Endpoint exchange.

For \(u=\eta_{\ell+1}-\eta_\ell\) and \(S_\ell\) the parameter swap (positions \(\ell,\ell+1\)), (3) moves the box \(\check R_\ell(u)\) through the row and intertwines it with the swapped row. Indeed the crossing parameters are differences of successive visiting lines’ labels (right minus left for the ordered inputs, exchanging labels at outputs), with auxiliary label \(-\lambda\): at the two columns (take \(1\le\ell<N\)) formula (3) uses the ordered labels \(-\lambda,\eta_\ell,\eta_{\ell+1}\), sliding the column exchange from below the two auxiliary crossings to above them. Thus \(\check R_\ell(u)\Psi(\boldsymbol\eta)=\Psi(S_\ell\boldsymbol\eta)\) by uniqueness and preservation of the empty projection. The one-arc coordinates satisfy a closed system. If one of \(i,j\) equals \(\ell\) or \(\ell+1\) and the other lies outside this pair, then \[T_u H_{ij}(S_\ell\boldsymbol\eta)= A_u H_{ij}(\boldsymbol\eta)+ C_u H_{i'j'}(\boldsymbol\eta),\] where the prime switches \(\ell,\ell+1\) on indices. Moving one port across the neighbor preserves the oriented interval type. For the two neighboring interval types the equations are \[ \begin{split} T_u H_{\ell,\ell+1}(S_\ell\boldsymbol\eta)&=D_u H_{\ell,\ell+1}(\boldsymbol\eta)+B_u,\\ T_u H_{\ell+1,\ell}(S_\ell\boldsymbol\eta)&=D_u H_{\ell+1,\ell}(\boldsymbol\eta). \end{split} \tag{4}\] These equations use only one-arc coordinates. Outside the exchanged pair, occupancies are unchanged; inside it, a box cannot remove an additional arc while retaining an active output port. A cap on an already paired input pair creates a loop and gives zero. Insertion from the vacuum creates only the short interval \(\ell\to\ell+1\), explaining the source term \(B_u\).

Commuting transports.

Set \(Y_1=Y\) and \(Y_{l+1}=g_l^{-1}Y_lg_l^{-1}\). We first establish their commutation relations directly from the braid relations. \(Y\) commutes with \(g_j\), \(2\le j<N\), by rotating and braiding \(g_{j-1}\) through \(g_{N-1}\cdots g_1\). More generally \(g_j\) commutes with \(Y_i\) for ordinary \(j\ne i-1,i\): for \(j\ge i+1\) this follows already by recursion, and \(g_j\) passes through \(g_{j+1}^{-1}g_j^{-1}\) as \(g_{j+1}\), commutes then with \(Y_j\), and passes back through \(g_j^{-1}g_{j+1}^{-1}\) as \(g_j\). This handles \(Y_{j+2}\) and thus later indices. \(Y_1Y_2=M^2\) commutes with \(g_1\): it is \(\Omega^2 (g_{N-2}\cdots g_1)(g_{N-1}\cdots g_2)\), and \(g_1\) passes through the rotation as \(g_{N-1}\), then through the two words as \(g_1\). For the last step note their product begins \(g_{N-2}g_{N-1}\) times the analogous product one size smaller; braid and induct down to the ordinary braid at size three. Thus \(Y_2Y_1=g_1^{-1}(Y_1Y_2)g_1=Y_1Y_2\).

All \(Y_i\) commute by induction: commutation at \(j,j+1\) implies it at \(j,j+2\) by \(g_{j+1}^{-1}\), then at \(j+1,j+2\) by \(g_j^{-1}\), and distant pairs follow similarly. Finally \(g_l\) commutes with \(Y_l Y_{l+1}\) and \(Y_l+Y_{l+1}\) by the recursion and \(g_l-g_l^{-1}=\kappa\) (for the product use \(g_lY_{l+1}=Y_l g_l^{-1}\), \(g_l^{-1}Y_l=Y_{l+1}g_l\) and the commuting pair). Hence the elementary symmetrics in the \(Y_i\) commute with all generators (use \(Y_1\) and the \(g\)’s to express \(\Omega\)).

Let \(\mathbf e\) be the constant function one, and let \(\mathcal W\) be the span of all words in \(g_1,\ldots,g_{N-1},\Omega\) applied to \(\mathbf e\). The common \(q^{-1}\) eigenspace of the ordinary \(g_l\) is exactly the line \(\mathbb C\mathbf e\): because \(A\ne0\) for our small distinct rapidities, those eigen-equations are precisely permutation symmetry. Each coefficient of \(\prod_i(y-Y_i)\) commutes with the generators, so it preserves that line. It follows that there is a scalar monic polynomial \(p(y)\) of degree \(N\) with \[\prod_i(y-Y_i)=p(y)I\quad\hbox{on }\mathcal W.\] As the \(Y_i\) commute, substitution gives \(p(Y_i)=0\) on \(\mathcal W\). We will identify the roots of \(p\) without assuming that \(\mathcal W\) is already the full orbit space.

Reduction to one coordinate.

Let \(\mathcal V_1\) be the \(N\)-dimensional space of functions of \(x_1\) alone. It is the common \(q^{-1}\) eigenspace of \(g_l\), \(2\le l<N\), which express symmetry in sites \(2,\ldots,N\). Since \(Y\) commutes with these operators, it preserves \(\mathcal V_1\). We coordinatize this space by values \(f(\xi_i)\) where \(\xi_i=e^{2\mathrm i(i-1)\epsilon}\); write \[P(z)=\prod_i(z-\xi_i),\quad d(z)=\prod_i\frac{z+q\xi_i}{\xi_i+qz},\qquad (L f)(\xi_i)=q^{1-N}d(\xi_i) f(q^2\xi_i).\] In \(L\), the input \(f\) is interpreted as the interpolation polynomial of degree less than \(N\). With \(J_i=d(\xi_i)\xi_i P'(\xi_i)\) as diagonal components one has \[ Y|_{\mathcal V_1}=\operatorname{diag}(J)\ L^\mathsf{T}\ \operatorname{diag}(J)^{-1}. \tag{8}\] In fact for \(h_w(z)=z/(z-w)\), \[g_l h_w(x_l)=q^{-1}\frac{x_l-q^2 w}{x_l-w}\,h_w(x_{l+1}),\quad Y h_w(x_1)=q^{1-N}d_1\,\frac{P(q^2 w)}{P(w)} h_{q^2 w}(x_1).\] Taking residues at \(w=\xi_j\) gives \(Y_{ij}=q^{1-N}d(\xi_i)P(q^2\xi_j)\xi_i/[\xi_j P'(\xi_j)(q^2\xi_j-\xi_i)]\), proving (8) by interpolation.

A positive limiting moment problem.

We claim \(L\) has \(N\) distinct eigenvalues near positive numbers (as \(\epsilon\to0\)), and \(\mathbf e\) is cyclic for (8) on \(\mathcal V_1\). Write \(D(z)=\prod_i(z+q\xi_i)\); then \(d(z)=q^N D(z)/D(q^2 z)\). The partial fractions \(f(z)/D(z)=\sum_j v_j/(z+q\xi_j)\) for the input polynomial reduce \(L f=y f\) to \[\mathcal C^\mathsf{T} v=y\mathcal C v,\qquad \mathcal C_{ij}=\frac1{\xi_i+q\xi_j}.\] To take the confluence write \(\eta_i^0=(i-1)\epsilon\). Extract the common matrix factors \[(e^{-\mathrm i\lambda}/2)e^{-\mathrm i\eta_i^0}e^{-\mathrm i\eta_j^0}.\] The entries of \(\mathcal C,\mathcal C^\mathsf{T}\) respectively become \[\sec(\eta_i^0-\eta_j^0-\lambda),\qquad \sec(\eta_i^0-\eta_j^0+\lambda).\] Use \[ \sec z=\int_{\mathbb R} e^{z t}w_0(t)\,dt,\qquad w_0(t)=\frac1{2\cosh(\pi t/2)},\quad |\Re z|<\pi/2. \tag{9}\] For real \(z\) this follows by shifting the contour in \(t\) up by \(2\mathrm i\) and taking the residue at \(\mathrm i\) (the exponential decay controls the ends, and for the integral \(I\) one gets \((1+e^{2\mathrm i z})I(z)=2e^{\mathrm i z}\)); the rest by analyticity. Thus \(\alpha(t)=\sum v_j e^{-\mathrm i\eta_j^0}e^{-\eta_j^0 t}\) paired against the rows \(e^{\eta_i^0 t}\) solves the moment eigenproblem for weights \(e^{\lambda t} w_0\) relative to \(e^{-\lambda t} w_0\). In polynomial-like bases \[U_m^\epsilon(t)=((e^{\epsilon t}-1)/\epsilon)^m,\quad V_m^\epsilon(t)=((1-e^{-\epsilon t})/\epsilon)^m \qquad(0\le m<N)\] for rows and for \(\alpha\), the two matrices converge by dominated convergence to positive definite real symmetric polynomial moment matrices (fixed \(N\); each powered difference quotient bounded by a polynomial times \(e^{N\epsilon |t|}\)). This positive definite limiting pair has positive generalized eigenvalues and is diagonalizable.

Each nonzero real limiting eigenpolynomial \(r\) satisfies that \(r(t)(e^{\lambda t}-y e^{-\lambda t})\) is orthogonal to all polynomials of degree less than \(N\) for \(w_0\). It must have at least \(N\) real sign changes (otherwise multiply by the product of linear factors at the sign changes). Since the exponential difference for \(y>0\) has just one zero, \(r\) hence has degree \(N-1\) with all simple real zeros and \(r(\mathrm i)\ne0\). If two linearly independent eigenpolynomials had the same eigenvalue, a linear combination would cancel their leading coefficient, contradicting the required degree \(N-1\). Thus every eigenvalue is simple.

By convergence of that finite matrix pair (invert the nonsingular denominator), for small \(\epsilon>0\) the eigenvalues are distinct, tending to those simple limits, and \(\alpha(\mathrm i)=\sum v_j/\xi_j\ne0\) for any eigenvector (normalize coefficients in \(V^\epsilon\); any converging eigenvector subsequence gives a nonzero scalar multiple of a limiting real eigenpolynomial).

Cyclicity and the full orbit.

For a left eigenvector of (8) obtained from \(f\), its pairing with \(\mathbf e\) is \[\sum_i f(\xi_i)/J_i =\frac{q^{1-N}}{y}\sum_i\frac{f(q^2\xi_i)}{\xi_i P'(\xi_i)} =-\frac{q^{1-N}}y\frac{f(0)}{P(0)}\ne0\] by interpolation and \(f(0)=D(0)q^{-1}\alpha(\mathrm i)\). Since every left eigenline has nonzero pairing with \(\mathbf e\) and the eigenvalues are distinct, this proves the cyclicity (polynomial interpolation in \(Y\)).

Cyclicity has placed the whole space \(\mathcal V_1\) inside \(\mathcal W\). Since \(p(Y)=0\) there, the degree-\(N\) polynomial \(p\) has exactly the \(N\) distinct eigenvalues just found as its roots. For small enough \(\epsilon\), none has square \(q^2\), and no two have ratio \(q^{\pm2}\): their limits are distinct positive numbers, whereas \(q^{\pm2}=\pm\mathrm i\).

The commuting operators \(Y_i\) on \(\mathcal W\) therefore simultaneously diagonalize. On a joint eigenvector, the identity \(\prod_i(y-Y_i)=p(y)I\) says that its eigenvalue tuple is a permutation of the \(N\) roots, each used once. Suppose neighboring entries are \(a,b'\). Commutation with every other \(Y_i\) and with \(Y_l+Y_{l+1}\) and \(Y_lY_{l+1}\) implies that \(g_l\) can send this vector only into the joint eigenspaces with the original tuple or the tuple with these entries swapped. If the swapped component vanished, the relation \(g_lY_{l+1}=Y_lg_l^{-1}\) would imply \[b'g_lv=a(g_l-\kappa)v.\] Together with (7), this forces \(b'/a=q^{\pm2}\), which was excluded. Thus the swapped component is nonzero.

Starting from any joint eigenvector and successively swapping adjacent entries produces all \(N!\) distinct tuples in \(\mathcal W\). The ambient space of functions on the orbit has dimension \(N!\). Hence \(\mathcal W\) is that entire space and every joint eigenspace is one-dimensional. An operator commuting with the generators is diagonal on these joint eigenspaces; the nonzero adjacent connections force all its diagonal entries to agree. Its value is therefore scalar. The same spectrum also proves that \(q^2-Y^2\) is invertible.

Lost arcs and saturation.

On lifting (fixing the first port), an arc counted by \(h_N(m)\) is exactly a planar half-plane excursion from \(e_2/2\) to \(e_2/2+m U\) which projects to a self-avoiding path mod \(N U\); the choice of positive cut interval specifies this lift displacement rather than \((m-N)U\). Since \(\sum_{m<N,\,m\ge1} h_N(m)\le\sum_{m\ge1}h(m)\), (2) and (11) thus imply equality \(c\sum_{m>0}h(m)=1\), with deficit of order \(N^{-1/4}\) from the lost arcs (endpoint spacing at least \(N\) or intersecting translates). Let \(\mu_+\) denote the measure with weights \(x^{\#\mathrm{vertices}}\) on half-plane excursions from \(e_2/2\) ending strictly to the right on the bottom. If \(\mathcal R\) is horizontal reflection, the full-wall measure in the introduction is \(\mu=\mu_++\mathcal R_*\mu_+\). Thus \(\mu(E)=2\mu_+(E)\) for reflection-invariant events, including the height and diameter restrictions below. Use horizontal \(z\) and upward \(y\) coordinates from the start, endpoint displacement \(X>0\), extrema \[M=-\min z,\quad P_+=\max z-X,\quad W=X+M+P_+,\quad H=\max y.\] Reversing the path and reflecting \(z\mathrel{\pdfliteral page{/Span << /ActualText <FEFF21A6> >> BDC}\mapsto\pdfliteral page{EMC}}X-z\) (a lattice symmetry for this \(X\)) exchanges \(M,P_+\). Every excursion with \(X\ge N|U|\) is lost in the comparison at period \(N U\), and every lost excursion has \(W\ge N|U|\). Taking \(N\) on the appropriate side of \(t/|U|\) therefore gives, with \(p=1/4\), \[ \mu_+(X>t)\ \lesssim\ t^{-p},\qquad \mu_+(W>t)\ \gtrsim\ t^{-p}\qquad(t\to\infty). \tag{12}\]

The lost-arc bounds distinguish endpoint displacement from total horizontal width. To obtain a height estimate, we compare the domain with two wedges whose slanted walls respond differently to height.

The two wedge identities.

Half-plane saturation and (1) show that the mass of exits on the receding sides of the half-plane exhaustion tends to zero: bottom real parts increase to one, and every other real part controls its positive mass by the same strictly positive cosine bound.

Add a slanted left boundary starting at horizontal coordinate \(-t\) on the bottom and making angle \(\alpha\) with the rightward direction, where \(\alpha=\pi/3\) or \(2\pi/3\). Retain the wedge to its right. Choose \(t>0\) on the admissible arithmetic progression for that angle; the wall normals are \(e_1,e_0\), respectively. The wedge exhaustion has vanishing mass on its receding faces as well, since those exits are a subset of the corresponding half-plane exits.

The turn for an exit on the new slanted side is \(\alpha\): the positive boundary tangent has rotated by \(\pi+\alpha\) from the source. The old bottom exit turns remain \(-\pi\) to the right and \(+\pi\) to the left. Subtracting the two domain identities gives \[e^{-\mathrm i\pi s}E_R+e^{\mathrm i\pi s}E_L =e^{\mathrm i\alpha s}C,\] where \(E_R,E_L\) are the lost rightward and leftward bottom masses, and \(C\) is the mass exiting the new side. Define \[L_\alpha=\max(y\cot\alpha-z),\qquad A_\alpha(t)=\mu_+(L_\alpha>t),\qquad B_\alpha(t)=\mu_+(X+L_\alpha>t).\] The first lost mass is \(E_R=A_\alpha(t)\). Translating and reversing the leftward arches gives \(E_L=B_\alpha(t)\). At an admissible cut a bottom arch cannot attain its maximum on the slant, since the line contains no vertex or bottom port. Rotate the last complex identity by \(e^{-\mathrm i\alpha s}\) and take imaginary parts. With subscripts \(+\) and \(-\) denoting \(\pi/3\) and \(2\pi/3\), respectively, this yields \[ A_\alpha(t)=c_\alpha B_\alpha(t),\qquad c_\alpha=\frac{\sin(s(\pi-\alpha))}{\sin(s(\pi+\alpha))}, \qquad 0<c_-<c_+<1. \tag{13}\]

Upper tails.

First \(A_+(t)\lesssim t^{-p}\). Indeed on allowed cuts, \(A_+(t)\le c_+[A_+(a t)+O(t^{-p})]\) for fixed \(0<a<1\) by (12). Choose \(a_0<a<1\) so \(c_+ a_0^{-p}<1\), and round \(a t\) down to an allowed cut still \(\ge a_0 t\) for large \(t\). Induction and monotonicity prove the bound. Since \(L_-\le M\le L_+\), symmetry for \(P_+\) and (12) now give upper tails \(\lesssim t^{-p}\) also for \(M,W\), for \(B_\pm(t)\), and for \(H\) by \(H/\sqrt3\le L_++\max z\).

Lower height and endpoint tails.

For a tail function \(F\), write \(\bar F=R^{-1}\int_R^{2R}F(t)\,dt\). Then \(\bar B_+\gtrsim R^{-p}\) since \(X+L_+\ge X+M\) and \(\mu_+(X+M>t)\ge\mu_+(W>2t)/2\) by symmetry. The grid restriction in (13) costs \(O(R^{-1-p})\) on taking averages (round down/up; each tail is monotone with upper bound as above). Hence \[\bar A_+-\bar A_-=(c_+-c_-)\bar B_+ + c_-(\bar B_+-\bar B_-)+O(R^{-1-p})\gtrsim R^{-p}.\] To bound this difference in the other direction, observe that \(0\le L_+-L_-\le2H/\sqrt3\). For each arch with \(H\le\varepsilon R\), the set of \(t\in[R,2R]\) lying between \(L_-\) and \(L_+\) has length at most \(2\varepsilon R/\sqrt3\), and is empty unless \(L_+>R\). Integrating the indicator difference therefore gives \[\bar A_+-\bar A_- \le (2\varepsilon/\sqrt3)A_+(R) +\mu_+(H>\varepsilon R).\] The first term is at most a fixed constant times \(\varepsilon R^{-p}\). Choosing \(\varepsilon>0\) sufficiently small proves the matching lower height tail.

Likewise (13) gives \(\bar B_+-\bar A_+\gtrsim R^{-p}\). The interval between \(L_+\) and \(X+L_+\) has length \(X\). Splitting according to \(X\le\varepsilon R\) now gives \[\bar B_+-\bar A_+ \le\varepsilon B_+(R)+\mu_+(X>\varepsilon R),\] and the same absorption proves the endpoint lower tail. Thus \[ \mu_+(H>t)\asymp\mu_+(X>t)\asymp t^{-1/4}. \tag{14}\] In particular \(\mu_+(\operatorname{diam}>t)\asymp t^{-1/4}\) for Euclidean diameter, by the height lower tail and the width and height upper tails. For a strip of height \(T\) with top line in the cut family (parallel to bottom), let \(B(T)\) be the total excursion weight from the fixed start to that top. By the same vanishing of side masses and (1), now top turn zero, \(B(T)=c\,\mu_+(H>T)\asymp T^{-1/4}\).

Fusion maps.

Fusion replaces a special adjacent pair by a single site, or deletes that pair. On the spin spaces the corresponding insertion maps are \(J:\mathbb C^3\to(\mathbb C^3)^{\otimes2}\) and \(K:\mathbb C\to(\mathbb C^3)^{\otimes2}\): \[J0=00+b_0 \mathsf c,\quad Js=b_0(0s+s0),\qquad b_0=(2\cos\lambda)^{-1};\qquad K(1)=00+\mathsf c .\] Here \(J0\) is the image of a vacancy, while \(Js\) is the image of an occupied spin. Both maps are injective. The identities we need say that the braid at the fusion parameter factors through the smaller space, and that an auxiliary strand can pass the inserted pair: for \((L,h)=(J,\lambda),(K,3\lambda/2)\), respectively, \[ \check R(2h)=LL^t,\qquad \check R_{12}(u-h)\check R_{23}(u+h)(L\otimes1)=(1\otimes L)R_f, \quad R_f=\begin{cases}\check R(u),&L=J,\\ I,&L=K.\end{cases} \tag{16}\] The reflected identity also holds (reverse sites and simultaneously reverse all spins). Embeddings between differently ordered tensor products always use the displayed order. The first identity follows by substitution. For the second, (3) forces the left-hand image into \(1\otimes L\), since \(L^t\) is onto. Projection of the third output to vacancy is injective on the image of \(J\), up to its nonzero diagonal coefficients; for \(K\), project both inserted outputs to vacancy. This recovers the induced operator on the smaller space. Its coefficients are checked as follows: writing \(+,-\) for \(u+h,u-h\), the needed coefficients reduce to \[ \begin{array}{ll} B_-+b_0 A_+D_-=b_0 B(u), &C_+ A_-+b_0B_+C_-=b_0 A(u),\quad C_+ C_-+b_0B_+A_-=C(u),\\ b_0(A_-+A_+ C_-)=A(u), & C_-+A_+A_-=C(u),\\ b_0(B_++C_+B_-)=B(u),& C_+D_-=D(u),\quad C_+E_-+B_+B_-=E(u) \end{array} \tag{17}\] for \(h=\lambda\); for \(3\lambda/2\) only \(C_+C_-+B_+A_-=1\) besides the vacuum. For example for occupied pair input of \(R_f\), projecting the third site after the first crossing gives \(b_0(C_+ st+B_+\mathsf c_{st}00)\); the next crossing gives the last three coefficients. For the scalar check put \(s_j=P_{u+j\lambda},\ t_0=2\cos\lambda\); the ordered identities for \(h=\lambda\), after substitution/cancellation, are \[\begin{array}{l} t_0s_{-1}s_3+s_6s_5=s_0s_1,\qquad t_0s_6+s_{-1}=s_5,\qquad s_6s_{-1}+b^2/t_0=s_0s_5,\\ s_4s_3+s_6s_{-1}=t_0s_5s_1,\qquad s_{-1}s_4s_3+b^2s_6=s_0s_5s_1,\\ s_1s_2+s_6s_{-1}=t_0s_0s_4,\qquad D=D,\qquad s_6s_{-2}+b^2=s_0s_4 . \end{array}\] They follow by \(s_{j+8}=-s_j,\ s_{j+1}+s_{j-1}=t_0s_j\) and product to sum (for the fifth, use the fourth and \(b^2-s_{-1}^2=-s_1s_{-3}\)). The \(3\lambda/2\) identity similarly reduces to \(b^2-s_{-3/2}^2=s_{1/2}s_{9/2}\). Thus (16) holds per diagram.

The transfer vector and the induction statement.

Consider the normalized disk transfer vector \(\psi_N\) near homogeneity for the same cyclic row with auxiliary label \(-\lambda\). Its empty coordinate is one and it is the unique fixed vector with that normalization there. Indeed at homogeneity powers of the row applied to the empty state count the same multipaths as before, only forgetting the annular data. This bounds them by (2) (finitely many possible lifts up to translations for each arc); all disk states are reachable (use laminar cut intervals as before, which can even be chosen within a single period by linear ordering). So the same matrix convergence argument applies, including for two sites; at one site there is only vacancy. At size zero we just use the formal vacuum of scalar one, its row having the vacant auxiliary term only. At homogeneity it gives the actual arc-configuration masses by increasing finite heights as before. Thus the fixed vector continues rationally in the exponentials \(e^{iz_i}\), and is rotation-covariant in disk diagrams.

The two-site limit.

The scalar pairing below needs one further consequence of the construction: two adjacent sites separating from all the others must carry their own two-site vector. With an adjacent pair \(i,j\) of variables tending to zero proportionally (in linear order here, generic proportions/rest), \(\psi\) also factors to leading order into the two-site \(\psi_2\) inserted disjointly into \(\psi_{N-2}\). Indeed read (19) at both zero by the direct descending \(q^3\) specialization: for both occupied the constant term is \(\mu\prod_{k\ {\rm ext}}x_k^2\) times the lower \(\Phi\) with a cup inserted, for just one occupied it is zero. For both vacant the constant is zero and the linear term is \((x_i+x_j)\prod_{k\ {\rm ext}}x_k^2\) times lower \(\Phi\), as read on both \(q^3\) slopes. On the reversed slope one slides across all exterior sites; a vacant anchor forces vacant insertion by \(K\), and at leading order one simply transports this vacancy (\(A,B\to0,\ C\to1\)). This includes the empty coordinate \(p_N\); restoring gauges and dividing by it gives the claim. Similarly at infinity, put \(X=x^{-1}\) with half-powers \(e^{-iz}\) and work with \(\widetilde\Phi(X)=\Phi(x)\prod X_k^{N-1-a_k}\), polynomial by (19). Then \(\psi(z)=\prod X_k^{a_k/2}\widetilde\Phi(X)/p_N(X)\). On the same original sheets \(z_i-z_j=\pm3\lambda\), (18) for \(p_N(X)\) still has leading prefactor \((X_i+X_j)\prod_{k\ {\rm ext}}X_k^2\) there, and \((X_i+X_j)/(X_i^{1/2}X_j^{1/2})=\mu\); the \(K\) insertions and the vacant slide argument still apply. This gives the identical Taylor reasoning in \(X\).

Gluing into separating polygons.

We now identify the physical observable represented by the pairing. Set \(k=l=n\ge3\) and \(z_1=\cdots=z_{2n}=0\). The angle \(\theta\) is an auxiliary parameter, and the polygon fugacity is \(\eta=2\cos\theta\); it is nonnegative for \(\theta\in[0,\pi/2]\). In particular, angle zero means fugacity two, whereas \(\theta=\pi/2\) means fugacity zero. Geometrically at these homogeneous parameters \(V_{n,n}\) counts collections of full-cylinder vertex-disjoint finite simple honeycomb polygons separating two points \(f,f'=f+nU\), unoriented and unrooted, with weight \(x^{|\gamma|}(2\cos\theta)\) per polygon \(\gamma\), \(|\gamma|\) its vertex count. Here \(x\) is the homogeneous vertex weight from the boundary part, period \(2nU\), and \(f\) is a point on the bottom line \(y=-1/2\) in the gap just before the first port. Indeed reflect one half-cylinder in that line (\(\mathbf a_{jk}\) maps to \(\mathbf b_{j+k,-1-k}\)) and glue the multipaths with identical occupancies to form disjoint polygons. Every separator must cross this line and splits uniquely into such arcs. Sum orientations by giving spin \(+1\) at upward crossings and \(-1\) at downward crossings (to both factors); these are exactly the compatible assignments. Termwise gluing and sums here are absolutely convergent by the finite multipath masses at fixed period.

Compactifying the two cylinder ends gives two disks with the seam as their common boundary (Figure 2). The annular data forgotten by a disk state record which interior face contains its filled cylinder end. The marks \(f,f'\) remain on the seam, so forgetting those interior end marks preserves their separation test and the block twist.

For an oriented polygon, its arc factors in the two vectors together multiply to \(q^2\) if \(f\) lies to its left, \(q^{-2}\) otherwise (sides can be taken on the sphere compactifying both ends of the cylinder). Indeed flatten that sphere minus \(f\) to the oriented plane with the seam as horizontal line in increasing port order, orthogonality at crossings maintained. For instance in complex plane coordinates of the cylinder use \(\xi=\exp(2\pi i(Z-f)/L)\) with period \(L=2n|U|\), then \(i(1+\xi)/(1-\xi)\). Each upper or lower simple arc has total turn \(-\pi\) times its earlier-endpoint spin, by closing against the line; joins go straight. The polygon turn is \(-2\pi\) if its unbounded side is on its left and \(+2\pi\) otherwise, giving the asserted product of \(q^{s_{\rm earlier}}\). The sum of spins in the initial block for this polygon equals its \(f\)-left indicator minus its \(f'\)-left indicator (the indicator drops across an upward crossing moving positively on the seam). Thus if it separates, the twist \(m\) with the arc product gives \(e^{\pm i\theta}\); otherwise the two orientations cancel since \(q^2+q^{-2}=0\). This proves the interpretation, in particular \(V_{n,n}=1\) at \(\theta=\pi/2\) and positivity and monotonicity in \(2\cos\theta\ge0\). Every separator crosses the seam segment from \(f\) to \(f'\) at one of its \(n\) ports. Disjoint polygons use distinct ports, so there are at most \(n\) separators. Thus the partition function is a polynomial in \(2\cos\theta\), with finite nonnegative coefficients. This physical interpretation is only asserted at balanced homogeneity; the rapidity-dependent pairing remains an algebraic object.

Scalar reductions.

We return to generic rapidities. The pairing is separately symmetric by exchange, transpose symmetry of the noncyclic boxes, conservation of total spin there and unitarity. Multiplying by \(p_N(x)^2\) gives polynomial of degrees \(\le2N-2\) each by (19), inversion and pairing equal occupancies. Within a block, paired reductions give simple deletion at difference \(3\lambda\) or merging at \(2\lambda\) for \(V\): order adjacently descending in the block. One vector inserts \(L\); the other reduces on application of \(L^t\), since its exchange by \(LL^t\) gives \(L\) of the reduced vector and \(L\) is injective. The twist passes through \(L\) by spin conservation. There is single-site deletion also at zero and infinity, and clustering the adjacent sites on either side of the internal block boundary (say at zero) factors off \(V_{1,1}\).

For clarity, we record these normalized scalar rules explicitly. Fix \(\theta\) and write \(V_{k,l}(X_A;X_B)\), where the subscripts give the lengths of the two variable lists; put \(N=k+l\) and \(X=(X_A,X_B)\). All specializations below are at generic nonzero values of the remaining variables. For \(k\ge2\), the two same-block rules in \(A\) are \[\begin{aligned} V_{k,l}(X_A,q^3y,y;X_B)&=V_{k-2,l}(X_A;X_B),\\ V_{k,l}(X_A,qu,u/q;X_B)&=V_{k-1,l}(X_A,u;X_B). \end{aligned}\] Here \(X_A\) has length \(k-2\) on both left-hand sides; the identical rules apply in \(B\). These identities concern the normalized pairing \(V\). After multiplication by \(p_N^2\), their scalar factors are the corresponding factors in (18), squared. For \(k\ge1\), single-site deletion is \[\lim_{x\to0}V_{k,l}(X_A,x;X_B) =\lim_{x\to\infty}V_{k,l}(X_A,x;X_B) =V_{k-1,l}(X_A;X_B),\] and likewise in \(B\). For \(k,l\ge1\), the two sites adjacent across the block boundary satisfy, for fixed generic \(u,v\), \[\lim_{\varepsilon\to0} V_{k,l}(X_A,\varepsilon u;\varepsilon v,X_B) =V_{k-1,l-1}(X_A;X_B)V_{1,1}(u;v).\] The same \(\theta\) is used in all factors. Separate symmetry permits the sites involved in a same-block rule to be moved next to each other.

An explicit scalar gas.

The preceding reductions characterize the scalar pairing. We now give a gas with those reductions and verify all of them. Its labels \(r_i\in\{-1,0,1\}\) are gas variables, not the spin coordinates \(s_i\) of the disk vector. Label zero is a vacant gas site; the two nonzero labels will later become contour residues. The gas sum imposes signed neutrality \(\sum_A r_i-\sum_B r_i=0\). Its equality with the twisted contraction will follow from the scalar characterization, rather than a direct identification of gas labels with spins.

We use a centered version of the fixed scalar polynomial, so that its pair factors depend on rapidity differences. Put \(\mathfrak p_j(w)=2\sin(w-j\lambda)\) for a rapidity argument; write \(\mathcal P_N=p_N(x)\prod_i x_i^{-(N-1)/2}\) for the centered Pf polynomial. For labels \(r,s=-1,0,1\) in that order use factors \(g_{t,r,s}(w)\), row-column entries with \(t=0\) (same block) or \(1\) (opposite blocks), argument later rapidity minus earlier: \[g_0= \begin{pmatrix} \mathfrak p_2\mathfrak p_3\mathfrak p_5\mathfrak p_6&-\mathfrak p_3\mathfrak p_5\mathfrak p_6/\mathfrak p_0&\mathfrak p_5\mathfrak p_6/(\mathfrak p_0\mathfrak p_1)\\ \mathfrak p_2\mathfrak p_3\mathfrak p_5/\mathfrak p_0&\mathfrak p_2\mathfrak p_3\mathfrak p_5\mathfrak p_6/(\mathfrak p_1\mathfrak p_7)&-\mathfrak p_3\mathfrak p_5\mathfrak p_6/\mathfrak p_0\\ -\mathfrak p_2\mathfrak p_3/(\mathfrak p_0\mathfrak p_7)&\mathfrak p_2\mathfrak p_3\mathfrak p_5/\mathfrak p_0&\mathfrak p_2\mathfrak p_3\mathfrak p_5\mathfrak p_6 \end{pmatrix},\qquad g_1= \begin{pmatrix} 1&\mathfrak p_5^2&\mathfrak p_5^2\mathfrak p_6^2\\ \mathfrak p_3^2&\mathfrak p_4^2&\mathfrak p_5^2\\ \mathfrak p_2^2\mathfrak p_3^2&\mathfrak p_3^2&1 \end{pmatrix}.\]

Matching the scalar gas with contour residues.

At balanced homogeneity (20) is precisely \[p_{2n}(1,\ldots,1)^2 V_{n,n}=(2+d)^{2n(n-1)}4^{n^2}\mathcal Z_n(\theta),\] The large factor on the right is the product of all vacant pair weights. In particular, the physical identity \(V_{n,n}(\pi/2)=1\) fixes a nontrivial value of \(\mathcal Z_n(\pi/2)\); it does not normalize that contour gas to one. To prove the residue identity, abbreviate \(K_0=K_{AA}=K_{BB}\) and \(K_1=K_{AB}=K_{BA}\); the normalization above is \(c_\star=4\cos\lambda\). Extract the product of all \((0,0)\) pair factors from (20) and associate each nonzero label to a letter at \(v_i=z_i-r_i\lambda/2\). Put \(u_{t,r}=g_{t,0,r}/g_{t,0,0}\). Direct substitution gives, for nonzero charges, \[\begin{aligned} \frac{g_{t,r,s}(w)g_{t,0,0}(w)}{g_{t,r,0}(w)g_{t,0,s}(w)} &=K_t(w-(s-r)\lambda/2)^{rs},\\ u_{0,r}(w)u_{1,r}(w)&=F(w-r\lambda/2)^r,\\ r c_\star\operatorname{Res}_{w=0}u_{0,r}(w)&=\mu . \end{aligned}\] For example \(u_{0,+}=-\mathfrak p_1\mathfrak p_7/(\mathfrak p_0\mathfrak p_2),\ u_{1,+}=\mathfrak p_5^2/\mathfrak p_4^2,\ u_{t,-}(w)=u_{t,+}(-w)\); for \((r,s)=(-,+)\) the left quotient is \(\mathfrak p_2\mathfrak p_0/(\mathfrak p_3\mathfrak p_1^2\mathfrak p_7)\) if \(t=0\), \(\mathfrak p_6^2\mathfrak p_4^2/\mathfrak p_5^2\) if \(t=1\), with equal-charge quotients just \(K_t(w)\).

Use identical generic small rapidity lists \((\zeta_j)_{j\le n}\) in each block, replacing \(F(v)^{nr}\) under the integral by \(\prod_j F(v-\zeta_j)^r\). Small circles then enclose simple individual poles at the shifted sites; all pair factors with indicated powers are regular on these small polydisks. Repeating a site within the same block costs a zero (of \(K_0\) at 0 for equal charges, or of \(1/K_0\) at \(\pm\lambda\) for opposite). Thus residues choose nonzero labels on distinct sites in each block, orderings canceled by the factorials. For an active site \(i\), the missing same-block self factor is supplied as \(r_i c_\star\operatorname{Res}u_{0,r_i}=\mu\), while the opposite-block site at the same rapidity contributes the ordinary \(u_{1,r_i}(0)\). The other \(u\) factors and the pair corrections in the identities then give exactly (20) divided by the extracted product. Take the lists back to zero by continuity (each order with more than \(2n\) letters already vanishes separately); the extracted factor is regular there with same-block value \((2+d)^2\) per pair and cross-block value \(4\).

Oscillator traces and the initial contour deformation

We represent the elliptic gas as a trace so that a current reflection can exchange the interaction matrix with its inverse. The momentum lattice is \[\mathbf p_A=(\mathbf p_s+\mathbf p_d)/2,\qquad \mathbf p_B=(\mathbf p_s-\mathbf p_d)/2, \qquad (\mathbf p_s,\mathbf p_d)\in\mathbb Z^2.\] Thus the two label momenta are either both integers or both half-integers. Use the shifts \(V_R|\mathbf p\rangle=|\mathbf p+R\rangle\), and at each level \(m\ge1\) two ordinary oscillators \([a_m,a_m^*]=I\) (on oscillator polynomials: multiplication \(a^*\), derivatives \(a\), orthogonal monomials of norms squared the multifactorials). All independent variables commute. With \(z=e^{\omega v}\) insert fields in smaller-radius-left order: \[E_R(v)=\omega V_R z^{-2R\cdot\mathbf p-1} \exp\left(-\sum_{m>0}\sqrt{2/m}\,z^{-m}R\cdot a_m^*\right) \exp\left(\sum_{m>0}\sqrt{2/m}\,z^m R\cdot a_m\right).\] The momentum transfer in \(U\) is \(|\mathbf p\rangle\mathrel{\pdfliteral page{/Span << /ActualText <FEFF21A6> >> BDC}\mapsto\pdfliteral page{EMC}}w(\mathbf p)|(-\mathbf p_B,-\mathbf p_A)\rangle,\ w=\exp[-\omega\lambda(c_s\mathbf p_s^2+2\mathbf p_d^2)]\). On oscillator monomials it is the induced action of \(r_m=(M-I)(M+I)^{-1}|_{p=m\omega}\). Use \(\operatorname{Tr}(U\,\cdots)/\operatorname{Tr}U\).

To check, the oscillator factor (including \(\omega^N\) for \(N\) letters) on the axis at distinct points has log-modulus \[ N\log\omega-\sum_{m>0}\frac1m\left( \sum_{ij}R_i\cdot M(m\omega) R_j\cos(m\omega(y_i-y_j))-N\right). \tag{22}\] There are three parts to the trace calculation. First separate the radii slightly, commute annihilations past creations, and pass to boundary values. For a scalar oscillator under transfer \(r^k\) on degree \(k\), the normalized trace of \(e^{h a^*}e^{b a}\) equals \(\exp(hb r/(1-r))\) by summing the geometric series and its derivatives. Second, keep the phases and momentum shifts that the log-modulus does not record. The ordered-commutator phase per \(i<j\) is \(\exp[i R_i\cdot R_j(\omega(y_i-y_j)-\pi\operatorname{sign}(y_i-y_j))]\) on one period by the sine series. The sign product under neutrality is \((-1)^{\mathfrak q^2-N/2}=\prod r_i\). Combining linear parts with all the field powers gives \(\exp[-i\omega D\cdot(2\mathbf p+Q)]\), using initial momentum \(\mathbf p\) on the right. Since the fields first change the rightmost momentum from \(\mathbf p\) to \(\mathbf p+Q\), the trace closes exactly when \[(\mathbf p_A,\mathbf p_B) =(-\mathbf p_B-Q_B,-\mathbf p_A-Q_A).\] Equivalently \(Q=(\mathfrak q,\mathfrak q)\) and \(\mathbf p_s=-\mathfrak q\), while \(\mathbf p_d\) is free. Gaussian Poisson summation of \(\mathbf p_d\) gives the inserted theta ratio times \(\exp[-\omega D_d^2/(8\lambda)-\omega\lambda c_s \mathfrak q^2]\). Third, compare this answer with the periodized pair and self energies. The Gaussian factors just obtained supply exactly their zero and quadratic modes: \(\mathcal K^T-4P_d(s^2/T+T/6)\) on \([-T,T]\) is periodic with modes \(-M(m\omega)/m\) at \(\pm m\), by the transform and the Fourier series of \(|s|-s^2/T-T/6\). The regular mean and the quadratic part, applied to \(Q_d=0\), give the two extra exponents; point-splitting the \(i=j\) singular term gives \(\log\omega\) by the cosine series. This proves the representation using \(F_T(v)^{nr}\eta_X^r E_R(v)\,dv/(2\pi i)\), where \(\eta_A\eta_B=e^{i\theta}\). It also shows the full neutral integrands have imaginary period \(iT\) in each variable.

Take outer axes at \(\pm b\lambda\) in real part of \(v\), for \(b>1/2\) sufficiently close to \(1/2\), and central axes at small \(\pm\epsilon_1\lambda\). Deform the circles coefficientwise to differences of ascending axes: for a plus letter, center (at \(-\epsilon_1\lambda\)) minus left outer axis; for a minus, right outer minus center (at \(+\epsilon_1\lambda\)). Indeed they bound cylinders enclosing the desired respective source poles; no pairwise poles intervene (at relative distance \(\lambda\) between opposite charges one can have zeros). We spell out convergence estimates below. Per label put \(f_X=\eta_X F_T^n\). Let \(C_+\) and \(C_-\) be the ascending central lines at real parts \(-\epsilon_1\lambda\) and \(+\epsilon_1\lambda\), respectively, traversed for one imaginary period. For one label define the ordered contour exponentials by their power series: \[\mathcal E_\pm(g) =\exp\!\left(\int_{C_\pm}g(v)E_{\pm e_X}(v)\,\frac{dv}{2\pi i}\right).\] At this stage identities are coefficientwise; convergence of the summed expressions is proved below. The central product can be written \[\mathcal E_+(f_X)\mathcal E_-(-1/f_X) =W(f_X)\mathcal E_+(-f_X),\qquad W(f)=\mathcal E_+(f)\mathcal E_-(-1/f)\mathcal E_+(f).\] Operators on different labels commute in this manipulation.

The Weyl action and reflected one-body factor

This is the level-one lattice current construction of Frenkel–Kac and Segal [6, 20]; an explicit classical realization is given in [5]. The calculation here fixes the action on momentum vacua and the signs that determine the reflected interaction.

We compute the current reflection on each energy shell, and then its action on the full gas. The energy operator is \(\mathcal H=\mathbf p^2+\sum m a_m^*\cdot a_m\). On one label write \(X_\pm(z)=E_{\pm e_X}/\omega\), and \(e,f_-\) for their integrated zero modes (\(dz/(2\pi i z)\)), without multipliers. They preserve finite-dimensional energy shells, since the \(z\)-power in a field matrix element is energy in minus energy out. On these shells \(e^*=f_-\), by reversing the normal product on the unit circle. Set \[H(z)=2\mathbf p_X-\sum_{m>0}\sqrt{2m}(z^{-m}a_{Xm}^*+z^m a_{Xm}).\] Then \([e,X_-(z)]=H(z),\ [f_-,X_+(z)]=-H(z)\); same-sign commutators vanish. In fact the ordered product \(X_+(\zeta)X_-(z)\,d\zeta/\zeta\) after contraction has prefactor \(z(z/\zeta)^{2\mathbf p_X}d\zeta/(z-\zeta)^2\), times the combined oscillator normal product. For the commutator take the negative residue (inner minus outer contour). The calculation is coefficientwise between finite shells, where the radial expansions of the contraction times Laurent polynomials suffice. Also \([H(z),e]=2X_+(z), [H(z),f_-]=-2X_-(z)\) directly by oscillator commutation (the Laurent delta against a field, before taking its zero mode). Thus \([e,f_-]=2\mathbf p_X\), and field coefficients transform in the adjoint representation. Finite-dimensional \(\mathfrak{sl}_2\) on the energy shells gives \(W(1)=\exp(e)\exp(-f_-)\exp(e)\) acting by sign flip on \(H\) (including oscillators) and by \(X_\pm\mathrel{\pdfliteral page{/Span << /ActualText <FEFF21A6> >> BDC}\mapsto\pdfliteral page{EMC}}-X_\mp\). Its action on a momentum vacuum of this boson sends \(|h\rangle\) to \(\sigma(h)|-h\rangle\); the vacuum field matrix element between \(h,h+1\) gives \(\sigma(h+1)=-\sigma(h)\). Here \(\sigma(0)=1,\ \sigma(-1/2)=1\) from the singlet and defining doublet, respectively (on those minimal-energy vacua the root mode actions are read directly from \(X_\pm\)). Thus the product Weyl sign on the two vacuum labels is \((-1)^{\mathbf p_d}\). For integer momenta the two signs multiply to \((-1)^{\mathbf p_A+\mathbf p_B}=(-1)^{\mathbf p_d}\). For half-integer momenta they multiply to \((-1)^{\mathbf p_A+\mathbf p_B+1}\), with the same value because \(\mathbf p_s\) and \(\mathbf p_d\) then have opposite parity.

We now insert the nonconstant source by a Cartan conjugation. Unlike the zero-mode reflection, this conjugation mixes oscillator levels; its analytic use will be justified after the contraction calculation. Define \[\begin{gathered} \mathcal D=\left(\prod_X\eta_X^{\mathbf p_X}\right) \exp\left[-\frac n2\sum_{X,m>0}\sqrt{2m}(L_m a_{Xm}^*+L_{-m} a_{Xm})\right],\\ L_m=T^{-1}\widehat L(m\omega),\qquad L(y)=\log F(iy). \end{gathered}\] Fixed compatible logarithms can be chosen in the powers. Use the odd imaginary decaying branch \(L\), so that \(\log F_T=L^T\) (periodized) on the axis. Explicitly \[\widehat L(p)=-\frac{2\pi}{p}J(a),\qquad J(a)=\frac{\sinh(7a/2)-\sinh(5a/2)-2\sinh(a/2)}{\sinh4a},\qquad L_0=0.\] Indeed on \(y>0\), \(L=2i\sum_{h>0}e^{-2hy}[(1+2(-1)^h)\sin(\lambda h)-\sin(3\lambda h)]/h\), with limit zero at 0 by the sine series. In transforming the odd extension, split \(4p/[h(4h^2+p^2)]= (4/p)(1/h-4h/(4h^2+p^2))\); the bracket’s \(1/h\) sums cancel, and sum the other terms by differentiating the periodic exponential formula above (or its Abel limit). Cartan commutation now gives \(\mathcal D E_R\mathcal D^{-1}= f_X^r E_R\) near the axis, and hence \(\prod_X W(f_X)=\prod_X W(1)_X\,\mathcal D^{-2}\).

Push this joint \(W(f_X)\) insertion all the way to the left (immediately after \(U\)), converting left outer plus fields times \(f_X\) to minus fields times \(-1/f_X\). The remaining central insertion consists only of plus fields, which can now be taken at 0 in real part, with the two outer axes carrying minus fields. The reflection has three effects on the trace. First, under the new transfer \(U\prod_X W(1)_X\) the oscillator matrix is \(-r_m\), hence \(M\) is replaced in (22) by \(M'=M^{-1}\), with trace ratio \(\prod_{m>0}\det M(m\omega)^{-1}\). Second, the leftmost \(\mathcal D^{-2}\) changes the source and contributes a common factor \(\exp(n^2 A_T)\), where \[ A_T=2\sum_{m>0}\frac{m\,L_m L_{-m}}{M_+(m\omega)}. \tag{23}\] Indeed its unordered oscillator exponential at level \(m\) has creation and annihilation vectors \[n\sqrt{2m}\,\mathbf 1 L_m,\qquad n\sqrt{2m}\,\mathbf 1 L_{-m},\] respectively. Their log self-contraction uses \(M'/2\); against a field on the right they give \[nr[(1/M_+-1)L_m z^m-(1/M_++1)L_{-m}z^{-m}].\] Thus after including \(F_T^{nr}\) the nonzero modes of the one-body exponent are those of \(nr\log G_T\), with \[G(v)=\frac{\cos(\nu v)-\cos(\pi/6)}{\cos(\nu v)+\cos(\pi/6)},\qquad G_T(v)=\prod_{j\in\mathbb Z}G(v+ijT).\] In fact \[\frac{J(a)}{M_+(p)}=\frac{\sinh a}{\cosh(3a/2)}.\] The same sine-pair transform, now with spacing \(\nu\), gives the transform of \(\log G(iy)\), for \(p>0\), as \[-\frac{2\pi\sinh a}{p\cosh(3a/2)}.\] Third, the momentum constraint changes. The reflected transfer swaps rather than negates the two labels, so closing the trace requires \[(\mathbf p_A,\mathbf p_B) =(\mathbf p_B+Q_B,\mathbf p_A+Q_A).\] Equivalently, \(Q_s=0\) and \(\mathbf p_d=-Q_d/2\), while \(\mathbf p_s\) ranges over all integers. The constraint \(Q_s=0\) also cancels the missing common mean in the replacement of the source. The Weyl vacuum sign, applied after the fields, is \((-1)^{\mathbf p_d+Q_d}=(-1)^{\mathbf p_d}\), independent of the remaining momentum sum. Combining the field multipliers with the \(\eta\)-powers of \(\mathcal D^{-2}\) gives \(\prod_X\eta_X^{-2\mathbf p_X-Q_X}=e^{-i\theta\mathbf p_s}\). Since \(2\mathbf p+Q=(\mathbf p_s,\mathbf p_s)\), the linear position phase is \(e^{-i\omega D_s\mathbf p_s}\). Poisson summation of this Gaussian momentum sum therefore gives \[ \Theta_T(\theta,D_s):= \sum_{j\in\mathbb Z} \exp[-(\theta+2\pi j+\omega D_s)^2/(4\omega\lambda c_s)]. \tag{24}\]

We collect the factors for later use. Put \[\Delta_T=\prod_{m>0}\det M(m\omega)^{-1},\qquad \mathcal N_T= \frac{\sqrt{\pi/(\omega\lambda c_s)}} {\sum_{h\in\mathbb Z}e^{-2\omega\lambda h^2}}.\] For a fixed reflected letter list \(\mathcal L\), let \(\mathcal C_{M'}(\mathcal L)\) denote the right side of (22) with \(M'=M^{-1}\). On distinct physical-axis positions \(v_i=iy_i\), with \(T>0\) real and \(Q_s=0\), the modulus of its scalar coefficient before integration and factorials is \[e^{n^2A_T}\Delta_T\mathcal N_T \exp\!\left(\mathcal C_{M'}(\mathcal L) -2\omega\lambda(Q_d/2)^2\right) \prod_i G_T(iy_i)^{nr_i} \left|\Theta_T(\theta,D_s)\right|.\] Here \(G_T(iy)>0\); the contour and ordering signs have modulus one. The signed coefficient itself is the trace contraction just computed. Its extension away from the axis is obtained by meromorphic continuation of those factors, not by continuation of a modulus.

After collecting the linear phase, the axis logarithms of the signed pair product are the cosine series in (22). The remaining ordering signs are constant on each ordered region. High-mode coefficients expanded in \(x=e^{-a}\) are integral, since \(M\) and \(M^{-1}\) have integral Taylor coefficients. On any finite-width strip, extracting finitely many elementary factors associated with \(\log(1-e^{-\omega k\lambda\pm i\omega s})\) leaves an exponentially convergent remainder. Thus the continued pair factors have integer orders of zeros and poles. In Section 6.2 we estimate this continuation together with the source and momentum factors above.

Convergence and the operator meaning of reflection

The algebraic reflection must also hold for the integrated trace. We justify it first for nearby contours, where energy damping gives absolute convergence, and then continue the finite outer-point expressions. Fix a finite list of unintegrated outer points on the two appropriate sides of the center. Initially place them very close to the central axes, all within the Cartan Fourier strip, and take strict radial order very near the unit circle. The old transfer at fixed real \(T\) decays exponentially in energy; indeed \(\|r_m\|\le \exp(-c m\omega\lambda)\) follows by positivity and continuity, with the given small/large-frequency asymptotics. Choose the small central displacements (and the starting positions in this argument) well inside the scale allowed by that decay and by the Cartan coefficients.

The common scalar factor

The reflection has isolated the factor \(\exp(n^2A_T)\). To remove the periodization later, we need its value at infinite period and analyticity in the small nome. We compute both now.

The quantity \(A_T\) itself extends analytically in the nome \(e^{-2T/3}\) at zero. Put \(l=\omega\lambda\). Then (23) is \(-l\sum_{m\in\mathbb Z} H(ml)/(ml)\), where the even function \(H(a)/a\) vanishes at 0 and \[H(a)=J(a)\sinh a/\cosh(3a/2) =\frac{x(1-x^2)(1-x)(1-x^3)^2}{(1-x^8)(1+x^3)} =\frac{\sum_{j=1}^{23} c_j x^j}{1-x^{24}},\quad x=e^{-a}.\] Coefficients \(c_1,\ldots,c_{12}=1,-1,-1,-2,3,3,1,-4,-3,-1,3,2\), with \(c_j=c_{24-j}\) thereafter, follow by multiplication. Poisson summation (periodize the decaying integrand) now has image frequencies \(kT/\lambda\). Shifting Fourier contours past the simple poles \(i\pi j/12\), with bounded residues away from 0, gives a convergent power series in the indicated nome: horizontal contours midway between poles have uniformly exponentially decaying rational factor in real direction. The constant term is \[-2\int_0^\infty H(a)\,\frac{da}{a}=\sum_j c_j\log\sin(\pi j/24)=A_\infty .\] Indeed use complete geometric-series packets (\(\sum c_j=\sum j c_j=0\), giving summable integrated packet bounds) and \(\int(e^{-u a}-e^{-v a})\,da/a=\log(v/u)\); pairing reflected indices gives the sine product. We use here the elementary product \(\sin(\pi h)/(\pi h)=\prod_{k\ge1}(1-h^2/k^2)\): for imaginary \(h\) it follows by logarithmic differentiation from the same periodized exponential sum giving \(\sum_{k\in\mathbb Z}1/(k^2+t^2)=\pi\coth(\pi t)/t\), then by analytic continuation. For \(w=\pi/24\), the half-exponent sine product above reduces by double angles to \(\sin^2(6w)\cos^2w\sin^2(5w)/(2\sin^2(4w)\sin^4(8w)\cos^2(3w))=2(2+\sqrt2)/9\).

Residue particles and source decay

Let \(N\) denote the number of elementary letters in one term of the reflected expansion; it is independent of the source exponent \(n\). Move the outer minus axes from \(\pm b\lambda\) to \(\pm\beta\lambda\), where \(\beta=3/2+\kappa\) and \(\kappa>0\) will be small and fixed. The central plus axis stays at zero. For either label \(A\) or \(B\), write \(\alpha_t\) for a plus letter at offset \(t\lambda\); a minus sign reverses its charge. Residues bind letters at a common longitudinal coordinate. The possible compound particles are \[ \gamma_J=\alpha_0-\sum_{t\in J}\alpha_t\quad(J\subset\{-1,1\}),\qquad -\alpha_{\pm\beta},\qquad \delta=-\alpha_{-3/2}-\alpha_{3/2}. \tag{25}\] Here each compound has a single label. The list can be read directly from the singularities crossed during the two contour moves. In fact the high-frequency expansion is \[M'=I(1+x)+H_3 x^3+O(x^4),\qquad H_3=\begin{pmatrix}-1&2\\2&-1\end{pmatrix}.\] Its coefficients at \(x^k\) for \(k\ge1\) are the plus-plus orders at displacement \(\pm k\lambda\), by the log expansions above; order at 0 is \(2I\). The source \(G_T^{-n}\) has no pole between old and new outer radii. Work first at real \(T\). Move left minus fields first: the simple pole at \(-\lambda\) from a center of the same label can bind one such field (a second has cancellation from a double zero). Likewise right minus fields can bind at \(+\lambda\); and they can bind to a free left minus field at distance \(3\lambda\), same label, yielding the across-pair \(\delta\), initially centered at offset \(-\kappa\) and then translated to the indicated symmetric position.

These moves may be done one point at a time with generic spectators and generic small cut parameters. In the left moves the unbound right minuses are still at \(+b\), too near for the order at distance 3 in offset units; in the right moves any already center-bound left minus is at \(-1\), again too near. A bound left minus does not interfere with right binding at the same center (zero logarithmic order at distance 2).

During the right shifts an already formed across-pair contributes cancellation at a potential aligned binding position by an end-point zero. During translation of across-pairs a minus at an aligned end interacts with nonnegative combined order; two aligned across-pairs give order \(2\) on the same label or \(4\) across labels. They stay at noninteger offsets from center compounds. Two center compounds on the same label have mutual order \(2-|J|-|J'|+2|J\cap J'|\ge0\); the order across labels is zero there. Thus resulting lists have no point-to-point poles on their streams (also for intermediate bound lists in the one-variable moves); the generic residue identities with simple poles integrate directly, with no differentiated factors.

Here one uses a common representative of the period for the longitudinal points at a binding (in the periodic Poisson sum). At elementary order \(N\) diagram and combinatorial costs are at most \(\exp(O(N\log(N+2)))\), since each unbound-point move branches to at most \(O(N+1)\) terms. Infinite-order convergence is only needed on the pre- and post-deformation sides when summing these coefficient identities.

The functional identities \[G(v-\lambda)G(v+\lambda)=G(v),\qquad G(v-3\lambda/2)G(v+3\lambda/2)=1\] also determine the source carried by each compound. Write \(G_{T,u}\) for \(G_T\) evaluated at offset \(u\lambda\), with the same longitudinal coordinate as the compound. Then \[\begin{array}{c|c|c|c} \text{particle}&\text{elementary letters}&\text{scalar charge}&\text{source}\\ \hline \gamma_\varnothing&1&1&G_{T,0}^{\,n}\\ \gamma_{\{t\}},\quad t=\pm1&2&0&G_{T,-t}^{\,n}\\ \gamma_{\{-1,1\}}&3&-1&1\\ -\alpha_{\pm\beta}&1&-1&G_{T,\pm\beta}^{-n}\\ \delta&2&-2&1 \end{array}\] Call the first two rows good particles. If there are \(K\) compounds, their scalar charges sum to \(Q_s=0\). Each nongood particle has charge at most \(-1\), whereas a good one has charge at most \(1\), so at least \(K/2\) are good. Since each compound contains at most three letters, \(K\ge N/3\); there are therefore at least \(N/6\) good particles. This is the proportion needed in the final particle-count estimate.

The scale at which the source confines these particles is \[ \Re T=B=(2/\nu)\log n,\qquad |\Im T|\le C_{\mathrm{im}} \tag{26}\] for sufficiently large fixed \(C_{\mathrm{im}}>3\pi/2\); allowing \(B\) to vary by order one also in continuation. Use streams \[y=s+i(h(s)-u\lambda),\quad -B/2\le s\le B/2,\] where \(u\) is the offset and \(h(s+B)=h(s)+\Im T,\ h(\pm B/2)=\pm\Im T/2\). Take \(h=0\) in the period except within a sufficiently large fixed distance of its ends, with arbitrarily small fixed Lipschitz bound, and bounded on the period. The sources of other particles are bounded there by \(\exp(O(1))\) each. Those of good particles are bounded by \[\exp(O(1)-c\,e^{\nu d_s}),\qquad d_s=B/2-|s|,\quad c>0.\] Indeed on good offsets the central \(G\) has modulus strictly less than one (\(\Re\cos(\nu v)>0\) on those straight lines) and its real log on the tail is negative of order \(e^{-\nu|s|}\); on outer offsets the modulus is greater than one. Bending near the ends and noncentral images have arbitrary phases but cost bounded error including the power \(n\).

A coercive quadratic form on the real circle

Start with a real period \(T=B\). Here is the bound we will prove. Partition the circle into bins of lengths bounded above and below by fixed positive constants, and let \(n_l\) count the compounds in bin \(l\). For a fixed residue list, let \(W_j\) be its integrand with the source factors and the two common factors \(\exp(n^2A_T)\) and \(\prod_{m>0}\det M(m\omega)^{-1}\) removed. It retains the Gaussian normalization, the Poisson branch \(j\), and the binding and self finite parts. For every fixed \(0<\epsilon<1\), \[ \sum_{j\in\mathbb Z}|W_j| \le C\exp\left\{ -\frac{(1-\epsilon)B(\theta')^2}{8\pi c_s\lambda} -c\sum_l n_l^2+CN\log(B+N+2)\right\}. \tag{27}\] This holds uniformly for real \(\theta'\) in a fixed compact subinterval of \((-\pi,\pi)\), for all sufficiently large \(B\). The constants may depend on that interval, \(\epsilon\), and a sufficiently small fixed \(\kappa>0\). We will keep a positive fraction of the bin-count term for the later complex-contour comparison. The proof occupies the present subsubsection and the smearing argument that follows.

We retain two parts of the interaction: a symmetric total-charge pole for the electric cost, and a strictly positive remainder for the local particle counts. A cross-label term that is positive only after integration must first be separated from that remainder.

Smearing, the electric cost, and particle-count bounds

We now convert the quadratic-form estimate into a bound on the actual integrand. The comparison is one-sided: after collecting each pair of compound particles, its point interaction is bounded above by the interaction of suitably smeared charges, with a controlled loss at internal finite parts. Work first on the real circle. Replace elementary point rows by smeared measures in the \(s\)-variable using the circular Poisson kernel \(\mathcal P_\rho\) of width \(\rho=(B+N+2)^{-4}\) (convolution multiplier \(e^{-|m|\omega\rho}\), positive density proportional to \((1-2e^{-\omega\rho}\cos(\omega s)+e^{-2\omega\rho})^{-1}\), mass one). We compare the original finite parts with the smeared energy. The total error is \(O(N\log(B+N+2))\), uniformly in particle order. The estimate must be applied to whole compound particles: elementary pairs can have negative logarithmic coefficients before their residues are combined.

Here are the quantitative details of the smearing comparison. Fix the contour separation \(\kappa>0\). For every used offset difference \(t\), put \(C_t=\lim_{p\to\infty}\Xi(p;t)\). The pole at zero and the folded exponential expansion imply, with constants allowed to depend on \(\kappa\), \[\|\Xi(p;t)-C_t\|\le \begin{cases}C/p,&0<p\le1,\\ Ce^{-cp},&p\ge1.\end{cases}\] Indeed the only terms of the folded expansion that do not decay are those whose shifted distance is zero; these are precisely \(C_t\). The finite set of other distances has a positive minimum after \(\kappa\) is fixed. For \(B\ge2\), \(\omega=2\pi/B\), the smooth remainder of each pair kernel is \[R_{B,t}(s)=-2\sum_{m\ge1} \frac{\Xi(m\omega;t)-C_t}{m}\cos(m\omega s), \qquad \|R_{B,t}'\|_\infty\le C(1+\log B).\] The last estimate follows by splitting the differentiated series at \(m\omega=1\): the first part is bounded by a harmonic sum and the second by a geometric sum.

The circular Poisson kernel \(\mathcal P_\rho\) is a probability density and satisfies \(\mathcal P_\rho*\mathcal P_\rho=\mathcal P_{2\rho}\). Writing \(|s|_B\) for circular distance to zero, for \(0<\rho\le1\) we have \[\int |s|_B\mathcal P_{2\rho}(s)\,ds \le C\rho\bigl(1+\log(B/\rho)\bigr).\] Consequently replacing a smooth pair remainder by its twice-smeared version changes it by at most \[C\rho\bigl(1+\log(B/\rho)\bigr)(1+\log B).\]

For the singular part put \(L_B(s)=\log|1-e^{i\omega s}|\). Its exact Poisson average and a one-sided comparison are \[(\mathcal P_{2\rho}*L_B)(s) =\log|1-e^{-2\omega\rho+i\omega s}|, \qquad L_B(s)\le(\mathcal P_{2\rho}*L_B)(s)+\omega\rho.\] For the inequality, square both sides after exponentiation and use \(e^{2\omega\rho}+e^{-2\omega\rho}\ge2\). Apply this inequality after collecting all elementary interactions between a pair of compound particles. Their total coincident-log coefficient is nonnegative; individual elementary interactions need not have this property. All constituents of one compound have the same longitudinal coordinate, so the collection uses one and the same \(L_B(s-s')\).

For a diagonal or binding finite part, subtraction is in the additive coordinate: \(L_B(s)-\log|s|\to\log\omega\) as \(s\to0\). In contrast, the smeared singular part at zero is \(\log(1-e^{-2\omega\rho})=\log\omega+\log(2\rho)+O(\omega\rho)\). Their difference is therefore \(O(1+|\log\rho|)\), with the \(\log\omega\) terms cancelling. There are \(O(N)\) such internal terms and \(O(N^2)\) external pairs, with uniformly bounded constituent multiplicities and log coefficients. On taking \(\rho=(B+N+2)^{-4}\), the total comparison error is \[O\!\left(N(1+|\log\rho|) +N^2\omega\rho +N^2\rho(1+\log(B/\rho))(1+\log B)\right) =O\bigl(N\log(B+N+2)\bigr).\] This compares finite parts as well as the off-diagonal kernel and preserves the nonpositive signs of all noncentral even-row measures. The same comparison applies at fixed real \(T\) on the initial lines: independent same-sign letters have nonnegative coincident orders, and the other line separations are nonsingular.

Continuation to complex periods

The real estimates have now justified the initial contour sums and the current reflection. To obtain the bound at nome zero, we need them also for bounded imaginary parts of the period in (26). We compare with the real energy; no positive quadratic form is asserted on the bent contours.

The oscillator determinant and the nome limit

The oscillator determinant is \[\det M=\frac{(1+x^3)^4(1-x^3)^2}{(1+x)^2(1-x^8)^2}.\] Integrating its logarithmic series gives \[\begin{split} -\int_0^\infty\log\det M\,da &=-4\frac{\pi^2}{36}+2\frac{\pi^2}{18} +2\frac{\pi^2}{12}-2\frac{\pi^2}{48}\\ &=\frac{\pi^2}{8}. \end{split}\] Here we used \(\int_0^\infty\log(1-e^{-ka})\,da=-\pi^2/(6k)\) and \(\int_0^\infty\log(1+e^{-ka})\,da=\pi^2/(12k)\). Riemann summation, uniform also with the bounded imaginary period parts, gives log-modulus of \(\prod_{m>0}\det M(m\omega)^{-1}\) equal to \(B\pi^2/(8\cdot2\pi\lambda)+o(B)\). Since \(B=(2/\nu)\log n\), the determinant contributes \(\pi/(8\lambda\nu)=3/8\) on the logarithmic scale. The electric coefficient is \(1/(4\pi c_s\lambda\nu)=2/(3\pi^2)\). Together with the particle bound, these give the log-modulus divided by \(\log n\), after removing \(\exp(A_T n^2)\), at most \[\frac38-\frac{2}{3\pi^2}\big((1-\epsilon)(\Re\theta)^2-(\Im\theta)^2\big)+o(1)\] on (26), uniformly for bounded complex \(\theta\) near the real segment. The analytic factor \(A_T\) from (23) agrees by continuation with its nome series there. Put \(q_T=e^{-2T/3}\). At fixed \(\Re T=B\), an interval of imaginary parts of length \(3\pi\) traverses the entire circle \(|q_T|=e^{-2B/3}\); our choice \(C_{\mathrm{im}}>3\pi/2\) therefore supplies its maximum. The initial small-circle prescription is holomorphic at \(e^{-2T}=q_T^3=0\), and \(A_T\) is holomorphic in \(q_T\). The maximum principle applied to \(\mathcal Z_{n,T}\exp(-A_Tn^2)\) gives the same upper bound at \(q_T=0\), where it equals \(\mathcal Z_n\exp(-A_\infty n^2)\). For each fixed \(\epsilon>0\) the error tends to zero uniformly on the stated compact angle sets. Take the upper limit as \(n\to\infty\), then let \(\epsilon\downarrow0\). This proves (21).

Recovering two representations.

The excursion and arm live on opposite sides of \(B\), so concatenate simply at the port (weight multiplicative before thinning), giving an ordered arch in the \(A\) half-plane through \(v\) of diameter \(\le C'R\). For each such arch \(\xi\) define \(m_D(\xi)\) for offset triple \(D\) to equal the thinning scalar if represented at that triple, zero otherwise. Indeed at a fixed triple the crossing of the \(B\) line is unique, so both pieces are uniquely specified by \(\xi\). Write \(m=\sum_D m_D\). Over distinct resulting arches, with unthinned weight \(w_\xi\), \[ \sum_\xi w_\xi m(\xi)\ge R^{3+z-p-o(1)},\qquad \sum_\xi w_\xi m(\xi)^2\le R^{5+z-2p+o(1)}. \tag{51}\] For the second bound take two triples; they need the same \(A\) line, and order them so the first \(B\) offset is at most the second. The crossing at the first precedes or equals the crossing at the second (at each the path goes from strict interior to strict exterior for that \(B\) wall). The prefix through \(v\) up to the first crossing is a first-triple triangle chord and costs \(\le R^{z+o(1)}\) in total. The segment between crossings if distinct stays between the parallel lines and is a strip bridge, costing \(\le C(1+\Delta)^{-p}\) from a fixed first crossing by the strip bound at (14), where \(\Delta\) is the offset difference (if zero there is no segment). The remainder is the second-triple arm and costs \(c_1 R^{-p}\) including its thinning, given the second crossing. Dropping other restrictions and the first thinning gives the desired bound after summing: there are \(O(R)\) choices for the common \(A\) offset, \(O(R^2)\) for the two offsets of the remaining third side, and the sum of \((1+\Delta)^{-p}\) for \(B\) offsets is \(O(R^{2-p})\). Thus (50) gives distinct arch mass at least \(R^{1+z-o(1)}\).

All these bounds are uniform over the indicated \(v\); all produced arches have height from their \(A\) line at least \(R/2\) since they visit \(v\). Sum over the \(\asymp R^2\) choices of \(v\): the aggregate arch mass with a marked visited vertex is at least \(R^{3+z-o(1)}\), allowing all of the \(O(R^2)\) possible starts with their oriented walls (triangle ports within distance \(O(R)\) of the origin). For any one such start and wall the mass is bounded by the full unnormalized length sum in a congruent standard half-plane with that height constraint and diameter \(\le K_0H\), by lattice symmetry and taking \(K_0\) sufficiently large. So this latter sum per fixed start is at least \(R^{1+z-o(1)}\), completing (49).

Three choices of the incoming leg.

First \(H_{ab}\) is the sum with \(\beta\) present of the simple exit terms from \(v\) in its complement, by the prefix form of (45). Physical exits cost \(\sum P_{d\mid ab}\); other exits hit \(\beta\), forming tripod terms \(C_I\). Alternatively take an existing spin path \(v\to b\) in \(D\); run the non-backtracking experiment from \(a\) stopping also at first hit to that path. This gives \(A_b=\sum_{d\ne a,b} u_{ad}P_{b\mid ad}+C_{II}\) by physical exits, exterior self-repeat cancellation as in (1), and hits. Similarly \(A_a=\sum_{d\ne a,b}u_{bd}P_{a\mid bd}+C_{III}\).

On a hit-tree branching at a vertex other than \(v\), all three options exist with the same magnitude. Compare option II to I oriented for its through chord \(a\to b\) (to give the chord spin weight in I one multiplies I by \(u_{ab}\)). Switching which incoming leg (from \(v,a\)) passes toward \(b\) then changes the branch turn, giving a ratio \(e^{\pm 2\mathrm i\lambda}\) according to the ray order. For III compare to I oriented as \(b\to a\), giving \(e^{\mp 2\mathrm i\lambda}\) (turns between consecutive rays in counterclockwise order are \(-\pi/3\), between reverse-ordered ones \(+\pi/3\)); other directed turns agree in each comparison. Thus \(u_{ba}C_{II}+u_{ab}C_{III}\) contributes \(c_* C_I\) there. Hits branching at \(v\) occur only in II,III and form the marked chords in opposite orientations, with a missing weight and turn at \(v\); together their normalized factor after removing the full chord phase is \(2\cos\lambda/x=2+c_*\). We get \[ \sum_{d\ne a,b}\big[c_*P_{d\mid ab} -u_{ba}u_{ad}P_{b\mid ad}-u_{ab}u_{bd}P_{a\mid bd}\big] = c_*H_{ab}+(2+c_*)K_{ab}-u_{ba}A_b-u_{ab}A_a . \tag{53}\]

Marking the boundary exploration.

Define \[\widetilde P_{d\mid ab} =\sum_e u_{ed}U_{ab,de}-K_{ab} =h_{ab}A_d-\sum_e u_{ed}R_{ab,de}.\] The second equality uses the conjugate of (1). We do not assert \(P_{d\mid ab}=\widetilde P_{d\mid ab}\). What we need, and now prove, is the equality of the three-term combinations \[\begin{aligned} &c_*P_{d\mid ab}-u_{ba}u_{ad}P_{b\mid ad}-u_{ab}u_{bd}P_{a\mid bd}\\ &\qquad=c_*\widetilde P_{d\mid ab} -u_{ba}u_{ad}\widetilde P_{b\mid ad} -u_{ab}u_{bd}\widetilde P_{a\mid bd}. \end{aligned}\] With a positive existing chord \(a-b\), run the complex exploration from \(d\), stopping also on its first hit to that chord. Give every occurrence of the vertex weight at \(v\), in the existing chord or the exploration, an extra fugacity and differentiate at fugacity one. The derivative at fugacity 1 of the total terminal weight is \(K_{ab}+\overline P_{d\mid ab}\). The first term differentiates the existing chord, after which the undeformed continuation has total weight one. The second differentiates the transition at a fresh, nonterminal \(v\). At fugacity one the sum of the two differentiated outgoing weights is one, and so is the total weight of all subsequent continuations. What remains is the first-arrival prefix at \(v\), whose reversal gives \(\overline P_{d\mid ab}\). Exterior self-repeats still cancel, physical exits give derivative \(\sum_e u_{de}U_{ab,de}\), and remaining hits give tripod terms carrying the mark on their union (once if \(v\) is there, including at a branch where it is the chord that counts the vertex). For any one such marked tripod, write \(C_i\) for the weight when the leg from \(i\) is the orphan exploration (the other two forming the positive chord). If \(i,j,k\) are the boundary ports in counterclockwise order then \(C_i/C_j=u_{ij}e^{\mathrm i\lambda}\): the magnitudes agree and joining the two legs into \(i\to j\) adds the turn \(-\pi/3\) at branching by planar ray order. Thus \[c_*\overline C_d=u_{ba}u_{ad}\overline C_b+u_{ab}u_{bd}\overline C_a .\] For instance in order \(a,b,d\), for increasing indices write \(u_{ij}=p_j/p_i\,e^{-\mathrm i r}\) with \(p_i=e^{\mathrm i s\phi_i}\) for lifted outward normal angles by (1). Then \(\overline C_b/\overline C_d=p_b/p_d\,e^{2\mathrm i\lambda}\), \(\overline C_a/\overline C_d=p_a/p_d\,e^{4\mathrm i\lambda}\); exchange \(a,b\) for the other order. This cancels the conjugated hit derivatives in the stated replacement.

Collecting the one-path terms.

Substituting in (53), the terms with \(A\)’s on the left sum by (1) to \(c_*h_{ab}T_v-(c_*-1)h_{ab}(A_a+A_b)-u_{ba}A_b-u_{ab}A_a\). Taking real parts gives (52).

A supply of chords intersecting each marked chord.

For the positive terms left over, given any chord \(\beta\) from \(a\) through \(v\), there is mass \(\gtrsim H^\alpha\) of chords with distinct unused endpoints separating \(a\) from \(v\), hence intersecting \(\beta\), giving the needed lower from \(G_\beta R_{\beta,\eta}\). Here are details. Near a corner use a smaller equilateral triangle there of small fixed proportional size, sharing the two adjacent faces (cut the third on the allowed grid), not containing \(v\). Chords between its two shared faces, with endpoints bounded away from corners by another sufficiently small proportional amount, have total mass \(\gtrsim H^\alpha\). Indeed in that triangle subtracting (1) from half-plane saturation loses bottom arches of mass \(\gtrsim H^{-p}\) per port by long displacements (14), so this lower bound holds for all different-side chords summed, and then for each pair of sides by cyclic symmetry. The grid triangles have this symmetry: after rotation by \(120^\circ\), any cut-offset differences are \((3/2)\)-integer triples summing to zero and are restored by lattice translation. Endpoint trimming costs arbitrarily little of the bound by the different-side upper just used. Thus for \(a\) sufficiently close proportionally to the shared corner these chords cut it off in a corner region away from \(v\). For \(a\) bounded away from corners proportionally, use same-side half-plane arches bracketing \(a\): take starts on one side of \(a\) along its wall at distance between \(\gamma H\) and \(2\gamma H\), ending towards and past \(a\) with gap \(>3\gamma H\) and diameter \(\le C\gamma H\), cost \(\gtrsim H^{-p}\) per start by (14) and the diameter upper choosing \(C\) large. For sufficiently small fixed \(\gamma\) they stay triangle chords and close off \(a\) in a region away from \(v\). In both constructions removing specified endpoints of \(\beta\) costs only \(O(1)\) mass. This justifies the positive lower in (54); on the right of (52) discard the last two nonpositive terms.

Chords of a prescribed diameter.

To estimate the first right term of (54), classify ordinary chords by dyadic \(1+\operatorname{diam}\sim L\) (\(1\le L\lesssim H\)). Uniformly per triangle their total at this scale with \(G\) included costs \(O(H L^{-p-\tau})\). Indeed different-side chords with dyadic gap \(1+|b-a|\sim g\lesssim L\) have both ports within \(O(g)\) of their corner, hence cost unweighted \(O(g^\alpha)\) as above, sufficient after summing \(g\). For the same-side bound, let \(a_{g,L}\) be the half-plane mass from a fixed left endpoint to right endpoints at these gap and diameter scales. Complete each such upper arch by an opposite-half-plane arch with the same endpoints. Its mass, with endpoints fixed, is \(\asymp g^{-1-p}\) by the pointwise bound for \(h\). The two arches form a simple polygon.

Let \(m(\gamma)\) count representations of its unoriented translation class. A representation specifies a horizontal cut meeting the polygon exactly twice; translating the left crossing to the standard port then recovers both arches. Thus its first moment satisfies \[E_1:=\sum_\gamma x^{|\gamma|}m(\gamma) \gtrsim a_{g,L}g^{-1-p}.\] For two such cuts, normalize the higher one. Their height difference \(\Delta\) is \(O(L)\), because both designated upper arches have diameter \(O(L)\). The high-cut upper arch costs \(a_{g,L}\). Between the cuts there are two strip bridges, costing at most \(B(\Delta)^2\) from the revealed crossings. Below them remains a fixed-end arch, costing at most \(Cg^{-1-p}\); the lower cut must also have endpoint gap of order \(g\). At coincident cuts use factor one. Summing the height gap gives \[E_2:=\sum_\gamma x^{|\gamma|}m(\gamma)^2 \lesssim a_{g,L}g^{-1-p}\sum_{0\le\Delta\lesssim L}(1+\Delta)^{-2p} \lesssim a_{g,L}g^{-1-p}L^{1-2p}.\] Every eligible polygon has diameter at least \(cL\), since it contains a designated upper arch of that diameter. The lower completing arch is unrestricted and may make the polygon much larger. Sum the bound in Proposition 12 over all diameter dyads above \(cL\) to obtain \[S:=\sum_{\gamma:m(\gamma)>0}x^{|\gamma|} \lesssim\sum_{j\ge0}(2^jL)^{-2}=O(L^{-2}).\] The inequality \(E_1^2\le SE_2\) now yields \(a_{g,L}\lesssim g^{1+p}L^{-1-2p}\). For \(\tau<1-p\), the dyadic sum satisfies \[\sum_{g\lesssim L}g^{-\tau}a_{g,L} \lesssim L^{-1-2p}\sum_{g\lesssim L}g^{1+p-\tau} \lesssim L^{-p-\tau}.\] There are \(O(H)\) starting ports; reversal accounts for the other endpoint order. This proves the required \(O(HL^{-p-\tau})\) bound.

The averaged loss under deletion.

Shrink every side inward by a grid amount \(m_L\asymp L\), choosing its constant larger than the diameter bound for this dyad, and call the remaining vertex set \(D_-\). A chord at scale \(L\) has an endpoint on one original side; every vertex of that chord lies within \(O(L)\) of that side. It therefore avoids \(D_-\), so \(D_-\subset D\setminus V(\beta)\). By filled-domain monotonicity their contribution in the difference term of (54) is bounded by their \(G\)-weighted mass times \(T(v,D)-T(v,D_-)\) (take the latter \(T(v,D_-)=0\) if \(v\notin D_-\)). To average this difference, shift the three coordinates one at a time. For one coordinate, write its consecutive allowed levels as integers \(a\le d\le b\), and write \(q=O(L)\) for the shift in grid steps. For any nonnegative monotone sequence \(F\), \[\sum_{d=a}^b\big(F(d)-F(d-q)\big) =\sum_{d=b-q+1}^bF(d)-\sum_{d=a-q}^{a-1}F(d) \le q\sup F.\] Extend the visit sum by zero when the domain no longer contains \(v\). This remains monotone. The uniform bound \(\sup F\le H^{z+o(1)}\) follows from (46); each offset window has order \(H\) levels. The three-coordinate difference therefore averages to \(O(L/H)H^{z+o(1)}\). The averaged right-hand deletion term in (54) is therefore at most \[\sum_{L\lesssim H} H L^{-p-\tau}\frac LH H^{z+o(1)} \lesssim H^{z+1-p-\tau+o(1)}.\] The sum is dyadic and \(1-p-\tau=\alpha-\tau>0\). Divide by \(H^\alpha\) in (54) to obtain (55).

Retaining separated endpoints.

We now derive the retained endpoint estimate for every fixed \(\epsilon>0\). It suffices to consider \(0<\epsilon<1\); the assertion for larger \(\epsilon\) follows by monotonicity of the retained endpoint set. On the set \(|b-a|<H^{1-\epsilon}\), the weight \(G_{ab}=(1+|b-a|)^{-\tau}\) is at least a constant times \(H^{-\tau(1-\epsilon)}\). Equation (55) therefore gives \[\left\langle\sum_{|b-a|<H^{1-\epsilon}}K_{ab}\right\rangle \le H^{z-\tau\epsilon+o(1)}.\] The total average \(\langle\sum_{a,b}K_{ab}\rangle\) is \(H^{z+o(1)}\), uniformly under the size and depth assumptions above, by (43), (46), and the positive cosine comparison. Since \(\tau\epsilon>0\), the small-gap contribution has a strictly smaller exponent. Hence \[ \left\langle\sum_{|b-a|\ge H^{1-\epsilon}}K_{ab}\right\rangle =H^{z+o(1)}\qquad(\epsilon>0\text{ fixed}). \tag{56}\] The sums are over ordered endpoint pairs, and the average is over the same three independent offset windows as in (55).

A plateau of fixed-end masses.

Consequently as a function of allowed endpoint displacement \(b_L\) has a plateau of lateral width \(\gtrsim L\) centered at some \(u_L=O(L)\), where it is everywhere \(\gtrsim L^{-1-p}\), and also such a plateau centered at \(-u_L\) by reflection. Indeed order \(L\) sites in a fixed tightness range reach a small absolute constant times \(L^{-1-p}\), by the total lower and (58), and by the variation bound the number of clusters containing them separated by dips below half that threshold is bounded. Even requiring bridges to stay within lateral distance \(K' L\) of their start we retain (say) half the plateau bound except at a set of at most an arbitrarily small fixed constant times \(L\) sites in the plateau, by choosing a fixed large \(K'\) via the same tightness bound.

Geometric slabs.

Subdivide at geometric scales into \(k=O(1+\log(L/g))\) slabs with spans comparable to nominal sizes \(l_i\) increasing from scale \(g\) by factors of order two to scale \(L\) (e.g. interior depths \(2^j g,\ j\ge0,\ 2^j g\le L/2\)). Thus successive sizes are comparable by absolute factors. Vary the interior cuts at tops of slabs \(i<k\) in small windows of lengths comparable to \(l_i\), and the endpoints there in small windows on the same scales about the interpolation. Take the endpoint-window proportions small enough that each fixed-end slab has slope of magnitude at most \(\delta\); since \(l_i\asymp g+y\) at depths in that slab, taking them and the sub-tube widths sufficiently small allows the preceding ordinary straight-tube estimate while satisfying the desired growing width throughout. All scales tend to infinity and constants are uniform. The mass per cut vector is then \(\ge c^k L^{-1}\prod_i l_i^{-p}\), including the counts of intermediate positions.

The cost of varying the cuts.

For two vectors the common paths split as in the straight-tube proof, with mass \(\le C^k L^{-1}\prod_i l_i^{-p}\prod_{i<k}(1+\Delta_i)^{-p}\) (use the supremum bound on a scale-\(L\) main interval). Each summed gap pair costs \(O(l_i^{2-p})\), and the number of single vectors is \(\ge c^k\prod_{i<k}l_i\). In the quotient of (50), the factors from the \(k-1\) variable cuts cancel all \(l_i^{-p}\) except the final one, whose scale is \(L\). The result is at least \(c_1^kL^{-1-p}\). Since \(k\le C_1+C_2\log(L/g)=O(1)+O(\epsilon\log R)\), this is \(R^{-1-p-O(\epsilon)}\). All constants here depend on the fixed tube geometry, while \(\epsilon\) affects only the number of scales.

Supplying the pieces.

Insert small admissible equilateral triangles near the bend vertices with face outward normals for entry and exit \(-n_{\rm prev},+n_{\rm next}\), at angle \(120^\circ\). Take side distances from the bend of a tiny fixed order \(hR\), then vary the locations of both used faces independently in tiny macroscopic windows (sizes also small relative to \(hR\)). Within each triangle the total mass of chords connecting the two specified faces, with both ports trimmed away from corners by a small fixed fraction of face length, is \(\gtrsim R^\alpha\), \(\alpha=1-p\), as in the proof of (54) by different-side comparison and cyclic symmetry (valid for either triangle orientation). Connect intervening gaps in the route by fixed-end normal-slab bridges using narrow tubes as above (the expanding one for the first). Indeed each slab’s lateral offset comes from the nominal line (possible small first tilt), terminal window and deviations of ports of size \(O(hR)\), allowing small slopes. We can take \(h\) and the terminal window sufficiently tiny before choosing tube widths. In the first segment the bend-port shift (or terminal shift if \(j=0\)) changes the slope from the prescribed one by an arbitrarily small amount, so the growing tube stays within arbitrarily small fixed multiples of \(g+y\) about the intended ray, up to the bend (or terminal). Thus per bend-cut vector the product mass is at least \[R^{-1-p-2jp-O(\epsilon)}\] by \(j\) triangle masses and \(j+1\) straight estimates.

Disjointness and localization of all possible seams.

Here the constituent paths concatenate disjointly. The bridges incident to a bend are on the outside of its respective cuts; moreover each stays away from the other used face’s possible seam ports (trimmed and with small cut windows), on the strictly interior side of that face’s supporting lines near and along the bridge. Indeed a trimmed incident port has such a margin already of fixed proportional order, and the ray heading back or onward away from the triangle moves deeper relative to the other face (dot product of the two outward normals \(-1/2\), robust to small slopes); take tube widths sufficiently small relative to the margins. Thus the two incident bridges are also apart. Nonincident pieces stay away by the macroscopic layout. In particular for any path representation only the appropriate incident straight bridge and that bend’s chord can visit the possible entry seam ports of the bend (across all its varying cuts); similarly for exit ports. For fixed cuts the designated crossings among such ports are unique.

Comparing two representations.

For two bend-cut vectors representing the same path, the more exterior entry-line crossing at a bend occurs before the inner one if different: it must lie on the incoming bridge under the other representation by the localization just given, and conversely the inner crossing on the other bend chord (coincident cuts give a single crossing). The exit comparison is reversed. Every entry here is before every exit even across representations by the same localization. Thus the segment between latest entry and earliest exit is a chord in the intersection triangle, between the two specified sides, cost \(O(R^\alpha)\) with endpoints summed as in (54). Segments between the compared crossings are bridges in the parallel gap strips (part of one straight bridge and the other chord, hence between the cuts), cost \(O((1+\Delta)^{-p})\) with inner port given. Across consecutive bends the latest exit is before the earliest next entry (under the representation with that exit the other entry is on the connecting bridge or the next chord). Remaining segments between exterior cuts are therefore fixed-end strip bridges, lying on the shared straight portions, costing \(O(R^{-1-p})\) each by (58) after gap bridges have fixed their ports (including first and last pieces here). This decomposition contains \(j\) common bend chords, \(j+1\) fixed-end straight pieces, and \(2j\) possible gap bridges. Its mass is at most \[C\underbrace{R^{j(1-p)}}_{\text{bend chords}} \underbrace{R^{-(j+1)(1+p)}}_{\text{straight pieces}} \prod_{h=1}^{2j}(1+\Delta_h)^{-p} =C R^{-1-p-2jp}\prod_{h=1}^{2j}(1+\Delta_h)^{-p}.\] Let \(m\) count the valid bend-cut vectors. There are order \(R^{2j}\) single vectors, while each paired window contributes \(O(R^{2-p})\). Consequently the first moment is at least \(R^{-1-p-2jp+2j-O(\epsilon)}\), and the second at most \(C R^{-1-p-2jp+2j(2-p)}\). Equation (50) gives distinct mass \(R^{-1-p-O(\epsilon)}\) per fixed endpoint. When \(j=0\), the widening-tube estimate already gives this conclusion.

Different incident sides.

If \(A\ne B\), write \(n=e_A,\ m=e_B\). From the \(B\) port take the ray \(+m\) to the upper wall. From the \(A\) port take the ray \(+n\) for \(R/4\), then bend to \(-m\) to the lower wall (\(n\cdot m=-1/2\)).

The same incident side.

If \(A=B\), write \(n=e_A,\ t\) its counterclockwise perpendicular unit vector, \(m=n/2+(\sqrt3/2)t,\ n_-=n/2-(\sqrt3/2)t\). Order ports left to right by \(t\)-coordinate, reversing the marked chord if necessary. Take initial rays \(n-\kappa t,\ n+\kappa t\) respectively, to forward depth \(R/8\) along \(n\), for sufficiently small fixed \(\kappa>0\). Bend the right one to \(+m\) to the upper wall. Bend the left one to \(n_-\) until \(m\)-coordinate \(s_-+R/8\), then bend to \(-m\) to the lower wall.

Figure 3 records the two layouts. The central chord is retained, while the added arms occupy disjoint exterior neighborhoods; the bends are the locations where the preceding triangle sewing is used.

In the first layout put \(d=a-b\), where \(a,b\) are the \(A,B\) ports. Since \(n\cdot d\ge0\), \(m\cdot d\le0\), and \(n\cdot m=-1/2\), the initial rays satisfy \[|d+yn-t'm|^2\ge |d|^2+y^2+(t')^2+yt'\qquad(y,t'\ge0).\] Their separation is therefore at least a constant times \(g+y+t'\). The later \(A\)-arm has macroscopic separation in \(n\)-projection from the \(B\)-ray and from the central \(A\) line, towards its exterior. In the second layout the initial separation in \(t\)-projection is at least \(g+\kappa(y+t')\); thereafter there is macroscopic separation in this projection, with the two arms continuing away from each other (also each later piece is apart from the opposite initial piece). Both beyond their first rays stay macroscopically to the exterior of the \(A\) line: the long \(n_-\) piece moves farther out. The first bend already has normal depth \(R/8\) outside that line, and the final length \(R/8\) toward \(-m\) reduces that depth by only \(R/16\). Thus a margin of at least \(R/16\) remains. The left arm’s nonconsecutive pieces are apart in \(t\)-projection. Thus these are uniformly simple routes of the bend type just discussed, with target-wall clearance except at the respective aligned terminal pieces, and we can ensure for the actual paths that each arm stays strictly outside its incident central line and the two are disjoint. Indeed first straight bridges remain strictly exterior to their starting line and opening widths and slope deviations can be made arbitrarily small relative to the displayed initial separation; tiny fixed-proportion bend and tube neighborhoods suffice afterwards. Nominal depths/lengths are comparable to \(R\), with \(c\) small as stipulated, and all subsequent parameter choices for the sewing estimates can be uniform over the central locations and independent of \(\epsilon\).

Normalizing an arm using only its own initial port.

Fix the ordered side pair, the target strip, and each arm’s role once and for all. In the equal-side case these roles are left and right along \(t\); otherwise they are the \(A\) and \(B\) sides. For a fixed triangle \(D\) and role, the route and its permitted narrow neighborhoods depend only on that arm’s own starting port \(a\). The second port is needed only to verify the separation of the two already chosen neighborhoods. All tube widths, bend sizes and terminal windows are chosen uniformly in the allowed ports and independently of \(\epsilon\).

Let \(Q_{D,\mathrm{role}}(a)\) be the mass of this one-arm family, with its terminal port summed over the fixed target window. The bend estimates give \(Q_{D,\mathrm{role}}(a)\ge R^{-p-K_{\mathrm{loss}}\epsilon}\) for a uniform \(K_{\mathrm{loss}}\) and large \(R\). Multiply every weight in this family by \[\lambda_{D,\mathrm{role}}(a) =\frac{R^{-p-K_{\mathrm{loss}}\epsilon}}{Q_{D,\mathrm{role}}(a)}\le1.\] Its thinned total mass is exactly \(R^{-p-K_{\mathrm{loss}}\epsilon}\). In particular, this scalar does not depend on the other endpoint of the central chord. Concatenating with the chord gives bridges through \(v\); the central interval is uniquely determined given the triangle by the interior condition and the exterior arm constraints. Over full distinct bridges visiting \(v\) write \(m_D\) for the product of thinning factors when represented for that triple, zero otherwise. Thus \[\sum_\xi w_\xi\sum_D m_D(\xi)\ \ge\ R^{3+z-o(1)-2p-2K_{\mathrm{loss}}\epsilon}.\]

The two-representation decomposition.

For two offset triples representing the same bridge the central intervals both contain \(v\) (visited only once). Entry at the more exterior \(A\) line precedes entry at the inner one if distinct (both before \(v\), and the former point cannot be inside the other chord); likewise exits in reverse order at side \(B\). Equal entry or exit offsets give the same corresponding crossing. Overlap between inner crossings is a marked excursion of the intersection triangle, total mass \(\le R^{z+o(1)}\). Segments between crossings at offset gaps \(\Delta_A,\Delta_B\) are strip bridges (within the more exterior chord and the other extension which stays strictly outside its central wall), costing the corresponding \(O((1+\Delta)^{-p})\) given inner ports. Remaining segments are full arms for the respective exterior-crossing representations. For this second-moment estimate the pair \(D,D'\) is fixed. Select the representation with the exterior entry crossing and the one with the exterior exit crossing, making a fixed choice at a tie. These known representation labels, roles and recovered own initial ports fix the two arm families and their thinning factors. The two selections may come from different offset triples. Retain these two factors from \(m_Dm_{D'}\), and drop the unused factors, which are at most one. Each recovered full arm then costs at most \(R^{-p}\), by its prescribed thinned total mass.

Figure 4 shows a mixed ordering when \(A\ne B\). It records order along the path; the preceding localization argument is what makes the marked interval a chord in the intersection triangle. When \(A=B\), the offset order is the same at entry and exit, so one triangle supplies both exterior arms. The decomposition still has two possible gap bridges, now with the same normal width.

The joint second moment bound is \[\sum_\xi w_\xi\big(\sum_D m_D(\xi)\big)^2 \le R^{6+z+o(1)-4p}.\] Here is the offset sum in both cases. If \(A\ne B\), the two used side offsets vary independently and each has \[\sum_{d,d'}(1+|d-d'|)^{-p}=O(R^{2-p}).\] The unused side gives \(O(R^2)\) further pairs. If \(A=B\), the two gaps coincide, and instead \[\sum_{d,d'}(1+|d-d'|)^{-2p}=O(R^{2-2p}),\] while the other two side offsets give \(O(R^4)\) pairs. The latter sum uses \(2p<1\). Thus in either case the offset sum is \(O(R^{6-2p})\). The marked chord and the two normalized exterior arms supply the remaining factor \(R^{z-2p+o(1)}\), proving the second-moment bound.

Apply (50) to obtain distinct bridge mass through \(v\) at least \(R^{z-4K_{\mathrm{loss}}\epsilon-o(1)}\). Sum this for \(\asymp R^2\) vertices \(v\), with only boundedly many oriented strip choices and \(O(R)\) starting ports total; use lattice symmetry to translate to standard strip bridges from a fixed port. This proves the confined length-sum lower of (57), since \(\epsilon\) was arbitrary. The constant \(K_{\mathrm{loss}}\) affects only the exponent. The corridor width was fixed by the layouts and tube widths independently of \(\epsilon\), so letting \(\epsilon\downarrow0\) keeps one fixed corridor.

For the upper, in convex bounded truncations of the strip the positive mass of bridges through any vertex \(v\), allowing all starts, is at most \(C T_v\) by (1). Uniformly \(T_v\le R^{z+o(1)}\) by (46): polygons around an incident face with dyadic diameter \(L\ge R\) cost only \(O(R/L)\) total mass in each such dyad, by Proposition 12 and at most \(O(RL)\) translations per shape (normal translation strip-constrained, lateral shift limited by enclosure). The large loops thus cost bounded partition factor and the others are in a ball of order \(R\) for (43). Exhaust to count all finite bridges. To make the last translation count explicit, let \(\mathcal F_R\) be one set of vertex representatives in the strip modulo translation by \(U\), and let \(A_R(v)\) be the positive mass of bridges through \(v\) with the bottom starting port also summed. Then \[M(R)=\sum_{v\in\mathcal F_R}A_R(v),\qquad |\mathcal F_R|=O(R).\] Indeed translate a bridge with one marked visited vertex so that its starting port is the fixed standard port. Conversely a bridge from that port, together with any one of its vertices, has a unique translate that places the marked vertex in \(\mathcal F_R\). All weights are preserved. The bound \(A_R(v)\le R^{z+o(1)}\) therefore proves the upper bound in (57).

We can now assemble Theorem 1. The lateral tightness proved at the start of this section makes \(B(R)-B_C(R)\) an arbitrarily small fixed fraction of \(B(R)\) when the fixed constant \(C\) is large. Together with the boundary estimate, this gives \(B_C(R)\asymp B(R)\asymp R^{-1/4}\). The two-arm construction gives the confined first-moment lower bound with a corridor constant independent of \(\epsilon\); the preceding upper bound and \(M_C(R)\le M(R)\) give both \(M_C(R)=R^{13/12+o(1)}\) and \(M(R)=R^{13/12+o(1)}\). Dividing either first moment by its corresponding mass yields the mean exponent \(13/12+1/4=4/3\).