Historical context.

Nienhuis’s analysis of the dilute \(O(n)\) model predicted the planar self-avoiding-walk exponents, including the length–distance power \(4/3\) [11]. Duminil-Copin and Smirnov proved the honeycomb connective constant using a local winding relation and a boundary identity [3]. Their mid-edge convention is the port convention used here. For critical strip mass, Beaton, Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann proved decay to zero in their work on surface adsorption [2]. Glazman and Manolescu gave a shorter proof and a logarithmic subsequence bound, and proved boundary two-point invariance for columnwise rhombic half-planes with angles in \([\pi/3,2\pi/3]\) [6]. Krachun and Panagiotis subsequently proved a polynomial upper bound for strip-crossing mass and a quantitative sub-ballistic estimate for uniform honeycomb walks [9]. These results distinguish critical activity, decay of crossing mass, and length–distance exponents; the finite inputs below specify the stronger estimates used here.

The irreducible-bridge construction goes back to Kesten [8]; its honeycomb mid-edge form appears in [2]. On the square lattice, the infinite critical law and its relation to strip-spanning laws are developed by Lawler–Schramm–Werner [10] and Dyhr–Gilbert–Kennedy–Lawler–Passon [4]. Gilbert’s appendix records Kennedy’s heuristic connecting strip mass of order \(h^{-1/4}\) to an irreducible height tail of order \(h^{-3/4}\) and renewal heights of order \(n^{4/3}\) [5]. Here whole irreducible bridges are independent increments; height, length and lateral displacement within one increment remain dependent.

The quantitative starting point is the sharp finite marked-polygon and bridge estimates of [16], together with the renewal estimates of [15]; Section 2 states exactly what we use. These are stronger inputs than the preceding critical and sub-ballistic results. The terminal-free fixed-height probability law and the prescribed half-plane arch probability law already appear in [15]. We give a different proof of the latter, through common-level pair renewals and arch-kernel regularity, and develop the local estimates needed to assemble the prescribed-endpoint laws in the other geometries. The mean estimates require additional length-weighted control and retain separate finite inputs.

The three endpoint mechanisms.

For two ports on one line, grow independent bridge sequences upward from the two ports and stop at their common renewal levels. Their common height transform has exponent \(1/2\). A separate lateral window estimate gives \(O(m^{-2})\) atoms for the difference after \(m\) common blocks. This supplies both regularity of the arch kernel and control of a long final joining arch; the complete argument is in Section 4.

For parallel cuts one must match height and lateral displacement at once. A height-tail bound alone does not pay for one prescribed height. Section 5 bounds all fixed-order differences of the strip mass, starting from the finite projection representation in [16]. This local regularity gives the one-piece height atom bound in Section 6. Together with two-dimensional concentration and a sum of heightwise maximal atoms, it pays the parallel endpoint restriction in Section 7. For nonparallel cuts, small normal prefixes initially exhaust only a sum over terminal ports. A local visit estimate and an artificial-arc boundary identity then control the exceptional mass at the prescribed terminal port. Each case thus pays for its actual endpoint restriction before dividing by the chord mass; Section 8 completes the nonparallel case.

The means use two further arguments. In Section 9, the calibrated finite estimates of [12] and their renewal consequences control marked arms at both ends of an arch. The resulting length-weighted contributions are summable over arch heights at each prescribed gap. In Section 10, the finite strip moments of [14] feed a renewal-reward identity. A two-bridge boundary comparison controls variation between terminal ports after averaging over heights, leading to the density-one and aligned-mixture mean laws rather than an all-height pointwise claim.

Interior connectors

The diameter bound gives the upper normalization, and summing Proposition 4 over order-\(R\) terminal ports gives the lower one. We prepare the length estimates next; the proof of the theorem is completed at the end of this section.

Bridge length and displacement windows

The fixed-height conclusion also appears in [15]. We give the argument from the finite and renewal inputs selected in Section 2.

We next arrange a displacement that points inward at both chord ends. This is needed when a translated chord is attached back to its original supports. Distances in the following geometric statement are physical; the bridge height \(h\) is still in row units.

The cost of two translation choices

Fix pure normals \(n,m\) with \(m\ne-n\). For a lattice translation \(z\) with \(n\cdot z>0\), let \(D_n(z)\) be the critical mass of strict \(n\)-bridges between the prescribed positions separated by \(z\). For \(n\cdot z<0\), set \(D_n(z)=D_n(-z)\); thus the bridge is always oriented in increasing \(n\)-level order. At zero height set \(D_n(0)=1\) and \(D_n(z)=0\) for nonzero \(z\) with \(n\cdot z=0\).

Recoverable attachments

Consider the two inward support normals \(n,m\) at a source \(a\) and a terminal interval whose distances from \(a\) are comparable to \(R\). When the supports differ, assume the endpoints have order-\(R\) clearance to the other support, as in the geometries above. We treat \(m\ne-n\); opposite supports will use Lemma 6 directly. For identical supports their common half-plane replaces the two-cut domain.

Choose a small fixed \(c>0\) and a large scale \(s\le cR\). Translate an input chord by a lattice vector \(v\) with \(|v|\le Cs\) and both \[g_1=n\cdot v/d,\qquad g_2=m\cdot v/d\] between fixed positive multiples of \(s\). Attach a height-\(g_i\) bridge from each translated endpoint out to its original support. Restrict the two bridges to diameter at most \(C's\) and length between \(s^{4/3-\delta}\) and \(s^{4/3+\delta}\). Such restrictions retain fixed fractions of the available bridge mass, by the preceding lemmas and the diameter tail. The branches stay in separated endpoint neighborhoods. They avoid the translated chord by its supports, and, for distinct supports, each stays beyond the translated support at the other end by endpoint clearance and the small choice of \(c\).

Proof of the interval length law

Common blocks and lateral concentration

We use the intersection-renewal construction discussed in [1], proving the required identities directly. Distances along the base are physical distances, with port spacing one; heights are in row units. Run two independent sequences of upward irreducibles of law \(p\), translating each new piece to the current endpoint. Stop at their next common positive renewal height. Let \(r\) be the resulting mass on finite pairs of terminating sequences, initially possibly a subprobability, and let \(T\) be their common height.

A pair block contains one complete bridge segment from each sequence, possibly made of several irreducibles. Let \(D_*\) be the maximum of the diameters of these two complete segments, and let \(Y\) be the second segment’s lateral displacement minus the first’s. In particular, \(|Y|\le2D_*\). Successive common blocks regenerate independently: testing a prescribed pair of terminating sequences uses only the pieces up to their termination.

Regularity of the joining kernel

We next attach short prefixes and recover them from the completed arch. Figure 1 shows this construction.

Concentration under the prescribed-gap law

The finite strip input

Put \(b=3\lambda\), \(C_t(x)=\cosh(t\sqrt x)\), \(\mathcal D(x)=2C_\lambda(x)-1\), and \(a_*=\mathcal D(-4)=2\cos(2\lambda)-1>0\). Define the positive measure \(\mu\) on \((0,\infty)\) by \[ d\mu(x)=\frac{z\mathcal D(z^2)}{\sinh(4\lambda z)}\,dz, \qquad x=z^2,\quad z>0. \tag{7}\] For \(N\ge1\), set \[\begin{split} M_N&=\operatorname{span}\{1,x,\ldots,x^{N-1}, \partial_t^j C_t|_{t=\lambda}:0\le j<N\},\\ c_1&=C_{\lambda/2},\quad c_3=C_{3\lambda/2},\quad h_0=\mathcal D^{-1},\quad h_1=c_1/\mathcal D,\\ g&=\int_0^\lambda C_t\,dt, \quad F_{0,N}=M_N+\mathbb R c_1, \quad G_N=M_N+\mathbb R g. \end{split}\] Let \(\Pi_Nh\in F_{0,N}\) be the unique oblique projection preserving all pairings against \(G_N\) in \(L^2(\mu)\), and write \(r_Nh=h-\Pi_Nh\). The projection is also defined for \(c_3\), whose pairings with \(G_N\) are finite although \(c_3\notin L^2(\mu)\).

We use the following finite facts from the strip analysis in [16].

1. The above projection exists. Whenever the denominator is nonzero, \[ B_N=2a_*\left[ r_Nh_0(-4)- \frac{r_N(c_3-h_1)(-4)\,r_Nh_0(0)}{r_N(c_3-h_1)(0)} \right]. \tag{8}\] The corresponding undivided statement is that there is a scalar \(s_N\) such that \(2B_N=4a_*[r_Nh_0(-4)+s_Nr_Nh_0(0)]\) and \(r_N(c_3-h_1)(-4)+s_Nr_N(c_3-h_1)(0)=0\).

2. For \(0<t<\lambda\), put \(G_{N,t}=M_N+\mathbb R C_t\). On ordered positive nodes \(X=(x_1<\cdots<x_{2N+1})\), the product of evaluation determinants for \((F_{0,N},G_{N,t})\), times a positive base weight at every node, defines a positive density whenever its moments are finite. Its mean interpolant is the oblique projection against \(G_{N,t}\). If \(e_0c_3\) denotes the interpolation residual at zero, then \(-e_0c_3\) is positive and increasing in each ordered node. Multiplying the base weight by an increasing positive function increases its expectation. Replacing \(G_{N,t}\) by \(G_N\) is a positive mixture of these densities.

3. Let \(F_N=F_{0,N}+\mathbb R c_3\), \(A_N=\{f\in F_N:f(0)=0\}\), and \(U_N=\{f\in M_N:f(0)=0\}\). For either determinant-product density \((F_{0,N},G_{N,t})\) or \((A_N,G_{N,t})\), expectations of \(\prod_i r(x_i)\), with \(r(x)=(x+\alpha)/(x+\beta)\) and fixed \(\alpha,\beta>0\), differ by bounded factors from the corresponding symmetric expectations on \((M_N,M_N)\) or \((U_N,U_N)\). The factors are uniform in \(N,t\) and remain so after any fixed number of such rational changes of the base weight. This is the interlacing reduction in the proof of [16]: its positive weight and exponential-domination hypotheses persist under each such change, and the bound uses only the range of \(\log r\), not \(t\). The two symmetric expectations themselves differ by a bounded factor. These statements follow from interlacing of the positive roots of the compressed multiplication forms; a codimension-one compression changes the logarithmic product by at most the range of \(\log r\).

4. For the probability measure \(\mu_c\) proportional to \(dz/\cosh(cz)\), \(x=z^2\), let \(\alpha_c=\pi/(2c)\) and \(w_t=\tan^2(\alpha_ct)\). Then \[ A_t(x)=\cos(\alpha_ct)C_t(x),\qquad \int A_tA_s\,d\mu_c=\frac1{1-w_tw_s}. \tag{9}\] Thus \(A_t=\sum_{j\ge0}P_j(x)w_t^j\) generates orthonormal polynomials \(P_j\) of degree \(j\). The identity and its differentiated forms hold when \(|t|+|s|<c\).

All base weights below have the required exponential domination. In particular these inputs apply to \(\mu\), to \(dW=z\,dz/\sinh(bz)\), and to \(r^jW\) for fixed \(j\) and fixed positive rational function \(r\) of the indicated form. We will prove both the estimates and the nonvanishing needed to use (8).

Transforming the residual space

Set \[v(\omega)=\tan\frac{\pi\omega}{2b},\qquad J(\omega)=v'(\omega),\qquad Q(v)=\frac{v^2(v^2-v(\lambda)^2)}{1-v^2v(\lambda)^2}.\] The strip \(|\operatorname{Re}\omega|<b/2\) maps bijectively to the unit disk. The rational function \(Q\) is an even finite Blaschke product; its zeros are \(0\) with multiplicity two and \(\pm v(\lambda)\) with multiplicity one. For an even holomorphic function \(A\) on the disk, define \[(T_NA)(\omega)=J(\omega)Q(v(\omega))^N A(v(\omega)).\] The factor \(Q^N\) encodes exactly the orthogonality conditions against \(M_N\).

Indeed, the cosine transform identifies \(h\in M_N^\perp\subset L^2(\mu)\) with \[ \int h(x)C_\omega(x)\,d\mu(x)=T_NA(\omega). \tag{10}\] The transform is even, has a zero of order \(2N\) at zero, and zeros of order \(N\) at \(\pm\lambda\). Dividing by \(JQ^N\) therefore gives an even holomorphic \(A\) on the disk. Conversely these zero conditions imply the required orthogonality.

Here is the Hilbert-space content of this identification. Let \(w_\mu(z)\) and \(w_W(z)\) be the even extensions of the \(z\)-densities of \(\mu\) and \(W\). The Fourier datum of the transform \(F=T_NA\) is \(f(z)=h(z^2)w_\mu(z)\), so \[F(\omega)=\frac12\int_{\mathbb R}f(z)e^{\omega z}\,dz, \qquad \|h\|_{L^2(\mu)}^2 =\frac12\int_{\mathbb R}\frac{|f(z)|^2}{w_\mu(z)}\,dz.\] We compare norms while holding \(F\), and hence \(f\), fixed. In the comparison measure \(W\), the same transform is represented by \(h_W=h\,d\mu/dW\); its squared norm replaces \(w_\mu\) in the last integral by \(w_W\). Moving to a parallel line multiplies the Fourier datum by \(e^{z\operatorname{Re}\omega}\). Integrating Plancherel’s identity across the strip supplies the reciprocal weight \(\sinh(bz)/z=1/w_W(z)\). The conformal Jacobian \(J\) then gives the transported comparison norm \[ \mathcal G_N^0(A,A) =c_b\int_{|v|<1}|Q(v)|^{2N}|A(v)|^2\,d^2v, \qquad c_b>0. \tag{11}\] For completeness, the converse Fourier representation follows by Fourier transforming the Cauchy–Riemann equation locally in frequency: the data on parallel lines must have the same exponential multiplier. Strip square integrability gives the stated weighted \(L^2\) data, and division by the even weight gives \(h\). Thus no unproved surjectivity of the transform is being used.

Write \(\mathcal G_N(A,A)=\|h\|_{L^2(\mu)}^2\) for the norm in (10). The density ratio of \(\mu\) to \(W\) is bounded above and below by positive constants, so \(\mathcal G_N\asymp\mathcal G_N^0\) uniformly in \(N\). More precisely, their difference is a fixed bounded sesquilinear form on the two traces \(T_NA(\pm d_{\mathrm{tr}}+iy)\), for a fixed \(d_{\mathrm{tr}}<b/2\) sufficiently close to \(b/2\). To see this, the difference of the reciprocal Fourier weights is, up to a fixed factor, \[z^{-1}\left\{\frac{\sinh(4\lambda z)}{\mathcal D(z^2)} -\sinh(bz)\right\}.\] It is bounded at zero and is \(O(e^{2d_{\mathrm{tr}}|z|})\) at infinity: the terms of largest exponential order cancel. Plancherel on the two parallel lines therefore bounds the difference form by their \(L^2\) trace norms.

If \(A(v)=\sum_{j\ge0}a_jv^{2j}\), both transported norms are equivalent to \[ \|A\|_N^2=\sum_{j\ge0}\frac{|a_j|^2}{N+j}. \tag{12}\] The boundary phase speed of the finite Blaschke product \(Q\) is smooth and strictly positive. Hence \(|Q(re^{i\theta})|^{2N}\) is bounded above by \(e^{-cN(1-r)}\), and bounded below by \(e^{-CN(1-r)}\) for \(r\) near one. Angular orthogonality of the powers, followed by radial integration, proves the equivalence.

Every fixed \(\ell\)th forward difference of \(\mathcal G_N^0\) costs \(O_\ell(N^{-\ell})\) as a form in (12), since it inserts \((|Q|^2-1)^\ell\) in the area integral. The same is true for \(\mathcal G_N\). On each of the fixed interior traces, \(v\to\pm i\) nontangentially at the two ends and \(Q(\pm i)=1\); thus \(|1-Q|\le C(1-|Q|)\). Each difference of \(T_N\) inserts \(Q-1\), and comparison with \(T_{\lfloor N/2\rfloor}\) costs \(O(N^{-1})\). Interior trace norms are bounded by strip area norms, by the mean-value inequality for holomorphic functions. The discrete product rule therefore gives the same bound for the correction form. All norms with finitely shifted indices are equivalent. Differencing \(\mathcal G_N^{-1}\mathcal G_N=I\) consequently shows that every \(\ell\)th difference of the inverse form costs \(O_\ell(N^{-\ell})\) as well.

The model evaluation kernel

Denote by \(K_N^0(v,u)\) the evaluation kernel for \(\mathcal G_N^0\), analytic in \(v\) and anti-analytic in \(u\). Thus, with the inner product linear in its first argument, \(A(u)=\mathcal G_N^0(A,K_N^0(\cdot,u))\) for every even function in the space and every \(|u|<1\). The following bounds are needed also at boundary points, provided the two variables do not meet on the even diagonal.

The model has now isolated the only large part of the evaluation kernel. The remaining change of measure contributes a bounded vector, which is small enough for every residual estimate below.

Let \(K_N\) denote the kernel for \(\mathcal G_N\). If \(u=v(t)\), \(0\le t\le\lambda\), or \(t\uparrow b/2\), then \[ \|\Delta^\ell(K_N(\cdot,u)-K_N^0(\cdot,u))\|_N \le C_\ell N^{-\ell}. \tag{15}\] At \(t=b/2\), where \(u=1\), the difference of columns has a limit in \(\|\cdot\|_N\). Neither full boundary column is asserted to be a finite-norm vector. We define \(K_N(\cdot,1)\) as the model boundary column plus this finite-norm correction, with evaluations in the open disk obtained as limits from inside. Indeed the pairs formed by a point on either fixed interior trace and these \(u\) are uniformly off the even diagonal. On the traces, \[|J|\asymp e^{-\pi|y|/b},\qquad 1-|Q|\gtrsim e^{-\pi|y|/b}.\] Lemma 14 and discrete product rules therefore give \[\|\Delta^\ell(T_NK_N^0(\cdot,u))\|_{L^2(\{\pm d_{\mathrm{tr}}+iy\})} \le C_\ell N^{-\ell}.\] For example the undifferenced integrand is bounded by \[CN e^{-\pi|y|/b}\exp\{-cNe^{-\pi|y|/b}\}.\] Its squared integral is bounded; each difference supplies its stated additional factor. Apply the bounded correction form to this column and solve with \(\mathcal G_N^{-1}\). The resolvent identity and the inverse difference bounds prove (15). Dominated convergence on the traces gives its endpoint interpretation.

Evaluation at a fixed interior disk point costs \(O(\sqrt N)\) in (12). Consequently, for \(v,u\) corresponding to \(0\le t\le\lambda\), the actual kernel has the same strictly positive leading limit after division by \(N\), with \(\ell\)th differences \(O(N^{1-\ell})\). The same holds when exactly one variable is \(1\).

Residual estimates at the two evaluation points

We prove the following bounds, which are the full analytic input to the quotient in (8): \[ \begin{split} |\Delta^\ell r_Nh_j(-a)| &\le N^{-1+b\sqrt a/\pi-\ell+o(1)}\quad(j=0,1),\\ |\Delta^\ell r_Nc_3(-a)| &\le N^{b\sqrt a/\pi-\ell+o(1)}\quad(a=0,4),\\ -r_Nc_3(0)&\ge c>0. \end{split} \tag{16}\] All orders \(\ell\) are fixed. The estimates with \(o(1)\) mean that each fixed positive exponent slack is allowed, with its own constant.

First take the orthogonal residual against \(M_N\) alone. For input \(h=h_j\) its transform is \(T_NA_{h,N}\), where \(A_{h,N}\) represents the functional obtained by pairing \(h\) with the residual space. Write \[h(z^2)=\int_{\mathbb R}f_h(y)\cos(yz)\,dy.\] The function \(f_h\) is integrable against \(e^{u|y|}\) for each \(u<\pi/(3\lambda)=\pi/b\). This follows by Fourier inversion and shifting within closed narrower strips: the first zeros of \(\mathcal D(z^2)\) have imaginary part \(\pi/(3\lambda)\), and the functions decay exponentially along every such horizontal line. Thus the functional on \(A\) is \(\int f_h(y)(T_NA)(iy)\,dy\).

On \(\|A\|_N\le1\), the area mean-value bound and comparison to \(T_{\lfloor N/2\rfloor}\) give \[|(\Delta^\ell T_NA)(iy)| \le C_\ell N^{-\ell} \exp\{-cNe^{-\pi|y|/b}\}.\] Indeed \(1-|Q(v(iy))|\gtrsim e^{-\pi|y|/b}\) and \(|1-Q|\lesssim1-|Q|\) there. Integrating against \(f_h\), then using the inverse-form differences, yields \[ \|\Delta^\ell A_{h,N}\|_N\le N^{-1-\ell+o(1)}. \tag{17}\] For an input \(C_s\) with \(0\le s<b/2\), the representing vector is instead \(J(s)Q(v(s))^NK_N(\cdot,v(s))\). As \(s\uparrow b/2\), all finite projection pairings and the residual transforms on compact subsets of the strip converge. Thus the formula extends to \(c_3\) using the boundary column just constructed and \(Q(1)=1\); it does not assert a bounded Hilbert-space evaluation functional at \(u=1\).

We now impose the one remaining oblique constraint. Put \(u_0=v(\lambda/2)\). The transform of \(r_Nh\) is \(T_N\) applied to \[ A_{h,N}-K_N(\cdot,u_0) \frac{\chi_N(A_{h,N})}{\chi_N(K_N(\cdot,u_0))}, \tag{18}\] where \(\chi_N\) averages values at \(v(t)\), \(0\le t\le\lambda\), with probability proportional to \(J(t)|Q(v(t))|^Ndt\). To check the formula, orthogonality to \(M_N\) is already encoded in \(T_N\), and \(\int_0^\lambda C_tdt=g\). On this interval \(Q\) has constant sign away from its zeros, so this last condition is precisely \(\chi_N=0\).

Every fixed \(\ell\)th difference of the probabilities defining \(\chi_N\) is \(O_\ell(N^{-\ell})\) in total variation. Divide their exponential base by its maximum. Differencing inserts powers of the gap to one. Integrals at exponents \(N/2\) and \(N\) are comparable: the gap vanishes to finite order at its finitely many maxima, by analyticity, and the one-dimensional radial Laplace estimate applies in disjoint neighborhoods of them. Normalization and the quotient difference rule give the claim. Kernel positivity and its leading limit now show \[\chi_N(K_N(\cdot,u_0))\ge cN, \qquad |\Delta^\ell\chi_N(K_N(\cdot,u_0))| \le C_\ell N^{1-\ell}.\] For \(h_j\), the subtracted vector in (18) has the same bound as (17): use the \(O(\sqrt N)\) interior evaluation bound, the \(O(\sqrt N)\) column norm, and the denominator of order \(N\), with their corresponding difference bounds. For \(c_3\), the scalar multiplier of the interior column has \(\ell\)th differences \(O_\ell(N^{-\ell})\). The model columns at \(1\) and \(u_0\) have \(T_N\)-images on the imaginary axis bounded by \(O(N^{-\ell})\) after \(\ell\) differences, by the off-diagonal kernel estimate; their finite-norm corrections satisfy the same bound. We have proved, with \(F_{h,N}(\omega)=\int(r_Nh)C_\omega\,d\mu\), that \[ |\Delta^\ell F_{h,N}(iy)|\le N^{a_h-\ell+o(1)},\qquad a_{h_0}=a_{h_1}=-1,\quad a_{c_3}=0, \tag{19}\] uniformly for \(y\in\mathbb R\).

A uniform nonzero denominator

It remains to prove \(-r_Nc_3(0)\ge c\). An upper estimate for the residual would not suffice to divide by it. First replace \(G_N\) by \(G_{N,t}\), \(0<t<\lambda\), and use the comparison measure \(W\) in both projection and transform. Its kernels are exactly \(K_N^0\), and \(\chi_N\) becomes point evaluation at \(v(t)\). Apart from the fixed positive factor \(J(b/2)\), the residual transform is \(J(iy)Q(v(iy))^N\) times \[K_N^0(v,1)-K_N^0(v,u_0) \frac{K_N^0(v(t),1)}{K_N^0(v(t),u_0)}, \qquad v=v(iy).\] After division by \(N\), Lemma 14 gives a limit uniform on the imaginary diameter. At \(v=\pm i\), its sign is the sign of \[\frac12- \frac{1-v(t)^2u_0^2}{(1+u_0^2)(1-v(t)^2)},\] which is strictly negative, uniformly for \(0\le t\le\lambda\). Indeed the second term is at least \(1/(1+u_0^2)>1/2\). The remaining factors \(O\) are positive on both diameters, with uniform positive bounds on the compact sets in question.

Fourier inversion at zero integrates the bracket divided by \(N\) against \[cNJ(iy)Q(v(iy))^Ndy,\qquad c>0.\] This positive measure has mass bounded above and below and concentrates at \(y\to\pm\infty\): on the imaginary diameter \(Q\ge0\), \(1-Q\asymp e^{-\pi|y|/b}\), and \(J\asymp e^{-\pi|y|/b}\). The boundedness supplied by Lemma 14 and the negative endpoint limit therefore give the desired lower bound for \(W\), uniformly in \(t\).

To pass to \(\mu\), its exact density ratio to \(W\) is \[\frac{d\mu}{dW} =\frac{(2u-1)^2(2u+1)}{4u(2u^2-1)}, \qquad u=\cosh(\lambda\sqrt x).\] It is positive and smooth on \([0,\infty)\) and eventually increasing. For \(r(x)=(x+2)/(x+1)\), choose a fixed integer \(j>0\) so that \(\mu/(r^jW)\) is increasing: on a compact interval the negative part of the logarithmic derivative is bounded, while \(-(\log r)'\) is strictly positive; outside it the original derivative is already nonnegative. The ordered association input then bounds the expectation of \(-e_0c_3\) for \(\mu\) below by that for \(r^jW\).

Each multiplication of the base weight by \(r\) changes this expectation by only a bounded factor. Indeed it multiplies it by the ratio of the characteristic-product expectations of \(\prod_i r(x_i)\) under \((A_N,G_{N,t})\) and \((F_{0,N},G_{N,t})\) at the current base weight. The interlacing input compares the first to \((U_N,U_N)\) and the second to \((M_N,M_N)\); symmetric codimension-one compression compares these last two. The bounded range of \(\log r\) gives a uniform constant at each step, independent of \(N,t\). After the fixed number \(j\) of steps, the lower bound therefore remains positive. Finally integrate the positive \(G_{N,t}\) mixture to obtain it for \(G_N\). This proves the last line of (16).

An atom of the one-piece height

The extra information supplied by strip regularity is one full power of height decay beyond the tail bound. This is the estimate that rules out bridges made from too few irreducibles at one prescribed endpoint.

Two forms of displacement smoothing

We need both a planar atom bound and a bound summed over possible heights. The latter is stronger than merely summing the planar bound over all heights, since the height distribution has an infinite tail.

Visits near the prescribed target

The prefix construction has so far controlled only the sum of exceptional masses. To pass to a single target, we need a bound for paths that approach that target before ending a specified distance away. We give a local visit estimate. For \(q\le cR\), consider paths from \(a\) to points \(P\) on the target ray at distance between \(q\) and \(2q\) from \(Q\), also coming within \(t\) of \(Q\), \(1\le t\le q\). Their total mass satisfies \[ H_q(t)\ \lesssim\ R^{-1/4}(t/q)^{3/8}. \tag{27}\] Here and below \(c>0\) can be taken small relative to the compact geometry bounds. Use the fixed-source construction of Proposition 9, with the side-1 outer landing exactly at \(a\) and side-2 outer landing free. Take translation scale a small fixed multiple of \(q\), so all bridge portions and translations have excursion less than (say) \(q/20\), with gaps between original and translated levels at each end of order \(q\). If \(q\) is bounded the estimate already follows from diameter. The first summed indicator mass is at least \(c_1 q^{1/2}H_q(t)\), using \(\gtrsim q^{3/4}\) summed bridge mass from \(a\) to its translates and \(\gtrsim q^{-1/4}\) on side 2. The target measure (paths from \(a\) to the other ray) has mass \(O(R^{-1/4})\).

For simultaneous translations \(v,v'\), split at both support crossings on each branch in level order as in the chord construction. The orders may differ at the two ends. All initial crossings still precede both terminal crossings, by localization of the two exterior branches. Write \(P_v,P_{v'}\) for the terminal ports of the two input chords; each lies in the bin at distance between \(q\) and \(2q\) from \(Q\). For \(t\le q/10\), a visit near either translated landmark \(Q+v\) or \(Q+v'\) cannot lie on an exterior branch. The initial branches remain at distance of order \(R\) from these landmarks. Each terminal branch has excursion less than \(q/20\) from its endpoint \(P_v+v\) or \(P_{v'}+v'\); the endpoint bin and the small translation scale keep every such endpoint at distance comparable to \(q\) from both landmarks. Thus both visits occur after both initial crossings and before both terminal crossings.

This common central portion stays above both target cuts. Its visit within \(t\) of the landmark on the lower cut therefore forces \(|m\cdot(v-v')|\le Ct\). Moreover, the chord between the innermost cuts retains the visit to the landmark belonging to its own terminal cut. Its terminal point is the translated \(P\) from that same list, so its distance to the landmark is still in the prescribed bin; its initial point is also prescribed. The endpoints and landmark have only shifted by \(O(q)\), so this mixed-cut wedge has comparable compact geometry bounds. If the visit estimate is known with exponent \(\alpha\) for these geometries, its central mass is consequently at most \(CR^{-1/4}(t/q)^\alpha\). The diameter bound starts this argument at \(\alpha=0\).

The mixed difference factor now sums to \(O(t^{1/2})\): \[\sum_{z:\ |m\cdot z|\le C t}D_n(z)(1+|m\cdot z|)^{-1/4}\le C' t^{1/2},\] with \(D_n\) defined before Lemma 8. The projected concentration and negative-quarter-moment bounds in (4) bound the contribution from \(j\) pieces by \(C\min(j^{-1/3},(1+t)^{3/4}j^{-4/3})\). Summing in \(j\) by renewal, including ties, proves the restricted estimate. The outer side-2 bridge costs \(O(q^{-1/4})\); the side-1 outer cost \(D_n(v_{\rm out})\) sums to \(O(q^{3/4})\), where \(v_{\rm out}\) gives the side-1 outer crossing. Drop mutual avoidance constraints between the difference, center and outer portions and multiply their upper bounds. The summed second moment is at most \[C q^{1/2}t^{1/2}\,R^{-1/4}(t/q)^\alpha .\] Cauchy–Schwarz proves the assertion with exponent \(1/4+\alpha/2\) (the remaining \(t\ge q/10\) always follows by diameter). Applying twice proves the displayed estimate; the shifts of supports in this finite bootstrap are \(O(q)\) each, so one can enlarge the compact bounds a little and decrease the allowed relative scale \(c\) at each step.

An endpointwise comparison

We adapt the exterior-arc comparison from [16] to the nonparallel supports. Take two individual endpoints \(P,Q\), with \(P\) nearer the apex and \(|P-Q|\in[q,2q]\). Add a formal connection from \(P\) to \(Q\) in the exterior of the target half-plane: a semicircle tangent to the normal rays at its anchors. This arc is not a honeycomb walk. Assign it modulus weight one, no visited-center length, and the phase \(e^{i\sigma\Delta W}\) of its geometric turn. Sum only paths using the arc exactly once; all their other edges and vertex weights are ordinary. The arc’s exterior lies outside \(W\), so it meets no ordinary path away from its anchors. Figure 2 records its orientation. Arrange the coordinates so the initial ray points right, and the target ray lies counterclockwise at angle \(\theta\). Thus \(a\to P,a\to Q\) turn by \(\theta\); the arc \(P\to Q\) turns by \(-\pi\). For a path \(\omega:a\to Q\) define \[f_W(P;\omega)=\frac1{2\cos(\sigma\pi)}- \sum_{b'\in(P,Q)}\ \sum_{\substack{\zeta:P\to b'\ \text{in }W\\ \zeta\text{ disjoint from }\omega}} \kappa^{L(\zeta)}.\] For paths to \(P\) use \(f_W(Q;\omega)\) analogously with continuation from \(Q\) and the same open interval. Then (integrals denoting critical weighted sums) \[\left|\int_{a\to Q} f_W(P;\omega)-\int_{a\to P} f_W(Q;\omega)\right| \le C R^{-1/4}(K_Q+K_P).\] Indeed the flux delivered by arc traversals before further continuation equals the outgoing flux using the arc once. Prefix cancellations at all ordinary vertices preserve the count of arc uses under unfinished-loop reversal; the whole arc is well after the initial cut so the source cannot be inside a loop. Use first a large convex truncation; distant contributions vanish by ordinary diameter bounds. After rotation by \(e^{-i\sigma\theta}\) the imaginary input is \(\sin(\sigma\pi)(K_Q-K_P)\). Outputs back to the initial support have modulus cost at most \(C R^{-1/4}(K_Q+K_P)\). Outputs on the target support from an initial traversal \(a\to Q\), then the arc to \(P\), must end nearer than \(Q\) by noncrossing. If ending even before \(P\) their relative phase after rotation is 1, while in the open interval it has imaginary part \(\sin(2\sigma\pi)\) (final arch positive turn \(\pi\)). Similarly from \(a\to P\) the only target-side terms with imaginary part end in \((P,Q)\), with opposite sign. Neither anchor can be such a final exit (used triangles). Dividing by \(\sin(2\sigma\pi)\) proves the comparison.

Reflection and the half-plane contour identity give total arch mass \(1/(2\cos(\sigma\pi))\) in each direction. Together with the half-plane arch estimates, this gives, for paths to \(Q\), \[ c' q^{-1/4}\le f_W(P;\omega)\le C(1+\min(q,\operatorname{dist}(P,\omega)))^{-1/4} \tag{28}\] and analogously after exchanging the endpoints. Indeed all completions counted there are among half-plane arches in one direction, but omitting at least the long-offset tail of mass \(\gtrsim q^{-1/4}\). All sufficiently small diameter ones up to scale the indicated minimum are included, by clearance and disjointness, and the diameter tail bounds the loss.

With \(q\) a sufficiently small fixed fraction of \(R\), the comparison absorbs the \(O(R^{-1/4})K_Q\) term. On the other side average over the order \(q\) choices of \(P\). The visit bound \(H_q(t)\), with exponent \(>1/4\), pays for the upper estimate on \(f_W(Q;\omega)\) by dyadic summation. More explicitly the total over \(P\) of these upper integrals costs \(O(R^{-1/4}q^{-1/4})\), including unit and smaller distances and also paths always at distance \(>q\). This yields the uniform bound \(K_Q\lesssim R^{-5/4}\).

Removing the exceptional mass

The bound for each individual kernel is now available. Apply the same comparison to the exceptional set left by the prefix construction. The same absolute comparison holds with the two left integrals restricted to exceptional paths and the right error still \(C R^{-1/4}(K_Q+K_P)\), with \(C\) independent of \(q\le cR\). To see this, subtract the comparison (in signed-difference form with its error) in \(W'\), starting at \(a'\), integrated over small prefixes. Its right error costs at most the same amount by the prefix injection. And the difference of the two \(f\) factors for a central path and full constructed path costs only \(O(R^{-1/4})\) each, since central completions in the sum stay beyond the prefix cut and avoid the prefix too, and any discarded full-domain completions must reach that cut from clearance of order \(R\). The constants are uniform by smallness of the prefixes.

Fix \(q\) of order \(\rho R\) for small fixed \(\rho>0\), and put \[e_R=R^{1/4}\sum_U E_U\longrightarrow0.\] Let \(I_R(q)\) be the sum over the order-\(q\) choices of \(P\) of \(\int_{a\to P,\,\mathrm{exceptional}}f_W(Q;\omega)\). To control its possibly singular weight, fix \(0<\delta<1/2\). Paths with \(\operatorname{dist}(Q,\omega)\ge\delta q\) contribute at most \(C(\delta q)^{-1/4}\sum_P E_P\), by (28). For the remaining paths, discard the exceptional restriction and sum (27) over dyadic distance bins below \(\delta q\). The exponent \(3/8-1/4=1/8\) is positive, so, including the final bounded-distance bin, for \(R\) sufficiently large at fixed \(\rho,\delta\) this gives \[I_R(q)\le C R^{-1/4}q^{-1/4} \bigl(\delta^{-1/4}e_R+\delta^{1/8}\bigr).\] First let \(R\to\infty\) at fixed \(\rho,\delta\), then let \(\delta\downarrow0\). Thus \(I_R(q)=o(R^{-1/4}q^{-1/4})\) for each fixed \(\rho\). Averaging the exceptional comparison over its order-\(q\) ports, using the lower bound in (28) and the individual kernel upper bound, now yields \[q^{-1/4} E_Q \le o(R^{-1/4}q^{-5/4})+C R^{-1/4} R^{-5/4}.\] Here the little-oh may depend on \(\rho\), but the comparison error constant \(C\) does not. In particular \[\limsup_{R\to\infty}R^{5/4}E_Q\le C\rho^{1/4}.\] Letting \(\rho\downarrow0\) proves \(E_Q=o(R^{-5/4})\).

The prescribed length law

Finally count lengths violating either power bound \(R^{4/3\pm\epsilon}\). Up to mass \(o(R^{-5/4})\), the prefix construction applies at the prescribed \(Q\). Under the iid law its prefix length has exponent \[\frac{3/4-\eta}{9/16}=\frac43-\frac{16\eta}{9}\] in probability. On small prefixes the center factor is at most \(CR^{-5/4}\), uniformly in their displacements. Therefore an iid event of probability \(o(1)\) contributes constructed mass \(o(R^{-5/4})\), without asserting independence after fixing \(Q\). Choose \(\eta<9\epsilon/16\). This proves the short-length bound and also makes prefixes longer than \(\frac12R^{4/3+\epsilon}\) negligible.

For the appended chord, any specific prefix displacement costs \(O(k^{-8/3})\) by Proposition 16. Align all new wedges by lattice translations \(\tau_w\); their apex intersections are triangular lattice vertices. In the common wedge the endpoints are \[x_w=a+w+\tau_w,\qquad y_w=Q+\tau_w, \qquad w=(x_w-y_w)-(a-Q).\] Thus each endpoint pair determines its prefix displacement. Since the angles are fixed and \(|w|=O(R)\) for small prefixes, the apex shift \(\tau_w\) is also \(O(R)\). In particular there are only \(O(R)\) possible aligned source ports, all at distance \(O(R)\) from the apex.

We may now sum freely over the endpoint pairs. For appended chords of diameter above \(S_*=R^{1+\lambda}\), the one-source diameter bound and the number of sources give \(CRS_*^{-1/4}\). All smaller-diameter chords lie in a common convex pure truncation of diameter \(O(S_*)\). The all-endpoint length-mass bound there shows that those of length above \(\frac12R^{4/3+\epsilon}\) have total mass at most \(CR^{-4/3-\epsilon}S_*^{25/12+o(1)}\). Multiplying these two estimates by \(k^{-8/3}\), respectively, gives \[CR^{-5/4-\lambda/4+8\eta/3},\qquad R^{-5/4-\epsilon+25\lambda/12+8\eta/3+o(1)}.\] Choose first \(\lambda>0\) so that \(25\lambda/12<\epsilon\), then \(\eta>0\) sufficiently small that both exponents are strictly below \(-5/4\), while retaining \(\eta<9\epsilon/16\). Both appended-chord exceptions are consequently \(o(R^{-5/4})\).

Restriction to a hexagon in Theorem 1 can only decrease these absolute exceptional masses. Its own normalizer is at least \(cR^{-5/4}\), by Proposition 4 and the endpoint clearances. Dividing proves that both tails vanish in probability in the restricted domain as well as in the full wedge.

This proves Theorem 18. Together with Theorem 10 and Theorem 17, it proves Theorem 1: the two side supports are identical, parallel, or nonparallel, and the connector lower bound supplies the normalizer in each finite hexagon.

The two arms and the dyadic target

Split an arch at a chosen highest vertex. Its two real incident edges go downwards. Extending both arms to the directly adjacent top wall port makes them strict downward bridges, duplicating just that vertex. Their critical product weight therefore differs from the arch weight by a fixed factor. We count the two shapes from a reference top port, impose equal heights and transverse displacement difference \(g\), and translate the first bottom endpoint to its prescribed position. This entails no additional placement sum.

The two ends of this decomposition impose different constraints. At the top the arms share a turn, so Proposition 20(ii) applies. At the bottom they start at distinct ports a distance \(g\) apart; we will prove a separate weak-order estimate there. To sum length, mark one irreducible on one arm with its length, and reserve independent pieces away from both ends and the mark to match height and displacement. Adding the two choices of rewarded arm bounds the full length weight.

Put \(d_*=4/3\) and \(a=3/4\). For a dyadic scale \(H\), let \[\mathcal M_g(H)= \sum_{\substack{\gamma:\,0\to g\text{ a half-plane arch}\\ H\le h(\gamma)<2H}} L(\gamma)\kappa^{L(\gamma)},\] where \(h(\gamma)\) is the integral height of the adjacent top wall just described. The target estimate is \[ \mathcal M_g(H)\le C_\eta g^{1/4}H^{d_*-2a+O(\eta)} \qquad\bigl(H\ge g/\log^2g\bigr). \tag{29}\] The smaller height scales will be negligible by strip confinement. Since \(d_*-2a=-1/6\), summing this bound for sufficiently small \(\eta\) gives total length mass \(g^{1/12+o(1)}\). Division by \(K_g\gtrsim g^{-5/4}\) then gives the required mean upper bound.

A local truncated reward

The first ingredient bounds the cost of the piece carrying the length weight.

Survival of the order at separated bottom ports

The shared-turn estimate does not apply at two different starting ports. We therefore prove the separate weak-order estimate required at the bottom ends of the arch. This estimate uses the calibrated finite inputs throughout, independently of the off-axis construction in Section 4.

Summing the marked-arm decomposition

The following terminal-summed consequence also appears in the proof of [15]. We include its normalization and first-moment argument to distinguish this confined mixture from the individual-gap law.

Endpoint variation from two disjoint bridges

First put \(t_h\) equal to the total critical product mass for two vertex-disjoint bridges with adjacent starts 0,1 in that order. Write \(J_h(u,v)\) for its refinement by ordered endpoints \(u<v\) on top. We claim \[b_h(y)=\sum_{u\le y<v}J_h(u,v).\] Indeed take a bridge spectator from 1 to \(y'\) on top and grow an active exit from 0 with phase \(e^{i\sigma W}\), stopping instead just before first contact with the spectator vertices. Take also the balance reversing the roles of 0 and 1 (spectator now from 0 to \(y'\)), multiplied by \(s_0=e^{6i\theta}\), \(\theta=\pi/8\). Revisit pairs cancel; convergence at fixed width in the vertical coordinate (the unbounded horizontal strip) follows by the finite strip exponential lateral cutoff.

At a first contact the three arms from \(0,1,y'\) form a tripod with the central weight included once, with bijections from the contact terms of each balance. The three branches have the same turn phases at the first contact, so their turn phases measured before turning at the center satisfy \(e^{i\sigma W_{y'}}=e^{i\theta}e^{i\sigma W_0}=e^{-i\theta}e^{i\sigma W_1}\). Hence the contact terms cancel in this combination (\(s_0=-e^{-2i\theta}\)).

The remaining active arches in the first balance can only go left, separated by the bridge from exits beyond the adjacent port (also no exit at it); in the second only right. Their phases including the combination factor are thus both \(\ell=e^{3i\theta}\). Project by \(X\mapsto \Im(\ell^{-1}X)/\Im(\ell^{-1})\), killing them and giving coefficients \(1,-1\) on the two types of bridge terms. This yields \[b_h(y'-1)-b_h(y')=\sum_{u<y'}J_h(u,y')-\sum_{v>y'}J_h(y',v).\] Sum up to \(y'=y\).

Consequently \(\sup_e b_h(e)\le t_h\) and the sum of absolute first differences of \(b_h(\cdot)\) on its coset is at most \(2t_h\). Also \(t_h\le C(A_{h+1}-A_h)\): reflect so the adjacent ports are on the top of the old strip, join them by the bounded minimal arch in the added one layer, and translate horizontally to put the resulting bottom arch’s starting port at the fixed source (use the ordered ends). A strip-exchanging lattice isometry with any needed offset can be used before this translation. Recovery has bounded multiplicity since the crossing ports of the old top line are exactly the two joined ports, adjacent and prescribed before translation. Thus by flat strip flux, for fixed \(0<c_1<c_2<\infty\), \[\sum_{c_1 H\le j\le c_2 H} t_j \le C b_{\lfloor c_1H\rfloor}=H^{-\alpha+o(1)}\quad\text{(as an upper bound)} .\]

A pointwise lower bound at every height

The variation bound is available only after summing heights. Nevertheless it will yield a lower bound at each height by averaging internal renewal cuts. Here is a uniform pointwise lower bound at all large heights for any fixed \(\delta>0\): \[b_h(e)\ \ge\ h^{-1-\alpha-o(1)} \qquad (|e|\le h^{1-\delta},\ e\in\mathbb Z+h/2).\] First take even \(h\). For a bridge \(\gamma\in\mathcal B_h(e)\), let \(N_h(\gamma)\) count the integers \(j\in[h/8,h/7]\) for which all three levels \(j,2j,h/2+j\) are single-crossing cuts. Put \(k=h/2-j\). Splitting at those cuts gives four bridges of heights \(j,j,k,k\), so, with convolution here only in the offset, \[\sum_{\gamma\in\mathcal B_h(e)}N_h(\gamma)\kappa^{L(\gamma)} =\sum_{h/8\le j\le h/7}(b_j*b_j*b_k*b_k)(e).\] For all but \(o(h)\) of these internal levels \(j\), the summed variation estimate gives \(t_j\le h^{-1-\alpha+\delta_0/8}\), where \(\delta_0=\min(\delta,1/2)\). The external height \(h\) remains arbitrary. Write \(A=b_j(\cdot)*b_k(\cdot)\). Its total is \(b_j b_k\), concentrated on \(|x|\le Ch\log^2 h\) up to negligible tails by the exponential horizontal strip bounds; its sup norm and total variation on its coset are bounded by \(t_j b_k,2t_j b_k\) respectively. Thus \[\sum_x A(x)A(e-x) =\sum_x A(x)A(x-e)\ \ge\ \|A\|_2^2-2|e|t_j^2 b_k^2, \qquad \|A\|_2^2\ge h^{-1-o(1)}(b_jb_k)^2.\] For good \(j\) the subtracted term is negligible. Summing gives first moment at least \(h^{-4\alpha-o(1)}\).

For the second moment, diagonal terms cost at most \(h^{-4\alpha+o(1)}\) in sum (use \(\sup b_j(\cdot)\le t_j\) on one factor). Off-diagonal terms \(j<j'\) are cut up by the union of both triples of levels, yielding endpoint convolution masses of seven bridges: four of heights comparable to \(h\), including \(j\), and three of heights \(\Delta,2\Delta,\Delta\), \(\Delta=j'-j\). Bound pointwise using the sup at \(j\) and total masses for all others. Since \(\sum_{\Delta\le h}b_\Delta^2 b_{2\Delta}\le h^{1-3\alpha+o(1)}\), summation of \(t_j\) gives \[\sum_{\gamma\in\mathcal B_h(e)}N_h(\gamma)^2\kappa^{L(\gamma)} \le h^{1-7\alpha+o(1)}.\] Weighted Cauchy–Schwarz therefore gives \[b_h(e)\ge \frac{\bigl(\sum_{\gamma\in\mathcal B_h(e)} N_h(\gamma)\kappa^{L(\gamma)}\bigr)^2} {\sum_{\gamma\in\mathcal B_h(e)} N_h(\gamma)^2\kappa^{L(\gamma)}} \ge h^{-1-\alpha-o(1)}.\] Odd heights follow by a bounded one-layer extension from even height with the appropriate half-step endpoint shift, using a lower-bound window with a smaller positive \(\delta\).

Inserting length between two bridge kernels

Write \(K=b*b\) in both height and offset. It follows that \[K_h(e)\ge h^{-2\alpha-o(1)} \qquad (|e|\le h^{1-\delta})\] again for admissible offsets. Indeed sum over heights \(h/3\le l\le 2h/3\) and offsets \(|x|\le h^{1-\eta}\) of the first factor, for arbitrarily small \(0<\eta<\delta/2\); both factors use the preceding lower bound with sublinear power windows. Send \(\eta\) down. On the upper side we have \[\sum_{j\le 2H}\sup_x K_j(x)\le H^{2\beta-1+o(1)} .\] Order the two factor heights by size, larger \(l\) and smaller \(r\). Apart from zero total height use \(\sup b_l(\cdot)\le t_l\) and \(\sum_{r\le l} b_r\le l^{\beta+o(1)}\) (up to constants); summing dyadically gives the bound since \(\beta-\alpha>0\).

The length insertion is exactly \(b^{(L)}=K*i^{(L)}\). Uniformly in the same windows we obtain the pointwise lower bound \[b_h^{(L)}(e)\ge h^{d_*+\beta-2-o(1)}.\] In fact take the middle piece from \(i^{(L)}\) with height \(\le m\log^2 m\) and absolute displacement \(\le m\log^4 m\), \(m=h^{1-\varepsilon}\), for small fixed \(\varepsilon>0\). Its available total mass is at least \(m^{d_*-\beta-o(1)}\), by the exponentially height-discounted first moment and tail truncations in Proposition 26. The remaining \(K\) uses height comparable to \(h\). Since \(m\log^4m\le h^{1-\varepsilon/2}\) for large \(h\), its offset still lies in a sublinear power window for each fixed \(\delta,\varepsilon>0\). Use the lower bound for \(K\) and send \(\varepsilon\) to zero. Also \[\sum_{H\le h\le 2H}\sup_e b_h^{(L)}(e)\le H^{d_*+\beta-1+o(1)}\] by the upper \(K\) estimate and the truncated \(i^{(L)}\) upper bound up to height \(2H\).

One density-one set and the aligned mixture

Put \(U_h=\sup_e b_h(e)\) and \(V_h=\sup_e b_h^{(L)}(e)\), with the suprema over the actual row coset. For dyadic \(H\), choose \(\rho_H\to\infty\) with \(\rho_H=H^{o(1)}\) so that the two summed upper bounds give \[\sum_{H\le h<2H}U_h\le\rho_H H^{-\alpha},\qquad \sum_{H\le h<2H}V_h\le\rho_H H^{d_*+\beta-1}.\] In each such interval retain precisely the heights satisfying \[U_h\le\rho_H^2 H^{-1-\alpha},\qquad V_h\le\rho_H^2 H^{d_*+\beta-2},\] and let \(G\) be the union of the retained heights. At most \(2H/\rho_H=o(H)\) heights are removed from each interval. Summing over dyadic intervals shows that \(G\) has density one. On \(G\), \[\sup_e b_h(e)\le h^{-5/4+o(1)},\qquad \sup_e b_h^{(L)}(e)\le h^{1/12+o(1)}.\] The construction of \(G\) uses global endpoint suprema and does not depend on \(\delta\) or \(\varepsilon\). For each fixed \(\delta>0\), the all-height lower bounds give, uniformly on its admissible window, \(b_h(e)\ge h^{-5/4-o(1)}\) and \(b_h^{(L)}(e)\ge h^{1/12-o(1)}\). Combining the lower numerator with the upper denominator, and then the upper numerator with the lower denominator, yields \[h^{4/3-o(1)}\le\frac{b_h^{(L)}(e)}{b_h(e)} \le h^{4/3+o(1)}\qquad(h\in G,\ |e|\le h^{1-\delta}).\] This proves Theorem 24 with its stated quantifiers.

For completeness, the mixture in Theorem 25 follows directly from the estimates before that deletion of heights. At each large even height, the aligned lower bounds are \(b_h(0)\ge h^{-5/4-o(1)}\) and \(b_h^{(L)}(0)\ge h^{1/12-o(1)}\). Summing over the order \(H\) even heights in \([H,2H]\) gives \(Z_H\ge H^{-1/4-o(1)}\) and \(M_H\ge H^{13/12-o(1)}\). The dyadic sums of the endpoint suprema give the reverse bounds \(Z_H\le H^{-1/4+o(1)}\) and \(M_H\le H^{13/12+o(1)}\). Their ratio proves the mixture mean at every large scale. No upper estimate at an individual discarded height has been used.