Critical weights and the amplitude problem

Nienhuis’s dilute \(O(n)\) calculation identified the critical point and the associated critical-exponent predictions for planar loop models [15]. Duminil-Copin and Smirnov proved that the honeycomb connective constant is \(\sqrt{2+\sqrt2}\), using a parafermionic observable whose local cancellation controls boundary path sums [4]. We use that local cancellation in a finite exploration form. Including the tangent factor in their observable of spin \(5/8\) gives the flux exponent \(1-5/8=3/8\) used here. Its phases will be kept explicit, because changing from an exterior source to an interior slit changes the surviving loop weight.

The local transfer matrices belong to the dilute integrable-model setting. Their exchange and fusion identities permit changes of row arguments and of circumference. Grimm and Pearce [11] give the dilute braid–monoid construction; Zhou and Batchelor [19] analyze periodic spectra and the seam weight in the dilute \(O(n)\) and Izergin–Korepin setting. Glazman and Manolescu prove boundary two-point invariance for columnwise rhombic half-plane tilings with angles in \([\pi/3,2\pi/3]\) and Yang–Baxter walk weights [9]. These results explain the usefulness of deforming local weights while preserving finite identities. In this paper the local matrix identities are proved in the precise gauge used by the boundary states.

The main analytic obstacle is a boundary coefficient. If a finite transfer matrix \(T\) has a simple dominant eigenvalue \(\lambda\), then a matrix element between two fixed boundary states has leading form \[\Lambda T^H\Psi=c\lambda^H+\text{smaller terms}.\] Knowing \(\lambda\) does not determine \(c\): the coefficient depends on both boundary states. Earlier integrable honeycomb adsorption boundaries were studied by Batchelor and Yung [1]; the caps considered here have their own contraction and normalization. Here \(\Psi\) and \(\Lambda\) are vacuum states that suppress closed loops outside the marked region. Their twists differ from the twist of the rows between them. The residue formula and the physical selection argument determine this coefficient in the required normalization.

The positive observables behind the proof

Two polygon sums connect this coefficient to Theorem 1. The first is planar. Fix a vertex \(v\) of the triangular tiling, equivalently a face centre of \(\mathbb H\). Let \(\mathcal Z_v(r)\) be the sum over finite collections of pairwise vertex-disjoint simple honeycomb polygons, each enclosing \(v\) and lying within Euclidean distance \(r\) of \(v\). A polygon \(P\) has weight \(2x^{|P|}\), where \(|P|\) is its number of edges, and the empty collection has weight one. Such collections are necessarily nested.

For the cylinder, orient the tiling with horizontal sides and group its triangles into rows of \(\pi/3\)-rhombi, whose sides are parallel to \(0\) and \(\pi/3\). The row index increases by one under translation by \(e^{i\pi/3}\). For an integer \(N\ge1\), identify horizontal translates by \(N\). Choose a seam along the row direction between neighboring columns, with tips at the cuts of integer heights \(0\) and \(H\ge0\).

Let \(\mathcal G_N(H)\) be the sum over finite collections of pairwise vertex-disjoint simple honeycomb polygons on this cylinder of two kinds: winding polygons separating the tips, and contractible polygons enclosing exactly one tip. Each polygon \(P\) has weight \(2x^{|P|}\), and the empty collection has weight one. Define the sum by exhaustion in both row directions.

For comparison, let \(L_N(H)\) be the winding-only partition function between the height cuts \(0\) and \(H\), with no polygon crossing either cut. Its polygons are contained in the slab, have the same weights, and the empty collection is included. Write \[\lambda_N:=\lim_{H\to\infty}L_N(H)^{1/H}.\] Section 12 proves that this limit exists and identifies it with the positive row growth factor. The following result states the cap amplitude and planar nesting estimates in these conventions.

The factor two between these exponents has a geometric meaning. When the seam tips are far apart, each tip supports a planar nest, while the winding polygons lie between them. The transfer coefficient records the two tip contributions after the extensive winding growth \(\lambda_N^H\) has been removed. Proving this comparison requires height bounds: a cylinder polygon can be too tall, or its planar lift too wide, to belong to a small neighbourhood of one tip. We establish moments of every fixed order before making the projection.

A complete route through the argument

The proof begins with positive path sums. Section 2 proves the local matrix identities, and Section 3 converts them into two estimates: a cylinder boundary deficit of order \(N^{-1/4}\) and winding-polygon mass of order \(N^{-1}\) per highest row. The deficit is the eventual source of the chord normalization \(R^{3/4}\).

Sections 4–8 determine the coefficient selected by the two caps. We first construct their normalized vectors and their polynomial denominator. A scalar residue formula then expresses products of row transfers between the caps, with an explicit normalization for the empty product. The root twists are initially independent. Resolving collisions proves that the scalar trace is holomorphic near a generic common twist, where colliding roots may contribute derivatives as well as ordinary evaluations. Adding and removing fused pairs of sites then extends the identity from distinct site arguments to arbitrary repeated row arguments by meromorphic continuation. This step is needed to compute powers \(T^H\), not just a finite collection of special matrix elements.

Sections 9–11 identify the physical term in that formula. A real counting system has a unique branch in explicit quantile boxes, including the two extreme roots. Polynomial orthogonality identifies the vacuum end of the branch. Positivity of the finite annular transfer and a separate estimate on the two boundary caps select its ordinary residue coefficient at generic parameters. The finite annular gap and cap estimate make the selected coefficient analytic near each point of this real branch, so the identity continues to the homogeneous physical endpoints. The coefficient is a ratio \(W/J\) of a root product and a Jacobian determinant, normalized at zero winding fugacity. Its logarithmic asymptotics give the exponent \(1/6\).

Section 12 obtains arbitrary fixed height moments from higher twist derivatives, projects the cylinder polygons to the plane, and proves Theorem 2. Section 13 completes the positive walk argument. An edge-slit exploration compares the weight of chords visiting an interior contact with the surrounding polygon partition. That contact weight is at most \(R^{1/12+o(1)}\) everywhere and has this order throughout a central region. Summing order \(R^2\) contacts gives the unnormalized length mass \(R^{25/12+o(1)}\). The boundary deficit gives \(Z_R^{\rm chord}\asymp R^{3/4}\), and their ratio is the exponent in Theorem 1.

The angle \(l=\pi/8\) is fixed throughout; integer shifts inside bracketed trigonometric formulas denote multiples of \(l\), whereas geometric row heights are integers in the physical lattice.

The local operator

Set \(l=\pi/8,\ x=(2\cos l)^{-1},\ q=e^{2il},\ p=q^{-1}\). Let \(V\) have basis \(|m\rangle\), \(m=0,+1,-1\), with \(s|m\rangle=m|m\rangle\). The state \(0\) is vacant and the states \(\pm1\) are occupied. All transposes below are taken in these bases, without complex conjugation. Put \(C=\sum_{m=\pm1}p^m|m,-m\rangle\) and \(P|mk\rangle=|km\rangle\), and define the five scalar weights by \[(a,b,h,d,e)(t)= \frac{(\sin2l\sin(3l-t),\,\sin2l\sin t,\,\sin(3l-t)\sin t,\, \sin(3l-t)\sin(2l-t),\,\sin(t-l)\sin t)}{\sin(2l+t)\sin(3l+t)} .\] Define \(R(t):V^{\otimes2}\to V^{\otimes2}\) by the following rules, where \(m,k=\pm1\) and the scalar weights have argument \(t\): \[\begin{split} |00\rangle&\mapsto |00\rangle+b C,\qquad |m0\rangle\mapsto h|m0\rangle+a|0m\rangle,\quad |0m\rangle\mapsto h|0m\rangle+a|m0\rangle,\\ |mk\rangle&\mapsto d|km\rangle+{\bf1}_{m=-k}p^{-m}(b|00\rangle+eC). \end{split}\] The rules conserve \(s_1+s_2\). Write \(\check R=R P\).

Rank reductions and inversion

At the two arguments \(2l\) and \(3l\), the range of \(R\) comes from a smaller space. For \(r=2\), let \(E=E_2:V\to V^{\otimes 2}\) take \(|0\rangle\) to \(|00\rangle+xC\) and \(|m\rangle\) to \(x(|m0\rangle+|0m\rangle)\) for \(m=\pm1\). For \(r=3\), let \(E=E_3:\mathbb C\to V^{\otimes 2}\) take 1 to \(|00\rangle+C\). The rank reduction and its compatible action on a third site are \[R(rl)=EE^t P,\quad R_{23}(t)R_{13}(t+rl)(E\otimes I)=(E\otimes I)S,\quad S= \begin{cases}R(t+l)&r=2,\\I&r=3.\end{cases}\] Here \(S\) acts on the source of \(E\) and the third site; for \(r=3\) it is the identity on that third site. We also have the inverse relation \(R_{12}(t)R_{21}(-t)=I\). All these are meromorphic identities in their arguments.

To verify the rank reductions, substitute \(t=rl\) in the five weights and use \(C^t C=0\). We give the coefficient checks for the compatible action, since its scalar normalization will be used later. Use subscripts \(0,j,*\) for the arguments \(t,t+rl,t+l\). For \(r=2\), expand both sides on the inputs \(|00\rangle,|m0\rangle,|0m\rangle,|mk\rangle\) of \(V\otimes V\). The resulting coefficient equalities are grouped in that order in the following table: \[\begin{array}{ll} b_0+x a_jd_0=x b_*,& b_j a_0+x h_j h_0=x,\quad b_j h_0+x h_j a_0=x b_*;\\ h_j+b_j b_0=h_*,&h_j b_0+b_j e_0=0,\quad a_j a_0+h_0=h_*,\\ x(a_jh_0+a_0)=a_*,&b_j d_0=a_*;\\ h_j h_0+x b_j a_0=h_*,& h_j a_0+x b_j h_0=x a_*,\quad a_j+x e_j b_0=x a_*,\\ e_j d_0=h_*,& a_j b_0+x d_j d_0+x e_j e_0=0;\\ d_j h_0+a_j a_0=d_*,&d_j a_0+a_j h_0=0,\quad h_j d_0=d_*,\\ x(h_j b_0+b_j)=b_*,& b_j b_0+h_j e_0=e_*,\quad e_j a_0=b_*,\quad e_j h_0=e_* . \end{array}\] The four groups occupy \(1,2,2,2\) rows. For example, the first group compares the three output types \(\sum p^m|0m,-m\rangle,\ \sum p^m|m,-m,0\rangle,\ \sum p^m|m0,-m\rangle\) on input \(|00\rangle\); the extra cap terms vanish because \(p^2+p^{-2}=0\). Each scalar equality follows on inserting the fractions and using \(2\sin A\sin B=\cos(A-B)-\cos(A+B)\) and \(8l=\pi\). For \(r=3\), the same substitution leaves the first and third groups, with the explicit \(x=h_*=1,\ a_*=b_*=0\); these are the required identities when site 3 is vacant and occupied, respectively.

The inverse relation reduces in the same way to \[dd^-=1,\quad aa^-+hh^-=1,\quad ah^-+ha^-=0,\quad b+d b^-=0,\quad b b^-+d e^-+e d^-=0,\] where a superscript minus denotes the argument \(-t\). Direct substitution proves all five equalities.

Interchange and row identities

We next derive the interchange identity from the two rank reductions. Transpose the fusion relation and then conjugate by the operation reversing all spins. This combined operation leaves \(R\) unchanged, while spin reversal takes \(E\) to \(PE\). It therefore gives the opposite-product intertwining with \(E^t P\). Combining the two intertwinings with \(R(rl)=EE^tP\) proves \[R_{12}(u-v)R_{13}(u)R_{23}(v)=R_{23}(v)R_{13}(u)R_{12}(u-v)\] when \(u-v=rl\), \(r=2,3\). It also holds at \(u-v=0\), since \(R(0)=P\). Multiplication by the inverse for the \(23\) pair identifies this equation with the \(132\) relation at differences \(u,u-v,-v\). The same three special cases therefore prove it when \(u=0,2l,3l\).

These six specializations determine the identity for every \(u\). Indeed, after clearing denominators, the powers of \(e^{iu}\) in each entry range from \(-4\) to \(4\) and have one parity: changing \(u\) by \(\pi\) conjugates by the sign of occupation at site 1. After removing a monomial factor, an entry is therefore a polynomial of degree at most four in \(e^{2iu}\). For generic \(v\), the six specializations are distinct modulo \(\pi\), so interpolation makes every entry zero. Meromorphic continuation gives the asserted identity for all parameters.

For a list of labelled site parameters \(\mathbf b=(b_1,\ldots,b_N)\) and \(\zeta\in\mathbb C^*\), define the spin row transfer by \[T^\zeta(u;\mathbf b)=\operatorname{Tr}_a\big(\zeta^{s_a} R_{a1}(u-b_1)\cdots R_{aN}(u-b_N)\big).\] For a fixed twist \(\zeta\), these operators commute at all regular row arguments. To see this for \(N>0\), conjugate the trace over two auxiliary spaces by their \(R\), then use interchange at each site and conservation of the auxiliary spins. The same assertion holds for the empty list with the evident convention. Interchange also gives the adjacent exchange relation \[T^\zeta(u;\ldots,A,B,\ldots)\check R_{j,j+1}(A-B) =\check R_{j,j+1}(A-B)T^\zeta(u;\ldots,B,A,\ldots).\] The analogous cyclic operation is as follows. Rotate the tensor factors one place to the left and multiply by \(\zeta^{s_1}\) on the site sent to the end. This map intertwines the row with the cyclically rotated list, by conservation in slot 1 and cyclicity of the auxiliary trace for entries at separate sites. Shifting a site parameter by \(\pi\) conjugates the row by the sign of occupation at that site.

Finally, specialize adjacent entries to \(d_0+rl,d_0\). The row intertwines through \(E_r\) with the shorter list obtained by replacing this pair with \(d_0+l\) for \(r=2\), or by deleting it for \(r=3\). This follows by inverting the fusion identity with site 3 as the auxiliary space and \(t=d_0-u\). The row relations, like the local ones, are meromorphic identities in all parameters.

Finite boundary flux and half-plane summability

Consider a convex finite domain tiled by triangles, and fix an entrance port \(a\). Give a path of total signed turn \(W\) the complex weight \(e^{icW}x^m\). The sum of these weights over paths from \(a\) to all exit ports is \(1\). We recall the local cancellation, including its phases, because the sign changes when the source lies in an interior slit later in the paper.

Introduce one unit of inward flux at \(a\), and stop a partial path whenever it reaches a side midpoint, recording whether it points into a triangle or out of one. At a fresh triangle the two extensions turn by \(\pm\pi/3\). Their total multiplier is \[2x\cos(c\pi/3)=1.\] At a previously visited triangle, pair the two stopped paths obtained by reversing the cycle segment. To check their phases, suppose that the earlier arrival-to-exit turn was \(+\pi/3\). The closing turn of the cycle is then \(-\pi/3\). The prefix from the boundary source arrives from the exterior of that cycle, so the cycle has rotation \(-2\pi\): in the counterclockwise alternative the reflex interior sector would contain the arriving prefix. Including the earlier turn gives a net increment \(-4\pi/3\), which becomes \(+4\pi/3\) on reversal. The pair cancels because \[2\cos(4\pi c/3)=0.\] The reflected case has the same cancellation. Thus all interior stops cancel and the input flux equals the boundary output.

For an exit port \(b\) following \(a\) counterclockwise, let \(\phi_a,\phi_b\) be the lifted counterclockwise boundary tangents. Closing the path along the boundary gives \[W=\phi_b-\phi_a-\pi.\] Convexity puts \(W\) in \([-\pi,\pi]\), so every real output weight is positive. Exhaust the lower half-plane by increasing triangles symmetric about \(a\). The two directions along their horizontal base have phases \(e^{\pm ic\pi}\), both with real part \(\cos(3l)\). Monotone convergence for these base paths, with the other positive exits discarded, gives \[ 2\cos(3l)\sum_{g>0}K_\infty(g)\le1. \tag{1}\] This finite bound is the input for the annular cap construction. The equality and the endpoint tail estimate will follow after the deficit has been evaluated.

Annular diagram rows and the zero-fugacity cap

We first distinguish the two row operators that enter the argument. The spin row \(T^\zeta(u;\mathbf b)\) of Section 2 acts on \(V^{\otimes N}\), with a twist \(\zeta^{s_a}\) in the auxiliary trace. The diagram row defined below acts instead on annular arch patterns. The two descriptions give the same scalars when the open diagram ports are closed with the specified caps and the orientations are summed; we do not identify their state spaces.

A tile has two incoming legs, auxiliary from the right and quantum from below, and the opposite outgoing legs. Its scalar weights are the functions \(a,b,h,d,e\) of Section 2. An empty tile has weight \(1\). A single connection from one input to the other stream’s output has weight \(a\), a single straight connection has weight \(h\), and a cup or cap joining the two inputs or the two outputs has weight \(b\). When all four legs are occupied, the double pairing of \(a\)-type has weight \(d\), and the other double pairing has weight \(e\). Contractible closed loops have weight zero, and simple winding loops have weight \(n\). At the physical argument \(t=l\), \[(a,b,h,d,e)(l)=(x,x^2,x^2,x^2,0).\] These are exactly the weights of dual self-avoiding paths in the two triangles of a split rhombus.

Here is the orientation comparison with the spin matrix. Choose \(\zeta\ne0\) with \(n=\zeta+\zeta^{-1}\) and orient each occupied arc. The spins \(+1,-1\) mean respectively along and against the positive stream direction. Straighten the positive stream tangents at the tile ports to parallel forward directions, with the auxiliary stream to the right of the quantum stream at entry and to its left at exit. A through connection then has net turn zero. A clockwise cup or cap has phase \(p=e^{-i\pi/4}\), and a counterclockwise one has phase \(p^{-1}\), exactly as in the terms involving \(C\) and \(p^{-m}\) in \(R(t)\). These spin phases use one quarter of the turn angle, separately from the boundary flux phase \(e^{icW}\) above. The changes of tangent used in straightening cancel at each join, so a closed curve of signed rotation \(\Theta\) has phase \(e^{i\Theta/4}\). Equivalently, the phase counts signed cups and caps and is unchanged by isotopy fixing the port tangents.

A contractible simple loop has rotation \(\pm2\pi\); its two orientations therefore contribute \(i\) and \(-i\), with sum zero. A simple winding loop has rotation zero in the flat cylinder frame and seam flux \(\pm1\); its orientation sum is \(\zeta+\zeta^{-1}=n\). This remains true for a diagram closed by arches in earlier rows. For each such arch one chooses an actual representative below the row, with the prescribed port tangents and homotopy, and includes that representative’s own turn and seam factors. Representatives above a row are handled in the same way.

We also need the interchange and inversion identities with open connections. For interchange, place streams \(1,2,3\) from right to left at entry, all pointing forward, and in reverse order at exit. For fixed occupied boundary slots in a disk, the oriented tensors of distinct noncrossing matchings are linearly independent. Indeed, after straightening the port tangents, the nonzero arc phase depends only on the oriented connectivity. For a chosen matching, assign inward spins to its left endpoints and outward spins to its right endpoints in a linearized boundary order. The result is a Dyck word. At every gap, a matching compatible with this assignment has at least the Dyck height many open arcs; equality at every gap singles out the chosen matching. Ordering matchings by these height tests gives a triangular test matrix with nonzero diagonal. The spin identities of Section 2 therefore imply the corresponding identities of open diagrams.

Sliding a crossing of two auxiliary streams around the \(N\) sites now proves commutation of periodic diagram rows: insert the crossing and its inverse, use interchange at each site, and cancel them after the trace. Any loops formed during this operation are annular loops with the prescribed weights. The proof applies at generic regular parameters and extends as a meromorphic identity.

Write \(\mathsf T_n(t)\) for the homogeneous diagram row with argument \(t\) and winding weight \(n\). A state specifies, at the bottom boundary circle, which of the \(N\) slots are vacant and how the occupied slots are joined by noncrossing arches below the circle, including their homotopy classes. No strand runs to infinity. A chosen lift of each arch has endpoint span of absolute value less than \(N\), since an arch disjoint from its translates cannot have interlacing lifted endpoints. There are consequently only finitely many states. To act by \(\mathsf T_n(t)\), attach a row above the input arches, evaluate closed loops, and read the output arch pattern. The preceding diagram interchange gives, for the same winding weight, \[\mathsf T_n(u)\mathsf T_n(v)=\mathsf T_n(v)\mathsf T_n(u).\]

Let \(|0\rangle\) be the all-vacant state and let \(\langle0|\) select its component. Set \(T(t):=\mathsf T_0(t)\), retaining this shorthand for the zero-fugacity diagram row. Every nonempty configuration with empty output contains a closed loop, so \[\langle0|T(t)=\langle0|.\] We next prove, for each fixed \(N\), \[ T(l)^H\longrightarrow|\Psi\rangle\langle0| \qquad(H\to\infty),\qquad \langle0|\Psi\rangle=1. \tag{2}\]

The one-winding derivative, fusion, and its polynomial

Keep the cap \(\Psi\) at zero fugacity and define \[ J_N(t):= \left\langle0\left| \left.\partial_n\mathsf T_n(t)\right|_{n=0} \right|\Psi\right\rangle . \tag{3}\] The derivative selects exactly one winding loop closed by the new row: \(\partial_n n|_{0}=1\), while higher powers of \(n\) vanish. The cap below contains ordinary open arches and no closed loops. At \(t=l\) the new row is the unique highest occupied row of the resulting polygon. Thus each actual unoriented quotient polygon is counted once, without a marked vertex or a division by horizontal translations, and \[J_N(l)=P_N.\] The fugacity-\(2\) gas used later instead assigns \(2x^{|\gamma|}\) to each polygon.

The strip operator

Use the strip coordinate \(s=2t-3l\), and write \[\alpha_{\rm s}:=\frac{\pi}{6l}=\frac43,\qquad \beta_{\rm s}:=\alpha_{\rm s}^{-1}=\frac34,\qquad S_a:=\{s\in\mathbb C:|\Re s|<a\}.\] The local straight weight in this coordinate is \[h(t)=\frac{\cos s-\cos3l}{\cos s+\cos l} \quad\text{when }s=2t-3l.\] Define \[j(s):=f_N\left(\frac{\cos s-\cos3l}{\cos s+\cos l}\right), \qquad G(s):=j(s-2l)+j(s+2l)-j(s).\] Then \(j(2t-3l)=J_N(t)\), and \(G(2t-l)\) is the left side of (5). Both \(j\) and \(G\) are even, so \(G\) has zeros of order at least \(N\) at \(s=\pm l\). Also \(j(i\infty)=G(i\infty)=1\).

The following representation depends only on the polynomial degree, not on fusion. This distinction will allow the same operator to be used for higher winding derivatives. For each fixed \(N\) and any polynomial \(f\) of degree at most \(N\), put \[j_f(s)=f\left(\frac{\cos s-\cos3l}{\cos s+\cos l}\right), \qquad G_f(s)=j_f(s-2l)+j_f(s+2l)-j_f(s).\] Define the even map and the convolution \[B(s):=\frac{\cos(\alpha_{\rm s}s)-d_0} {\cos(\alpha_{\rm s}s)+d_0}, \qquad d_0:=\cos(\alpha_{\rm s}l)=\frac{\sqrt3}{2}, \qquad (\mathcal C_g F)(s):=\int_{\mathbb R}g(y)F(s+iy)\,dy,\] \[g(y):=\frac1{2\pi} \left(\frac{2^{-1/2}}{\cosh y-2^{-1/2}}-\frac1{\cosh y}\right).\] The map \(B\) identifies the quotient of \(S_{3l}\) by \(s\mapsto-s\) conformally with the disk. It sends \(\pm l\) to \(0\), and \(B'(l)\ne0\). The kernel \(g\) is real and even. With \(\widehat g(k)=\int g(y)e^{iky}\,dy\), residues over the \(2\pi i\) period give \[1-\widehat g(k) =1-\frac{\sinh(6lk)-\sinh(4lk)}{\sinh(8lk)} =(2\cosh(2lk)-1)\frac{\sinh(6lk)}{\sinh(8lk)}.\] The expressions at \(k=0\) are interpreted by continuity. Furthermore, \[\int_{\mathbb R}g=\frac14,\qquad p_g:=\int_{\mathbb R}|g|<\frac12.\] For the strict inequality, the negative part is zero on \(|y|\le1\) and is bounded by \((1-2^{-1/2})/(2\pi\cosh y)\) on \(|y|>1\); integrating this bound and adding twice the negative mass to \(1/4\) gives \(p_g<1/2\).

With these definitions, the degree bound gives the representation \[ \begin{gathered} G_f(s)=(I-\mathcal C_g)q(B(s))+\lambda \quad(s\in S_{3l}),\\ q(z)=\sum_{m=1}^N q_mz^m, \end{gathered} \tag{7}\] for some \(q\) and constant \(\lambda\).

To prove (7), first Fourier transform \(j_f(iy)-f(1)\), which decays exponentially at both ends. Shifting the contour through one real period \(16l\) crosses the mirrored poles at \(7l,9l\), each of order at most \(N\). Their residues give, on \(S_{7l}\), \[j_f(s)=f(1)+\int_{\mathbb R} \frac{H(k)}{\sinh(8lk)}e^{ks}\,dk, \qquad H(k)=P_e(k)\sinh(lk)+P_o(k)\cosh(lk),\] where \(P_e\) is even, \(P_o\) is odd, and both have degree at most \(N-1\). The displayed symbol therefore gives \[G_f'=(I-\mathcal C_g)F,\qquad F(s):=\int_{\mathbb R}\frac{kH(k)}{\sinh(6lk)}e^{ks}\,dk.\]

Here \(F\) lies in the holomorphic odd Bergman space on \(S_{3l}\), with area norm \(\int_{S_{3l}}|F|^2\,dA\). This space explains the degree bound in (7). Plancherel on the vertical lines \(s=r+iy\), followed by integration over \(-3l<r<3l\), gives the Fourier-side norm multiplier \(\sinh(6lk)/k\), up to the Fourier normalization. The line transforms satisfy \(\partial_r\widehat F=k\widehat F\) on interior lines in the sense of distributions, by the Cauchy–Riemann equation. In this norm the span of the representers for jets of orders \(0,\ldots,N-1\) at \(l\), restricted to odd functions, consists exactly of the preceding numerators \(kH(k)\): even jet orders give the \(\sinh(lk)\) terms and odd orders the \(\cosh(lk)\) terms.

Equivalently, every odd holomorphic function on the strip has an even primitive, which is a function of \(B\). For a primitive \(p(B(s))\) with \(p(z)=\sum_{n\ge1}p_nz^n\), conformal change of variables gives \[\int_{S_{3l}}|(p(B))'|^2\,dA =2\pi\sum_{n\ge1}n|p_n|^2.\] This parametrizes the whole odd Bergman space. Since \(B'(l)\ne0\), the jets of \(F=(p(B))'\) of orders \(0,\ldots,N-1\) at \(l\) are triangularly equivalent to the first \(N\) coefficients of \(p\). Their representer span is therefore exactly the set of \((q(B))'\) with \(\deg q\le N\) and \(q(0)=0\). Thus \(F=(q(B))'\), and integration in \(s\) proves (7).

For later use, express the convolution on these coefficients. Because \(g\) is even, \(\mathcal C_g B^m\) is even and hence a holomorphic function of \(z=B(s)\). With the local inverse branch \(s(0)=l\), set \[\begin{aligned} \Phi_m(z)&:=(\mathcal C_g B^m)(s(z)),& M_{nm}&:=[z^n]\Phi_m(z)\quad(n,m\ge1),\\ M_{0m}&:=\Phi_m(0)=(\mathcal C_g B^m)(l).&& \end{aligned}\] Since \(|B(l+iy)|<1\), the defining convolution also gives \(|M_{0m}|\le p_g\). The matrix \(M\) is real and self-adjoint on \[\ell^2(n):=\left\{v:\sum_{n\ge1}n|v_n|^2<\infty\right\}, \qquad \|M\|_{\ell^2(n)\to\ell^2(n)}\le p_g.\] Indeed \((\mathcal C_g B^m)'=\mathcal C_g(B^m)'\), so differentiation identifies its action with \(\mathcal C_g\) on the odd Bergman space. Imaginary translations are unitary on the full strip Bergman space; their average against the real even kernel is self-adjoint, has norm at most \(\|g\|_1\), and preserves oddness. In particular, \[nM_{nm}=mM_{mn}.\] This Hilbert-space contraction does not assert contraction on \(\ell^1\).

Apply (7) to the first-derivative function \(G\). Its order-\(N\) zero at \(B=0\) gives \[q_n=\sum_{m=1}^N M_{nm}q_m\quad(1\le n<N), \qquad \lambda=\sum_{m=1}^N M_{0m}q_m.\] The compressed Hilbert contraction makes \(I-(M_{nm})_{n,m<N}\) invertible. Define its unique normalized solution by \[Q_N(z):=\sum_{m=1}^N c_mz^m,\qquad c_N=1,\qquad c_n=\sum_{m=1}^N M_{nm}c_m\quad(1\le n<N).\] If \(q_N=0\), the contraction would force \(q=0\), and the zero at \(l\) would force \(G=0\), contrary to \(G(i\infty)=1\). Thus \(q\) is a nonzero multiple of \(Q_N\). For a polynomial with zero constant coefficient define \[\begin{aligned} (\mathcal Tq)(s)&:=(I-\mathcal C_g)q(B(s)) +(\mathcal C_g q(B))(l),\\ \mathcal E(q)&:=(\mathcal Tq)(i\infty) =\frac34q(1)+\sum_mM_{0m}q_m. \end{aligned}\] Writing \(\widetilde G:=\mathcal TQ_N\) and \(E_N:=\mathcal E(Q_N)\), we obtain \[ E_N\ne0,\qquad G(s)=\frac{\widetilde G(s)}{E_N}. \tag{8}\] The nonvanishing here holds for each fixed \(N\), by the existing function \(G\). We next need bounds on this normalizer as \(N\) varies.

Uniform control of the normalizer

The Hilbert contraction has determined \(Q_N\), but a two-sided estimate for \(G\) requires a strict bound on its lower coefficients. The following estimates for \(M\) will give that bound.

Recovering the two positive masses

The two conclusions of Proposition 3 come from different tails of the strip calculation. The inverse shift kernel at \(s=-l\) gives \(P_N\), while the slower tail of \(g\) determines \(f_N'(1)=ND_N\).

The inverse of the shift symbol is \[(2\cosh(2lk)-1)^{-1}=\frac{\cosh(lk)}{\cosh(3lk)}.\] Its Fourier kernel can be written \[\mathcal K(y)= \frac{\operatorname{sech}(\alpha_{\rm s}y+i\pi/6) +\operatorname{sech}(\alpha_{\rm s}y-i\pi/6)}{12l}.\] It is analytic for \(|\Im y|<2l\). Uniformly for \(v\) in any closed inner subinterval of \((-2l,2l)\), its tails are \[\begin{aligned} \mathcal K(u+iv)&\sim c_h e^{-\alpha_{\rm s}u}e^{-i\alpha_{\rm s}v} &&(u\to+\infty),\\ \mathcal K(u+iv)&\sim c_h e^{\alpha_{\rm s}u}e^{i\alpha_{\rm s}v} &&(u\to-\infty). \end{aligned} \qquad c_h=\frac{\sqrt3}{6l}>0.\] The endpoint behavior \(j(s)-1=O_N(e^{-|\Im s|})\) on closed inner substrips of \(S_{7l}\) justifies Fourier inversion of \(j-1\) and \(G-1\). The constant mode has multiplier \(1\), and \(\int\mathcal K=1\). Contour displacement to the imaginary axis, followed by taking real parts, therefore gives \[P_N=j(-l)=\int_{\mathbb R} \operatorname{Re}\mathcal K(y-il)\,G(iy)\,dy.\] Here \(G(iy)\) is real, and evenness permits the opposite sign of the displacement as well. Since \(\alpha_{\rm s}l=\pi/6\), the real part of the displaced kernel is positively comparable to \(e^{-\alpha_{\rm s}|y|}\) for large \(|y|\). On the other hand \[B(iy)=\frac{\cosh(\alpha_{\rm s}y)-d_0} {\cosh(\alpha_{\rm s}y)+d_0} =1-\Theta(e^{-\alpha_{\rm s}|y|})\in(0,1).\] On any bounded \(y\)-interval, (12) is exponentially small in \(N\). On either tail the same equation, followed by the substitution \(v=e^{-\alpha_{\rm s}|y|}\), gives \[\int_{|y|>Y}e^{-\alpha_{\rm s}|y|}B(iy)^N\,dy \asymp \int_0^{e^{-\alpha_{\rm s}Y}}e^{-\Theta(Nv)}\,dv \asymp N^{-1}\] for fixed sufficiently large \(Y\). The positive tails dominate the bounded middle, proving \(P_N\asymp N^{-1}\).

For the deficit, define \[\kappa:=2(\cos l+\cos3l)(2\cos2l-1)>0,\qquad \gamma_g:=\frac{1-2^{-1/2}}{\pi}>0.\] At fixed \(N\), expansion of the rational expression for \(j\) at imaginary infinity gives \[\lim_{y\to\infty}e^y(1-G(iy))=\kappa f_N'(1).\] For \(Q_N\), put \[D_Q(u):=Q_N(1)-Q_N(B(iu)),\qquad I_m:=\int_{\mathbb R}e^u(1-B(iu)^m)\,du.\] The constant terms cancel in the endpoint difference of (8), leaving \[1-G(iy)=\frac1{E_N} \left(D_Q(y)-\int_{\mathbb R}g(t)D_Q(y+t)\,dt\right).\] For fixed \(N\), \(D_Q(u)=O_N(e^{-\alpha_{\rm s}|u|})\). Also \(g(v)\sim-\gamma_g e^{-|v|}\) and \(e^y|g(u-y)|\le Ce^u\). Since \(\alpha_{\rm s}>1\), the direct term tends to zero after multiplication by \(e^y\), and dominated convergence in the convolution gives the exact fixed-\(N\) limit \[\kappa f_N'(1)=\frac{\gamma_g}{E_N} \sum_{m=1}^N c_m I_m.\] No limit in \(N\) has been interchanged with this endpoint limit.

The integrals \(I_m\) are nonnegative and increase with \(m\), because \(0<B(iu)<1\). From \(1-B(iu)\asymp e^{-\alpha_{\rm s}|u|}\) at infinity, \[I_m\asymp m^{\beta_{\rm s}}.\] For example, on \(u\ge0\) the integrand is comparable to \(e^u\min\{1,m e^{-\alpha_{\rm s}u}\}\); splitting at \(u=\beta_{\rm s}\log m\) gives the stated power, and the negative half-line is bounded. Since \(c_N=1\), \[\left(1-\sum_{m<N}|c_m|\right)I_N \le\sum_{m=1}^N c_m I_m \le\left(1+\sum_{m<N}|c_m|\right)I_N.\] The strict margin in (11) and the positive bounds on \(E_N\) show \[f_N'(1)\asymp N^{\beta_{\rm s}}=N^{3/4}, \qquad D_N=\frac{f_N'(1)}N\asymp N^{-1/4}.\]

These comparisons have been proved for sufficiently large \(N\). For every remaining \(N\), the cap sum is finite and the full horizontal polygon has positive weight \(h(l)^N\), so \(P_N>0\). Also a half-plane path from \(0\) to \(N\) exists with positive weight. Together with (1) and \(K_N(g)\le K_\infty(g)\), this gives \[D_N\ge 2\cos(3l)\sum_{g\ge N}K_\infty(g)>0.\] The finitely many remaining circumferences can therefore be absorbed into the comparison constants, completing Proposition 3.

Finally the deficit also saturates the half-plane bound. If \(C_0=2\cos3l\), then \[1-D_N=C_0\sum_{g<N}K_N(g) \le C_0\sum_{g<N}K_\infty(g) \le C_0\sum_{g>0}K_\infty(g)\le1.\] Since \(D_N\to0\), the last sum equals \(1\), and the same comparison gives \[C_0\sum_{g\ge N}K_\infty(g) =1-C_0\sum_{g<N}K_\infty(g) \le D_N=O(N^{-1/4}).\] Thus \[2\cos(3l)\sum_{g>0}K_\infty(g)=1,\qquad \sum_{g\ge N}K_\infty(g)=O(N^{-1/4}),\] as required by the later boundary comparisons.

The disk matching space

We use the spin transfers \(T^\zeta(u;\mathbf b)\) of Section 2, with their labelled and possibly inhomogeneous site parameters, and take \(\zeta=i=p^{-2}\). The annular diagram space of Section 3 retains the homotopy of each arc around the cylinder. The space used here is instead a subspace of \(V^{\otimes N}\) defined by ordinary disk matchings, with no annular winding data.

For each noncrossing partial matching of the ordered sites in a disk, form a tensor by placing \(|0\rangle\) at each unmatched site and inserting \(C\) at the left and right endpoints of each arc, in that order. Let \(D\) be the span of these tensors. Every vector in \(D\) has total spin zero. The matching tensors are linearly independent: first separate them by their occupancy sets, then test a matching against the spin word that assigns \(+\) to each opening endpoint and \(-\) to each closing endpoint. Any matching compatible with that word has at least its word height many unclosed arcs at each cut, with simultaneous equality only for the chosen matching. Ordering by these heights gives a triangular test with nonzero diagonal. We call the coordinate on \(|00\ldots0\rangle\) the empty coefficient.

The adjacent operators \(\check R\) preserve \(D\). Their nonempty contractions use \(C^t\): this kills an arc \(C\), and otherwise joins its two partners with the prescribed phases, since cup-cap straightening uses \(p^m p^{-m}=1\). New arcs are inserted as \(C\). The last arc therefore cannot be capped off with nonzero weight; because the empty-to-empty entry is 1, \(\check R\) also preserves the empty coefficient. Twisted left rotation preserves matching tensors as well, by \(p^m\zeta^m=p^{-m}\).

The full row has the same two properties. To express its twisted trace using disk matchings, append an auxiliary and a spectator in \(|00\rangle+C\) at the right. Pass the auxiliary from last to first through the sites using adjacent \(\check R\)’s, and then contract the first and last exterior legs, namely the auxiliary and spectator, by \(\langle00|-C^t\). This gives exactly \(T^\zeta\), because \(-p^{2m}=\zeta^m\) for \(m=\pm1\). If the occupied exterior legs are joined to each other, their contraction is zero. If they are joined inward to partners, the contraction forms a new arc whose left spin is \(-m\), with phase \(-p^{3m}=p^{-m}\). Thus the output remains in \(D\). Again a last arc cannot disappear with nonzero weight, so only a completely empty input contributes to the empty output, with coefficient 1. Let \(D'\subset D\) be the invariant subspace of vectors with empty coefficient zero.

Construction of the generic fixed vector

We claim that for generic site parameters there is a unique \(\Psi_N(\mathbf b)\in D\) with empty coefficient 1 such that \(T^\zeta(u)\Psi_N=\Psi_N\) for every regular \(u\). Its components are rational in the half variables \(y_j=e^{ib_j}\). Set \(\Psi_0=1\) and \(z_j=y_j^2\). The proof finds finitely many row arguments for which a linear combination of \(T^\zeta-I\) is invertible on \(D'\); the empty normalization then gives the fixed vector by linear equations.

For that purpose, let the sites have very separated scales in \(\mathbb C^*\): successive ratios of their moduli, in increasing order, tend to infinity. Evaluate the row at intermediate scales \(z=e^{2iu}\), one in each gap, with its ratios to the scales on either side tending to their respective extremes. For \(z/z_j\to\infty\), the local weights satisfy \(a,b\to0,\ h\to1,\ d\to q^3,\ e\to q^{-3}\). The limit preserves occupancy on both legs, and the auxiliary spin can only increase or stay fixed: on \(|1,-1\rangle\), the opposite-spin off-diagonal weight is \(q^3+p^{-2}q^{-3}=0\). Its diagonal entries are \(q^{3s_a s_j}\). For \(z/z_j\to0\), the limits of \(d,e\) are exchanged; the auxiliary spin can only decrease or stay fixed, and the diagonal entries are \(q^{-3s_a s_j}\).

We will also need the same tests when the sites form nonempty blocks of comparable scales, with successive blocks very separated. Place the test scales between those blocks. The limiting rows preserve every site’s occupancy and are simultaneously triangular on the full spin-zero sector, in any linear ordering compatible with the componentwise order of the block prefix sums. The blocks are ordered by increasing scale, which need not agree with their order around the disk. For a cut before the test gap, every spin in its prefix can only decrease. For a cut after the gap, every spin in the complementary suffix can only increase, so total-spin conservation again makes the prefix sum nonincreasing. If every block sum is preserved, all changes within a block have the same sign and hence every site spin is unchanged.

At the cut after block \(k\), let \(S_k\) be the spin in the smaller blocks. A contribution on the diagonal has constant auxiliary spin, so the diagonal entry of the limiting row is \(1+i q^{6S_k}-i q^{-6S_k}\). All the tested diagonal entries can equal 1 only when every block spin is even. Consequently, on every occupancy sector containing an occupied singleton block, a generic linear combination of the limiting \(T^\zeta-I\) is invertible, with one choice working for all such sectors. This is the interblock exclusion we shall use below.

For a fully separated list, every occupied sector has an occupied singleton block. Only the empty tensor remains, so the same combination is invertible on \(D'\), also before the limit when the scales are sufficiently separated. For \(N=1\), \(D'=0\) directly. Choose one regular configuration where invertibility holds and keep its test arguments and coefficients fixed. Rationality then makes invertibility generic in the site parameters. The combination maps \(D\) into \(D'\) and is invertible on \(D'\); its kernel in \(D\) is therefore one-dimensional and contains exactly one vector of empty coefficient 1. Since all rows commute with the combination and preserve the empty coefficient, this vector is fixed by every row. Solving the finite linear equations proves rationality in \(\mathbf y\) as well as the claimed uniqueness.

The row identities of Section 2 now transfer to the normalized vector by uniqueness. It is invariant under a common translation of the site parameters, covariant under a period shift at one site by the sign of occupation there, and covariant under twisted cyclic rotation. Its adjacent exchange equation is \[\Psi_N(\ldots,A,B,\ldots)=\check R_{j,j+1}(A-B)\Psi_N(\ldots,B,A,\ldots).\]

Counting the poles

Existence and uniqueness determine the vacuum as a rational vector. To use its entries in the residue formula, we need its least common denominator, including the multiplicities of its factors. We first count poles while varying one site. The endpoint limits will then determine the leading coefficients, and symmetry and total degree will leave the fusion factors to be fixed.

For each component, divide by \(\prod_{j:\,\text{occupied}}y_j\). Sign covariance makes the result rational in \(\mathbf z\). Let \(\mathscr D_N(\mathbf z)\) be the least polynomial common denominator of these rational components on \((\mathbb C^*)^N\), chosen with no coordinate factor. It is unique up to a nonzero constant. We will prove that \[\deg_{z_j}\mathscr D_N=N-1\quad(1\le j\le N),\qquad \deg\mathscr D_N=\frac{N(N-1)}2,\] that \(\mathscr D_N\) is symmetric and homogeneous, and that none of the divisors \(z_B/z_A=q^{\pm2},q^{\pm3},1\) is a factor. These divisor assertions concern generic points of each divisor; their intersections require separate justification when used.

Fix all sites except \(B\) and vary \(z_B\). Choose the fixed parameters generically, outside the loci where specialization creates cancellation, a degree drop, or a new coordinate factor, and then let their scales become very separated within this set. Such choices form a Zariski open set: a one-variable gcd identity keeps the reduced common fractions reduced after specialization. At a pole of the normalized vector, no combination of regular row transfers minus identity can be invertible on \(D'\), since the normalized linear equations would otherwise solve holomorphically there.

Consider any sequence of such poles as the fixed scales become more separated. After passing to a subsequence, the scales separate into singleton blocks except possibly for one pair \(A,B\) with comparable modulus. If no such pair remained, the fully separated tests would give invertibility on \(D'\) and rule out the poles. We may therefore assume that \(z_B/z_A\) has a nonzero finite limit, and pass to further subsequences to fix the square-root ratios. The interblock exclusion rules out all occupancy sectors with an occupied site outside this pair.

It remains to examine the pair. Evaluate additional rows at arguments for which \(e^{iu}/y_A\) is fixed and generic. In the limit these rows preserve the outside occupancies; on the subspace with empty spectators they reduce to the two-site transfer. The outside-occupancy decomposition also splits \(D\) and \(D'\). Hence degeneracy on \(D'\) requires a simultaneous eigenvalue 1 on the two-site \(D'\), which is the line spanned by \(C\). This follows either by commutation or by taking a generic combination of all the tests.

Localization and existence of the simple poles

Call the earlier site of the pair in disk order \(F\), and the other \(G\). In the two-site problem, evaluation at the first site \(u=b_F\) gives \(R_{FG}(b_F-b_G)\zeta^{s_F}\). On the disk matching space this acts by \[C\mapsto d(b_F-b_G)C,\qquad |00\rangle\mapsto |00\rangle+b(b_F-b_G) C.\] At a regular first-site evaluation, subtracting the numerator and denominator of \(d(t)\) shows that \(d(t)=1\) requires \(\sin(2t)=0\). The case \(t=0\bmod\pi\) is excluded by evaluating instead at \(u=b_F+l\). Up to the sign gauges, the eigenvalue on \(C\) is then \(2x^4\ne1\): at \(l\) we have \(e=0,\ h=d=x^2\), and for outgoing pair spins \(m,-m\), the vacant traced auxiliary contributes \(p^m h^2\), while the occupied ones contribute \(\zeta^{-m}p^{-m}d^2=p^m d^2\).

This test also excludes a limiting pair for which the first-site evaluation itself has a denominator pole. Multiply the two-site operator by its two scalar sine denominators. At \(u=b_F\), its restriction to \(D'\) minus the cleared identity is nonzero at any such pole, because the numerator of \(d\) is nonzero there. It is consequently nonzero at some nearby regular row argument for that limiting pair. The only remaining possibility is \(t=\pi/2\bmod\pi\), or \(z_B/z_A\to-1\).

We now count the poles near each possible limit \(-1\). Take the interblock tests together with the first-site test \(u=b_F\), allowing that argument to move with the varying site. On a small disk in \(z_B/z_A\) about \(-1\), the transfers are analytic on each square-root sheet for sufficiently large separations and converge uniformly to the limiting operators. On \(D'\), those limits preserve the outside occupancies. Choose a generic linear combination of their differences from identity. On sectors with occupied spectators, the interblock exclusion makes it invertible at the center; the pair test can be given a sufficiently small nonzero coefficient to preserve this. On the remaining one-dimensional sector, the interblock differences vanish and the pair test has the simple zero of \(d-1\). The determinant of the combination on \(D'\) therefore has one simple zero at the center in the limit. The argument principle and the normalized linear equations show that, for sufficiently separated scales, the vector has at most one pole in the disk, and that pole is simple.

There is in fact a pole in each disk. On the whole spectator-empty pair sector in \(D\), the interblock transfers are the identity: both pair spins lie on the same side of a test, so neither can change, and their sum is zero. On a sufficiently small circle about \(-1\), the normalized vector therefore converges to the unique normalized two-site solution. Its coefficient of \(C\) is \(b/(1-d)\), which has nonzero residue. Contour integration forces a pole of the vector inside the circle. The two square-root signs give equivalent gauges. Passing to subsequences makes the argument apply along any sufficiently separated sequence, since the limiting tests do not depend on the extreme ratios.

There is thus exactly one simple pole near each antipodal value \(z_B=-z_A\), for the \(N-1\) fixed sites \(A\), and none away from those disks. Generic specialization gives \(\deg_{z_j}\mathscr D_N=N-1\) for every \(j\). The same tests show that \(\mathscr D_N\) has none of the divisors \(z_B/z_A=q^{\pm2},q^{\pm3},1\): make this pair comparable and the other sites separated, and use the pair tests away from \(-1\). They also show that a combination is invertible on \(D'\) at generic points of each such divisor. For \(N=1\), the degree and these assertions hold directly. The absence of the equal-site divisor is only a generic statement on that divisor; it does not by itself establish regularity when all sites coincide. Section 10 proves that regularity for the sufficiently large sizes used below, from the projected annular row and its spectral gap.

Endpoint limits, symmetry, and degree

The pole count controls finite nonzero site values. We next let one site tend to zero or infinity while the others remain generic and fixed. In either limit, \(\Psi_N\) is bounded and converges to the embedded \(\Psi_{N-1}\) with that site vacant.

To prove this, evaluate one row at an intermediate scale separating the single departing site. Its triangular limit preserves occupancy and excludes an occupied singleton. Other fixed generic row arguments also preserve that site’s occupancy in the limit and reduce on its vacant block to the shorter transfers. Together they give an invertible combination on \(D'\). On the vacant-site spin-zero block, the intermediate-scale test is the identity: monotonicity prevents any other site from changing, and the relevant prefix spins vanish. Thus the normalized solution of the limiting equations is precisely the embedded shorter vacuum. Invertibility gives boundedness and convergence to that solution. Throughout these arguments, the rows are evaluated at regular arguments before and in the indicated limits, and degeneracy is tested by finite linear equations.

Exchange and inversion now imply that \(\mathscr D_N\) is symmetric. Away from the exceptional ratio divisors, the exchange matrix is regular and invertible, so it preserves all polar divisors and their multiplicities. The exceptional divisors themselves are absent by the pole tests. Primitivity therefore makes a permutation act on \(\mathscr D_N\) by a scalar; an alternating sign is impossible because the equal-site divisor is absent. A common translation of the site parameters gives a common dilation of \(\mathbf z\). Translation invariance of the vector and primitivity similarly make \(\mathscr D_N\) homogeneous.

Its total degree is \(M_N=N(N-1)/2\). Choose rationally independent real slopes \(w_j\) in any increasing order and let \(z_j=\exp(Lw_j)\), \(L\to\infty\). Nonzero exceptional polynomials arising in fixed-site specializations stay nonzero for large \(L\). The root count and locations then give \(z_j\partial_{z_j}\mathscr D_N/\mathscr D_N\) tending to the number of smaller sites. On the other hand, rational independence makes one monomial of \(\mathscr D_N\) uniquely dominant along this path, and these limits are its exponents. Their sum is \(0+1+\cdots+(N-1)=M_N\), which is the total degree by homogeneity.

Finally, regard \(\mathscr D_N\) as a polynomial in one site and pass to the endpoint limit in the divided components that are vacant at that site. Its leading coefficient times each limiting reduced component is a Laurent polynomial. Thus the leading coefficient clears every reduced denominator, and minimality of \(\mathscr D_{N-1}\), with no coordinate factors, makes it divisible as a polynomial by \(\mathscr D_{N-1}\). The total degree makes the quotient a constant. Normalize \(\mathscr D_0=\mathscr D_1=1\), and recursively choose this leading coefficient to be \(\mathscr D_{N-1}\). Symmetry makes the choice independent of the site.

Fusion normalization

With this normalization, define \[Z_N=\mathscr D_{N}(\mathbf z)\prod_j y_j^{-(N-1)},\qquad \mathcal U_N=Z_N\Psi_N .\] Both are Laurent polynomials in \(\mathbf y\). For adjacent slots \(d_0+rl,d_0\), \(r=2,3\), let \({\rm red}\) denote the shorter list in which the pair is replaced by \(d_0+l\) for \(r=2\), or deleted for \(r=3\). We claim the recurrences \[ \Psi_N|_{\rm pair}=E_r\Psi_{\rm red},\quad Z_N|_{\rm pair}=2\cos(rl)\prod_{j\ {\rm other}}\prod_{h'=1}^{r-1} 2\cos(b_j-d_0-h'l)\ Z_{\rm red}. \tag{13}\] Here \(E_r\) is inserted in the indicated positions. The first formula follows at generic points of the pair divisor: \(E_r\) maps matching tensors into \(D\), preserves the empty normalization, and intertwines the transfers, so generic uniqueness applies there.

To prove the second formula, first locate the extra zeros on a pair divisor. All specializations of Laurent polynomials in this argument take place on the torus, with a chosen square-root sheet for the half variables. An ordered adjacent triple \(c+6l,c+3l,c\) forces \(\mathcal U_N=0\): polynomial continuation from the two pair divisors puts it in both \(E_3\) images, and these images have zero intersection because the two-site vector has matrix rank three. This wheel zero persists in cyclic orders with generic spectators inserted, by moving those spectators away through regular exchanges and then rotating. It also persists modulo \(\pi\) by the sign gauges.

On the specified adjacent pair divisor, the wheel zero therefore occurs at every spectator value \(b_j=d_0+h'l+4l\bmod\pi\). For \(r=3\), these values complete either end of the ordered wheel. For \(r=2\), the cyclic triple is \(d_0+2l,d_0,d_0+5l\), which modulo the period rotates to \(d_0+8l,d_0+5l,d_0+2l\). Since \(E_r\) is injective, the first fusion formula and polynomiality of \(\mathcal U_N\) show that \(\mathscr D_N|_{\rm pair}\Psi_{\rm red}\) is regular on the torus and vanishes on each of these extra divisors. Thus \(\mathscr D_N|_{\rm pair}\) is divisible as a polynomial by \[\mathscr D_{\rm red}\prod_{j\ {\rm other}}\prod_{h'=1}^{r-1}(z_j+q^{h'}z_{d_0}),\qquad z_{d_0}=e^{2id_0}.\] This divisibility includes multiplicities even if one of the extra factors coincides with a divisor of \(\mathscr D_{\rm red}\). Indeed, at any divisor the minimum component order of \(\Psi_{\rm red}\) is minus the order of \(\mathscr D_{\rm red}\), by minimality of the denominator and the empty normalization. Adjoining a dummy variable or taking square roots on the torus does not change that assertion. Regularity supplies the order from \(\mathscr D_{\rm red}\), and the wheel zero supplies the additional order.

The quotient after division has total degree 1 and no spectator degree, so it is proportional to \(z_{d_0}\). After centering, this gives the displayed recurrence up to a constant. Compare the highest terms in each spectator in succession. The leading coefficient of \(y_j^{N-1}\) in \(Z_N\) is \(Z_{N-1}\prod_{k\ne j}y_k^{-1}\), and the removed pair product \(e^{i(2d_0+rl)}\) equals the product of the half variables contributed by the replacement site, when present, and the \(h'\)-factors. The comparison reduces the constant to the two-site case, where symmetry and normalization give \(Z_2=2\cos(b_1-b_2)\). This proves the second recurrence in (13).

An explicit component

The fusion recurrences also determine the component with two consecutive blocks of opposite spins. This component will fix the normalization of the cap moments in Section 7.

For \(N=2m\) and \(\epsilon=\pm1\), write \(\epsilon^m(-\epsilon)^m\) for the spin word consisting of \(m\) copies of \(\epsilon\) followed by \(m\) copies of \(-\epsilon\). Its component is \[ (\mathcal U_N)_{\epsilon^m(-\epsilon)^m} =(p^\epsilon\,2\cos 3l)^m\! \prod_{\substack{i<j\\\text{same spin}}} H(b_i-b_j),\qquad H(t)=(2\sin(2l-t))(2\sin(3l-t)). \tag{14}\] To prove the formula, use the scalar exchange weight \(d\) within each block, interchanging generic intermediate sites of that block. It gives one factor \(H\) for each same-spin pair, so all the indicated factors divide the component on the half-variable torus. At either endpoint, an occupied component of \(\Psi_N\) vanishes; sign covariance gives at least one half-variable order of vanishing. After multiplication by \(Z_N\), the highest and lowest degrees of this component of \(\mathcal U_N\) therefore lie between \(-(N-2)\) and \(N-2\) in each variable. The quotient by the displayed product is constant. Fuse the middle pair at \(d_0+3l,d_0\). The two cosine factors for each spectator are precisely the removed \(H\), so (13) determines the constant recursively. Twisted rotation gives the other cyclic versions.

The marked forms and their collision orders

A finite-jet representation of the specialized trace

The collision limit need not be a sum of ordinary evaluations. The next corollary records the additional derivatives that can remain, and the relations satisfied at every point that contributes. It is used at generic \(K\) satisfying the preceding hypotheses. Section 10 reaches the physical values \(K=i\) and \(K=1\) by analytic continuation of the physical branch and its coefficient, after using this corollary at generic nearby twists. Write \(\mathbf p=(p_1,\ldots,p_N)\), where \(p_i=e^{2it_i}\), and let \[\mathcal A=\mathbb C[p_1^{\pm1},\ldots,p_N^{\pm1}]\] be the Laurent polynomials regular on the entire root torus.

Resolving the finite spin words

For an ordered block \(I=(i_1,\ldots,i_r)\), define its diagonal auxiliary entries by \[A^{I}_{\pm}(w)=\langle\pm|_a R_{a i_1}(w-b_{i_1})\cdots R_{a i_r}(w-b_{i_r})|\pm\rangle_a.\] Thus \(A_+(w)\) below means \(A^X_+(w)\) and \(A_-(w)\) means \(A^Y_-(w)\). We resolve the cap contraction by inserting the commuting spectral projectors of these auxiliary diagonal entries in the partial ordered row monodromies. Entries with a fixed extremal auxiliary spin commute at different arguments (fully polarized Yang–Baxter component, both as row and column). Within each block they conserve total spin and are triangular (rightmost changed label increases), with labels \(\sigma=s\) on \(X\), \(\sigma=-s\) on \(Y\); diagonal signatures are products over its sites of \(d,h,e\) at argument \(w-b\) for \(\sigma=+,0,-\), respectively. These signatures separate simple eigenlines by numerator zeros in each site’s shift orbit on using a word-independent common denominator. The term of a joint word vanishes or both block label sums equal some \(r\), with multiplier \(\eta^r\), by conservation. Empty blocks have the obvious scalar conventions. It remains to evaluate \(\Lambda\Pi\Psi\) for each joint projector \(\Pi\).

The next reduction removes zero labels while preserving this contraction. Split each zero site into the ordered pair \(b+1,b-1\), replacing its label by \((+,-)\). The monodromy fusion identity intertwines the block entries through the spin injection \(E_2\), and the signature matches there because \(d(t-l)e(t+l)=h(t)\). The signature remains simple on the fusion locus: in the common pair denominator representation the first numerator choices have zeros (relative to center) \((4,3),(4,1),(2,1)\), the second \((2,1),(2,-1),(0,-1)\) modulo 8. Separation at a generic test argument per block therefore makes the projector regular there. Applying vacuum fusion on both sides proves that \(\Lambda\Pi\Psi\) is unchanged. We can consequently compute the all-nonzero word on an enlarged generic list, performing all exchanges there before restricting the split pairs.

For an all-nonzero word, the extremal orders identify the selected eigenlines. Within each block reorder rapidities to the minimum word (+ then -), or maximum word (- then +), keeping internal same-label orders fixed. Adjacent exchanges transport the projectors via Yang–Baxter. For fixed sum these words are extremal even allowing zeros, so the left eigenline in minimum order is the pure minimum covector, and the right eigenline in maximum order the pure maximum ket (simultaneously for both blocks). If \(U\) is the product of adjacent swaps taking maximum list to minimum, this gives the contraction \[\frac{\Psi(\min)_{\min}\ \Lambda(\max)_{\max}} {\langle\min|U|\max\rangle},\qquad \langle\min|U|\max\rangle=\prod_{\text{within block }(+,-)} e(b_+-b_-).\] Indeed pass each + from first to last in their relative order across the minuses; to arrive still + it could not have donated any label spin, since each encountered minus is initially minimal; induction leaves the minuses unchanged. All these operations precede restriction to fused values.

Matching spin words with frozen slots

The projector calculation has reduced the matrix element to explicit products. We now assign a frozen-slot term to each word and check its phase and denominator. Recall that the label is \(\sigma=s\) on \(X\) and \(\sigma=-s\) on \(Y\), where \(s\) is the actual eigenword spin. Encode an original site’s spin by the slot length \(j=1+s\): its slots are \(x-a\) on \(X\) or \(y+a\) on \(Y\), for \(a=1,\ldots,j\). Since both block label sums are \(r\), the numbers of occupied slots are \(k+r\) on the left and \(m-r\) on the right, hence \(N\) in total.

To calculate the word’s contribution, let its enlarged plus/minus \(\sigma\)-label counts on \(X,Y\) be \((A,B),(C,D_0)\); in particular a plus label on \(Y\) is an actual minus spin. Conservation gives \(M=A+D_0=B+C\). The minimum ket has actual spins \(+^A-^M+^{D_0}\), and the reversed negated maximum word has actual spins \(+^C-^M+^B\). Rotate the first plus segment to the end in each and use the two-block component formula (14); the phases are \(p^{2A-M+2C-M}=p^{2r}\), converting \(\eta^r\) to \(K^{-r}\). Besides \(K^{-r}\), the result has \(C_0=2\cos 3l\) per enlarged site divided collectively by \(Z_{\rm enlarged}^2\). Within each block equal labels contribute \(H(t)H(-t)\), \(t\) the difference; opposite labels contribute \(1/e(b_+-b_-)\). Across blocks, equal actual spins \(n\) give \(H(-n(x-y))^2\); unequal spins give 1.

Applying the fusion recurrence to \(Z_{\rm enlarged}^2/Z_N^2\) reorganizes this contribution as \[\frac{K^{-r} (C_0\alpha)^N}{Z_N^2}\, \prod_{\{b,b'\}\subset\mathbf b}H(b-b')H(b'-b)\prod_{x,y}d(x-y) \ (-1)^{km+m}\operatorname{Res}_{\mathcal L} W\prod_{i,x}S(t_i-x)^{-1}\prod_{i\ {\rm left}}D_i ,\] where \(\alpha=4\cos l\) and \(\mathcal L\) uses any one labeling of the occupied slots.

Here are details of the elementary reorganization. Besides the base pair products displayed, each occupied slot \(p'\) with another original site \(b\) has a factor \(V_{\rm pair}(p'-b)\) within a block and \([p'-b\mp1]^{-2}\) across (upper sign left slot). Slots at distinct sites interact by \(G(u)\) within block and \(([u][u-2]/[u-1])^2\) across, \(u=\) left minus right there. One verification uses offset/spin lists \(B_0=((0,-)), B_1=((1,+),(-1,-)), B_2=((0,+))\) (reflect offsets on Y). Explicitly put \(F_{ij}(t)=[t+3]^2[t+4]^2[t+5]^2\) for both indices 1, \([t+4]^2\) for just one, 1 otherwise (squared fusion factors). For \(z=t+a-b\) in the first identity, \(z=t+a+b\) in the second, \[\begin{gathered} \prod_{(a,n)\in B_i,(b,n')\in B_j} \begin{cases}H(z)H(-z)&n=n'\\H(-nz)/([nz-1][nz])&n\ne n'\end{cases}\\ =F_{ij} H(t)H(-t)\prod_{c=1}^i V_{\rm pair}(t-c)\prod_{h=1}^j V_{\rm pair}(-t-h)\prod_{c,h}G(t-c+h),\\[4pt] \prod_{(a,n)\in B_i,(b,n)\in B_j} H(-nz)^2 =F_{ij} H(t)^2\prod_c[t-c-1]^{-2}\prod_h[t-h-1]^{-2}\\ {}\times\prod_{c,h}\left(\frac{[t-c-h][t-c-h-2]}{[t-c-h-1]}\right)^2 . \end{gathered}\] Both follow by inserting the three displayed lists, canceling with \([-v]=-[v], [v+8]=-[v]\). Use reflection of all rapidities for within Y. Single-site constants are \(C_0, C_0^2/((2\cos 2l)^2 e(2l))=1,C_0\); equivalently \(C_0\alpha^j\) times the successive left-slot residues of \(\prod V_{\rm pair}\prod G\) at that site alone (residues \(1,1/(2[2l]),[l]^2/(4[2l]^2)\) in ordinary angle coordinates), or their reflected right-slot residues multiplied by \((-1)^j\). Remaining factors match the displayed integrand since across blocks \(V_{\rm pair}S^{\pm1}\) and \(G R_{\rm pair}\) in oriented arguments give the stated cross corrections each with a minus. The total sign including reflection is \((-1)^{(k+r)m+(m-r)k+(m-r)(k+r)+m-r}=(-1)^{km+m}\).

Finally \(K^{-r}=K^k\prod_{i\ {\rm left}} K^{-1}\) and \(\prod_x L(x)/d_{\rm full}(x)=(-1)^{Nk}K^k\prod_{xy}d(x-y)\prod_{i,x}S(t_i-x)^{-1}\). Each unlabelled frozen assignment has exactly \(N!\) labelings. At diagonal twists the residue lemma therefore turns the sum of the displayed word contributions into (39): its sign agrees because \(N+Nk\equiv km+m\pmod2\), and its prefactor is \[\frac{(C_0\alpha)^N}{N!\,Z_N^2} \prod_{\{b,b'\}\subset\mathbf b}H(b-b')H(b'-b), \qquad C_0\alpha=(2\cos3l)(4\cos l)=2\sqrt2.\] For \(X=\varnothing\), the cap contraction is one, proving (40). In the independent-twist residue lemma this same empty subset has only right slots and no left twist multipliers. Hence \(\tau_N(1)\) is already independent of every \(K_i\) and equals its generic diagonal value. Substitution of the normalization now proves (41).

A scalar two-site reduction

Let \(\mathbf b_{\rm red}\) be the ordered list of \(n\ge0\) sites obtained by deleting two neighboring sites \(x,y\), in that order, from a larger list \(\mathbf b_\epsilon\), where \(x=y+3+\epsilon\). All other sites are generic. Use subscripts \(\epsilon\) and \({\rm red}\) on \(\tau,P,d_{\rm full},W,D_i,J\) to distinguish the two lists; a subscript \(0\) on a regular coefficient denotes its specialization at \(\epsilon=0\).

Removing the auxiliary pairs

The positive roots at the vacuum endpoint

Set \(l=\pi/8\) and \(q=e^{2il}\). For this subsection take additive sites \(b_a=i\eta_a\), so their multiplicative coordinates \(\xi_a=e^{2ib_a}=e^{-2\eta_a}\) are positive. Repeated sites are allowed. For \(z=e^{2iw}\), put \[B(z)=\prod_{a=1}^N\prod_{j=3,4,5}(z-q^j\xi_a).\] A polynomial is said to have the missing modes if its coefficients of \(z^{4m+1}\) all vanish.

At homogeneous sites \(\xi_a=1\), these phase labels form a contiguous ordered solution at \(\delta=1/4\). The containment estimate in the proof of Proposition 13 puts every such solution in its quantile rectangle, where uniqueness identifies it with that proposition’s solution. For \(N\ge N_0\), the implicit function theorem then continues the same roots along \[K=e^{i\theta},\qquad \delta=\theta/(2\pi),\qquad 0\le\theta\le\pi/2.\] Indeed \(T(y_j)=j-\tfrac12+\delta\) is equivalent to \(D_j=-K\). The roots remain distinct and positive in multiplicative coordinates. The branch is analytic locally in \(K\) and under small complex changes of the additive sites.

The isolated ordinary term at generic parameters

Return to additive site notation \(\mathbf b\). On the selected local root branch write \[a(\mathbf b,K)=\frac{W}{J},\qquad J=\det\bigl(\partial_{t_j}\log D_i\bigr),\] and let \(F(u;\mathbf b,K)\) be its normalized TQ function. On a solution it can be evaluated as \(P(u)/d_{\rm full}(u)\), using the Laurent polynomial part from Section 6; this expression has already cancelled the apparent poles at shifted roots.

The selected quotient \(a=W/J\) also extends across an unshifted root–site coincidence on the positive slice. Such a point was excluded from the deliberately small regular configuration locus, but \(S(0)=-1\), \(V(0)=1/[l]^2\), and the marked-pole site numerators retain nonzero \([l][2l]\) factors there. With distinct roots and \(J\ne0\), the equations and \(W\) are holomorphic, and the implicit function theorem gives the same local inverse branch as the limits of generic independent-twist points. Along the whole homogeneous real branch \(W\ne0\) and \(J\ne0\), so \(a\) is analytic and nonzero locally at every point.

The finite-jet representation in Lemma 15 is used only at generic common twists where the toric ends have been excluded. The endpoint identity below will instead concern the analytic continuation of this selected quotient and of the independently defined cap coefficient.

The fixed-\(N\) annular gap and the cap ends

Let \(\mathsf M_N(\mathbf b,u,K)\) be the finite annular diagram row of Section 3, with local arguments \(u-b_a\), contractible loop weight zero, and winding weight \(n=K+K^{-1}\). Let \(\mathsf A_N\) be its principal submatrix on nonempty annular states. At homogeneous sites and \(K=e^{i\theta}\), its weights are nonnegative for \(l\le u<2l\) and \(0\le\theta\le\pi/2\).

We describe precisely the finite sums at the ends of \(G_h\). Expand the caps in their finite disk-matching bases and the rows in tile diagrams, summing the orientations of each connected component. A horizontal cut is empty when all its quantum ports are vacant. Let \(E^-_a\) be the sum from the bottom cap through \(a\) rows to an empty cut, with every earlier cut nonempty. Let \(E^+_b\) be the sum from an empty cut through \(b\) rows to the top cap, with every later cut nonempty. Set \(E^-_0=E^+_0=1\), the empty coefficients of the normalized caps. Let \(C_h\) be the sum of diagrams in \(G_h\) with every cut nonempty, including the two end cuts. These are finite sums whenever the cap entries are regular.

Put \(Z_m^{00}=\langle0|\mathsf M_N^m|0\rangle\). At an empty cut the orientation sums split, and the portion between two empty cuts has exactly the ordinary \(0,n\) loop weights. Choosing the first and last empty cuts therefore gives the exact identity \[ G_h=C_h+\sum_{\substack{a,b\ge0\\a+b\le h}} E^+_b\,Z_{h-a-b}^{00}\,E^-_a . \tag{61}\] The case \(a+b=h\) has a single empty cut and \(Z_0^{00}=1\). At \(h=0\), null-loop contraction gives \(C_0=0\), consistently with \(\Lambda\Psi=1\).

Continuation to the homogeneous physical parameters

The generic trace term and the cap estimate now identify the same analytic coefficient. The following statement retains the distinct bulk and cap twists.

The next section estimates this quotient uniformly as \(N\to\infty\). The annular gaps and analytic neighborhoods used here were established separately for each fixed \(N\); they supply the exact coefficient identity and make no uniform claim about those gaps.

Bulk limits and the twist derivative

Write \[\boldsymbol h=H^{-1}\boldsymbol1=\partial_\delta(x_i)_i, \qquad \kappa_i=\min(i,N+1-i).\] The equality follows by differentiating the counting equations, and Proposition 13 bounds \(h_i\) uniformly. We first establish the following meaning of a bulk limit: for every sequence \(N\to\infty\), \(|\delta|\le1/4\), and \(\kappa_i\to\infty\), \[ \int_{\mathbb R}\varphi(z)\bigl(E(y_i+z)+s\bigr)\,dz\longrightarrow0 \quad(\varphi\in C_c^\infty(\mathbb R)),\qquad h_i\longrightarrow\frac43=\frac{ds}{d\delta}. \tag{66}\] The assertion allows \(\delta\) and the row \(i\) to vary with \(N\).

Here is the quantization argument for the first limit. On every fixed rapidity window about a bulk \(y_i\), \[f(y):=T'(y)=N\rho(y)+(r'*E)(y)\asymp a_i\longrightarrow\infty, \qquad |f^{(m)}(y)|\le C_m a_i\] for fixed \(m\). The correction \(r'*E=r'*(E+s)\) is bounded, and the assertions for \(N\rho\) follow from the density bounds in Section 9. Thus \(T\) is increasing there for large \(N\). The quantile boxes put atoms on both flanks of a slightly smaller window. Let \(y(q)\) be the inverse coordinate for \(q=T(y)-\delta\) on the larger window. The atoms occur at the exact consecutive values \(j-\tfrac12\). On a complete unit cell between successive atoms, \[\#\{j:y_j\le y\}-q=\lfloor q+\tfrac12\rfloor-q\] has integral zero in \(q\). For a smooth test function supported in the smaller window, the derivative of its transformed weight is \[\frac{d}{dq}\left(\frac{\varphi(y(q)-y_i)}{f(y(q))}\right) =O(a_i^{-2}).\] Summing the zero-mean cancellation over \(O(a_i)\) complete cells and bounding the two end cells by \(O(a_i^{-1})\) proves that \(\#\{j:y_j\le y\}-(T(y)-\delta)\) tends weakly to zero on the window.

Take a weak-star subsequential limit of the bounded translated discrepancies and a limit \(\delta_*\) of the twists. Since \[\#\{j:y_j\le y\}-(T(y)-\delta)=E-r*E+\delta,\] the preceding cancellation and the integrable tails of \(r\) show that the limit \(E_*\) on the whole line satisfies \[E_*=r*E_*-\delta_*.\] The bounded solution is uniquely the constant \(-s_*=-4\delta_*/3\): subtracting that constant leaves \(U=r*U\), and \(\|r\|_1<1\) forces \(U=0\). This proves the first assertion of (66) for every bulk sequence. Kernel decay then upgrades it to \[ (r^{(m)}*E)(y_i+z)\longrightarrow0 \quad\hbox{locally uniformly in }z,\qquad m\ge1. \tag{67}\] One first restricts the convolution to a compact window and uses weak convergence, then removes the tails using the bound on \(E+s\); the same argument for the next derivative gives local uniformity.

For the limit of \(h_i\), write \(d_j=f_j/a_j\). The linearized equations are \[d_j h_j=1+\sum_k r(y_j-y_k)\frac{h_k}{a_k}.\] On a bulk window \(d_j\to1\) uniformly. The sums on the right, viewed as functions of the rapidity in place of \(y_j\), have uniformly bounded derivatives and uniformly negligible tails by (51). They therefore have locally uniform subsequential limits. At the atoms, the linearized equation and \(d_j\to1\) make \(h_j\) converge to one plus that continuous limit. The unit quantile cells form a rapidity grid of vanishing mesh in the window, as in the bulk Jacobian estimate. Passing to the limit in the sums gives a bounded continuous function satisfying \[h_*=1+r*h_*.\] Its only bounded solution is the constant \((1-\int r)^{-1}=4/3\), again by \(\|r\|_1<1\). This proves the second assertion of (66).

Set \[\Phi_N(\delta)=\log|W|-\sum_i\log a_i.\] Since \(\partial_\delta y_i=h_i/a_i\), differentiation and (65) give \[ \begin{aligned} \Phi_N'(\delta)&=\sum_i\frac{h_i}{a_i}B_i,\\ B_i&=\sum_{j\ne i}g'(y_i-y_j) -N\,\operatorname{pv}\!\int_{\mathbb R}g'(y_i-y)\rho(y)\,dy -\frac{\rho_i'}{\rho_i}. \end{aligned} \tag{68}\] We next prove that \(B_i\) is uniformly bounded and that, along every bulk sequence, \[ B_i=-4s+o(1). \tag{69}\]

Choose an even smooth compactly supported function \(\chi\) equal to one near zero. The singularity of \(g'\) is \(2/y\), so \[g'(y)=b(y)+k(y),\qquad k(y)=\frac{2\chi(y)}y,\] where \(b\) is smooth and odd. The explicit formulas for \(L_a\) give \(g'(y)=\pm2+O(e^{-2|y|})\) and \(g''(y)=O(e^{-2|y|})\) at the two ends. Hence \(b'\) is integrable and \(\int b'=4\). The value \(b(0)=0\) lets us include the self atom in its sum. Integration by parts against the discrepancy gives its entire contribution to the difference in (68): \[\sum_j b(y_i-y_j)-N\int b(y_i-y)\rho(y)\,dy =\int b'(y_i-y)E(y)\,dy.\] It is uniformly bounded; by (66) and the integrable tails of \(b'\), it tends to \(-s\int b'=-4s\) in the bulk.

For a fixed edge layer \(\kappa_i\le M\), the singular contribution is also uniformly bounded. On the fixed support window of \(\chi(y_i-\cdot)\), the density \(N\rho\) and its first derivative are bounded by a constant depending on \(M\). The quantile gaps are at least \(0.58\), so the rapidity gaps between atoms in this window have a positive lower bound depending on \(M\); there are only boundedly many such atoms. This bounds the lattice sum. For the principal value integral, subtract the constant \(a_i=N\rho_i\), whose integral against the odd kernel \(k\) is zero, and use \[\left|\operatorname{pv}\!\int k(t)N\rho(y_i-t)\,dt\right| \le 2\int\frac{|\chi(t)|}{|t|} |N\rho(y_i-t)-N\rho_i|\,dt.\] The first-derivative bound makes the right side finite uniformly on the layer. The remaining term \(\rho_i'/\rho_i\) is uniformly bounded by the density estimate.

The determinant contribution

It remains to show that the leading logarithmic contribution of \(\det H\) is independent of \(\delta\). Factor \[H=\operatorname{diag}(d_i)(I-A),\qquad d_i=\frac{f_i}{a_i},\qquad A_{ij}=\frac1{d_i}\frac{r(y_i-y_j)}{a_j}.\] The strict margin (50) and the uniform upper bound on \(d_i\) imply \[d_i\ge\epsilon,\qquad \sum_j|A_{ij}|\le q_0<1\] with absolute \(\epsilon,q_0\), uniformly throughout the rectangles. The diagonal logarithms are negligible at the scale \(\log N\). In fact \(f_i-a_i=(r'*E)(y_i)\) is uniformly bounded, and the logarithm is Lipschitz on the fixed interval containing all \(d_i\). Hence \[|a_i\log d_i|\le C|(r'*E)(y_i)|.\] The right side is uniformly bounded and tends to zero in every bulk limit by (67). The averaging rule (74), applied to \(z_{N,i}=a_i\log d_i\), gives \[ \frac1{\log N}\sum_i\log d_i=o(1) \tag{76}\] uniformly in \(\delta\).

The path \(I-tA\), \(0\le t\le1\), is invertible because \(\|tA\|_\infty<1\). Its real determinant remains positive, and the convergent trace series is \[\log\det(I-A)=-\sum_{k\ge1}\frac{\operatorname{tr}A^k}{k}.\] For all \(N,i,\delta\) and \(k\ge1\), its diagonal terms have the summable bound \[ a_i|(A^k)_{ii}|\le Cq_0^{\,k-1}. \tag{77}\] Indeed \(a_i|A_{ji}|=|r(y_j-y_i)|/d_j\le C\), while \(\sum_j|(A^{k-1})_{ij}|\le q_0^{k-1}\); multiply these two bounds in the final factor of \(A^k\).

For fixed \(k\), the bulk limit of the same quantity is \[ a_i(A^k)_{ii}\longrightarrow (r^{*k})(0). \tag{78}\] For \(k=1\) this is \(r(0)/d_i\to r(0)\). For larger \(k\), expand the product as \[a_i(A^k)_{ii} =\sum_{j_1,\ldots,j_{k-1}} \frac{r(y_i-y_{j_1})r(y_{j_1}-y_{j_2})\cdots r(y_{j_{k-1}}-y_i)} {d_i d_{j_1}\cdots d_{j_{k-1}}\, a_{j_1}\cdots a_{j_{k-1}}}.\] First restrict each rapidity step to a fixed bounded interval. All intermediate atoms then lie in a fixed bulk window, where \(d_j\to1\) uniformly and the weighted quantile grids converge to Lebesgue measure. Iterated quadrature gives the corresponding truncated convolution. The restriction can be removed uniformly: every row of \(|A|\) has negligible mass outside a large rapidity step by (51), and the terminal factor \(a_i|A_{ji}|=|r(y_j-y_i)|/d_j\) is bounded and decays exponentially with its step. For fixed \(k\), these bounds control the union of the finitely many omitted step ranges. This proves (78) along every bulk sequence.

Apply (74) for each fixed \(k\) to \(z_{N,i}=a_i(A^k)_{ii}\). Equations (77)–(78) give \[\frac{\operatorname{tr}A^k}{\log N} =\frac2{\alpha_{\rm r}}(r^{*k})(0)+o(1)\] uniformly in \(\delta\). Passing this limit through the trace series is justified by the same bound: \[\frac{|\operatorname{tr}A^k|}{\log N} \le Cq_0^{k-1}\frac{\sum_iw_i}{\log N} \le C' q_0^{k-1}\] for all large \(N\), uniformly in \(\delta\). After division by \(k\), the right side is summable. Thus \[\frac{\log\det H}{\log N} =-\frac2{\alpha_{\rm r}}\sum_{k\ge1}\frac{(r^{*k})(0)}k+o(1)\] uniformly in \(\delta\), where (76) was used. The series limit is independent of the twist.

Integrate (75) from the vacuum value \(\delta=1/4\) to \(\delta=\theta/(2\pi)\), and subtract the determinant contributions at those two values. Uniformity of both estimates gives \[\frac{\log\left|(W/J)_\theta/(W/J)_{\pi/2}\right|}{\log N} =-\frac4{\alpha_{\rm r}} \bigl(s_\theta^2-s_{\pi/2}^2\bigr)+o(1), \qquad 0\le\theta\le\frac\pi2.\] By (64) this is the logarithmic asymptotic of \(c_{\rm phys}(\theta)\). The bulk row has spin twist \(K^{-1}=e^{-i\theta}\) and winding fugacity \(K+K^{-1}=2\cos\theta\), while the two vacuum caps remain fixed. Since \(s_\theta=2\theta/(3\pi)\) and \(\alpha_{\rm r}=8/3\), we obtain \[ \frac{\log|c_{\rm phys}(\theta)|}{\log N} =\frac16-\frac{2\theta^2}{3\pi^2}+o(1), \qquad 0\le\theta\le\frac\pi2, \tag{79}\] with the error uniform on this interval. At zero winding fugacity, \(\theta=\pi/2\), the coefficient is exactly one. At physical fugacity \(2\), \(\theta=0\), it is \(N^{1/6+o(1)}\). The next section uses the geometric interpretation and height estimates to turn this coefficient into a planar nesting law.

Uniform bounds for every fixed twist derivative

The first winding derivative controls the total mass of one winding polygon. We now bound every fixed derivative. These bounds will control the heights of the polygons before passage from the cylinder to the plane. The analyticity radius in the winding fugacity may depend on the circumference; no uniform radius will be needed.

We use the following notation from Section 3: \[l=\frac{\pi}{8},\qquad \alpha_{\mathrm s}=\frac{\pi}{6 l}=\frac43,\qquad \beta_{\mathrm s}=\alpha_{\mathrm s}^{-1}=\frac34, \qquad S_a=\{s\in\mathbb C:|\Re s|<a\},\] \[h(s)=\frac{\cos s-\cos3 l}{\cos s+\cos l},\qquad B(s)=\frac{\cos(\alpha_{\mathrm s}s)-d_0} {\cos(\alpha_{\mathrm s}s)+d_0},\qquad d_0=\cos(\alpha_{\mathrm s} l)=\frac{\sqrt3}{2}.\] An even holomorphic function on \(S_{3 l}\) is a holomorphic function of \(z=B(s)\) on the disk. The point \(s= l\) maps to \(z=0\), and \(B'( l)\ne0\). Put \[(\mathcal C_g U)(s)=\int_{\mathbb R}g(t)U(s+it)\,dt, \qquad g(t)=\frac1{2\pi} \left(\frac{2^{-1/2}}{\cosh t-2^{-1/2}}-\frac1{\cosh t}\right).\] Thus \(p:=\|g\|_1<1/2\) and \(\int g=1/4\).

For precision, here are all the operator facts from the strip calculation in Section 3 that enter the argument. If \(j=f\circ h\) with \(\deg f\le N\), then, on \(S_{3 l}\), \[ j(s-2 l)+j(s+2 l)-j(s) =(I-\mathcal C_g )q(B(s))+\lambda, \qquad q(z)=\sum_{m=1}^N q_mz^m. \tag{80}\] Here \(q\) has no constant term. With the inverse branch \(s(0)= l\), define \[M_{nm}=[z^n](\mathcal C_g B^m)(s(z))\quad(n,m\ge1), \qquad M_{0m}=(\mathcal C_g B^m)( l).\] The matrix \(M\) is real and self-adjoint on the weighted space \(\ell^2(n)\) with squared norm \(\sum_{n\ge1}n|v_n|^2\), and its operator norm is at most \(p\). For one constant \(C\) independent of the indices, \[ |M_{nm}|\le\frac Cn \min\left\{\left(\frac nm\right)^{\beta_{\mathrm s}}, \left(\frac mn\right)^{\beta_{\mathrm s}}\right\}, \qquad \sum_{n\ge1}|M_{nm}|\longrightarrow p, \qquad M_{0m}\longrightarrow0. \tag{81}\] These are operator inputs, not assumptions on the higher derivatives. In particular, we will not assume that \(M\) is a contraction on \(\ell^1\).

From derivatives to polygon height moments

We now use the derivative bounds in positive path sums at the physical row \(t=l\). Throughout this subsection a polygon \(P\) has bare weight \(w(P)=x^{|P|}\), with no loop fugacity included. An open path still has \(w(\gamma)=x^{m(\gamma)}\). Thus a winding polygon in the gas with fugacity \(\nu\) has activity \(\nu w(P)\); in particular, \(P_N\) is the bare mass from Proposition 3, not the mass at fugacity \(2\).

For a path or polygon let \(r_-\) and \(r_+\) be its lowest and highest occupied rhombus rows. Write \[\operatorname{top}(P)=r_+(P),\qquad \operatorname{ht}(P)=r_+(P)-r_-(P)+1.\] For an arc below the cut at row \(0\), put \(\operatorname{dep}(\gamma)=-r_-(\gamma)\). Changing between row cuts and centre heights costs only an absolute number of rows. Constants denoted by \(C_{\rm lat}\) below allow for this fixed lattice convention, and \(u_+=\max\{u,0\}\).

The two seam tips and the planar comparison

The height estimates permit the projection to the plane. We first identify the cap matrix element as a positive two-tip partition and control the cost of horizontal slab walls.

At homogeneous sites let \(G_H=\Lambda(T^1(l))^H\Psi\), where the superscript \(1\) denotes trivial spin twist. The cap regularity argument in Section 10 represents the lower and upper caps as limits of empty boundaries extended by physical rows with twists \(i\) and \(-i\), respectively; the upper construction uses transpose, reversal, and spin negation. In a finite such approximation the seam twist is \(i\) below the tip at cut \(0\), \(1\) between cuts \(0\) and \(H\), and \(-i\) above the tip at \(H\), as indicated in Figure 1.

For an oriented loop let \(c_t\) be its signed crossing number of the upward seam ray from tip \(t\), with left-going auxiliary flow positive. For a counterclockwise contractible loop, \(c_t\) is one if its disk contains \(t\) and zero otherwise. Its turn phase is \(i\), and its total full-seam crossing number is zero. For a winding loop oriented to the left, \(c_t\) is one if \(t\) lies in its lower complementary component and zero otherwise; its turn phase is one and its full-seam phase for the constant twist \(i\) is \(i\). In either case the two successive twist ratios are \(1/i=(-i)/1=-i\). The sum of the two orientations is therefore \[2\Re\bigl(i(-i)^{c_0+c_H}\bigr) =\begin{cases} 2,&c_0+c_H=1,\\ 0,&c_0+c_H=0\text{ or }2. \end{cases}\] Thus precisely the contractible loops enclosing one tip and the windings separating the tips survive, each with activity \(2x^{|P|}\). Every finite cap approximation is the corresponding positive gas in its finite cylinder. Exhaustion proves \(G_H=\mathcal G_N(H)\).

Let \(\mathfrak b_N\) be the total bare weight of windings meeting a fixed horizontal cut. Counting their possible row placements and using (105) gives the uniform wall bound \[ \mathfrak b_N\le\sum_{\eta\in\mathscr W_N^0}w(\eta) (\operatorname{ht}(\eta)+C_{\rm lat}) \le CNP_N\le C. \tag{115}\] The winding-only slab partition in the introduction is \(L_N(h)=\Xi_{N,h}(2)\), after translating the slab. Stacking slabs and retaining only polygons internal to each gives the lower inequality below. For the upper inequality, keep the exclusions within the two slabs and discard all exclusions involving polygons that meet their common wall. Those polygons have total activity at most \(2\mathfrak b_N\), so their unrestricted gas is at most \(\prod_\eta(1+2w(\eta))\le e^{2\mathfrak b_N}\). Hence \[ L_N(h)L_N(j)\le L_N(h+j) \le e^{2\mathfrak b_N}L_N(h)L_N(j). \tag{116}\] The positive annular transfer at fugacity \(2\) is primitive by Section 10. Its dominant eigenvalue \(\lambda_N\) is therefore the exponential growth rate of its empty-to-empty entry \(L_N(h)\). Iterate (116) on multiples of \(h\) and take the growth rate to obtain \[ e^{-2\mathfrak b_N}\lambda_N^h\le L_N(h)\le\lambda_N^h, \qquad 0\le\log\lambda_N\le 2P_N. \tag{117}\] For the last inequality, discarding all exclusions gives \(\log L_N(h)\le2\sum_{\eta\in\mathscr W_{N,h}}w(\eta)\le2hP_N\). In particular, for \(H>2N\), \[ \frac{L_N(H-2N)}{\lambda_N^H} \ge e^{-2\mathfrak b_N}\lambda_N^{-2N} \ge e^{-2\mathfrak b_N-4NP_N}\ge e^{-C}. \tag{118}\]

Write \(Z(r)=\mathcal Z_v(r)\) for the planar nesting partition at either tip. Choose a small absolute \(a_0>0\) so that a disk of radius \(a_0N\) projects injectively to the period-\(N\) cylinder and, when centred at the lower tip, lies below the cut at \(N\); the analogous disk at the upper tip lies above the cut at \(H-N\). For large \(H\) the two disks are disjoint. Their nests coexist with every winding configuration internal to the reduced slab \([N,H-N]\), and all those windings separate the tips. Thus \[\mathcal G_N(H)\ge Z(a_0N)^2L_N(H-2N).\]

For the upper bound use the original slab \([0,H]\). A separating winding not internal to this slab must meet its lower or upper wall, so such windings have total activity at most \(4\mathfrak b_N\). A contractible polygon with centred lift of diameter at most \(D=N^{1+3\epsilon}\) lies within distance \(D\) of its enclosed tip: the tip lies in the convex hull of the polygon. The centred lifts at each tip form a planar nest, whose partition is bounded by \(Z(D)\). Finally, (114) bounds the total activity of the remaining contractible polygons at both tips by \(4q_{N,\epsilon}\). Retain the exclusions within each of the two small nests and within the slab-internal winding gas, and discard all other exclusions. In finite exhaustions this gives, and hence in the limit gives, \[\mathcal G_N(H)\le L_N(H)Z(N^{1+3\epsilon})^2 \exp(4\mathfrak b_N+4q_{N,\epsilon}).\] These bounds in particular make the positive full-cylinder sum finite. Equations (117)–(118) now give the comparison \[ e^{-C}Z(a_0N)^2\le \lim_{H\to\infty}\lambda_N^{-H}\mathcal G_N(H) \le e^{C_\epsilon}Z(N^{1+3\epsilon})^2. \tag{119}\] The limit exists by the physical cap expansion and equals \(c_{\rm phys}(0)=N^{1/6+o(1)}\). The left inequality in (119), with monotonicity between successive \(a_0N\), gives \(\limsup_{r\to\infty}\log Z(r)/\log r\le1/12\). The right inequality, with \(N=\lfloor r^{1/(1+3\epsilon)}\rfloor\), gives \(\liminf_{r\to\infty}\log Z(r)/\log r\ge1/(12(1+3\epsilon))\). Letting \(\epsilon\downarrow0\) proves \(Z(r)=r^{1/12+o(1)}\) and Theorem 2.

An independent criticality consequence

Let \(c_n\) be the number of ordinary centre-to-centre self-avoiding walks of \(n\) edges from a fixed centre. We show that the series \(\sum_n c_n z^n\) has radius of convergence \(x\). The decomposition is the decreasing-span unfolding method associated with Hammersley and Welsh [12], in the present port convention.

Along the normal to a horizontal row, centre heights lie on a uniform discrete mesh, and no dual bond is horizontal. Split a walk at its global minimum height and reverse the first part. Each of the two resulting walks is nonnegative relative to its starting height. In either part, take the segment ending at the last vertex of maximum displacement from its start, and repeat on the remainder with the sign reversed. Every segment has positive span and has its two ends at its two extreme heights. The spans strictly decrease after the first comparison, which may be an equality. Their sum is \(O(n)\), since the total variation of height along an \(n\)-edge walk is \(O(n)\). There are consequently \(O(\sqrt n)\) segments.

For a prescribed start, sign, and span, the total weight of such a segment is bounded by an absolute constant. To see this, add boundary legs at its two extremes to make it a crossing between parallel horizontal triangle-side lines. If the outward bond at an extreme is vertical, use its midpoint port. Otherwise first use a fixed slanted outward bond to reach a new triangle with a vertical outward bond. The additional centre lies strictly beyond the old extreme, so it is unvisited. The two boundaries and the source port are determined by the start, sign, and span up to finitely many lattice types, and deleting the added legs has bounded multiplicity and bounded weight cost. Exhausting the strip by finite convex parallelograms, the positive real flux of Section 3 bounds the total crossing mass by an absolute constant. The conversion between \(x\) per edge and \(x\) per visited centre costs one further constant for each segment.

At a fixed starting centre, the minimum centre has only \(O(n^2)\) possible positions. There are at most \(n^{O(\sqrt n)}\) possible span lists, and the product of the crossing bounds costs \(C^{O(\sqrt n)}\). Dropping endpoint and total-length constraints within the segment sums therefore gives \[c_nx^n\le \exp\bigl(O(\sqrt n\log(n+2))\bigr).\] The radius of convergence is at least \(x\).

For the opposite inequality, half-plane saturation and the definition of \(D_N\) give the positive omitted mass \[D_N=2\cos(3l)\left( \sum_{g\ge N}K_\infty(g) +\sum_{g=1}^{N-1}\bigl(K_\infty(g)-K_N(g)\bigr)\right) \asymp N^{-1/4}.\] Each omitted arc uses at least \(a_{\rm len}N\) centres for an absolute \(a_{\rm len}>0\). An arc in the first sum has endpoints separated by at least \(N\); one in the second meets a nonzero horizontal translate and hence has two vertices separated by at least \(N\). Deleting the port legs maps these arcs to centre walks with bounded multiplicity, so for large \(N\), \[D_N\le C\sum_{n\ge a_{\rm len}N/2}c_nx^n.\] If the counting series had radius larger than \(x\), choosing a fugacity strictly between them would make this tail exponentially small in \(N\). This contradicts the polynomial lower bound. Thus the radius is exactly \(x\), at either centre type by lattice symmetry. The denominator proof that follows uses the positive deficit and depth estimates directly.

The macroscopic chord denominator

Put \(\chi=\cos(3l)>0\) and write \[B_\infty=2\sum_{g>0}K_\infty(g)=\chi^{-1}\] for the total half-plane arch mass from one boundary port, including both endpoint directions. We will compare this saturated mass with the arches that fit in a strip or a triangle.

The edge-slit identity

To compute the length numerator, fix an interior side midpoint \(e\) of the tiling. Cutting the honeycomb edge at \(e\) creates two slit ports, one facing each adjacent triangle. For this subsection paths may have any diameter. Let \[T_e=\sum_{\substack{\gamma\text{ boundary-to-boundary}\\e\in\gamma}} x^{m(\gamma)}\cos(cW(\gamma)),\qquad c=3/8,\] where both endpoint orders are counted and \(W\) includes the first and last turns. Define \(Z_e\) to be the partition function of disjoint simple polygons in \(\Delta_R\) avoiding and enclosing \(e\), with weight \(2x^{|P|}\) per polygon. Define \(L_e\) by additionally marking one disjoint polygon through \(e\): this polygon has weight \(x^{|P|}\), while every other polygon still has weight \(2x^{|P|}\) and encloses \(e\).

The length numerator and the mean

Let \(A_e\) be the unsigned mass of all ordered boundary chords through \(e\). The planar nesting estimate and Lemma 19 give \[A_e\le R^{1/12+o(1)}\] uniformly over contacts in the triangle. If \(e\) lies in the concentric triangle of side \(R/3\), both adjacent primal vertices have distance comparable to \(R\) from the boundary. For fixed positive \(a_{\rm in},a_{\rm out}\), domain inclusion gives \[\mathcal Z_{v_i}(a_{\rm in}R) \le\mathcal Z_{v_i}^{\Delta_R} \le\mathcal Z_{v_i}(a_{\rm out}R),\qquad i=1,2.\] Monotonicity therefore gives \(A_e=R^{1/12+o(1)}\) uniformly throughout that central region. The distance from the concentric triangle of side \(R/3\) to the boundary of \(\Delta_R\) is \(\sqrt3R/9\). Thus every chord through a central contact has diameter at least \(\sqrt3R/9\), and in particular belongs to \(\mathcal C_R(\delta)\) for every fixed \(0<\delta\le1/50\).

There are order \(R^2\) central contacts and at most order \(R^2\) contacts altogether. A path visiting \(m\) centres crosses exactly \(m-1\) interior contacts. Define \(M_R(\delta)=\sum_{\gamma\in\mathcal C_R(\delta)}m(\gamma)w(\gamma)\). For each fixed \(0<\delta\le1/50\), \[ \begin{split} M_R(\delta) &=Z_R(\delta)+ \sum_e\sum_{\substack{\gamma\in\mathcal C_R(\delta)\\e\in\gamma}} w(\gamma)\\ &=R^{25/12+o(1)}. \end{split} \tag{128}\] The lower bound uses the central contacts, for which all chords are admissible. The upper bound allows all chords at every contact. Dividing (128) at \(\delta=1/100\) by (126) proves the port mean assertion in Theorem 1.

We finish with an explicit comparison for ordinary centre walks. Let \(\mathcal W_R(\delta)\) be the ordered centre-to-centre walks contained in \(\Delta_R\), with both endpoints on its inner boundary layer and diameter at least \(\delta R\). If \(n(\omega)\) is their number of edges, put \[\begin{gathered} Z_R^\circ(\delta)=\sum_{\omega\in\mathcal W_R(\delta)}x^{n(\omega)},\\ M_R^\circ(\delta)= \sum_{\omega\in\mathcal W_R(\delta)}(n(\omega)+1)x^{n(\omega)}. \end{gathered}\] Every such walk has at least one extension by boundary half-edges to two ports, and at most \(Q=9\) extensions, since each endpoint triangle has at most three boundary sides. Deleting the two half-edges recovers the walk. An extension has exactly \(m=n+1\) visited centres and weight \(x^{n+1}=x\,x^n\). Moreover, for an absolute \(b\), \[\mathop{\mathrm{diam}}(\omega)\le\mathop{\mathrm{diam}}(\text{port extension}) \le\mathop{\mathrm{diam}}(\omega)+b.\] Take the two fixed cutoffs \(\delta_0=1/100\) and \(2\delta_0=1/50\). For all large \(R\), deleting the half-edges from a chord of diameter at least \(2\delta_0R\) leaves a centre walk of diameter at least \(\delta_0R\); every extension of a centre walk at cutoff \(\delta_0\) is itself a chord at that cutoff. The weight and multiplicity bounds give the two squeezes \[ \begin{gathered} \frac{Z_R(2\delta_0)}{xQ}\le Z_R^\circ(\delta_0) \le\frac{Z_R(\delta_0)}x,\\ \frac{M_R(2\delta_0)}{xQ}\le M_R^\circ(\delta_0) \le\frac{M_R(\delta_0)}x. \end{gathered} \tag{129}\] Both cutoffs lie in the fixed range already proved. Hence \(Z_R^\circ(\delta_0)\asymp R^{3/4}\) and \(M_R^\circ(\delta_0)=R^{25/12+o(1)}\). Their ratio is the mean of \(n+1\), so the mean edge length is \(M_R^\circ(\delta_0)/Z_R^\circ(\delta_0)-1=R^{4/3+o(1)}\). This proves the centre-to-centre assertion.