter. Stop the single-word chain of the first walk at its first true boundary at or above that layer. Every row used by the weight lies below this boundary. Exact-word factorization therefore gives a fresh \(P^+\) suffix, independent of the removed path prefix under the weighted law. The weight may depend on infinitely many sites in those \(J\) layers; only their heights matter for this factorization.

For fixed \(J\), the height and size of this initial prefix are tight under \(P_x^+\), and hence under every one of the bounded weighted laws. Its height divided by \(n(r)\) tends to zero. Its displacement divided by \(r\) also tends to zero, as does its centering correction, since \(\theta_r/r\to0\). The suffix limit and its tight continuous modulus thus give the asserted limit under the approximating law. Send \(r\to\infty\) first and then \(J\to\infty\). The same proof applies to the second marginal, and tightness of the two marginals gives joint tightness. ◻

Proof. In the top layer choose the windows \[I_1=\{z:|z_2-x_2-k\theta_r|<ar/3\},\qquad I_2=\{z:|z_2-y_2-k\theta_r|<ar/3\}.\] They are disjoint. Lemma 19 and the normal tail bound show that the shared conditioned probability of either endpoint error has limsup at most \(Ce^{-ca^2/t}\), uniformly in the starts. Average the contact bound under the environment law with density \(q(x)q(y)/p_2(x,y)\). This gives the same bound for trace contact under \(P_{x,y}^{++}\).

To compare laws off contact, first use raw successful prefixes: both walks reach level \(k\) before dropping below the common base. For any two disjoint finite traces, their raw prefix weights agree after averaging in shared and independent environments, because their departure-row sets are disjoint. Passing from two finite successes to two infinite no-drop events removes raw mass at most \(2(u_k-p)\) under either pair law. Moreover \(p_2(x,y)-p^2\to0\) uniformly for the present starts, since their separation tends to infinity. Thus the conditioned prefix laws agree off contact up to a variation error that tends to zero. Their total masses are both one, so the contact mass on the independent side has the same upper bound up to this error. The contact estimate proves [eq:17]. ◻

Proof. Choose \(1<\alpha<\gamma<2\). The potential will be a flattened and capped version of \(u^\alpha\), where \(u\) is the absolute gap at an observed common cut. Its expected increase will pay for the number of near-diagonal cuts before the next observation. We need one estimate at nearly Gaussian radii and another below the largest remaining exceptional radius.

Proof. Take any subsequential continuous-path joint limit \(B_1,B_2\) of [eq:16] under \(P_{0,0}^{++}\), on fixed horizons (enlarge a horizon if needed at endpoints), and let \(\mathcal G_s\) be the joint past. By [eq:18] \[\int_0^T\mathbf1_{\{B_1(s)=B_2(s)\}}\,ds=0\quad\text{almost surely}. \tag{21}\] One can integrate continuous cutoffs of the gap supported in shrinking neighborhoods of zero; their grid sums give the same limiting expectations by joint tightness with continuous limits.

Conditional marginal increments. At a deterministic time \(s\), put \(H_r=\lfloor sn(r)\rfloor\) and restart the prelimit pair at \(\sigma(H_r)\). This is a stopping index of the chain of common words, not a stopping time for an individual walk. All first-hit coordinates of both paths at levels at most \(H_r\) are determined by the words traversed through this boundary. Its conditional suffix law is the fresh shared conditioned law from the terminal pair of sites.

By [eq:6], \(\sigma(H_r)-H_r\) is uniformly tight. Consequently its normalized time shift vanishes. Joint tightness with continuous limits, on a slightly enlarged horizon, makes the values at \(H_r+\lfloor tn(r)\rfloor\) and at \(\sigma(H_r)+\lfloor tn(r)\rfloor\) differ by \(o_{\Pr}(r)\) after subtracting the same linear centering. At the initial endpoints this also follows directly from [eq:6]; the centering correction is a tight integer times \(\theta_r/r\), which vanishes by Lemma 18.

Test against bounded continuous functions of finitely many joint past coordinates, represented at floor grid times in the prelimit. Condition at the common boundary and test the fresh normalized increment of length \(\lfloor tn(r)\rfloor\). Lemma 19 makes its conditional bounded continuous test expectation converge to the normal value uniformly in the random terminal pair of sites. The time replacements just justified do not change the limit. Passing to the joint limit, then using a monotone-class argument for the joint past, proves that for each \(t>0\), \[\mathcal L(B_i(s+t)-B_i(s)\mid\mathcal G_s) =N(0,\sigma_*^2t),\qquad i=1,2.\]

Off-diagonal cross increments. Moreover if \(g\) is a continuous \([0,1]\)-valued cutoff vanishing on \([-2a,2a]\), \(a>0\), and \(\delta_i=B_i(s+t)-B_i(s)\), then for bounded \(\mathcal G_s\)-measurable \(F\), \[\big|E[F g(B_2(s)-B_1(s))\delta_1\delta_2]\big| \le \|F\|_\infty\, t\,e_a(t),\qquad e_a(t)\longrightarrow0\quad(t\downarrow0). \tag{22}\] The bound can be uniform in \(s\). To see this, again restart as above; on the supported past gaps the restart state has absolute gap \(\ge a r\) except with vanishing error. Clip each future normalized increment symmetrically to the interval \([-w\sqrt t,w\sqrt t]\), for the moment fixed \(w,t\). By [eq:17] at the fresh state their product expectation differs from the independent-law one by at most \(Cw^2 t\exp(-c a^2/t)\) in limsup; the independent-law value tends to zero by the single Brownian limits. Test first against continuous bounded past functions and pass to the joint limit. Removing the clipping there costs at most \(t\,o_{w\to\infty}(1)\|F\|_\infty\), by the conditional marginal normals and Cauchy–Schwarz. Continuous past tests extend to bounded measurable ones, for instance by bounded approximation using \(E(|\delta_1\delta_2|\mid\mathcal G_s)\le \sigma_*^2 t\). Choosing \(w\to\infty\) slowly gives [eq:22].

Identification of the joint law. These facts force \(B=(B_1,B_2)\) to consist of independent Brownian motions. For a direct verification fix \(s_0\), bounded \(F\) of the joint past there and a real vector \(\xi\in\mathbb R^2\). Telescope the exponential of \(i\xi\cdot(B(s)-B(s_0))\) on an equal mesh of \([s_0,v]\), multiply by \(F\) and take expectation. Taylor remainders through order two total \(o(1)\) by the marginal Gaussian third absolute moment bounds. First-order terms vanish by the past-conditional laws, and diagonal second-order terms give the Riemann sum using their variances. Cross terms total \(o(1)\): away from gap zero by [eq:22], and the complementary terms are bounded by a constant times the mesh sum of step lengths times expected continuous cutoff near gap zero (by the past-conditional absolute product bound), which is negligible by continuity and [eq:21] as the cutoff shrinks. Thus \(f(v)=E[F\exp(i\xi\cdot(B(v)-B(s_0)))]\) satisfies \[f(v)=E F-\tfrac12\sigma_*^2|\xi|^2\int_{s_0}^v f(s)\,ds .\] This gives the Gaussian increment law independent of the joint past, proving the assertion. ◻

Proof. Let \(F\) be a bounded real continuous test on path space, and let \(M_r(\omega)\) be its expectation under the quenched conditioned path. Give the environment the probability law \[\widehat P(d\omega)=\frac{q(0)^2}{E[q(0)^2]}P(d\omega).\] Under this weighting, two conditionally independent quenched conditioned paths have exactly the annealed law \(P_{0,0}^{++}\). Proposition 22 and its marginal limits therefore give, with \(b=E[F(B)]\) for the limiting Brownian path, \[E_{\widehat P}M_r\longrightarrow b, \qquad E_{\widehat P}M_r^2\longrightarrow b^2.\] Thus \(M_r\to b\) in \(\widehat P\)-probability. Since \(q(0)>0\) almost surely, \(P\) is absolutely continuous with respect to \(\widehat P\); convergence in probability also holds under \(P\).

For [eq:23], use Lemma 18 to replace the linear centering by the medians. Choose a nonnegative continuous path test below the desired event, with a fixed bounded supremum and with the endpoint, throughout a small neighborhood of time one, strictly inside the window of radius \(\rho\). Such a test can be chosen with positive Brownian expectation: the time-one centered normal law assigns positive mass to a smaller window, and continuity permits the nearby-time and sufficiently large fixed supremum restrictions. Eventually this fixed supremum bound is smaller than \(b_*(r)\). The quenched no-drop-conditioned probability is consequently bounded below by a fixed positive constant with \(P\)-probability tending to one. Finally, on successful prefixes the total variation cost of replacing no-drop conditioning by conditioning on \(T_H<T_{-1}\) is at most \(1-q(0)/a_H\), which tends to zero almost surely. This proves the stated normalization by \(a_H\). Translation invariance gives the same conclusion at any translated start. ◻

Proof. Take a subsequence realizing the left \(\limsup\) and extract compactified weak limits \(\mu,\nu'\) in \([0,\infty]\) for the two laws. At each \(u>0\), \(\mu([u,\infty])\) is bounded by the limit-law probability of the open event above \(u/2\), hence by the corresponding prelimit \(\liminf\). By [eq:11] and [eq:9], this is at most \(C_1\) times a prelimit \(\limsup\) on the \(V\) side above \(u/(2C_1)\); bounding by the limiting closed tail and enlarging it once more gives \[\mu([u,\infty])\le C_1\nu'((u/(4C_1),\infty]).\] For the first inequality of [eq:25] simply integrate with weight \(2u\) up to \(L\), since capped squares are bounded continuous. For the second, first upper bound the left using \(X\ge A\) on the limiting law (upper semicontinuous test). This gives \(A^2\mu([A,\infty])\) plus the tail integral between \(A\) and \(L\). Transfer each term to bound the sum by a constant times the \(\nu'\)-expectation of the capped square on the open tail above \(A/(4C_1)\). By lower semicontinuity that last expectation is bounded by the prelimit \(\liminf\) on our subsequence. Enlarging constants gives [eq:25]. ◻

Proof of Proposition 24. We will concatenate a fixed number of short blocks. In each block we choose a probability law on a restricted set of successful prefixes, except on a rare failure event. The two requirements are different: the chosen endpoint displacements must have small variance, and the restriction must retain a fixed positive amount of the original kernel mass. The first requirement keeps the auxiliary first-hit process in a narrow tube; the second will transfer that conclusion back to the quenched walk.

Work along an arbitrary sequence \(r\to\infty\), and pass to a subsequence on which \[F(a)=\lim_{r\to\infty}\frac{m(ar)}{m(r)} \qquad(a>0)\] exists. This extraction is possible because the ratios are monotone in \(a\), and changing \(a\) by a factor changes \(m(ar)\) by at most its square. Thus they are equicontinuous on compact logarithmic intervals. In particular \(F(1)=1\), and the limit \(F_0=F(0+)\) exists in \([0,1]\). It suffices to prove the clipping assertion on every such further subsequence with some fixed positive mass bound. Otherwise there would be a sequence of scales at which every deterministic slope fails even for mass thresholds tending to zero.

Proof. Splitting one starting mass. First consider a single prescribed start. We will produce \(k\) ordered endpoint groups, with support gaps of order \(r\), each carrying a fixed positive mass. Use \(J_0\) blocks of length \(h=\lfloor n(r)\rfloor\), followed by a remainder, to reach \(H=C_*n(r)\), where \(J_0\) is a large fixed integer and \(C_*=2J_0+2\). Clipping through \(C_*n(r)\) supplies a common deterministic slope \(s_r\) and, with arbitrarily high environment probability, fixed positive mass within \(\eta r\) of that line. The tolerance \(\eta\) will be small compared with \(1/J_0\).

Endpoint spread, Equation [eq:13], supplies a designated sign of a deviation of size at least \(cr\) from \(hs_r\), with single-walk \(P^+\)-probability at least \(c>0\). These constants are independent of the clipping parameters. The corresponding raw \(K_h\)-mass has expectation at least \(pc\), by using paths on \(D\), and is bounded by one. It therefore exceeds a fixed positive threshold on an environment event of fixed positive probability. The same reasoning applies after averaging over any starting probability.

Maintain the groups as submeasures of the walked mass, normalizing within each group for its next test. If there are fewer than \(k\) groups and the designated sign is positive, take the rightmost group and a second-coordinate median. Preserve its half at or below the median by central clipping. From the half at or above the median, retain the designated jumping mass whenever it exceeds its fixed threshold; these endpoints form a new rightmost group. Preserve all other groups by central clipping as well. For a negative designated sign, perform the reflected operation at the leftmost group.

Both median halves have mass fraction at least one half. A median atom may be used in both tests. In the positive-sign case, the retained central endpoints have an upper bound given by the median plus \(hs_r+\eta r\), while the jumping endpoints have a lower bound given by the median plus \(hs_r+cr\). The negative-sign case has the reflected inequalities. Thus the new groups have disjoint supports with a gap of at least \((c-\eta)r\). Every old gap loses at most \(2\eta r\) under central propagation. Once there are \(k\) groups, only their central preservation is needed.

The initial-distribution clipping lemma makes all the central preservations succeed simultaneously with arbitrarily high conditional probability. Abort if a preservation fails. Subtracting this small error from the jump event shows that, whenever a split is still needed, preservation together with a split has conditional probability at least a fixed \(s_*>0\), independent of \(\eta,J_0\). The operations are measurable: take full restrictions to the indicated events and choose, for example, lower integer medians.

Choose \(J_0\) large, and then choose the preservation errors small compared with \(1/J_0\). The probability of no abort but fewer than \(k-1\) splits in these blocks is at most \[\sum_{i<k-1}\binom{J_0}{i}(1-s_*)^{J_0-i}.\] Indeed, on such a pattern a split is needed at every step, and each prescribed nonsplit without abortion costs at most \(1-s_*\) by conditioning on the previous blocks. Apply central clipping once more over the remaining height, which is at most \(C_*n(r)\). Taking \(\eta\) sufficiently small after fixing \(c,J_0\) leaves final gaps at least \(c'r>0\). The product of the finitely many retained mass fractions gives a fixed positive lower bound for each group in the original single-walk kernel \(K_H\). Every test and continuation uses only layers below its actual endpoint.

Choosing separated endpoints. For any starting tuple whose \(k\) starts each have these \(k\) options, the product kernel gives fixed positive mass to separated endpoint tuples. Choose \(2c_0<c'\) and expose endpoints successively. Every earlier endpoint excludes at most one group at the next start, since two distinct groups are separated by more than \(2c_0r\). Thus at the \(i\)th choice at least one of the \(k\) groups remains available. Each choice costs only its fixed positive group-mass lower bound. Averaging the favorable fraction over \(\pi\), as in Lemma 26, proves [eq:30]. ◻

Proof. Truncate each \(q\) upward to \(q\vee e^{-B_*}\), changing the product by at most \(e^{-B_*}\): if a factor changes, the entire new product is at most this quantity. By bounded product-field mixing at mutually separated sites, the expectation of the inverse \(\lambda\)-power of the truncated product is bounded by \(2(Eq(0)^{-\lambda})^k\) for sufficiently large separation. To see the required uniformity, at fixed \(k,B_*\) approximate the bounded single-site functional in \(L^1(P)\) by one using finitely many rows and having the same bound. Sufficient separation makes all its translates disjoint. On the event on the left of [eq:31], the truncated product integral is \(<3s\). Jensen’s inequality for its negative power and Markov’s inequality give [eq:31]. ◻

Proof. From the current layer in a subdivision, monitor the local path mass with budget \(d'/L'\), stopping at the first height where it is \(<s:=\kappa^{-k}e^{-x/L'}\). The assumptions give \(e^{-B_*}\le s\le1\). At any fresh start whose separation is still at least \(G-d'\), the probability of such a threat is bounded by the parenthesis in [eq:32]. Indeed the raw mass of all \(k\) infinite \(D\)-successes is exactly the current \(q\)-product integral. Its part violating the budget within \(H\) layers has expectation at most \(CkH/n(d'/(2L'))\), by [eq:8]–[eq:9] and a union over single walks on \(D\) with a median deviation greater than \(d'/(2L')\). The other path factors are bounded by one and can be dropped. If the product integral is at least \(2s\) and the violating mass at most \(s\), enough mass satisfies the budget at every first hit to prevent a threat. Lemma 28 and Markov’s inequality now give the claimed bound; the factor \(\kappa^{-k\lambda}\) is absorbed into \(e^{Ck}\). Neither \(D\) nor \(q\) is used to choose the stopping layer: the actual test is the finite-height mass, measurable from rows below that layer.

At a threat, extend the passing mass from one layer earlier by a direct upward step for every particle; at offset zero the passing mass is one. This preserves relative displacement errors and retains mass at least \(\kappa^ks=e^{-x/L'}\). Normalize at the actual stopping layer and restart with the fresh upper environment. A final piece with no threat also has mass at least \(e^{-x/L'}\).

If the horizon finishes within \(L'\) pieces, concatenation gives original mass at least \(e^{-x}\), with total relative-motion budget at most \(d'\). Otherwise each of the first \(L'\) pieces has detected a threat before completion. The conditional threat estimate applies at every restart: only lower rows have been revealed, and separation remains at least \(G-d'\). Multiplying these conditional bounds gives Equation [eq:32]. ◻

Proof. We construct one sequence of retained endpoint laws and use it at every target height. A stage starts with an extra signed gap, or buffer, between the walks. It usually increases this buffer by a fixed factor; on failure it reduces the buffer by a smaller factor. The resulting positive logarithmic growth limits the number of stages through height \(H\) to order \(\log(1+H)\), apart from an exponentially integrable random error. Each stage costs only a fixed amount of logarithmic mass.

Proof. The main construction gives several chances to create separated mass before any prescribed short interval of target heights. A failed chance reveals only layers below the next available chance, so the estimate can be repeated there. We first describe these chances and their probability bounds, and then choose the constants to cover all target heights. Throughout these estimates \(A\ge1\) and the integer \(k\ge2\) are provisional fixed parameters. Write \[\Lambda=\log16,\qquad j_*=\frac1{8A},\qquad m=\lfloor j_*b\rfloor.\] We take \(b\) large enough that \(m\ge1\). All uses of the rapid-loss bound [eq:32] will have cap \(B_*=2kb\); after fixing \(b\), increasing \(s_{\rm fl}\) will ensure its separation hypotheses.

Proof. Compactness gives a subsequential weak limit. The shift is continuous, and the occupation sum telescopes: for every bounded continuous \(F\), \[\left|\mathbb L_N[F\circ\tau]-\mathbb L_N[F]\right| \le \frac{2\|F\|_\infty}{E_{Q_N}M}\longrightarrow0.\] At the origin, [eq:40] bounds the occupation mean of \(J_i\). Truncation passes this bound to the limit, so \(J_0<\infty\) almost surely. The bounds [eq:38] then pass and give the stated integrability and finiteness. For the positive truncated-drop bound in [eq:40], put \(C_{L_0}=B_0/k+2bL_0\) and use the globally bounded continuous test \[(c_0\wedge C_{L_0})\mathbf1_{\{J_0\le L_0\}}.\] It agrees with the required drop on every supported array, by [eq:38]. Its expectation therefore passes to the limit and gives \(E_{\mathbb L}c_0>0\). The occupation law gives \(h_0\ge1\), and shift invariance gives all assertions at every integer index. Uniform ellipticity is a closed condition on the row coordinates and therefore also passes to the limit. ◻

Proof. We first record the elementary entropy facts used here. Relative entropy is \({\rm KL}(Q\mid P')=\int f_0\log f_0\,dP'\), where \(f_0=dQ/dP'\), and is infinite without absolute continuity. It has the variational formula \[{\rm KL}(Q\mid P') =\sup_{F\text{ bounded measurable}} \left\{QF-\log P'e^F\right\}.\] The upper bound follows from Jensen applied to \(e^F/f_0\). For equality, truncate \(\log f_0\) between \(\pm B\) and send \(B\to\infty\), or test a singular set. Thus projection cannot increase entropy. On compact metric spaces, bounded continuous tests suffice by bounded approximation in \(L^1(Q+P')\); the variational formula therefore gives lower semicontinuity under joint weak convergence of both laws.

For an event of respective probabilities \(s,t\), where \(0<t<1\), projection to its indicator gives \({\rm KL}(Q\mid P')\ge s\log(1/t)-\log2\). If a reference law is a data marginal times an independent fixed law, replacing that marginal by the controlled law’s own data marginal only decreases finite entropy. Indeed the new density is the old density divided by its conditional mean given the data, and the entropy difference is the nonnegative marginal entropy. These log-density identities are taken on sets of positive controlled density; negative parts are integrable, and conditional Jensen for \(f_0\log f_0\) justifies the marginal term.

Before the limit, at deterministic \(i\ge0\), put \(\mathcal H_i=\sigma(\mathcal F_{t_i},(U_{i,a})_a)\) and \(\Phi_i=(\Omega^{i,m,a})_a\); \(\mathcal A_i\) is \(\mathcal H_i\)-measurable. Include the same auxiliary sampling under \(P\). Given \(\mathcal H_i\), the spliced field above \(t_i\), in absolute horizontal coordinates, is still iid under \(P\). Indeed, at both bounded layer stopping times the layers at and above the stopping layer are fresh, while the current anchors use only lower-layer data. Keeping the slab up to the second stopping layer and replacing its tail by an independent copy therefore preserves the iid law. Thus \(\Phi_i\) has conditional law \(\mathbf R(\mathcal A_i,d\phi)\).

Let \(L_t=dQ_N|_{\mathcal F_t}/dP|_{\mathcal F_t}\). Under \(Q_N\), the conditional density of the information stopped at \(t_{i+m}\), together with the independent extension, relative to the corresponding \(P\)-law given \(\mathcal H_i\), is \(L_{t_{i+m}}/L_{t_i}\). The anchors at \(i\) use the same lower-data sampling kernel under both laws, and the extension costs no density. Conditional Jensen therefore gives, for every bounded \(F\), \[\begin{aligned} &E_{Q_N,{\rm aux}}[F(\mathcal A_i,\Phi_i)\mid\mathcal H_i]\\ &\quad\le E_{Q_N,{\rm aux}}[\log L_{t_{i+m}}-\log L_{t_i}\mid\mathcal H_i] +\log\int e^{F(\mathcal A_i,\phi)}\mathbf R(\mathcal A_i,d\phi). \end{aligned}\] Multiply by \(\mathbf1_{\{i<M\}}\), which is past-known, and average as for \(\mathbb L_N\). Jensen to pull the log outside this average gives the \(\mathbb L_N\) entropy bound at index zero (against its own current-data marginal and the kernel) of \[\frac{1}{E_{Q_N}M}\sum_{i\ge0}E_{Q_N}\big[\log L_{t_{i+m}}-\log L_{t_i}\big] \le \frac{mD_0\log N}{E_{Q_N}M}.\] Increments from \(i\ge M\) are zero. The displayed sum telescopes over deterministic indices, since \(t_i=N\) for \(i\ge N\); its subtracted initial expected log densities are nonnegative relative entropies. For passage to the limit, finite-class consistency is a closed condition: self-offsets vanish, finite offsets are symmetric with sign change, and specified finite offsets add on triples. Violations are detected by finitely many clopen coordinate events. On this support the reference kernel is weakly continuous. For any finitely many field coordinates, coincidence of the input sites is specified by a finite-offset equality, while different sites have independent rows. Finite-coordinate tests suffice by compactness. Consequently the reference laws converge weakly with their current-data marginals. Lower semicontinuity and [eq:40] yield the bound \(C_E m\); shift invariance gives every \(i\). ◻

Proof. Before the limit, the anchor draws at \(j\), excluding a possible own label, are conditionally independent samples given the environment and \(U_{i,a}\). The occupation indices and weights depend only on the environment, not on these draws. The mean-square error in [eq:42] is therefore \(O(2^{-n})\), including the possible own-label contribution. The finite-offset indicators are continuous, so these summable bounds pass to the limit. Borel–Cantelli proves [eq:42].

For distinct labels and any continuous function \(g\) on \([0,1]\), the same conditional sampling identity gives \[E_{\mathbb L}\!\left[ g(w_j^{ia}(z)) \bigl(\mathbf1_{\{O_{ia,jb}=z\}}-w_j^{ia}(z)\bigr)\right]=0.\] Approximate the indicator of zero by bounded continuous functions to obtain the second assertion. All the indices and sites are countable, so the assertions hold on one common probability-one event. ◻

Proof. Before the limit, local no-drop continuation is contained in the global survival stream. Dividing its endpoint inequality by the new global mass gives [eq:43], with \(C=\log(a_{t_i}/a_{t_{i+m}})\). At finite limiting marks, pass first the inequalities involving finitely many starting sites and bounded-length successful kernel paths. Strict violations of these finite-path inequalities are open, since each path weight is a finite product of row coordinates. Increasing the finite restrictions gives [eq:43]. Successful paths to layer \(H\) use no departure row at or above \(H\), so the auxiliary tail is irrelevant.

Before the limit, at a fixed index the average in [eq:44] has conditional mean at most \(e^{-C}\) given the environment, again by the global mass inequality. Its terms are independent bounded functions of the independently sampled current anchors; they do not use appended rows. Chebyshev bounds the conditional probability of exceeding \(e^{-C}+\epsilon\) by \(2^{-n}\epsilon^{-2}\), and this also holds after occupation averaging. For fixed finite \(H\), the function \(a_H\) is lower semicontinuous, being a sum of nonnegative continuous finite-path weights. Thus the strict upper-violation event is open at finite marks, and the same summable probability bound passes to the limit. Borel–Cantelli, followed by a countable sequence \(\epsilon\downarrow0\), proves [eq:44]. ◻

Proof. Dust is isolated. If a dust label \((i,a)\) had a distinct finite-offset neighbor \((j,b)\), with offset \(z\), profile consistency would give \(w_i^{jb}(-z)=w_i^{ia}(0)=0\). Apply the second assertion of Lemma 35 with \((j,b)\) as the reference label and \((i,a)\) as the sampled target. This is impossible. Thus dust labels are singleton classes even when all times are included.

At any fixed time, the dust indicators are exchangeable, because anchor-permutation symmetry passes to the limit. They have an asymptotic frequency \(\delta_i\), which is positive if any dust occurs. To see the latter assertion, apply the ergodic theorem to the stationary label sequence of dust indicators. On the invariant zero-frequency event, taking expectations shows that each indicator vanishes.

Fix \(i,m\). Under the reference law of Lemma 34, conditional on \(\mathcal A_i\), the singleton dust classes have independent upper fields. Their values \(q(0;\Omega^{i,m,a})\) are therefore iid with mean \(p\). Finite entropy gives absolute continuity, so the strong law on the dust subsequence also holds under \(\mathbb L\) whenever that subsequence is infinite. Since \(a_H\ge q\), including in every spliced field, [eq:44] yields \[e^{-C_{i,m}}\ge p\delta_i.\] This holds simultaneously for every \(m\). On \(\mathcal E\), its left side tends to zero, by [eq:41] and stationarity. Dust is therefore absent at all times on \(\mathcal E\).

There are finitely many proper classes. For fixed \(n\) and \(r_0>0\), whenever possible choose from \(\mathcal A_i\) labels in \(n\) distinct classes with self masses at least \(r_0\). On this selection event, when \(C_{i,m}>\zeta m\), [eq:43] and the total-mass bound for each output profile give \[\prod_{\rm chosen} q(0;\Omega^{i,m,a})\le r_0^{-n}e^{-n\zeta m}.\] The event of successful selection together with this deterministic product bound is measurable from the current data and the extension fields. Its reference probability is at most \[r_0^{-n\lambda}(E q^{-\lambda})^n e^{-\lambda n\zeta m},\] by [eq:4] and class independence. The binary entropy bound and Lemma 34 show that its \(\mathbb L\)-probability has \(\limsup_{m\to\infty}\) at most \(C_E/(\lambda n\zeta)\). On the selection event intersected with \(\mathcal E\), the product bound holds for all sufficiently large \(m\). This bounds the probability of that intersection by the same quantity. Send \(r_0\downarrow0\), then \(n\to\infty\), to exclude infinitely many proper classes.

The classes persist and have full support. Over any positive integer height, the local kernel has positive weight between any two horizontal sites: move horizontally at the starting layer and then ascend directly. Uniform ellipticity makes this finite script positive. Thus [eq:43] gives a full-support next profile for every current proper class. By [eq:42], every positive coordinate is represented by next-time labels. Distinct classes cannot merge, by the global finite-offset equivalence relation.

Conditional on the shift-invariant event \(\mathcal E\), the current number of classes is therefore nondecreasing, finite, positive, and stationary. It is consequently constant almost surely. Indeed, writing \(N_i\) for this count, stationarity and monotonicity give \[E[\min(N_{i+1},K)-\min(N_i,K)]=0\] for every integer \(K\). The nonnegative difference vanishes almost surely; letting \(K\) increase proves \(N_{i+1}=N_i\). This uses no moment of the number of classes. The class sets transport bijectively through time, and every profile has full support by the update from the preceding time. ◻

Proof. We use the two-tail statistic introduced at the start of this section. For an integer \(R\ge1\), recall that \[T_R(w)=\max_{j\in\mathbb Z} \min\{w(z_2\le j),w(z_2\ge j+R)\}.\] Here \(z_2\) means the second full-space coordinate of a horizontal site. Full support and finite mass imply \(0<T_R(w)\le1\), and a maximizing integer \(j_R\) exists. Choose one measurably. For each fixed full-support \(w\), its maximizing cuts satisfy \[j_R\longrightarrow-\infty,\qquad j_R+R\longrightarrow+\infty.\] For example, if \(j_R\) were bounded below along a subsequence, the left mass at both \(j_R\) and \(j_R-1\) would stay bounded below whereas the right mass would tend to zero. Moving the cut one unit left would strictly increase the smaller mass, contradicting maximality. The other endpoint is treated by moving the cut right. Thus the normalized restrictions \(\alpha_-\) and \(\alpha_+\) of \(w_0\) to these two tails eventually sample sites outside every fixed neighborhood of the representative origin.

Fix \(m\) throughout the next steps. Let \(Q_R\) be the mass of endpoint pairs \((u,v)\) with \(v_2-u_2\ge R\) under \[(\alpha_-K_{H_m}^{\Omega_m})\otimes (\alpha_+K_{H_m}^{\Omega_m}).\] Every starting pair already has signed gap at least \(R\). Its outward order event is therefore contained in the event just tested. Finite entropy transfers the probability-one event in [eq:33] to \(\Omega_m\), simultaneously for the countably many starting pairs and integer heights, so it applies at the random height \(H_m\). That estimate and Jensen’s inequality yield \[\begin{equation*} \tag{46} \begin{aligned} \log Q_R&\ge-A_\to\log(1+H_m)-Z_R,\\ Z_R&=\iint\mathcal Z((0,x),(0,y);\Omega_m) \,\alpha_-(dx)\alpha_+(dy). \end{aligned} \end{equation*}\] Write \(\nu_\pm=\alpha_\pm K_{H_m}^{\Omega_m}\). There is an integer \(j\) with \(\nu_-(u_2\le j)\ge Q_R/2\) and \(\nu_+(v_2\ge j+R)\ge Q_R/2\). Choose \(j\) minimal for the first inequality, so \(\nu_-(u_2<j)<Q_R/2\). If the second inequality failed, split ordered pairs into \(u_2<j\) and \(u_2\ge j\). The latter condition forces \(v_2\ge j+R\). Each part would then have mass strictly less than \(Q_R/2\), since both endpoint measures have mass at most one, contrary to the definition of \(Q_R\).

Each initial tail has mass at least \(T_R(w_0)\). Applying [eq:43] to the two tails at this cut gives \[\begin{equation*} \tag{47} \log T_R(w_m)-\log T_R(w_0) \ge C_m-A_\to\log(1+H_m)-Z_R-\log2. \end{equation*}\] This is the decisive inequality: normalization contributes \(C_m\), whereas preserving the order of the two tails costs only a logarithm of the traversed height and the local environmental remainder.

We next remove the environmental bias from that remainder. We claim \[\begin{equation*} \tag{48} \limsup_{R\to\infty}E Z_R \le M_\mathcal Z:=\sup_{x,y}E_P\mathcal Z(x,y)<\infty. \end{equation*}\] Write \(f_m(D,\Omega)\) for the density relative to [eq:45]. Under that product reference, additionally sample \(x,y\) from \(\alpha_-\otimes\alpha_+\) conditionally on \(D\) alone. Then \(EZ_R\) is the reference expectation of \(f_m(D,\Omega)\mathcal Z((0,x),(0,y);\Omega)\).

The exponential moment in [eq:33] is uniform in the sampled pair. Consequently the density and the remainder can both be truncated with errors uniform in \(R\). To see the only delicate part, on \(\{f_m>B\}\) Young’s inequality gives, for sufficiently small \(\lambda>0\), \[\lambda f_m\mathcal Z\le f_m\log f_m+e^{\lambda\mathcal Z}.\] The entropy term has vanishing tail. The reference probability of \(\{f_m>B\}\) is at most \(1/B\), and Cauchy–Schwarz bounds the exponential term on that event by \(O(B^{-1/2})\). Once \(f_m\) is bounded, truncating \(\mathcal Z\) is justified directly by its exponential moment.

Approximate the bounded density by a bounded nonnegative function \(g(D,\Omega)\) using only a fixed finite box of field rows around the origin, allowing arbitrary dependence on \(D\). With \(L^1\) error at most \(\varepsilon\), its reference integral is at most \(1+\varepsilon\). Approximate the capped remainder, uniformly in \(x,y\), by a bounded function using fixed-radius upper-row neighborhoods of those two sites. This follows from the local approximation in [eq:33]; boundedness converts convergence in probability to convergence in \(L^1\). As \(R\) increases, the two tail supports and their neighborhoods leave the fixed origin box. Under the product reference, conditional on \(D,x,y\), the two approximating factors then use disjoint iid row sets. Their product expectation is bounded by \((M_\mathcal Z+\varepsilon)E_{P_\uparrow}g(D,\cdot)\). The threshold in \(R\) may depend on \(D\); boundedness permits integration by reverse Fatou. Sending the approximation errors to zero and removing the truncations proves [eq:48]. This entire step keeps \(m\) fixed, so its approximation boxes need not be uniform in \(m\).

For fixed \(R,m\), the right side of [eq:47] is integrable. The entropy inequality and the same exponential moment show that \(EZ_R<\infty\). Let \(X=\log T_R(w_0)\) and \(Y=\log T_R(w_m)\). Translation invariance of \(T_R\) and stationarity make these finite nonpositive variables identically distributed. Their difference has integrable negative part by [eq:47]. Clipping both variables below at \(-B\) gives an integrable zero-mean difference; its positive and negative parts increase to the corresponding parts of \(Y-X\). Monotone convergence therefore shows that \(Y-X\) is integrable and \(E(Y-X)=0\). Taking expectations in [eq:47], and now sending \(R\) to infinity, yields \[mEc_0\le A_\to E\log(1+H_m)+M_\mathcal Z+\log2.\] Finally let \(m\) increase. With \(Z_i=\log(1+h_i)\), \[\log(1+H_m)\le\log m+\max_{0\le i<m}Z_i, \qquad E\max_{i<m}Z_i\le T+mE[Z_0\mathbf1_{\{Z_0>T\}}].\] Divide by \(m\), send \(m\) to infinity and then \(T\) to infinity. The assumed logarithmic moment gives \(E\log(1+H_m)=o(m)\), hence \(Ec_0\le0\). Nonnegativity of \(c_0\) proves the proposition. ◻

Completion of the proof of Theorem 1. Apply Proposition 38 to \(\mathbb L_*\). The profile construction supplies its update and moment hypotheses, and the marked field argument above supplies the finite entropy hypothesis. But on the shift-invariant event \(\mathcal E\), the stationary ergodic theorem gives \(E_{\mathbb L_*}c_0>\zeta\). This contradicts the proposition and excludes the bad-event sequence \(Q_N\). Thus [eq:5] holds. Proposition 7 gives finite mean regeneration duration and an almost-sure deterministic vector velocity with positive first coordinate. This vector lies on the deterministic ray of Corollary 5, whose projection on the original transient direction \(\ell\) is positive. Undoing the signed permutation of the coordinate axes therefore gives \(X_n/n\to v\) with \(v\cdot\ell>0\), as asserted in Theorem 1. ◻