The physical observables

Let \(\mathbb H\) be the honeycomb graph formed by the centers of the triangles of a regular triangular tessellation of side length one. Planar distances and diameters refer to this Euclidean embedding. A port is the midpoint of a tessellation side on the boundary of a region. A port-to-port walk traverses distinct triangle centers and stops at its first boundary exit. If it visits \(N\) centers, its critical weight is \(x^N\), where \[x=(2+\sqrt2)^{-1/2}.\] For a closed polygon, \(|\lambda|\) denotes its number of bonds, which is also its number of vertices. All polygon sums below count unoriented, unrooted cycles. A sum modulo translations counts each lattice translation class once. Two cycles are incompatible if they share a vertex; a cycle is incompatible with itself. Their basic pair weight is \(x^{|\lambda|+|\lambda'|}\).

Identify horizontal tessellation coordinates modulo \(M=2m\), with \(m\) even. Mark the face centers \(O=(0,0)\) and \(O'=(m,0)\) on a horizontal cut. Call the \(2m\) tessellation sides along this cut its sites. Write \(A\) for the \(m\) sites from \(O\) to \(O'\) in eastward order and \(B\) for the remaining \(m\) sites. Let \(G_m(g)\) be the sum over collections of disjoint simple cylinder cycles, each separating \(O\) from \(O'\), with weight \[\prod_{\lambda}g x^{|\lambda|}.\] The empty collection contributes one; Figure 1 shows the two site intervals. The cylinder is infinite in the vertical direction; equivalently take increasing finite cylinders with closed rims, meaning that every port on either rim is unused. The proof below establishes finiteness and identifies this limit. Because each cycle crosses the cut between the marks, \(G_m\) is a polynomial in \(g\) of degree at most \(2m\).

For a regular tessellation hexagon \(D_R\) of integer side length \(R\), let \(\mathcal Z(R)\) be the partition of disjoint polygons contained in \(D_R\) and surrounding its central face, each weighted by \(2x^{|\lambda|}\); the empty collection contributes one. Let \(\mathcal C_R\) consist of the ordered first-exit chords of \(D_R\) whose diameter is at least \(R/10\). Both boundary ports are free: the sum includes every starting port and every possible first-exit port, retaining the direction from start to exit. Define the finite partition and probability law by \[Z_R^{\rm chord}=\sum_{\gamma\in\mathcal C_R}x^{N(\gamma)},\qquad \mathbb P_R^{\rm chord}(\gamma) =\frac{x^{N(\gamma)}}{Z_R^{\rm chord}}\quad(\gamma\in\mathcal C_R).\]

For a positive quantity \(F(R)\), the notation \(F(R)=R^{a+o(1)}\) means \(\log F(R)/\log R\to a\). An upper bound \(F(R)\le R^{a+o(1)}\) means that for every \(\epsilon>0\) it is at most \(C_\epsilon R^{a+\epsilon}\). Uniform occurrences of this notation have constants and thresholds independent of the indicated geometric data.

The chord conclusions describe this finite law, which is normalized over all lengths allowed by the domain and the diameter cutoff. The visit estimate concerns the stated bulk vertices. The final inequality sums closed polygons modulo translations. Its proof uses the positive mass of single polygons passing through two specified bonds on a tilted cylinder. Theorem 29, stated at the start of Section 12, gives the precise restrictions on the two site lists and the uniform estimate used for this purpose.

Earlier work

Nienhuis’s analysis of the dilute \(O(n)\) model predicted the two-dimensional self-avoiding-walk exponents [14]. The conventional spatial prediction is \(\nu=3/4\): the mean-square displacement of a uniform length-\(n\) walk is expected to be \(n^{2\nu+o(1)}\) [3]. The reciprocal \(1/\nu=4/3\) motivates the length exponent in our finite chord law, whose normalization and boundary conditions are specified above. Duminil-Copin and Smirnov proved the exact honeycomb connective constant by a parafermionic local cancellation and a boundary-flux argument [3]. Their theorem determines the critical activity used here. The local cancellation also gives a positive bound on a convex boundary partition; the chord law requires quantitative control of the part carried by macroscopic exits. The vanishing of critical bridge mass was proved by Beaton, Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann [2]. For honeycomb bridges, Glazman and Manolescu subsequently obtained a logarithmic subsequence bound. For their specified columnwise rhombic half-plane tilings they also proved invariance of boundary two-point functions [8]. Krachun and Panagiotis obtained a polynomial upper bound on the critical bridge partition and used it to prove quantitative sub-ballisticity for uniform length-\(n\) honeycomb walks [13]. These results concern distinct normalizations and precision levels; our boundary proof establishes the radius-arch exponent with arbitrary exponent slack.

The extension to rhombic weights connects this cancellation to Yang–Baxter methods [8]. Our local-weight and fusion calculations keep the physical normalization explicit.

Grimm and Pearce developed a two-colour braid–monoid formulation of dilute loop models and its Yang–Baxterization [9]. The trigonometric weights underlying the rhombic model trace back to Nienhuis [15, 8]. Finite-size corrections to periodic dilute-loop transfer-matrix spectra have integrable-model predecessors [18]. For an open-strip dilute \(O(1)\) model, Fehér and Nienhuis derive fusion and current recursions [4]. Their current interpolation in Section 6 uses rapidity-symmetry hypotheses stated in Section 5; Appendix E supplies a boundary fusion construction and explains why that construction does not establish the current symmetries. The finite observable here is different, and the symmetry within each of its two site groups is proved directly. These earlier spectral and current formulas do not by themselves identify its marked vacuum pairing or physical normalization.

The fermionic contractions and bosonic oscillator conversion belong to the classical Wick and boson–fermion methods [17, 5, 6]; we compute their twist, charge conventions and analytic limits in the normalization needed here.

Proof strategy

The cylinder calculation begins by orienting occupied edges, with the three states empty, forward and backward. Phases attached to turns, seam crossings, and crossings of \(A\) make the two orientations of a loop cancel unless it separates the marks; a separating loop contributes \(g\). Long stacks with empty rims converge to vectors indexed by spin configurations on the cut, whose pairing with the cut insertion gives the partition. Finiteness of the total half-plane arch weight, proved in Section 2, justifies their limits. Section 3 then uses local exchange and fusion, a spectral projector, and polynomial interpolation to obtain an exact finite sum over three-valued site spins. It temporarily allows arbitrary consecutive lists \(A,B\) and nonzero complex spectral variables at their sites. The balanced honeycomb case is recovered by choosing equal list sizes and setting all site spectral variables to \(1\).

Individual terms in this spin sum have poles when these variables coincide, although the combined expression is regular. Sections 4 and 5 replace the sum by contour residues, separate the coincident sites, and express the contractions as an angular fermion trace. Evaluating that trace in oscillator coordinates gives a gas on a horizontal circle of auxiliary length \(T\). Its particles have two species, corresponding to \(A\) and \(B\), and carry integer charges. The gas records a running total charge, an integer that changes by minus each particle’s total charge as particles are crossed. The resulting weights are signed or complex. For the balanced cylinder, the limit \(T\to\infty\) is taken at each fixed site count; the estimates in the site count come afterward.

Section 6 controls the absolute gas weights by a quadratic cost in the number of particles in each box and in this running total charge at box boundaries. Section 7 uses this estimate to form a compact block propagator, with a contracting auxiliary coordinate carrying the interaction with earlier blocks exactly. The site factors suppress certain particle types on a central interval of length proportional to \(\log m\). The limiting operator there omits those types; the exterior operator allows all types. Their propagation rates therefore produce a power of \(m\).

The exterior operator still has signed weights, and its eigenvalues of maximal modulus may have Jordan blocks. At fixed \(m\), the finite identity supplies a limit in which the perturbed trace minus the normalized physical value times the unperturbed trace tends to zero. After normalization by the spectral radius, Lemma 18 expands the \(n\)th power of the exterior operator into factors \(\zeta^n\) with \(|\zeta|=1\), multiplied by polynomials in \(n\), and compares the polynomial coefficients in the trace identity. This retains all Jordan terms and bounds the physical value without dividing by a trace that may cancel. The Pfaffian scalar computed in Section 8, the exact normalization \(G_m(0)=1\), and positivity of the coefficients of \(G_m\) then give the cylinder exponent and the incompatible-pair bound.

The chord mean also requires the size of its own finite partition \(Z_R^{\rm chord}\). The local boundary balance controls the total exit weight from a port; the chord partition requires the part carried by macroscopic exits. Write \(W_R\) for the total critical weight of first-exit same-boundary arches in a half-plane bounded by a horizontal tessellation line that start at a fixed port and reach distance at least \(R\). Section 10 determines \(W_R\) by an argument depending on the local transfer identities and independent of the compact propagator. The added-row expansion identifies the derivative of a one-row auxiliary cylinder increment with its circumference \(M\) times the deficit left in the boundary balance by half-plane arches that embed in that cylinder. Fusion gives functional relations for the increment, which determine the derivative through a positive density on ordered points \(0<k_1<\cdots<k_M\) whose additional factor is an average over unitary conjugates. Comparison with ensembles whose orthogonal polynomials can be evaluated gives the derivative exponent \(3/4\). The resulting deficit recovers the full mass in the boundary balance and gives an upper bound for the endpoint tail and a lower bound for the radius tail. A wedge comparison supplies the matching radius upper bound, hence \(W_R=R^{-1/4+o(1)}\). A boundary comparison transfers this estimate to macroscopic hexagon exits; summing over the free starting ports gives \(Z_R^{\rm chord}=R^{3/4+o(1)}\).

Proposition 26 applies the boundary balance after removing the triangle centered at a honeycomb vertex \(v\) not adjacent to a boundary port. It compares the raw weight of chords visiting \(v\) with the sum of the nesting partitions around its three incident faces. For vertices within \(R/10\) of the hexagon center, every such chord meets the diameter cutoff when \(R\) is sufficiently large. Summing over vertices bounds the mean below by a constant times \(R^2\mathcal Z(\lfloor R/2\rfloor)/Z_R^{\rm chord}\) and above by a constant times \(R^2\mathcal Z(3R)/Z_R^{\rm chord}+R\); the additive term accounts for vertices adjacent to boundary ports. Section 11 uses the \(O(\log m)\) incompatible-pair bound to show that the sum of \(x^{|\lambda|}\) over all noncontractible separating cycles and all contractible cycles whose disk lifts are not confined within a fixed multiple of \(m\) around their enclosed mark is \(O(\sqrt{\log m})\). Their contribution costs a subpower factor. The remaining cycles lift near one of the marks, giving upper and lower comparisons of \(G_m(2)\) with products of planar nesting partitions at comparable radii. This gives the exponent \(1/12\); the visit comparison and chord normalization then give the mean and bulk-visit exponents.

The length-square mass is handled through the positive sum of \(x^{|\lambda|}\) over single polygons using two specified bonds. After a lattice rotation or reflection, the marked estimate in Theorem 29 applies to cylinders whose cuts use horizontal tessellation steps and steps tilted by \(\pi/3\). Both marked bonds cross horizontal steps, and the proportions of the two directions in each unmarked list satisfy the restrictions in that theorem. Sections 12–14 derive the matrix and horizontal representations of this two-bond sum, including their scalar normalization and uniform suppression for those lists. If their sizes are \(m\) and \(n\), the two suppression intervals have lengths proportional to \(\log m\) and \(\log n\). On the parts covered by only the longer interval, the mixed propagator allows all types in one species and only the unsuppressed types in the other. Section 15 bounds this propagator by a separate calculation with one species. Monotonicity of the running total charge controls the central region where both suppressions act. The final counting argument sums the marked estimate over selected bond pairs whose displacements fit these cuts and separately bounds polygons for which too few such pairs are available. This proves (1).

An auxiliary cylinder and spectral equation

Take a cylinder obtained by identifying modulo \(M\) steps horizontally, tiled by rows of \(M\) rhombi each, with basis \(1,e^{i\theta}\) and potentially a different \(\theta\) per row. Connections inside a tile link pairs of bond midpoints on distinct sides, using each at most once, noncrossingly. The empty weight is 1, a straight connection between opposite sides weighs \(v\), the single turns of absolute angles \(\theta,\pi-\theta\) weigh \(u,d\) respectively (port directions normal to sides); and two simultaneous turns weigh \(a\) for the pairing left–down and up–right, or \(b\) for left–up and right–down. With \(\phi=3\theta/8\), \(s_k=\sin(\phi+k\alpha)\), \(h=\sin(2\alpha)\), use the trigonometric SAW weights \[(u,d,v,a,b)=\frac{(h s_5,h s_0,s_0s_5,s_5s_6,-s_0s_7)}{s_2s_3}.\] At \(\phi=\alpha\) these are \((x,x^2,x^2,x^2,0)\), exactly the original lattice rules (split each tile into the two equilateral triangles). The relation between rhombic geometry, parafermionic cancellation and integrable loop weights was developed by Ikhlef and Cardy [10]. Glazman’s corrected general loop-weight formulas [7] give a useful comparison; the five weights and their critical specialization here are checked directly below. See also the boundary two-point results in [8].

For a finite stack, close both rims (require unused ports). The partition \(Z\) of one noncontractible polygon only (unoriented, unrooted) is the linear coefficient in fugacity \(n\) of the full loop sum giving weight zero to contractible loops and weight \(n\) per noncontractible loop. For real angles \(0<\theta<\pi\), orient loops with factor \(\exp(-iT/4)\) and twist \(p\) to the power signed number of seam crossings (east positive); summing orientations effects precisely these loop weights for \(n=p+p^{-1}\). Indeed disjoint arcs can be drawn in tiles with the indicated tangent turns; an embedded loop in the flat cylinder is either contractible with turn \(\pm2\pi\) or has winding \(\pm1\) and turn zero. For instance the latter follows by mapping conformally by an exponential to a round annulus and using the plane turning formula there. These identities then extend algebraically.

The oriented matrix, with spins \(0,\pm1\) on horizontal-flow and descending-flow bonds (respectively across the sides parallel to \(e^{i\theta}\) and to \(1\)), has row indices (left, up), column indices (right, down), positive signs meaning left to right or up to down. Remove the phase \(e^{i\theta(\text{left}-\text{right})/4}\) (telescoping). The resulting \(R(\phi)\) conserves the sum, with blocks, \(\lambda=e^{-2i\alpha}\): \[\begin{array}{c|c} (s',s')\quad(s'=\pm1)& a\\ (s',0),(0,s')&\begin{pmatrix}v&u\\u&v\end{pmatrix}\\ (1,-1),(0,0),(-1,1)& \begin{pmatrix} b&d\lambda& b\lambda^2+a\\ d\lambda^{-1}&1&d\lambda\\ b\lambda^{-2}+a&d\lambda^{-1}&b \end{pmatrix} \end{array}\] (the turns connecting left to down have magnitude \(\theta\), those connecting left to up have magnitude \(\pi-\theta\)).

Local exchange and fusion

The matrix identities below have two uses. Exchange lets us interchange two rows of local matrices. Fusion identifies the action on a smaller auxiliary space; at a matching site the action on the quotient vanishes. We treat these as meromorphic identities, including at the rank-deficient fusion matrices.

Let \(V=\mathbb C^3\) with basis \(|0\rangle,|+\rangle,|-\rangle\), and use subscripts to indicate an ordered pair of tensor factors. On a copy \(V_i\), let \(s_i|k\rangle_i=k|k\rangle_i\). Write \(P_{ij}\) for the interchange of factors \(i,j\), and put \[\Delta_R(\psi)=\sin(\psi+2\alpha)\sin(\psi+3\alpha).\]

A Fourier identity for the scalar kernels

The finite vacuum normalization and the later boundary moment problem use the same elementary transform. Recording it here makes both calculations independent of their order.

The oriented observable and its vacuum limits

Put \(q=e^{2i\alpha}=\lambda^{-1}\), introduce site variables \(x_j=e^{2i\phi_j}\) (not vertex fugacities), and, with \(w=e^{2i\xi}\), set \[T_p(w)=\mathrm{Tr}_a\, p^{s_a} R_{a1}(\xi-\phi_1)R_{a2}(\xi-\phi_2)\cdots R_{aM}(\xi-\phi_M), \qquad p\in\{i,-i\}.\] Here \(s_j\) is the diagonal spin, index \(a\) the auxiliary space. We work on the total site charge zero subspace \[\mathcal H_M=\operatorname{Span}\{|\sigma\rangle: \sigma\in\{-1,0,1\}^M,\ \sum_j\sigma_j=0\}\] and denote its empty vector by \(|0\rangle\). Choices of square roots of the variables are understood. For each fixed \(p\), the rows commute by Lemma 3. The vacuum matrix element satisfies \[\langle0|T_{p_1}(w_1)\cdots T_{p_L}(w_L)|0\rangle=1\] for arbitrary words in the two twists, at regular parameters. Indeed the oriented phases in the blocks can still be drawn on the reference tiling with fixed angle \(\pi/3\), keeping the angle-independent phases after removing the telescoping factor and varying only the scalar weights of the tile pictures. Each contractible loop has phase \(\pm i\) before seam signs, and each noncontractible one has turn zero and winding \(\pm1\). Seam factors for either orientation are those for uniform twist \(i\), times the same sign from the layers of twist \(-i\). All loops therefore cancel.

We will use this word identity in two ways. First, every linear combination \(A\) of operators \(T_p(w)-I\), even with the two twists mixed, has zero as an eigenvalue: \(\langle0|A^k|0\rangle=0\) for \(k\ge1\), and the characteristic polynomial would contradict this if its constant term were nonzero. Second, fix one twist \(p\). Choose finitely many row deviations spanning the linear span of all \(T_p(w)-I\) in \(\operatorname{End}\mathcal H_M\) and simultaneously triangularize them. For every choice of coefficients, one diagonal character of the corresponding linear combination vanishes. A finite union of proper hyperplanes cannot cover the coefficient space, so one character vanishes identically. Thus this fixed-twist family has the simultaneous eigencharacter \(T_p(w)=1\).

For physical site variables all one, use twist \(p=-i\) above the cut and \(p=i\) below, with the insertion \[D_A=(ih)^{\sum_{j\in A}s_j}.\] To check the weight, orient a contractible simple cycle counterclockwise or a noncontractible one eastward, and let \(I_1,I_2\) indicate whether \(O,O'\) belong to its left component (interior or upper, respectively). The signed cut crossings in \(A\) (downward positive) give \(I_2-I_1\). Signed seam crossings above and below \(O\) (eastward positive) give \(-I_1,I_1\) in the contractible case and \(1-I_1,I_1\) otherwise. These numbers follow by moving along the corresponding transverse arcs from component to component. Including turn, seam and insertion, the phase is \((-i)(-1)^{I_1}(ih)^{I_2-I_1}\), with its inverse for reversed orientation. Thus it cancels unless the cycle separates the marks, and then sums to \(g\).

The top and bottom vacuum vectors tend to finite limits \(L_-^{\rm phys},R_+^{\rm phys}\); the subscripts \(+,-\) on vacuum vectors stand for twists \(i,-i\). Untouched closed loops separately cancel, leaving disjoint arcs connecting ports on the cut in each half-cylinder. Any lift of such an arc has port separation less than \(M\): otherwise its endpoints interlace those of a period translate, contradicting noncrossing. Products of finitely many half-plane arch sums \(H(d)\) from Section 2 therefore dominate the absolute weights.

These physical vectors have the simultaneous eigencharacter \(T=1\) for their respective twists. For example, insert a regular row \(T_i(w)\) just below the cut and commute it to the far closed rim. With the rim ports empty that row uses no double turn, so its absolute weights are bounded by the physical weights times a constant depending on \(M,w\). An arc affected by the row must reach the receding rim. Its total weight tends to zero by the same arch summability. The inserted row thus leaves the limit unchanged, and the top vector is treated in the same way. Both vacuum components are 1. This proves the height limit and finite coefficient sums (for example by taking \(g>0\)), with \[G_{A,B}=L_-^{\rm phys}D_A R_+^{\rm phys}.\]

Fusion and the algebraic vacuum projector

Write \(H_p=T_p-I\). We first record the behavior of a local matrix when \(z=w/x_j\) tends to zero or infinity. Directly from the sine weights, \[\begin{array}{c|c|c} & (a,b,v)&(u,d)\\ \hline z\to0&(q^{-3},q^3,1)&O(z^{1/2})\\ z\to\infty&(q^3,q^{-3},1)&O(z^{-1/2}). \end{array}\] The entries are holomorphic in \(z^{1/2}\) at zero and in \(z^{-1/2}\) at infinity. On a state with auxiliary spin \(r\) and site spin \(s\), the diagonal limit is \(q^{3rs}\) at infinity and \(q^{-3rs}\) at zero. The only surviving off-diagonal entry at infinity changes the auxiliary spin from \(-1\) to \(1\) and the site spin from \(1\) to \(-1\) (row minus column); at zero this move is reversed. In particular every surviving move at an inactive site preserves its vacancy parity.

Square-root factors occur only in the single turns, which change auxiliary vacancy parity. Their number in an auxiliary trace is even, so \(H_p(w)\) is rational in \(w\), with possible simple poles \(q^5x_j,q^6x_j\) at generic sites. The same observation for a site says that changing a square-root sign conjugates \(T_p\) by \(P_j=(-1)^{s_j^2}\). At either endpoint of the row parameter, a closed auxiliary trace cannot use a strictly one-directional off-diagonal move. On \(\mathcal H_M\) the remaining diagonal trace is \(\sum_{r=-1}^1p^r q^{\pm3r\sum_j s_j}=1\). Thus \(T_p(0)=T_p(\infty)=I\). With \(C_\ell(z,b)=z/(z-q^\ell b)\) we obtain \[H_p(w)=\sum_{j=1}^M\sum_{t=0}^1 C_{5+t}(w,x_j) U_{jt},\] where the coefficient operators commute. The basis functions vanish at zero and have distinct poles; the condition at infinity additionally gives \(\sum_{j,t}U_{jt}=0\).

The two embeddings and row fusion above give at any site \(x_j\), generically, \[T_p(x_j)T_p(q^2x_j)=T_p(qx_j),\qquad T_p(x_j)T_p(q^3x_j)=I.\] Indeed the combined auxiliary monodromy has local factors \(R_{2k}(\phi)R_{1k}(\phi+\delta\alpha)\), with the lower parameter on the left. Lemma 4 identifies their restriction to \(\operatorname{im}J_\delta\) and shows that the quotient factor at site \(j\) is zero: its full image is already in \(\operatorname{im}J_\delta\otimes V_j\). The quotient monodromy therefore has zero trace. On the restriction the twist is the fused twist, or the identity for \(\delta=3\), giving the two formulas. This is a trace of the restriction transported through \(J_\delta\), so no isometry factor enters. Square roots can be chosen consistently; shifting a whole row parameter by \(\pi\) leaves \(T_p\) unchanged.

Set \(T_j=T_p(x_j)\). Expanding the fusion formulas in the commuting coefficients \(U_{kt}\) gives a homogeneous system \(\mathcal L(T)U=0\). Its two rows for site \(j\), with column site \(k\), type \(t\), and \(C_\ell=C_\ell(x_j,x_k)\), are \[\begin{split} C_{5+t}-C_{4+t}+T_j C_{3+t},\\ C_{4+t}+T_j(C_{2+t}-C_{3+t}). \end{split}\] The second is the rank-one fusion equation minus the first. At \(T=1\) these rows are \(C_5+C_3-C_4\) on the same row and column list \(Y=(x_1,q x_1,\ldots,x_M,q x_M)\). Subtracting the constant \(1/2\) gives the skew kernel \[S(u,v)=\frac{(u-v)(2(u+v)^2-d(u,v))}{2(u+v)d(u,v)},\qquad d(u,v)=u^2+\sqrt2uv+v^2.\] The adjugate of an even skew matrix is skew. Adding a matrix with constant entries therefore does not change its determinant, and \(\det\mathcal L(1)=(\operatorname{Pf}S(Y))^2\). Whenever this scalar denominator is nonzero, \[\Pi_p=\frac{\det\mathcal L(T)}{\det\mathcal L(1)}\] is a projector onto the common fixed space. Indeed the determinant is taken in the commutative algebra generated by the rows. The adjugate identity for \(\mathcal L(T)U=0\) shows that \(\Pi_p\) annihilates every \(U_{jt}\), while on their joint kernel \(T_j=1\) and \(\Pi_p=1\). Hence its image is exactly that kernel and it is idempotent. Evaluating its polynomial numerator by the arbitrary-word vacuum identity also gives \[\langle0|\Pi_p|0\rangle =\langle0|\Pi_p\Pi_{p'}|0\rangle=1 \qquad(p,p'\in\{i,-i\}).\]

The scalar normalizers

The denominator of the projector is governed by three scalar functions. The factorization below separates the factor \(N\), which will clear the two selected vacuum vectors, from the additional factor \(N_2^*\) present for the full projector.

We use the Pfaffian of an even skew matrix: it is the signed matching sum, equivalently the coefficient of the oriented volume form in the normalized top power of its two-form. It transforms under simultaneous row and column change by the determinant, and its square is the determinant, as follows by reduction to two-dimensional skew blocks. For the site list \(X\), put \(\mathcal P(X)=\prod_{j<k}d(x_j,x_k)/(x_j-x_k)\) and define \[\begin{split} N(X)&=\mathcal P(X)\operatorname{Pf}_{j,k}\frac{x_j^2-x_k^2}{d(x_j,x_k)},\\ N_2^*(X)&=\mathcal P(X)^2\det_{j,k}\frac{(2+\sqrt2)x_jx_k}{d(x_j,x_k)},\\ \mathcal J(X)&=S(1,q)^{-M}\operatorname{Pf} S(Y)\prod_{j<k}P_*(x_j,x_k),\\ P_*(u,v)&=\frac{(u+v)(u^2+v^2)d(u,v)^2}{(u-v)^2(u^2-\sqrt2uv+v^2)} . \end{split}\] When \(M\) is odd, pad the Pfaffian defining \(N\) by a last index whose column entries above the diagonal are 1. All empty-list normalizations are 1. Write \(N_1=N\), \(N_2=N_2^*\), \(N_3=\mathcal J\), and set \[\ell_2(u,z)=z-q^5u,\qquad \ell_3(u,z)=(z-q^5u)(z-q^6u).\]

The fixed vacuum lines

We use widely separated site magnitudes to identify the fixed character and to locate the possible poles of its projection. The following description also records the common similarity needed when the sites of a scale are interspersed in the cyclic order.

Take all scales to be singletons and probe at each singleton’s swap point. On a diagonal block its eigenvalue is \(p_j^{S_j}\). If \(S_j=\pm1\), total charge zero implies \(\sum_{k\ne j}\operatorname{sgn}(j-k)S_k\) is odd. Since \(p=q^{2}\) or \(q^{6}\), the exponent of \(q\) in \(p_j\) is odd, and \(p_j^{S_j}\ne1\). Only the all-empty block can have the simultaneous character, with multiplicity one. A generic linear combination of these finitely many probe deviations therefore has a simple zero eigenvalue in the limit and for sufficiently separated actual sites; zero is always an eigenvalue by the vacuum argument. Rationality in square-root coordinates proves generic simplicity, with no Jordan extension at the fixed character.

Consequently the generic projector has rank one. Set \[L_p=\langle0|\Pi_p,\qquad R_p=\Pi_p|0\rangle.\] Vacuum normalization gives \(\Pi_p=R_pL_p\) and \(L_pR_{p'}=1\) for both twists. We use \(R_+=R_i\), \(L_-=L_{-i}\) and the corresponding shorthand for the other two vectors.

Selected vacuum vectors and polynomial degrees

The projector denominator contains both \(N\) and \(N_2^*\). We now show that \(N\) alone removes the poles of the particular column \(R_i\) and row \(L_{-i}\), after accounting for their square-root parity. This component statement will give the degree bound for the scalar pairing.

At any nonzero site list with \(N(X)\ne0\), this lemma gives componentwise continuation of \(R_i\) and \(L_{-i}\) through the polynomial formulas, including at simultaneous shifted-site coincidences. This is the continuation used later for tilted physical lists; the homogeneous determinant confluence above is a separate construction.

Fusion of the scalar observable

The function \(\mathscr G\) is symmetric within each group. At generic sites the Yang–Baxter similarity interchanges adjacent site parameters, preserves charge on that pair, and works for both twists. It therefore commutes with \(D_A\) for a same-group interchange. The fixed lines transform by this similarity; their normalization scalars cancel in \(L_-D_A R_+\) because \(L_-R_+=1\) before and after the interchange.

For a same-group pair \(u,q^\delta u\), \(\delta=2,3\), the function \(\mathscr G\) reduces to \(\mathscr G(X_{\rm red})\). By symmetry place \(q^\delta u\) immediately before \(u\). Equation (2) then intertwines the full row with a row whose site is \(qu\) for \(\delta=2\), or with both sites removed for \(\delta=3\). Full and reduced fixed lines are simple at generic points of this specialization: separate the other scales and use the active pair test, where \(a(q^\delta)=0\). Thus the specialization is taken by regular eigenprojection. The full right vector is \(J_\delta\) applied to the reduced one, since the vacuum row of \(J_\delta\) extracts the vacuum. The full left vector restricts along \(J_\delta\) to the reduced left vector; its scalar is 1 by the overlap \(L_-R_+=1\). The insertion commutes through \(J_\delta\) because the pair lies in one group. These generic evaluations, together with (4), give exact fusion evaluations of the cleared scalar polynomial.

Normalization on paired scales

When the groups have equal size, the fusion and deletion evaluations leave one scalar coefficient undetermined. We determine it by putting one site of each group on each separated scale, with generic fixed ratios. A charged scale block cannot carry the character. For an odd pair total \(S_j=\pm1\), the product of the two regular site probes is \(p_j^{S_j}\ne1\) by the parity test. Indeed the two transfers on active spaces 1,2 are \(R_{12}p_j^{s_1}\) and \(p_j^{s_2}R_{21}\) at opposite differences, and unitarity gives the product. For \(S_j=\pm2\), the first probe gives \(p_j^{\pm1}a\ne1\) at a generic ratio.

There is consequently an isolated simple character in the product of neutral pair blocks, with the original twists. In the vacuum matrix element only its vacuum grade block contributes. The insertion \(D_A\) is diagonal and commutes with the common diagonal similarity \(F\). That similarity therefore cancels between the two projections, and its entry on the empty vector is 1. The result is a product of pair projections. For a pair whose first coordinate \(u\) is in \(A\) and second coordinate \(v\) is in \(B\), the formulas for \(l,r\) and the sine weights give \[\frac{d_{\rm tile}^2}{(1-a)^2} =(2-\sqrt2)\frac{uv}{(u+v)^2}.\] Thus the paired-scale limit is \[ \mathscr G\ \longrightarrow\ \prod_{\rm pairs}\left(1+ (2-\sqrt2)(h+h^{-1})\frac{uv}{(u+v)^2}\right) \tag{5}\] (read at the limiting ratios).

The scalar characterization

Put \(\mathscr F_{A,B}=N(X)^2\mathscr G(X;A,B)\), and write \(e_k(u,v)=v-q^ku\). In a deletion or fusion, the remaining coordinates keep their groups and cyclic order; a fused site belongs to the group of its two predecessors.

Finite spin sum

We now verify an explicit family with the properties of Lemma 10. Use indices \(s,t=-1,0,1\) (symbols \(-,0,+\) in tables), and let \(\kappa_{st}\in\{-1,0,1\}\) be congruent to \(s-t\) modulo 3. Define \[p_{st}(u,v)=q^{-\kappa_{st}}P_{st},\qquad p'_{st}(u,v)=q^{\kappa_{st}}P'_{st}\] by the following products, listing ascending order; for descending use the swapped-index products with \(k\mapsto-k\) throughout. \[\begin{array}{c|cc} (st)&P&P'\\ \hline (--),(++)&e_2 e_3 e_5 e_6&1\\ (00)&e_2 e_3 e_5 e_6/(e_1e_7)&e_4^2\\ (-0),(0+)&e_3 e_5 e_6/e_0&e_5^2\\ (-+)&e_5e_6/(e_0e_1)&(e_5 e_6)^2 \end{array}\] Both phased products are symmetric under simultaneous swap of spins and arguments, since \(e_k(v,u)=-q^k e_{-k}(u,v)\). Set \(c=2-\sqrt2\) and \(d_{\rm a}=i\sqrt c,\ d_{\rm b}=-i\sqrt c\) on the two groups.

Defects and their pair kernels

We split the factors between nonzero spins into a Schur Pfaffian and a Gaussian quadratic form. The accompanying one-site factor will provide the poles whose residues select the sites carrying nonzero spins.

First let both groups have the same generic distinct real coordinates \((w_i)_{i=1}^m\). Regard nonzero spin as a defect labeled \(\delta=-s=\pm1\) with centered argument \(v_i=w_i+i\delta/2\) (in shifts \(i\) denotes the imaginary unit). Define \[\begin{split} V_-(w)&=\frac{e_1 e_7 e_3^2}{e_0 e_6 e_4^2},\qquad V_+(w)=\frac{e_1 e_7 e_5^2}{e_0 e_2 e_4^2}=\frac1{V_-(w+i)},\qquad e_k=1-q^k e^{\eta w},\\ L(d)&=\frac{(y-x)^2(y^2+x^2)}{xy(y^2-\sqrt2xy+x^2)},\qquad J(d)=\frac{xy(y+x)^2}{(y^2+\sqrt2xy+x^2)^2},\qquad y/x=e^{\eta d}. \end{split}\] Then the normalized sum uses one factor \[z^\delta\,\frac{p_0\sqrt c}{4}\prod_{l\ne i}V_\delta(w_i-w_l)\] per defect and pair factors \(L(v_j-v_i)^{\delta_i\delta_j}\) for the same group (species), \(J(v_j-v_i)^{\delta_i\delta_j}\) across species, subject to equal total \(\delta\).

Indeed remove powers \((xy)^{st}\) from same-group pairs of the table, \((xy)^{-st}\) across, and \(x^{s^2}\) per site (total 1). For \(s=+\) against two matched zeros the same-group ratio to baseline is \(q^{-1}e_1e_7/(e_0e_6)\) and the cross ratio \(q e_3^2/e_4^2\); for \(s=-\) reflect \(q\). At its own matched coordinate only the second contributes (\(p_0/4\)). Dividing the two-defect ratio by the two corresponding one-defect ratios gives \((\delta_i\delta_j)L(v_j-v_i)^{\delta_i\delta_j}\) or \((\delta_i\delta_j)J(v_j-v_i)^{\delta_i\delta_j}\) from the pair table (the argument shift multiplies \(y/x\) by \(q^{(s_i-s_j)/2}\)). Total defect number is even, so the signs multiply to \((-1)^{n_-}\) with \(n_-\) counting negative \(\delta\). This cancels the site phase \(i^{n_{\rm a}-n_{\rm b}}\) (\(n_{\rm a},n_{\rm b}\) the defect counts per group) by equal total charges.

Introduce \[\begin{gathered} a_*=\pi/6,\quad C(s)=\tanh(a_*s),\\ R^0(s)=C(s)^2/[C(s+i)C(s-i)],\\ G_d(s)=\log(R^0(s)/L(s)),\quad G_c(s)=-\log J(s). \end{gathered}\] Take the real logarithms on the real line, with the limiting value at \(s=0\), and then their analytic continuations. The matrix kernel \(G\) has diagonal \(G_d\) and off-diagonal \(G_c\), with \(G_d(0)=-\log A_*\), \(A_*=3/(2c)\). At infinity \(G_d(s)+\eta|s|,\ G_c(s)-\eta|s|\) decay exponentially. In Fourier notation \(\widehat f(k)=\int e^{-ikv} f(v)\,dv\), the transforms of \(G_d,G_c\) themselves at \(k\ne0\), in units of \(2\pi/k\), are \[g_d=\frac{\cosh4k+\cosh2k-\cosh3k}{\sinh4k} -\frac{\cosh3k-\cosh2k+\cosh k-1}{\sinh3k},\qquad g_c=\frac{1-2\cosh k}{\sinh4k}.\] The difference channel is used as covariance only with neutral tests. For instance \(\log[2(\cosh(\eta v)-\cos(\eta b))]\), \(0\le b\le4\), gives \(-(2\pi/k)\cosh((4-b)k)/\sinh(4k)\) off zero, by expanding the log after subtracting \(\eta|v|\) (sum of rational transforms is the elementary cosine series of \(\cosh((4-b)k)\) on \(0\le b\le4\)). The same rule at period 6 gives the terms of \(\log R^0\) using absolute values paired for conjugates. Constants at infinity fix transforms of integrable combinations; a residual linear \(|v|\) part uses the usual neutral kernel by integrating \((1-\cos(kv))/k^2\). The covariance is positive in both channels; for \(k>0,\ t=e^{-k}\), \[g_d+g_c=\frac{t^4(1-t)}{(1+t)(1+t^2)(1+t^4)(1-t+t^2)},\qquad g_c<0.\] Write \(\widehat G_\pm=\widehat G_d\pm\widehat G_c\). The same formulas give \[ \begin{aligned} \widehat G_+(k)&=\pi/4+O(k^2),& \widehat G_-(k)&=\pi/k^2+O(1) &&(k\longrightarrow0),\\ \widehat G_+(k)&=O(e^{-4|k|}/|k|),& \widehat G_-(k)&=O(e^{-3|k|}/|k|) &&(|k|\longrightarrow\infty). \end{aligned} \tag{7}\] Thus the common channel has the ordinary value \(\widehat G_+(0)=\pi/4\), whereas the difference channel is applied to tests of total mass zero. The neutral formulas continue analytically for differences of arguments in \(|\Im s|<3\).

For \(k\ne0\), define multipliers \[E(k)=\frac{\cosh(3k/2)}{2\sinh(k/2)\cosh k},\qquad D(k)=\frac{k}{2\pi}E(k),\] and decaying functions \(a,\lambda_0\) by odd real transforms, at \(k>0\) given by \[\widehat a(k)=\frac{2\pi}{k}\frac{t^{1/2}(1-t)(1-t^4)}{(1+t^3)^2},\qquad \widehat\lambda_0=D\widehat a=\frac{\sinh k}{\cosh(3k/2)}.\] Both transforms have value zero at \(k=0\). In particular \(a\) and \(\lambda_0\) are purely imaginary on the real line. Define \[U_\delta(w)=V_\delta(w)\exp\!\left(\delta[(G_d+G_c)*\lambda_0](w+i\delta/2)\right).\] We claim that, for \(|\Im v|<1/2\), \[U_-(v+i/2)=e^{a(v)},\qquad U_+(v-i/2)=e^{-a(v)}.\] For the first identity, note that \(\log V_-(v+i/2)\) is decaying with zero winding near the real \(v\) line: factors \(1-e^{\eta(v+i b)}\) at \(0<b<8\) have phase change \(\eta b-\pi\), and the signed multiplicities at \(1.5,7.5,3.5,0.5,6.5,4.5\) are respectively \((1,1,2,-1,-1,-2)\), balanced also in the sums of \(b\). In fact if a sum of such logs (plus a constant) has multiplicities \(n_b\) with \(\sum n_b=0\), its derivative decays exponentially at both horizontal infinities and is \(8i\)-periodic with residues \(n_b\) at \(-ib\) modulo period. Shifting the derivative transform down by \(8i\) (positive \(k\)) thus gives the log transform \(-(2\pi/k)\sum_b n_b t^b/(1-t^8)\) there. This off-zero calculation also applies for bounded logs with step limits by taking the transform via differentiation. For our decaying log it gives the expression \((2\pi/k)t^{1/2}(1-t)(1-t^3)^2/(1-t^8)\) (extended oddly). This equals \((1+D(\widehat G_d+\widehat G_c))\widehat a\). The Fourier bounds give the stated strip continuation, and \(V_+(w)=V_-(w+i)^{-1}\) gives the second identity. In particular \(U_+(w)=U_-(w+i)^{-1}\) meromorphically. The decays and strip continuations also follow by shifting Fourier integrals; the convolution in \(U\) continues well beyond \(|\Im|<2\).

A fixed-site periodic Pfaffian

Fix a horizontal period \(T>0\). For a decaying function, subscript \(T\) denotes the sum over translates by \(T\); for a meromorphic function tending to \(1\) exponentially at both horizontal infinities, it denotes the product over those translates. These sums and products converge locally uniformly on the strips used below, away from their stated poles. An integral \(\int_T dx\) is over one full period and is unnormalized. For \(k=2\pi n/T\), the coefficient of \(e^{ikx}\) in a periodized decaying function is \(\widehat f(k)/T\).

Let \(P_\pm\) be the orthogonal projections on the spans of \((1,1)^t\) and \((1,-1)^t\). Take centered real Gaussian oscillators \[\chi_\sigma(v)=\sum_{k=2\pi n/T\ne0}\chi_{\sigma,k}e^{ikv}, \qquad \chi_{\sigma,-k}=\overline{\chi_{\sigma,k}},\] whose covariance between the coefficients at \(k,-k\) is \(\widehat G(k)/T\), and whose other mode covariances vanish. Positivity follows from the two channels above. By (7), \[\mathbb E\sum_{k\ne0}|\chi_{\sigma,k}|e^{\rho|k|}<\infty \qquad(0\le\rho<3/2).\] Consequently these series are almost surely analytic on \(|\Im v|<3/2\), with absolute convergence on every smaller closed strip.

Choose a centered common Gaussian vector with covariance \(\widehat G_+(0)P_+/T\) and a uniform vector \((t_0,-t_0)\) with \(0\le t_0<2\pi\), independently of each other and of \(\chi\). Their sum is denoted by \((\varphi_{01},\varphi_{02})\). Independently of all preceding variables, choose \[\boldsymbol\ell=(\ell_1,\ell_2)=(j,-j),\quad j\in8\mathbb Z,\qquad \mathbb P(j)=D_\ell^{-1}e^{-4\pi j^2/T},\quad D_\ell=\sum_{j\in8\mathbb Z}e^{-4\pi j^2/T},\] and set \[\varphi_\sigma(v)=\varphi_{0\sigma}+(2\pi/T)\ell_\sigma v+\chi_\sigma(v).\] We write \(\varphi(v)=(\varphi_1(v),\varphi_2(v))^t\).

For each species use a Wick expectation: the expectation of an ordered list of an even number of fields is the Pfaffian of their two-point entries, and odd expectations are zero. Its two-point function is \[\langle\psi(u)\psi(v)\rangle_T=C^{[T]}(v-u),\qquad C^{[T]}(d)=\sum_{n\in\mathbb Z} \frac{\pi}{a_*T\sin(\pi(d+3i+6in)/T)}.\] For fixed \(T\) the image series converges locally uniformly away from its poles. It is odd, \(6i\)-periodic, horizontally antiperiodic, and has residue \(1/a_*\) at \(3i\).

To define the contour expression also at coalescing sites, choose \[0<r<\min\{1/8,T/8\},\qquad |w_l|<r/4\quad(1\le l\le m),\] initially with distinct real \(w_l\), and let \(\gamma_r\) be the positive circle \(|w|=r\). Put \(W=(w_1,\ldots,w_m)\) and \[\Lambda_W(x)=\sum_{l=1}^m\lambda_{0,T}(x-w_l).\] Define, at first as its Wick expansion, \[ \mathcal C_{\sigma,T}(W)= \left\langle\exp\left( a_*\sum_{\delta=\pm1}\oint_{\gamma_r}\frac{dw}{2\pi i} (-z^\delta)e^{i\delta\varphi_\sigma(w+i\delta/2)} \prod_{l=1}^m U_{\delta,T}(w-w_l)\, \psi(w+3i)\psi(w+i\delta)\right)\right\rangle_T . \tag{8}\] The same list \(W\) is used for both species. The normalized expression is \[ \mathfrak C_T(W;\theta)= \frac{\mathbb E\!\left[ e^{i\int_T\Lambda_W(x)(\chi_1+\chi_2)(x)\,dx} \mathcal C_{1,T}(W)\mathcal C_{2,T}(W)\right]} {\mathbb E e^{i\int_T\Lambda_W(x)(\chi_1+\chi_2)(x)\,dx}}. \tag{9}\] At \(W=0^m=(0,\ldots,0)\) write \(\mathcal C_\sigma=\mathcal C_{\sigma,T}(0^m)\) and \(\mathfrak C_{m,T}=\mathfrak C_T(0^m;\theta)\). For comparison with the finite formula put \[x_l=e^{\eta w_l},\qquad X(W)=(x_1,\ldots,x_m,x_1,\ldots,x_m),\qquad P_0(W)=\prod_{i<j}p_{00}(x_i,x_j)^2 \prod_{i,j}p'_{00}(x_i,x_j).\] This is the all-zero term for the matched list. The displayed domain avoids its zeros and poles, and its value at \(W=0^m\) is \(P_0\). The first and second copies in \(X(W)\) belong to \(A\) and \(B\), respectively.

The proposition takes the infinite-period limit at fixed \(m\), before introducing any infinite expansion of trace operators. The identities in the rest of this section hold for each fixed \(T>0\). Their use as an absolutely integrated horizontal gas will require the later stability estimate, which is needed only for sufficiently large integer block periods.

A conditional angular trace

We now fix \(T>0\), \(m\), \(\theta\), and one analytic sample of the fields, and use homogeneous sites. The contraction rule is the normal-ordering principle of Wick [17]; here we compute its periodic covariance and trace realization. Suppress the species index. Fold the imaginary coordinate into six sheets, indexed cyclically by \(j=1,\ldots,6\), and write \(j^*=j+3\). On the fermionic exterior space take \[\{\Psi_{j,r},\Psi_{l,s}\}=\delta_{j^*,l}\delta_{r,-s}, \qquad \Psi_{j,r}^*=\Psi_{j^*,-r}, \qquad r,s\in\tfrac12+\mathbb Z, \qquad \Psi_j(v)=\sum_r\Psi_{j,r}y^{-r},\quad y=e^{2\pi i v/T}.\] The modes with \(r>0\) annihilate the vacuum. Let \(L_0\) be total creation energy, let \(\mathbf q=e^{-2\pi/T}\), and let \(\mathcal U\) fix the vacuum and send \(\Psi_j\) to \(-\Psi_{j-1}\) under conjugation. Put \(\Gamma=\mathbf q^{L_0}\mathcal U\).

For the off-diagonal roots below define \[X_{bc}(f)=-\sum_{n\in\mathbb Z}f_n \sum_{r+s=n}\Psi_{b^*,r}\Psi_{c,s} =-\int_T\frac{dv}{T}\,f(v)\Psi_{b^*}(v)\Psi_c(v), \qquad J_{bc}=\mathrm e_{bc}-\mathrm e_{c^*b^*},\] where \(f(v)=\sum_nf_ny^n\) is analytic and periodic and the integral means its modewise Fourier integral. We only use \(b\notin\{c,c^*\}\). The notation \(X_{bc}(f;\gamma)\) uses a shifted horizontal line \(\gamma\); it gives the same Fourier integral when \(f\) is analytic between the lines. As a formal mode series, the commutator of \(X_{bc}(f)\) with the column of fields is \(fJ_{bc}\) times that column. Thus the matrices of conjugations multiply in the reverse order from the operators.

For the fixed sample set \[b(v)=z^{-1}e^{-i\varphi(v)}[U_{-,T}(v+i/2)]^m, \qquad b_\pm(v)=b(v\pm i).\] This function is periodic, since \(\ell\in8\mathbb Z\). For \(U_{-,T}(v+i/2)\), the nearest zero and pole occur at \(v=i/2\) and \(v=-i/2\) modulo \(T\), respectively; the next divisors occur at \(v=\pm3i/2\) modulo \(T\). Its exponential factor is nonzero wherever it is analytic. Combining these facts with the sample strip \(|\Im v|<3/2\) shows that \(b,b_+,b_-\) and their inverses are analytic on \(|\Im v|<1/2\), bounded on each smaller closed strip over one period. These are conditional bounds depending on the sample and the fixed parameters.

The seam operator and outgoing coefficients

The four-current trace can be reorganized so that one operator carries the exchange of sheets \(1,2\). Define \[W_b=e^{-X_{12}(b^{-1})}e^{X_{21}(b)}e^{-X_{12}(b^{-1})}.\] It acts on the formal field series by the matrix \(P_b\) whose blocks on sheets \(1,2\) and \(4,5\) are \[P_b|_{1,2}=\begin{pmatrix}0&-b^{-1}\\ b&0\end{pmatrix}, \qquad P_b|_{4,5}=\begin{pmatrix}0&-b\\ b^{-1}&0\end{pmatrix} =(P_b|_{1,2})^{-t},\] and which fixes sheets \(3,6\). The exact operator identity is \[ \begin{aligned} &e^{X_{23}(b_+^{-1})}e^{X_{21}(b)} e^{-X_{12}(b^{-1})}e^{-X_{16}(b_-)} \\ &\qquad= e^{X_{26}(bb_-)}e^{X_D}W_b e^{-X_{13}((bb_+)^{-1})},\\ &X_D=X_{12}(b^{-1})+X_{13}((bb_+)^{-1}) -X_{16}(b_-)+X_{15}(b_-/b_+). \end{aligned} \tag{16}\] We verify the scalar as well as the field action. Put \(A_{\rm r}=X_{23}(b_+^{-1})\), \(B_{\rm r}=X_{26}(bb_-)\), and \(C_{\rm r}=X_{12}(b^{-1})\). Isolating the defining three-root word for \(W_b\) and conjugating the left field of \(X_{16}\) by \(W_b\) writes the left side as \(e^{A_{\rm r}}e^{C_{\rm r}}e^{B_{\rm r}}W_b\). The roots \(A_{\rm r},B_{\rm r}\) commute, with no central term: their only possible single contraction leaves two copies of \(\Psi_5\), and two contractions are impossible. Thus the middle exponent after moving them past it is \(e^{-\operatorname{ad}B_{\rm r}}e^{\operatorname{ad}A_{\rm r}}C_{\rm r}\). In this conjugation \(\Psi_4\) stays fixed and its partner changes by \[\Psi_2\longmapsto \Psi_2+b_+^{-1}\Psi_3-bb_-\Psi_6+(bb_-/b_+)\Psi_5.\] Multiplying by \(b^{-1}\) gives exactly \(X_D\). Also \(W_b^{-1}A_{\rm r}W_b=-X_{13}((bb_+)^{-1})\). Every transformed partner in these steps is nondual to its first field, so normal ordering produces no scalar contraction. The graded convergence in Lemma 13 makes these conjugations identities of the operators used in the trace. The field calculation therefore fixes the scalar as well as the matrices \(J_{bc}\) and proves (16).

The part of \(W_b\) depending on the oscillators will be evaluated in Section 5. Periodizing \(U_-(v+i/2)=e^{a(v)}\) gives \(U_{-,T}(v+i/2)=e^{a_T(v)}\) on \(|\Im v|<1/2\). Hence write \[b=b_w e^{\mathsf A},\qquad b_w=b_0y^{-\ell},\qquad b_0=e^{-i(\theta+\varphi_0)}, \qquad \mathsf A=ma_T-i\chi=\sum_n\mathsf A_ny^n .\] Here \(\mathsf A_0=0\) because \(\widehat a(0)=0\) and \(\chi\) has zero spatial mean. For \(j=1,2,3\) use the normally ordered currents \[h_{j,n}=\sum_{r+s=n}:\Psi_{j,r}\Psi_{j^*,s}:, \qquad [h_{j,n},\Psi_j(v)]=y^n\Psi_j(v).\] Then \[ W_b=W_{b_w} \exp\!\left(\sum_n\mathsf A_n(-h_{1,n}+h_{2,n})\right). \tag{17}\] Indeed, with \(K=\frac12\sum_n\mathsf A_n(h_{1,n}-h_{2,n})\), conjugating the defining root word for \(W_{b_w}\) by \(e^K\) changes its coefficients to those of \(W_b\). The action of \(W_{b_w}\) interchanges the nonzero \(h_{1,n},h_{2,n}\), so \(W_{b_w}KW_{b_w}^{-1}=-K\) and \(e^KW_{b_w}e^{-K}=W_{b_w}e^{-2K}\). Only nonzero modes enter. At zero mode the Cartan difference would instead transform to \(-(h_{1,0}-h_{2,0})+2\ell\), which explains the use of \(\mathsf A_0=0\).

Finally we put each fixed coefficient of the remaining three exponentials on the outgoing side of \(W_b\). On the finite-energy core the mode identity is \[\Gamma\Psi_{j,r}\Gamma^{-1}=-\mathbf q^{-r}\Psi_{j-1,r}.\] Equivalently, as a formal mode series, \[\Gamma\Psi_j(v)\Gamma^{-1}=-\Psi_{j-1}(v+i).\] The corresponding current rule, including its scalar argument, is \[ \Gamma X_{bc}(f;\gamma)\Gamma^{-1} =X_{b-1,c-1}\bigl(f(\,\cdot-i\,);\gamma+i\bigr). \tag{18}\] The two field signs cancel and the orientation is unchanged. In a trace this identity is first applied to fixed modes; the movement estimates in the proof of Lemma 13, before factorial summation, give absolute sums for each fixed monomial and permit their reindexing and Fourier integration.

More explicitly, let \(Y_L,Y_D,Y_U\) be fixed monomials from the leftmost, middle, and rightmost exponentials of (16). Cyclicity gives \[\operatorname{Tr}_{\rm gr}(\Gamma Y_LY_DW_bY_U) =\operatorname{Tr}_{\rm gr} \bigl(\Gamma W_bY_U(\Gamma Y_L\Gamma^{-1}) (\Gamma Y_D\Gamma^{-1})\bigr).\] The denominator remains \(Z_\Gamma\). Call the rightmost root \(U\), the cycled leftmost root \(L\), and retain \(D\) for the middle sum. Their outgoing order is \(U,L,D\). One may initially put the unshifted \(U\) pairs just above folded height \(0\), and the incoming \(L,D\) pairs on lines \(-\rho_L,-\rho_D\) with \(0<\rho_D<\rho_L<1/2\). After (18) these latter lines have heights \(1-\rho_L,1-\rho_D\). Further infinitesimal separations realize strict order \[0<\eta_U<\eta_L<\eta_D<1\] for each fixed coefficient, within the original scalar analytic strips. Section 5 evaluates these ordered coefficients and establishes its further contour displacements from their paired analytic expressions.

Thus the Gaussian expectation of the total trace is defined by its equality to the finite contour polynomial. Passing that expectation through the reorganized infinite expansion requires the absolute estimate of Section 6. The conditional trace identities here hold for every fixed \(T>0\); the later integrated gas identity uses sufficiently large integer periods to which that estimate applies.

Charge sectors and the seam trace

Work first with one species. Write \(\Psi_j^+=\Psi_j\) and \(\Psi_j^-=\Psi_{j^*}\) for \(j=1,2,3\). The eigenvalues \(c=(c_1,c_2,c_3)\in\mathbb Z^3\) of the three zero modes \(h_{j,0}\) label charge sectors. In each sector the nonzero modes satisfy \[[h_{j,n},h_{l,s}]=n\delta_{n,-s}\delta_{jl},\qquad h_{j,n}^*=h_{j,-n},\] and generate an oscillator space from a ground state \(|c\rangle\) of energy \(|c|^2/2\). To define this ground state, for each \(j\) fill the creators of sign \(\operatorname{sgn}c_j\) up to energy \(|c_j|-1/2\), with the largest energy leftmost, and put the three lists in the order \(j=1,2,3\). The anticommutation relations give the displayed commutator: changing normal order contributes exactly \(n\) mode pairs for \(n>0\). The positive oscillator modes kill \(|c\rangle\).

These oscillator spaces account for all fermionic states. For one value of \(j\), encode a configuration by occupied positions in \(\tfrac12+\mathbb Z\), starting with the negative positions filled. A positive-sign creator inserts a particle at a positive position, and a negative-sign creator deletes one at a negative position. In charge \(c_j\), the occupied positions in decreasing order are \[c_j+\tfrac12-k+\lambda_k,\qquad k\ge1,\] for a unique partition \((\lambda_k)\). Every partition occurs, and its excess energy is \(\sum_k\lambda_k\). These are exactly the multiplicities generated by the negative oscillator modes.

Let \(R_j\) raise \(c_j\) by one, commute with nonzero modes, and act on the charge basis with Klein sign \((-1)^{\sum_{l<j}c_l}\). In matrix elements the field is \[ \Psi_j^\epsilon(v)=y^{1/2}R_j^\epsilon y^{\epsilon h_{j,0}} \exp\!\left(\epsilon\sum_{n>0}y^n h_{j,-n}/n\right) \exp\!\left(-\epsilon\sum_{n>0}y^{-n}h_{j,n}/n\right), \qquad y=e^{2\pi i v/T},\quad \epsilon=\pm1. \tag{19}\] Both sides have the same commutators with all nonzero \(h\) modes. Their matrix elements between ground states agree by inserting or deleting the largest filled mode, which fixes the remaining factor and sign. Powers of \(y\) use the indicated coordinate before exponentiation.

Recall from the preceding section that \[b=b_w e^{\mathsf A},\qquad b_w=b_0y^{-\ell},\qquad b_0=e^{-i(\theta+\varphi_0)},\qquad \mathsf A=m a_T-i\chi,\] where \(\mathsf A\) has zero spatial mean, and that \[W_b=W_{b_w}\exp\!\left(\sum_n\mathsf A_n(-h_{1,n}+h_{2,n})\right).\] We also retain \(Z_\Gamma=\mathrm{tr}\Gamma\) from the conditional trace. The action of \(W_{b_w}\) on a ground state is \[ W_{b_w}|c\rangle =(-1)^{c_1+c_1c_2}b_0^{\ell+c_2-c_1} |c_2+\ell,c_1-\ell,c_3\rangle, \tag{20}\] and it swaps oscillator labels 1 and 2. Here is a verification that also fixes its scalar. In the three-root word for \(W_{b_w}\), the modes with \(r+s=\ell\) form commuting blocks \((1+F)(1+E')(1+F)\), where \[F=b_0^{-1}\Psi_{4,r}\Psi_{2,s},\qquad E'=-b_0\Psi_{5,-s}\Psi_{1,-r}.\] When \(r,s\) have opposite signs the block fixes the vacuum. When both are positive it contributes \(E'\), and when both are negative it contributes \(F\). Put the species-1 creator first in each contributing block. The sign for grouping creators by species cancels the sign for reversing the second energy list, giving \(W_{b_w}|0\rangle=b_0^\ell|\ell,-\ell,0\rangle\). This calculation can be made on finite occupation sets, since only finitely many blocks act nontrivially there. Conjugation of fields sends \(\Psi_1\mapsto-b_0^{-1}y^\ell\Psi_2\) and \(\Psi_2\mapsto b_0y^{-\ell}\Psi_1\). With the Klein signs, increasing \(c_1\) changes the ground coefficient by \((-1)^{1+c_2}b_0^{-1}\) and increasing \(c_2\) changes it by \((-1)^{c_1}b_0\). Increasing \(c_3\) changes nothing. These ratios and the vacuum value prove (20); the evenness of \(\ell\in8\mathbb Z\) is used here.

Rotation sends a ground charge \(d\) to \((d_2,d_3,-d_1)\) with sign \((-1)^{\sum d_j+d_1(d_2+d_3)}\). The first part counts the minus sign on each creator, and the second moves the first species list past the other two. Applying rotation to (20) therefore gives \[ \Gamma W_{b_w}|c\rangle =\mathbf q^{|c''|^2/2}(-1)^{c_2+c_3+c_2c_3} b_0^{\ell+c_2-c_1}|c''\rangle, \qquad c''=(c_1-\ell,c_3,-c_2-\ell). \tag{21}\] On creation oscillators of frequency \(n>0\), in addition to this ground state action, it acts by \[ B_n=\mathbf q^n\operatorname{diag}(1,R),\qquad R=\begin{pmatrix}0&1\\-1&0\end{pmatrix}. \tag{22}\]

Consider a fixed coefficient in the three current exponentials on the two sides of \(W_b\). By the cyclic rule (18), its outgoing field pairs can be put to the right of \(W_b\) in the order \(U,L,D\). Here \(U\) denotes the rightmost root, \(L\) the cycled leftmost root, and \(D\) the middle exponential; their individual labels will be specified below. Each \(\Psi\) carries again the physical multiplier \((-2\pi i/(a_*T))^{1/2}\). Consequently a current \(X_{bc}(f)\) contributes \((a_*/(2\pi i))f(v)\,dv\) together with its pair of physical fields. For a field of sign \(\epsilon\) at folded coordinate \(v\), define its centered coordinate \(w\) by \[ w=v-i/2\quad(j=1,\text{ short}),\qquad w=v-i\quad(j=2,\text{ outer}),\qquad w=v\quad(j=3,\text{ outer}). \tag{23}\]

Let the total inserted charges of types \(1,2,3\) be \((S_1,E_2,F_3)\), and let \(n=(n_1,n_2,n_3)\) be the ground charge to the right of the fields. Applying (21) to \(c=n+(S_1,E_2,F_3)\) and imposing trace closure gives \[ S_1=\ell,\qquad n_2=(F_3-E_2-\ell)/2,\qquad n_3=(-F_3-E_2-\ell)/2. \tag{24}\] The last two values are integral: every current has two fields, so \(S_1+E_2+F_3\) is even, and \(\ell\) is even. The charge \(n_1\) remains free. The \(b_0\) factor becomes \(b_0^{(F_3+E_2-\ell)/2-n_1}\).

For later sign accounting, order all fields from left to right. If \(N_j\) is the number of fields of type \(j\), the Klein exponent apart from inversions is \(n_1(N_2+N_3)+n_2N_3\). Add the exponent in (21) and substitute (24). The result modulo two is \[ (F_3+E_2)/2+\#(3\text{ before }2) +\#(\text{outer before short}). \tag{25}\] For example, \(N_2+N_3\) is even and the non-inversion remainder reduces to \(n_2+F_3\equiv(E_2+F_3)/2\pmod2\). The initial ground-coordinate power on field \(i\) of type \(j\) is \(y_i^{1/2+\epsilon_i(n_j+\sum_{l>i,\,j_l=j}\epsilon_l)}\).

Oscillator contractions at fixed insertion order

For normalized creators with coefficient vector \(u\) followed by normalized annihilators with coefficient vector \(v'\), the oscillator trace with propagator \(B_n\), divided by its empty trace, is \[\exp\!\left({v'}^t B_n(I-B_n)^{-1}u\right).\] For one oscillator with multiplier \(t\), cyclicity and the commutator give \(\langle(a^*)^l a^l\rangle=l!(t/(1-t))^l\) and zero for unbalanced moments; diagonalizing the unitary part gives the vector formula. Reordering two exponentials adds the contraction of the earlier annihilator with the later creator. An unsplit exponential adds half of its internal contraction.

Apply these rules to the seam source \(\exp(\sum_n\mathsf A_n(-h_{1,n}+h_{2,n}))\). For \(k=2\pi n/T>0\) and \(s=(-1,1,0)^t\), (22) gives \[s^t\left(\tfrac12+B_n(I-B_n)^{-1}\right)s =\tfrac12\coth(k/2)+\tfrac12\tanh k =\frac{\cosh(3k/2)}{2\sinh(k/2)\cosh k}=E(k).\] Since \(D(k)=kE(k)/(2\pi)\), the source self-contraction is \(\exp(\tfrac12\int_T\mathsf A D\mathsf A)\). Its cross contractions dress a positive short and a positive outer field respectively by \[ f_s^{\rm osc}(w)=-\sum_{k=2\pi n/T\ne0} \frac{\mathsf A_n e^{ikw}}{2\sinh(k/2)},\qquad f_o^{\rm osc}(w)=\sum_{k=2\pi n/T\ne0} \frac{\mathsf A_n e^{ikw}}{2\cosh k}. \tag{26}\] A negative field receives the negative exponent. To check the centering in this formula, the creator contraction uses \(s^t(I-B_n)^{-1}\) and the annihilator contraction uses \(B_n(I-B_n)^{-1}s\). The short coordinate has \(y^n=\mathbf q^{n/2}e^{ikw}\); the two outer coordinates have \(y^n=\mathbf q^ne^{ikw}\) and \(e^{ikw}\). Substitution gives the two displayed multipliers. The empty oscillator trace is \[ D_W=\prod_{n\ge1}(1-\mathbf q^n)^{-1}(1+\mathbf q^{2n})^{-1}. \tag{27}\]

For fields \(i<l\) put \(d'=w_l-w_i\) and \(Z=e^{2\pi i d'/T}\). Each short field has self-factor \(H_s=\prod_{n\ge1}(1-\mathbf q^n)\), and its pair base is \[F_s(Z)=(1-Z)\prod_{r\ge1}(1-\mathbf q^rZ)(1-\mathbf q^r/Z).\] Each outer field has self-factor \(H_o=\prod_{r\ge1}(1-\mathbf q^{2r})^{(-1)^r}\), and its pair base is \[F_o(Z)=(1-Z)\prod_{r\ge1} [(1-\mathbf q^{2r}Z)(1-\mathbf q^{2r}/Z)]^{(-1)^r}.\] Every pair base is raised to \(\epsilon_i\epsilon_l\). Use \(F_o(Z^{-1})\) when a type-3 field precedes a type-2 field. For example, the outer block of \(B_n(I-B_n)^{-1}\) at \(t=\mathbf q^n\) is \((tR-t^2I)/(1+t^2)\); combining it with the two centered \(y\) powers gives this orientation. There is no oscillator pair contraction between a short and an outer field.

These formulas hold for the fixed current coefficient under discussion. Here are the convergence details needed for that assertion. Start with outgoing folded coordinates in strict strip order \(0<\eta_1<\cdots<\eta_a<1\), where \(\eta_i=\Im v_i\). The uncycled pairs start slightly above zero; the cycled pairs start slightly below one. Cyclicity may first be applied to individual mode pairs. At fixed analytic loop coefficients, a quadratic Fourier mode of energy increment \(\Delta N\) has only \(O(1+N+|\Delta N|)\) potentially acting terms on energy \(N\). Exponential Fourier decay and the graded kernel estimate of Lemma 13 therefore make the integrated sums absolute under \(\mathbf q^{L_0}\). Cycling reindexes these absolutely convergent energy and occupation sums.

At separated field coordinates, let \(N_0\) be the energy just left of the first field, \(N_i\) the energy to the right of field \(i\), and \(N_a\) the terminal energy. A field mode \(r\) changes energy by \(-r\), so its coordinate power together with \(\Gamma\) supplies the damping \[\exp\!\left[-\frac{2\pi}{T}\left( \eta_1N_0+\sum_{i=1}^{a-1}(\eta_{i+1}-\eta_i)N_i +(1-\eta_a)N_a\right)\right].\] The energy dimensions and the remaining graded kernels grow only subexponentially. Inside one bilinear no internal contraction is possible; if its two coordinates coincide, the intermediate energy is bounded by the sum of its bracketing energies and the number of mode splittings is polynomial. The same formula is therefore integrable on the original nearby lines, also by a coincident-coordinate limit within a bilinear.

One may equivalently sum the same finite-level matrix elements in oscillator bases and first truncate the oscillator modes. After replacing coefficients by their absolute values, \(B_n\) is bounded by \(\mathbf q^n\) times a permutation. Direct creator–annihilator sums have the ordered-radius gap, thermal wraps have the remaining annulus gap, and source terms have the Fourier decay of \(\mathsf A\). Their large-\(n\) logarithms are summable. The ground sums have Gaussian decay at this fixed field list and winding. This permits removal of the truncations and proves the oscillator formulas on the initial lines. Subsequent fixed-term moves use analytic continuation of these formulas. Absolute integration over the Gaussian samples will be proved separately below.

Changing the period

The product formulas that turn the angular contractions into horizontal ones are \[ \begin{split} Z^{-1/2}F_s(Z)/H_s^2 &=\frac{-i}{T}e^{-\pi d'^2/T} \frac{2\sinh\pi d'\prod_{r\ge1} (1-e^{-2\pi(rT+d')})(1-e^{-2\pi(rT-d')})}{H_*^2},\\ H_s&=\sqrt T\,e^{-\pi T/12+\pi/(12T)}H_*, \qquad H_*=\prod_{r\ge1}(1-e^{-2\pi Tr}),\\ Z^{-1/2}F_o(Z) &=-i\mathbf q^{-1/2}\tanh(\pi d'/4) \prod_{r\ge1}\tanh(\pi(rT+d')/4)\tanh(\pi(rT-d')/4),\\ H_o&=\sqrt{T/8}\,\mathbf q^{-1/4} \prod_{r\ge1}\tanh(\pi rT/4). \end{split} \tag{28}\] For the first identity, the two sides have the same simple zeros and multipliers under \(d'\mapsto d'+T,d'+i\). Their quotient is a pole-free doubly periodic function, and the derivatives at zero fix it. The scalar formula for \(H_s\) can be obtained in the same comparison: after removing the zero, compare the quadratic logarithmic coefficients. With \(U(p)=\sum_{r\ge1}p^r/(1-p^r)^2=\sum_{r\ge1}rp^r/(1-p^r)\), divisor interchange gives \[-\frac{\pi^2}{6T^2}+\frac{4\pi^2}{T^2}U(\mathbf q) =\frac{\pi^2}{6}-\frac\pi T-4\pi^2U(e^{-2\pi T}).\] Multiplying this equality by \(-1/(2\pi)\) gives \(\partial_T\log(H_s/H_*)=1/(2T)-\pi/12-\pi/(12T^2)\); equality at \(T=1\) fixes the scalar constant. For the outer identity, compare divisors and multipliers under \(+T,+4i\). The product of the left side and its \(2i\) translate is \(-\mathbf q^{-1}\), fixing the magnitude of the constant; a small positive real \(d'\) fixes its sign. The derivative at zero then gives the formula for \(H_o\).

Write \(\alpha_p=\theta+\varphi_0+2\pi p\) and \(g_w=2\pi\ell/T\). For the short fields let \(N_s\) be their count and \(X_s=\sum_i\epsilon_iw_i\). Extracting \(Z^{1/2}\) from their pair bases changes the coordinate powers to \(y_i^{\epsilon_i(n_1+\ell/2)}\). The identities \[2\sum_{i<l}\epsilon_i\epsilon_l=\ell^2-N_s, \qquad \sum_{i<l}\epsilon_i\epsilon_l(w_l-w_i)^2 =\ell\sum_i\epsilon_iw_i^2-X_s^2\] show that the net \(H_s\) power is \(\ell^2\) and describe the pair Gaussian. Completing the heat and coordinate powers leaves \(\mathbf q^{(n_1+\ell/2)^2/2+\ell^2/8}\) and \(e^{-i\alpha_p\ell/2}\). Since \(\ell/2\) is integral, Poisson summation gives \[\sum_{n_1\in\mathbb Z} \mathbf q^{(n_1+\ell/2)^2/2} e^{i(\theta+\varphi_0)(n_1+\ell/2)} e^{2\pi i(n_1+\ell/2)X_s/T} =\sqrt T\sum_{p\in\mathbb Z} e^{-T\alpha_p^2/(4\pi)-\alpha_pX_s-\pi X_s^2/T}.\] The \(X_s^2\) term cancels the pair Gaussian. Including physical field normalizations, the short contribution apart from \(\sqrt T e^{-T\alpha_p^2/(4\pi)}\) is as follows: \[\begin{gathered} \text{field dress: }\quad \epsilon_i(-\alpha_pw_i-g_ww_i^2/2),\\ \text{scalar: }\quad \exp[-\pi T\ell^2/12-\pi\ell^2/(6T)-i\alpha_p\ell/2],\\ \text{per-field factor: }\quad (2\pi/a_*)^{1/2}H_*. \end{gathered}\] The remaining short pair base is \(2\sinh\pi d'\) with its two image factors from (28). The possible phase \((-i)^{\ell^2/2}\) is one for \(\ell\in8\mathbb Z\).

For the outer fields put \(Y_i=e^{2\pi iw_i/T}\). Extract the inverse square root when type 3 precedes type 2 and the ordinary square root otherwise. The residual power of every \(Y_i\) is \(-\ell\epsilon_i/2\). The ground and coordinate exponent of \(\mathbf q\) before the modular factors is \[\frac{n_2^2+n_3^2}{2}+E_2n_2+\frac{E_2^2}{2} =\frac{(E_2+F_3)^2+\ell^2}{4}.\] The modular pair and self factors remove \((E_2+F_3)^2/4\). Orient every remaining odd outer base as \(d'=w_l-w_i\); this contributes the 3-before-2 sign. The outer contribution is \[\begin{gathered} \text{field dress: }\quad -i\epsilon_i(\alpha_p+g_ww_i)/2,\\ \text{scalar and phase: }\quad \mathbf q^{\ell^2/4}e^{i\alpha_p\ell/2} (-i)^{(E_2+F_3)^2/2},\\ \text{per-field factor: }\quad (\pi/(4a_*))^{1/2}\prod_{r\ge1}\tanh(\pi rT/4). \end{gathered}\] The remaining pair base is \(\tanh(\pi d'/4)\) with its images. Writing \(E_2+F_3=2h\), the displayed phase is \((-1)^h\). It cancels the charge sign in (25), and the reorientation cancels its 3-before-2 sign. The sign left is exactly \((-1)^{\#(\text{outer before short})}\). In combination with the odd like-family bases, it makes the coordinate expression alternating under exchange of any neighboring fields. Whole even current pairs may therefore be reordered without a sign. The total winding scalar per species is \[ \exp[-\pi T\ell^2/12-2\pi\ell^2/(3T)]. \tag{29}\]

The tilted field and the first particle lists

Return to the two species and put \(\mathbf1=(1,1)^t\). The source in the regulator is \(im\int_TDa_T\,\mathbf1\cdot\chi\), since \(\widehat\lambda_0=D\widehat a\). The curvature just computed satisfies \[\frac12\int_T(ma_T\mathbf1-i\chi)^tD(ma_T\mathbf1-i\chi) +im\int_TDa_T\,\mathbf1\cdot\chi =m^2\int_Ta_TDa_T-\frac12\int_T\chi^tD\chi.\] This is an identity for the common factor of each fixed current coefficient. The odd real transforms of \(a\) and \(\lambda_0\) make them purely imaginary on the real line; \(D\) is real and positive. Thus the remaining quadratic factor is a real Gaussian tilt. Including division by the original source expectation gives \[ \widetilde G=(I+\widehat GD)^{-1}\widehat G,\qquad D_g=\prod_{n\ge1}\det(I+E(k)g(k))^{-1},\qquad \kappa_T=\int_T a_TD(I+(\widehat G_d+\widehat G_c)D)a_T, \tag{30}\] with overall scalar \(D_g e^{m^2\kappa_T}\) and \(k=2\pi n/T\) in the product. The determinant has power \(-1\) because each positive frequency represents the two real modes \(k,-k\). The products converge at fixed period: the zero frequency is excluded and \(gE\) decays exponentially at high frequency. We henceforth let \(\chi\) have this centered real tilted law. This operation is performed on the common factor before summing the new integrated series.

Combine (26) with the mean/winding field dresses: \[f_s(w)=f_s^{\rm osc}(w)-\alpha_pw-g_ww^2/2,\qquad f_o(w)=f_o^{\rm osc}(w)-i(\alpha_p+g_ww)/2.\] Their multipliers give \[e^{f_s(v-i/2)-f_s(v+i/2)}=b(v)^{-1},\qquad e^{f_o(v-i)+f_o(v+i)}=b(v).\] Define \(H(v)=e^{f_s(v+i/2)-f_o(v-i)}\). The nonzero-mode multiplier in its exponent is \(-E\) on \(\mathsf A\), while the mean/winding difference is \(-\alpha_pv-g_wv^2/2+5g_w/8\). Therefore \[ \begin{gathered} H(v)=B_T(v)^m \exp(2\pi m/T+\xi(v)-\alpha_pv-g_wv^2/2+5g_w/8),\\ B(v)=\frac{\cosh(\pi v/3)-\sqrt3/2} {\cosh(\pi v/3)+\sqrt3/2},\qquad \widehat\xi(k)=iE(k)\widehat\chi(k)\quad(k\ne0). \end{gathered} \tag{31}\] Here \(B_T\) is the periodized product. Indeed \[\widehat{\log B}(k)=-(2\pi/k)\sinh k/\cosh(3k/2) =-E(k)\widehat a(k),\qquad \widehat{\log B}(0)=-2\pi.\] Removing this zero coefficient accounts for \(2\pi m/T\) in (31). The field \(\xi\) is real on the real line: \(E\) is real and odd. Its high-frequency covariance below shows that it is analytic on \(|\Im v|<3/2\). The displayed exponential recurrences continue from their defining strips. For example, for \(1/2<\Im v<3/2\), the upper value of \(H\) can also be written as \(e^{f_s(v-i/2)-f_o(v-i)}b(v)\); the first recurrence provides that continuation. The lower continuation is analogous.

We now specify the current pairs as particles. A short or outer entry \((a,\epsilon)\) means a field at \(w=x+ia\) of sign \(\epsilon\); an \(H\) entry \((a,r)\) means the full factor \(H(x+ia)^r\). All entries in one row belong to one species. The four rows \(P,M,O_+,O_-\) come from roots \(12,15,13,16\) in \(D\). The rows \(U,L\) come from the rightmost root and the cycled leftmost root respectively. The separate sign is their scalar coefficient sign. \[ \begin{array}{c|c|c|c|l} &\text{shorts}&\text{outers}&\text{sign}&H\text{-charges}\\\hline P &(1/2,1)&(1,1)&+&(0,1)\\ M &(1/2,-1)&(1,1)&+&(0,1),(1,-1),(-1,-1)\\ O_+&\varnothing&(1,1),(0,1)&+&(0,1),(1,-1)\\ O_-&\varnothing&(1,1),(0,-1)&-&(0,1),(-1,-1)\\ U &(d,-1)&(1/2+d,1)&-&(3/2+d,-1)\\ L &(-d,-1)&(-1/2-d,-1)&+&(-3/2-d,-1) \end{array} \tag{32}\] The internal order puts the height-\(1\) outer first in each \(D\) row, and the short first in \(U,L\). To verify the table, a cycled \(D\) pair has fields \(\Psi_3\Psi_{j-1}\) at \(x+i\) and its coefficient at \(x\). The recurrences give the four dresses \[H(x),\quad \frac{H(x)}{H(x+i)H(x-i)},\quad \frac{H(x)}{H(x+i)},\quad -\frac{H(x)}{H(x-i)}.\] For \(U\) at folded coordinate \(v=x+i(1/2+d)\) the dress is \(-H(v+i)^{-1}\); for \(L\) at its pre-cycling coordinate \(v=x-i(1/2+d)\) it is \(H(v-i)^{-1}\). These are exactly the listed heights and signs.

At this stage fix \(d=d_-<0\) with \(|d_-|\) sufficiently small. Starting from the outgoing ordered strip in the preceding subsection, move the folded \(U\) coordinates to imaginary part \(1/2+d_-\), the cycled \(L\) coordinates to \(1/2-d_-\), and the cycled \(D\) coordinates to \(1\) by a limit from below. Repeated pairs retain close separations until their analytic coincident limits are taken. These moves remain in the ordered strip. The original \(b\) coefficients have no poles there by their \(U_\delta\) products, and a possible dual-pair singularity between \(D\) pairs is cancelled by the zero from their common first field. In particular the extreme \(H\) heights have absolute value \(3/2-|d_-|<3/2\). The moves and the table thus hold for every fixed coefficient before Gaussian integration.

Let \(e_1,e_2\) be the species basis. For a particle \(j\) in species \(\sigma\) define \(q_j\) as its total short charge times \(e_\sigma\). Inspection of the table shows that the total of its \(H\)-charges is the same vector. The six values are \((1,-1,0,0,-1,-1)e_\sigma\). Consequently the trace constraints (24) imply, for every contributing term, \[ \sum_jq_j=\boldsymbol\ell=(\ell_1,\ell_2)\in8\mathbb Z(1,-1). \tag{33}\] This constraint is retained in all sums below.

Absolute integration on the first contours

We first record the covariance used in the absolute estimate. Write \(P_+\) for the common and \(P_-\) for the difference species projection. At nonzero frequency the covariance of \(\xi\) has symbol \(2\pi\mathsf h(k)/k\), where \[ \begin{gathered} \mathsf h=gE^2(I+gE)^{-1},\qquad \mathcal S=P_-+P_+/9,\qquad \mathcal L=I-\mathcal S=(8/9)P_+,\\ \mathsf h(k)/k=\mathcal S/k^2+c_0+O(k^2),\qquad c_{0,-}=-17/12,\quad c_{0,+}=-25/108. \end{gathered} \tag{34}\] For example, \(E=1/k+7k/12+O(k^3)\), \(g_+=k/8-43k^3/96+O(k^5)\), and \(g_-=1/(2k)-17k/24+O(k^3)\) give these expansions. With \(t=e^{-k}\) the exact channel symbols are \[\mathsf h_+=\frac{t^4(t+1)}{(1-t)(1+t^2)(1-t+t^2)(1+t+t^2)^2},\qquad \mathsf h_-=\frac{t^3(4t^4-7t^3+10t^2-7t+4)} {(1-t)(1+t)(1+t^2)(1-t+t^2)^3}.\]

Define \(K^r\) on the real line by transform \(2\pi(\mathsf h/k-\mathcal S/k^2)\). It is continuous, even, and exponentially decaying. More precisely, for \(s>0\), \[ K^r(s)=\sum_{j\ge1}P_j(s)e^{-\pi j s/6}, \tag{35}\] where the matrix polynomials have uniformly bounded degree and coefficients of at most polynomial growth. The series also defines \(K^r\) for \(\Re s>0\). To see this, the transform is removable at zero and its other poles lie on the imaginary \(i\pi/6\) grid with bounded order. Shift Fourier inversion to lines between successive grid points. On those lines \(\mathsf h\) is \(O(e^{-3|\Re k|})\) uniformly, while the subtracted rational term is integrable; the residues give (35). The principal parts of the rational functions of \(e^{-k}\) are periodic along the grid, which gives the stated coefficient bounds.

The form convenient at coincident horizontal coordinates is \[ K_0(s)=K^r(0)+\int_{\mathbb R}(\cos(ks)-1)\mathsf h(k)\frac{dk}{k}. \tag{36}\] It equals \(K^r(s)-\pi|s|\mathcal S\) for real \(s\). It is analytic on \(|\Im s|<3\). Since the leading high-frequency term is \(4P_-e^{-3k}\) for \(k>0\), its continuation to \(|\Im s|<4\) has the sole singular part \[ -4P_-\log(1+s^2/9). \tag{37}\] In particular the exponentials between integer species charges are meromorphic in that wider strip. The zero-mean periodic covariance is \[ K_T(s)=\sum_{j\in\mathbb Z}K^r(s+jT) +\pi(s^2/T-|s|+T/6)\mathcal S-2\pi c_0/T, \qquad -T<s<T. \tag{38}\] Fourier series proves this identity. At complex shifts the term involving \(K^r(s)-\pi|s|\mathcal S\) is read as \(K_0(s)\) and the other terms continue analytically.

The following calculation concerns one finite list of \(H\)-charges. Write \(r_i\) for its signed species vector, \(w_i=x_i+ia_i\), and \[X=\sum_i r_iw_i,\qquad X^{(2)}=\sum_i r_iw_i^2.\] All squares in analytic formulas are bilinear. Wick’s formula for this list is \(\exp(\tfrac12\sum_{i,l}r_i^tK_T(w_l-w_i)r_l)\). Under (33), \(\mathcal S\boldsymbol\ell=\boldsymbol\ell\). The quadratic and constant terms of (38) therefore contribute \[ \exp\!\left[ \frac\pi T(\boldsymbol\ell\cdot X^{(2)}-X^t\mathcal S X) +\frac{\pi T}{12}|\boldsymbol\ell|^2 +\frac{17\pi}{12T}|\boldsymbol\ell|^2\right]. \tag{39}\] The first term cancels the \(-g_ww_i^2/2\) dresses. The remaining winding terms cancel exactly: (29) cancels the term of order \(T\), and the coefficients of \(\pi|\boldsymbol\ell|^2/T\) from Wick, the \(5g_w/8\) dresses, the modular scalar, and the normalized regulator weight are respectively \[\frac{17}{12}+\frac{15}{12}-\frac8{12}-\frac{24}{12}=0.\] The common factor \(e^{2\pi m/T}\) in \(H\) also cancels, since \(\mathbf1\cdot\boldsymbol\ell=0\).

We apply this algebra first to an absolute moment on the \(d_-\) contours. Write \(r_i=t_i e_{\sigma_i}\) with \(t_i\in\mathbb R\). Because each \(\xi_\sigma\) is real on the real line, \[\left|e^{\sum_i t_i\xi_{\sigma_i}(x_i+ia_i)}\right| =e^{\sum_i t_i\Re\xi_{\sigma_i}(x_i+ia_i)}.\] Writing \(r@w\) for a signed species charge \(r\) at \(w\), its expectation under the tilted law is precisely the ordinary Wick exponential obtained by replacing each signed charge by two signed halves, \[ r_i@(x_i+ia_i)\quad\longmapsto\quad \tfrac12r_i@(x_i+ia_i)+\tfrac12r_i@(x_i-ia_i). \tag{40}\] No sign of \(t_i\) is changed. This is a finite real Gaussian moment: the largest height is \(3/2-|d_-|\), so the high-frequency variance has a positive exponential gap against \(\mathsf h_-(k)=O(e^{-3k})\). In channel \(\varepsilon=\pm\), the resulting height kernel is \(-\mathsf h_\varepsilon(k)\cosh(a_i k)\cosh(a_l k)\) in the negative logarithm of the residual magnitude. No continuation term is present.

The symmetrized list has the same total \(\boldsymbol\ell\), and its two moments are \[ \widetilde X=\Re X=\sum_i r_ix_i,\qquad \widetilde X^{(2)}=\Re X^{(2)}=\sum_i r_i(x_i^2-a_i^2). \tag{41}\] In particular the second moment retains \(-a_i^2\). Substituting these moments in (39) proves exactly the same winding cancellations for the absolute moment. Once those cancellations are made, \(\boldsymbol\ell\) remains fixed by (33); it is not an additional unrestricted summation index.

The heat sum is bounded before integration over constant modes. Write \(\theta=\vartheta+i\tau\), \(X=Y+iZ\), and \(\boldsymbol\alpha_{\mathbf p}=a_{\mathbf p}+i\tau\mathbf1\) with \(a_{\mathbf p}\) real. For each heat summand, \[ \left|e^{-T\boldsymbol\alpha_{\mathbf p}^2/(4\pi) -\boldsymbol\alpha_{\mathbf p}\cdot X}\right| =e^{-Ta_{\mathbf p}^2/(4\pi)-a_{\mathbf p}\cdot Y} e^{T\tau^2|\mathbf1|^2/(4\pi)+\tau\mathbf1\cdot Z}. \tag{42}\] The second factor costs \(e^{O_\theta(T+N)}\), where \(N\) is particle count, because the height lists are bounded. Apply the triangle inequality to the heat sum. At fixed \(k_0=p_1+p_2\), the opposite uniform mode and the remaining integer sum unfold to \(du/(2\pi)\) on \((u,-u)\). The common Gaussian has covariance \(\pi P_+/(4T)\). Its real completion, including \(e^{-\pi Y^t\mathcal S Y/T}\), gives \[ \frac{\sqrt{8/9}}{\sqrt{2T}}\sum_{k_0\in\mathbb Z} e^{-T\beta_\vartheta^t\mathcal L\beta_\vartheta/(4\pi) -\beta_\vartheta^t\mathcal L Y}, \qquad \beta_\vartheta=(\vartheta+\pi k_0)\mathbf1. \tag{43}\] Here \(Y_+=\mathbf1\cdot Y/\sqrt2\) and \(Y_-=(1,-1)\cdot Y/\sqrt2\). The opposite integral contributes \((2T)^{-1/2}e^{\pi Y_-^2/T}\), and the common integral contributes \(\sqrt{8/9}e^{\pi Y_+^2/(9T)}\) as well as the displayed terms. These two exponentials cancel the corresponding parts of \(-\pi Y^t\mathcal S Y/T\). The two factors \(\sqrt T\) from Poisson summation have been kept outside this calculation.

For this real majorant, order the real particle coordinates and put \(Q(x)=\sum_{x_j<x}q_j\). The real long-distance short and \(H\) terms are \[\pi\sum_{u<j}(x_j-x_u)q_u^t\mathcal Lq_j.\] For each \(k_0\) summand in (43), its exponent together with these terms is the exponent of the real state square below, with \(p(x)=p_0-Q(x)\) and \(p_0\in\mathbb Z^2\) having sum \(k_0\): \[\exp\!\left[-\pi\int_T (p+(\vartheta/(2\pi))\mathbf1)^t\mathcal L (p+(\vartheta/(2\pi))\mathbf1)\,dx\right].\] Closure holds modulo \(8\mathbb Z(1,-1)\) by (33). For each such path, the base real product consists of this state square, the short and outer local and residual magnitudes, and the local and residual Wick factors from the complete symmetrized \(H\) lists. It is the weight \(W^{\rm pre}_{d_-,\vartheta}(\mathcal Y,p)\) in Theorem 15, with state angle \(\vartheta=\Re\theta\). The explicit second factor of (42) is kept outside this product and bounded separately by \(e^{C_\Omega(T+N)}\) on a compact set \(\Omega\) of angles. The height phases have disappeared under the modulus, so this base product has no Berry factor. In particular no signed phase formula is applied to the symmetrized list, whose \(H\) height moment is zero.

At fixed \(m,T,d_-\) the deterministic \(B_T\) factors on these contours have a finite bound per particle. The pre-collision clause of Theorem 15, after absorbing the heat factor \(e^{C_\Omega(T+N)}\) and the deterministic linear count cost, therefore proves the required absolute summability on its domain of integer periods for every finite \(m\). The allowed periods are independent of \(m\); only the absorbed count cost changes. More explicitly, let \(\nu\) index a finite current monomial with specified species and roots, with the winding fixed by (33). Let \(I_{\nu,\mathbf p}\) be its full conditional scalar coefficient after the common factor in (30) is extracted. It includes the exponential factorial, modular field, pair, and image factors, table signs and \(H\) dresses, the \(\mathbf p\) heat factor, and the original winding probability. Then \[ \sum_\nu\sum_{\mathbf p\in\mathbb Z^2} \int_{\Gamma_-^\nu}\int d\mu_{\rm const}\, \mathbb E_{\rm tilt}|I_{\nu,\mathbf p}|<\infty. \tag{44}\] Here \(\Gamma_-^\nu\) is the product of the fixed \(d_-\) contours, integrated with product arclength, equivalently \(\prod_jdx_j\) on the shifted horizontal lines, and \(d\mu_{\rm const}\) is the original common-Gaussian and opposite-uniform constant-mode law. To see the final summation explicitly, if \(n_j\) counts particles in unit box \(j\) and \(p_{c,j}=(p_1+p_2)/\sqrt2\) is the state at its left endpoint, stability bounds the magnitude by \(\exp(CT-c\sum_j(n_j^2+p_{c,j}^2))\), up to the absorbed linear costs. For a finite label count \(J\), every box contributes at most a sum of the form \[\sum_{n\ge0}\frac{J^n}{n!}e^{-cn^2+A_{m,T,d_-}n}<\infty.\] The initial-state sum is Gaussian in \(k_0=\sqrt2p_{c,0}\) and has eight quotient classes for each \(k_0\). Dropping other configuration constraints only enlarges this bound. There is no further winding multiplicity. This proves (44), locally uniformly for \(\theta\) in compact sets. Its constants may depend on the fixed \(d_-\); it asserts no uniform estimate as \(d_-\uparrow0\).

Tonelli and Fubini now pass the conditional current identities through the Gaussian and constant-mode integrations. This is the point at which the regulator becomes the sum of the signed Wick-integrated terms on the first contours. The conditional trace calculation alone was not used for that interchange.

Wick integration and the collision residue

For each signed term furnished by (44), use the original integer \(H\)-charges in Wick’s formula. Equation (39) and its winding cancellations now hold with complex \(X,X^{(2)}\). The signed constant-mode integration is the analytic version of (43): \[ \frac{\sqrt{8/9}}{\sqrt{2T}}\sum_{k_0\in\mathbb Z} \exp[-T\beta^t\mathcal L\beta/(4\pi)-\beta^t\mathcal L X], \qquad \beta=(\theta+\pi k_0)\mathbf1. \tag{45}\] The two Poisson factors \(\sqrt T\) remain outside. For a fixed finite field list the full heat sum converges normally on compact contour regions, with a bound of the form \(e^{-cTk_0^2+C_N|k_0|}\). Retaining this sum gives the horizontal periodicity inherited from the original current expression. This periodic meromorphic expression is the one used in the contour moves; an individual heat summand need not be periodic.

We make the local normalization explicit before calculating the residue. For a basic type \(v\), let \(N_s(v),N_o(v)\) be its field counts, let \(I_v\) count outer-before-short pairs inside its displayed internal order, and let \(\tau_v\) be its table sign. The global outer-before-short sign from (25) can be assigned locally as \((-1)^{I_v}i^{N_s(v)\bmod2}\). Indeed a particle has the same parity of short and outer counts. If \(N_{\rm odd}\) particles have odd short count, the interparticle inversion exponent is \(N_{\rm odd}(N_{\rm odd}-1)/2\). The trace constraint makes \(N_{\rm odd}\) even in each species, and then \((-1)^{N_{\rm odd}(N_{\rm odd}-1)/2}=i^{N_{\rm odd}}\).

Put \(K_{\rm diag}(s)=(K_0(s))_{\sigma\sigma}\), the same for either species, and \(\kappa_j^{\rm loc}=K_{\rm diag}(ji)\) for \(j=0,1,2\). For each type \(v\), denote its ordered short, outer, and \(H\) lists by \(\mathscr S_v,\mathscr O_v,\mathscr H_v\). The constant local factor of a basic particle is \[ \begin{split} \gamma_v={}&\frac{a_*}{2\pi i}\tau_v(-1)^{I_v}i^{N_s(v)\bmod2} \left(\frac{2\pi}{a_*}\right)^{N_s(v)/2} \left(\frac{\pi}{4a_*}\right)^{N_o(v)/2}\\ &\times\prod_{i<l\text{ in }\mathscr S_v} [2\sinh(\pi i(a_l-a_i))]^{\epsilon_i\epsilon_l} \prod_{i<l\text{ in }\mathscr O_v} [\tanh(\pi i(a_l-a_i)/4)]^{\epsilon_i\epsilon_l}\\ &\times\exp\!\left(\tfrac12\sum_{i,l\text{ in }\mathscr H_v} r_ir_l K_{\rm diag}(i(a_l-a_i))\right). \end{split} \tag{46}\] The factor uses \(K_0\) within a particle; image factors remain outside it. Since \(a_* /(2\pi i)=-i/12\), the short and outer field norms are \(\sqrt{12}\) and \(\sqrt{3/2}\). Substitution gives \[ \begin{aligned} \gamma_P=\gamma_U&=-\frac{\sqrt2}{4}e^{\kappa_0^{\rm loc}/2}, &\gamma_L&=\frac{\sqrt2}{4}e^{\kappa_0^{\rm loc}/2},\\ \gamma_M&=-\frac{\sqrt2}{4}e^{3\kappa_0^{\rm loc}/2-2\kappa_1^{\rm loc}+\kappa_2^{\rm loc}}, &\gamma_{O_+}=\gamma_{O_-}&=-\frac18e^{\kappa_0^{\rm loc}-\kappa_1^{\rm loc}}. \end{aligned} \tag{47}\] For the \(O\) rows the outer base at height difference \(-1\) is \(-i\); its inverse for \(O_-\) accounts for the equality of the two constants. In particular \(P\) and \(M\) have nonzero local factors.

Choose a fixed \(d_+>0\) in the post-collision stability range, with \(|d_-|<d_+\); both are small enough for the strips used below. In each finite term move the \(U\) short-center contours upward first, from height \(d_-\) to \(d_+\). They cross the \(L\) short-center contours, initially at height \(-d_-\). Then move the remaining \(L\) contours downward to \(-d_+\) and move any fused centers to the real line. A lower horizontal contour minus an upper one is the positive boundary of the intervening cylinder strip. Thus a crossed pole contributes \(+2\pi i\) times its residue in the \(U\) center.

For a same-species \(U,L\) pair write \(\delta=z_U-z_L\), where \(z_U,z_L\) are their short centers. Their ordered fields are \[(z_U,-)_{s},\quad(z_U+i/2,+)_{o},\quad (z_L,-)_{s},\quad(z_L-i/2,-)_{o}.\] Their \(H\)-charges are \(-1\) at \(z_U+3i/2\) and \(z_L-3i/2\). The joint table sign is negative and the ordered union has one outer-before-short inversion, so those two signs cancel. Its two shorts have even parity. The current and field-norm factors together are \(-1/8\). Therefore the full local germ is \[ -\frac18\,2\sinh(-\pi\delta)\, \coth\!\left(-\frac{\pi(\delta+i)}4\right) \exp[K_{\rm diag}(0)+K_{\rm diag}(\delta+3i)]. \tag{48}\] Define \(K_{\rm diag}^{\rm reg}(s)=K_{\rm diag}(s)+2\log(1+s^2/9)\) by continuation from the inner strip. By (37) it is regular at \(3i\), and \[e^{K_{\rm diag}(\delta+3i)} =\frac{81e^{K_{\rm diag}^{\rm reg}(\delta+3i)}}{\delta^2(\delta+6i)^2} \sim-\frac9{4\delta^2}e^{K_{\rm diag}^{\rm reg}(3i)}.\] Using \(2\sinh(-\pi\delta)\sim-2\pi\delta\) and \(\coth(-i\pi/4)=i\), the residue of (48) in \(z_U\) is \[-\frac{9\pi i}{16}\exp[K_{\rm diag}(0)+K_{\rm diag}^{\rm reg}(3i)].\] The fused local activity is hence \[ \gamma_{F_{\rm pair}}=\frac{9\pi^2}{8}\exp[K_{\rm diag}(0)+K_{\rm diag}^{\rm reg}(3i)]>0. \tag{49}\] The singular power is the integer \(-2\), so it has no branch sign; also \(d\delta/dz_U=1\). The regular value is real by continuation along \(s=iy\) with \(y<3\), proving the last inequality.

Call this fused particle \(F_{\rm pair}\). Its internal order is \(U\) then \(L\), and its complete lists, at its real center \(x\), are \[ \begin{array}{c|c|c} \text{shorts}&\text{outers}&H\text{-charges}\\\hline (0,-1),(0,-1)&(1/2,1),(-1/2,-1)&(3/2,-1),(-3/2,-1). \end{array} \tag{50}\] These are the lists \(\mathscr S_{F_{\rm pair}},\mathscr O_{F_{\rm pair}}, \mathscr H_{F_{\rm pair}}\). The local activity has already taken the residue of the entire factor (48); no internal pair product is evaluated at its coincident short centers. External interactions and nonzero images simply evaluate at equality and combine into these union lists. The constant (49) is independent of \(T,m\) and the state. In particular the local germ uses \(K_0\), before any long-distance separation; the corresponding constant from \(K^r\) alone would have an extra phase.

Here are all possible divisor crossings in these moves. From (37), two extreme \(H\)-charges of sign \(-1\) in the same species give a double pole at height difference \(3\); in different species they give a double zero. The two negative shorts in a \(U,L\) pair give one zero, leaving exactly the simple same-species pole used above. Against a fused particle, \(U\) or \(L\) has two short zeros and at most one double extreme-height pole. Two fused particles have four short zeros and at most two such double poles. Thus these later encounters are regular and cannot create a second fusion of the same particle. A same-species \(U,U\) or \(L,L\) coincidence has like-charge short and outer zeros and equal \(H\) heights; a cross-species coincidence has only the regular equal-height \(H\) factor. Neither has a pole.

Among \(D\) rows, a \(P,M\) coincidence has one inverse short factor, cancelled by the zero of their common height-\(1\) outer field. An \(O_+,O_-\) coincidence has one inverse height-\(0\) outer factor, cancelled by the same common outer zero. The other \(D\) coincidences have no uncancelled inverse factor. Their \(H\) height differences are at most \(2\). Mixed \(D\) and \(U,L,F_{\rm pair}\) \(H\) differences are at most \(5/2+d_+<3\); the other outer differences remain below \(2\), and no other short divisor is crossed. Finally \[ B(v+i)B(v-i)=B(v),\qquad B(v+3i)=B(v)^{-1} \tag{51}\] gives the simplified \(D\) factors \(B(x),1,B(x-i),B(x+i)\) and makes the \(U,L\) factors cancel at fusion. The zeros of \(B\) are at \(i(\pm1/2+6n)\) and its poles are at \(i(\pm5/2+6n)\), \(n\in\mathbb Z\). The moving inverse \(U,L\) arguments have heights in \((1,2)\) and \((-2,-1)\), respectively, while the \(D\) arguments have heights \(0,\pm1\). Thus these factors meet no divisor during the collision moves. The same statements hold for their periodized products. They account for every pair of rows in (32) and (50).

At fixed particle number the preceding statements are identities of periodic meromorphic contour integrals. Each \(U\) may fuse with at most one \(L\), and fusions occur only within a species. For \(u,l\) original particles and \(r\) disjoint pairs the multiplicity is \[ \frac{\binom ur\binom lr r!}{u!\,l!} =\frac1{(u-r)!(l-r)!r!}. \tag{52}\] There is no additional \(1/2\), because \(U,L\) are distinct and the residue is always in the \(U\) center. Reordering whole bilinears has even parity, and each matching gives the same ordered union and sign. Thus the formula also counts the absolute values of the individual residue contributions.

The post-collision data and endpoint summability

From now on write \(d=d_+>0\). For a type \(v\) define its scalar charge and height moments by \[q_v=\sum_{(a,\epsilon)\in\mathscr S_v}\epsilon =\sum_{(a,r)\in\mathscr H_v}r,\qquad s_v=\sum_{(a,\epsilon)\in\mathscr S_v}a\epsilon,\qquad b_v=\sum_{(a,r)\in\mathscr H_v}ar.\] For a particle of this type in species \(\sigma\), the vectors used below are \(q_j=q_ve_\sigma\), \(s_j=s_ve_\sigma\), and \(b_j=b_ve_\sigma\). Let \(\omega_v\) be the product of its \(B\) factors with the signed \(H\) powers. The recurrences (51) give the complete table \[ \begin{array}{c|r|r|r|l} v&q_v&s_v&b_v&\omega_v(x)\\\hline P&1&1/2&0&B(x)\\ M&-1&-1/2&0&1\\ O_+&0&0&-1&B(x-i)\\ O_-&0&0&1&B(x+i)\\ U&-1&-d&-3/2-d&B(x+i(3/2+d))^{-1}\\ L&-1&d&3/2+d&B(x-i(3/2+d))^{-1}\\ F_{\rm pair}&-2&0&0&1. \end{array} \tag{53}\] The periodized product is denoted \(\omega_{v,T}\), with value \(1\) in the two constant rows. The original deterministic factor is exactly \(\omega_{v,T}(x)^m\). These simplified expressions also specify the value when separate factors in a ratio have cancelling zeros and poles. The physical fused particle has charge \(-2e_\sigma\); the normalized half-fused vector used in the matrix proof of stability is only a coordinate in that proof.

For positive real separation \(x\) and species-labelled types \(u,v\), define the forward residual factor using their complete lists by \[ \begin{split} \mathscr R_{uv}(x)={}& \prod_{\substack{(a,\epsilon)\in\mathscr S_u\\ (b,\epsilon')\in\mathscr S_v\\ \sigma_u=\sigma_v}} [1-e^{-2\pi(x+i(b-a))}]^{\epsilon\epsilon'}\\ &\times \prod_{\substack{(a,\epsilon)\in\mathscr O_u\\ (b,\epsilon')\in\mathscr O_v\\ \sigma_u=\sigma_v}} [\tanh(\pi(x+i(b-a))/4)]^{\epsilon\epsilon'}\\ &\times\exp\!\left[ \sum_{\substack{(a,r)\in\mathscr H_u\\(b,r')\in\mathscr H_v}} rr'e_{\sigma_u}^tK^r(x+i(b-a))e_{\sigma_v}\right]. \end{split} \tag{54}\] An empty product is one. The series (35) and the logarithmic short and outer expansions make its log exponentially decaying at large \(x\). On a periodic configuration, its forward product takes each particle in one period as source and every strictly later particle in the periodized sequence as target. In particular it includes all nonzero self-images. Only the true diagonal at separation zero is omitted.

We next justify summing the finite contour identities. For each summand of (45), write \(\beta=\beta_\vartheta+i\tau\mathbf1\) and \(X=Y+iZ\). Its modulus is the real heat summand with \(\vartheta\) and \(Y\), times \(\exp[T\tau^2\mathbf1^t\mathcal L\mathbf1/(4\pi)+\tau\mathbf1^t\mathcal L Z]\). On a compact \(\theta\) set the latter costs \(e^{O_\theta(T+N)}\). For distinct sorted particles let \(A_{uv}(x)\) be the direct pair factor obtained from \(\mathscr R_{uv}(x)\) by replacing each short base by \(2\sinh(\pi(x+i(b-a)))\) and each \(K^r\) by \(K_0\). The relation \(K_0(s)=K^r(s)-\pi s\mathcal S\) on \(\Re s>0\) gives directly \[|A_{uv}(x)|=e^{\pi xq_u^t\mathcal Lq_v}|\mathscr R_{uv}(x)|.\] All height terms in the ratio are phases, also after continuation across height \(3\) for the integer \(H\)-charges. The images are already residual factors. The real long-distance part therefore completes into the common state square used above, and the remaining pair magnitudes are exactly \(|\mathscr R_{uv}|\) from the actual integer height lists. The local magnitudes are those of (47) and (49). This is exactly the post-collision real pair/state/local weight estimated by Theorem 15. The bounded imaginary-angle costs are linear in the count and are absorbed there.

The deterministic factors require no added count cost on these contours. Writing \(z=\cosh(\pi(x+ia)/3)\) and \(r=\sqrt3/2\) shows \(|z-r|<|z+r|\) precisely when \(\cos(\pi a/3)>0\). Hence \(|B(x+ia)|<1\) for \(|a|<3/2\) and \(|B(x+ia)|>1\) just beyond those lines. Every nonconstant row of (53) has \(|\omega_v(x)|<1\), and this remains true image by image. The post-collision stability bound and the same count and state summation used in (44) therefore prove absolute summability at this fixed \(d_+\) for all supported periods.

Each finite term has already been deformed with its residues. Equation (52) shows that the sum of their absolute residue contributions is bounded by the ordinary post-collision factorial series, so the two endpoint bounds permit summation of all those identities. No estimate uniform between \(d_-\) and \(d_+\) is required. In particular the raw Gaussian field was evaluated only on the first contours; the contour crossing used its Wick-integrated meromorphic functions.

The exact horizontal gas

It remains to express each signed post-collision term in variables local to the horizontal direction. Order its distinct real centers in one period. For a positive separation \(s=x+i(b-a)\), the full short factor splits as \(2\sinh\pi s=e^{\pi s}(1-e^{-2\pi s})\), and \(K_0(s)=K^r(s)-\pi s\mathcal S\) on \(\Re s>0\). Keep the full \(K_0\) within a particle. For distinct particles \(u<j\), summing the separated linear terms from their lists gives \[\pi\sum_{u<j}(x_j-x_u)q_u^t\mathcal Lq_j +i\pi\sum_j(s_j-\mathcal S b_j)\cdot (Q(x_j-)+Q(x_j+)-\boldsymbol\ell), \qquad Q(x)=\sum_{x_j<x}q_j.\] The residual factors, including all images, are precisely (54).

Choose \(p_0\in\mathbb Z^2\) with \(p_{0,1}+p_{0,2}=k_0\) and put \(p(x)=p_0-Q(x)\). The real-coordinate linear term above and the heat and mean terms of (45) complete to \[-\pi\int_T(p+(\theta/(2\pi))\mathbf1)^t\mathcal L (p+(\theta/(2\pi))\mathbf1)\,dx.\] This follows by expanding the square, using \(\int_TQ\,dx=T\boldsymbol\ell-\sum_jq_jx_j\) and \(\mathcal L\boldsymbol\ell=0\). The imaginary terms become \[-i\pi\sum_j(s_j-\mathcal S b_j)\cdot(p(x_j-)+p(x_j+)) -i\theta\mathbf1^t\mathcal L\sum_j b_j.\] To check the discarded phase exactly, (53) gives \[s_v-b_v=(1/2,-1/2,1,-1,3/2,-3/2,0) \quad\text{for }P,M,O_+,O_-,U,L,F_{\rm pair}.\] It is congruent to \(q_v/2\) modulo integers, so its total is integral in each species by (33). The phase left by the substitution is \[2\pi p_0\cdot\sum_j(s_j-b_j) -\pi\boldsymbol\ell\cdot\sum_j(s_j-b_j),\] a multiple of \(2\pi\). Shifting \(p_0\) by \(8(1,-1)\) leaves each local weight unchanged. For a fixed sum \(k_0\), the eight classes modulo this shift have the same full-tour weight: shifting between them by \((1,-1)\) changes the product phase by \(2\pi\) times the same integral total. This explains the factor \(1/8\) in the normalization below.

This computation also checks the fused local phase. If \(U\) precedes \(L\) and \(s=z_L-z_U=x-2id\), their full short and \(H\) factors divided by their residual factors give \[e^{\pi s-\pi\mathcal S_{\sigma\sigma}(s-3i)} =e^{\pi\mathcal L_{\sigma\sigma}x} e^{i\pi(3\mathcal S_{\sigma\sigma}-2d\mathcal L_{\sigma\sigma})}.\] The first factor is their pair term in the state square. The second is the product of their local phases for successive states \(p,p+e_\sigma,p+2e_\sigma\); their \(\theta\) phases cancel since \(b_U+b_L=0\). At collision this phase is already part of the full \(K_0\) germ (48). The fused row has \(q=-2\), \(s=b=0\) and therefore requires no further local phase.

We can now define the post-collision gas. Let \[\mathcal V=\{1,2\}\times\{P,M,O_+,O_-,U,L,F_{\rm pair}\}.\] A configuration \(\mathcal Y=((x_j,v_j))_{j=1}^N\) consists of a finite number of particles with \(0\le x_j<T\) and \(v_j\in\mathcal V\); coincident real positions form a null set and are discarded. For \([p_0]\in\mathbb Z^2/8\mathbb Z(1,-1)\) let \(p\) have the jumps \(-q_j\) in increasing order, using a consistent lift during the period, and impose \(p(T)=p(0)\) in the quotient. Define \[ \begin{split} \mathcal Q_\theta(\mathcal Y,p)={}& \exp\!\left[-\pi\int_T (p+(\theta/(2\pi))\mathbf1)^t\mathcal L (p+(\theta/(2\pi))\mathbf1)\,dx\right]\\ &\times\prod_j\exp\!\left[ -i\pi(s_j-\mathcal S b_j)\cdot(p(x_j-)+p(x_j+)) -i\theta\mathbf1^t\mathcal L b_j\right]. \end{split} \tag{55}\] The weight with the deterministic site factors omitted is \[ W_{d,\theta}(\mathcal Y,p)=\mathcal Q_\theta(\mathcal Y,p) \prod_j\gamma_{v_j} \prod_{\substack{1\le j,k\le N,\ n\in\mathbb Z\\x_k+nT>x_j}} \mathscr R_{v_jv_k}(x_k+nT-x_j). \tag{56}\] Thus \(j=k,n=0\) is excluded and every \(j=k,n\ge1\) is included. The constants \(\gamma_v\) use (47) or (49), independently of the species. For the empty configuration the particle and residual products are one, while the state heat factor remains. For distinct positions the factors in (56) are continuous and nonzero: the local activities are nonzero, the short and outer residual bases have no zero or pole at positive real separation, and the remaining factors are exponentials.

For counts \((n_v)_{v\in\mathcal V}\) with \(N=\sum_vn_v\), use any fixed labeling of those particles in the integral and set \[ Z_{m,T}=\sum_{(n_v)\in\mathbb Z_{\ge0}^{\mathcal V}} \frac1{\prod_v n_v!} \sum_{[p_0]\in\mathbb Z^2/8\mathbb Z(1,-1)} \int_{[0,T)^N}\!\mathbf1_{\{p(T)=p(0)\}} W_{d,\theta}(\mathcal Y,p) \prod_{j=1}^N\omega_{v_j,T}(x_j)^m\,\prod_{j=1}^Ndx_j. \tag{57}\] The indicator uses quotient closure. All definitions are independent of the chosen lift by the preceding phase check. The absolute endpoint bound proves convergence of this series for the periods in the next proposition, locally uniformly for \(\theta\) in compact sets. The local factors use \(K_0\) and the forward product uses \(K^r\), so no complex convention for \(|s|\) is needed inside a particle.

The table now exposes the properties used by the following sections. There are finitely many types with bounded integer jumps; the nonzero \(P,M\) activities permit both signs of each species jump. The two types whose site multipliers are exactly one, \(M,F_{\rm pair}\), have strictly negative common charge. Every other site multiplier has modulus below one on the post-collision real line. The forward residual log has the exponentially decaying finite-degree expansion inherited from (35). Theorem 15 controls the absolute closed-sequence weights, and Section 7 uses these properties to build the horizontal operator.

The scalar rates

We finish by calculating the two scalar limits used in that operator argument. In terms of \(t=e^{-k}\), the eigenvalues of \(I+Eg\) are \[\frac{(1-t^6)^2}{(1-t^4)(1-t^8)},\qquad \frac{(1+t^3)^4}{(1-t^4)(1-t^8)}.\] Riemann sums have only logarithmic endpoint singularities. Using \[\int_0^\infty\log(1-e^{-ak})\,dk=-\frac{\pi^2}{6a},\qquad \int_0^\infty\log(1+e^{-ak})\,dk=\frac{\pi^2}{12a},\] gives the rates \(-13\pi/144\) for \(D_g\) and \(11\pi/144\) for each \(D_W/Z_\Gamma\). The Gaussian sum has \(D_\ell=\Theta(\sqrt T)\) by comparison with its integral, so it and the powers of \(T\) are subexponential. Consequently \[ \lim_{T\to\infty}\frac{\log c(T)}{T}=\frac\pi{16}. \tag{59}\]

The symbols in (30), with \(\widehat a(-k)=-\widehat a(k)\), similarly give \[ \lim_{T\to\infty}\kappa_T =-2\int_0^\infty \frac{t(1-t)(1-t^3)^2}{(1+t^2)(1+t^3)(1+t^4)}\frac{dk}{k} =2\log\frac{2(2+\sqrt2)}9. \tag{60}\] Here is the evaluation of the integral. Multiplying its rational function of \(t\) by \(1-t^{24}\) gives coefficients \(d_j=d_{24-j}\) for \(1\le j\le23\), whose first twelve values are \[(1,-1,-1,-2,3,3,1,-4,-3,-1,3,2).\] Their total after reflection is zero. Let \(R(t)\) denote the rational function in the integrand, so \((1-t^{24})R(t)=\sum_jd_jt^j\). The grouped partial sum through period \(N\) is \(R(e^{-k})(1-e^{-24(N+1)k})\). After division by \(k\) it is dominated by \(R(e^{-k})/k\), which is integrable because \(R(e^{-k})=O(k^3)\) at zero and \(O(e^{-k})\) at infinity. Dominated convergence therefore justifies the period expansion. Frullani’s log difference on each block gives its integral as \(-\sum_jd_j\log(24n+j)\). Pair \(j\) with \(24-j\), using exponent \(d_{12}/2\) at \(j=12\). The normalized product \[\frac{\prod_{n=0}^N(n+a)(n+1-a)}{N!(N+1)!} \longrightarrow a\prod_{n\ge1}(1-a^2/n^2)=\frac{\sin\pi a}{\pi}\] then gives the exponentiated sum of the paired logarithms as \(\prod_{j=1}^{11}\sin(\pi j/24)^{d_j}\). The sine product itself follows, for example, by taking the limit in finite odd sine multiplication \[\frac{\sin x}{N'\sin(x/N')}= \prod_{j=1}^{(N'-1)/2} \left(1-\frac{\sin^2(x/N')}{\sin^2(\pi j/N')}\right), \qquad N'\text{ odd}.\] Writing \(s_j=\sin(\pi j/24)\) and pairing complementary small angles reduces the required product to \[\frac{(s_{11}s_5)^2}{s_9^2s_8^4}= \frac{2(2+\sqrt2)}9,\] which proves (60). Proposition 14 combined with the fixed-site regulator limit of Section 4 therefore supplies the period limit needed later, always at fixed \(m\).

The matrix of signed height measures

Write \(\mu_i^s,\mu_i^o,\mu_i^H\) for the signed measures on real heights in the short, outer, and \(H\) columns of (32), for a label \(i\) in one species. For example, \[\mu_P^s=\delta_{1/2},\qquad \mu_P^o=\delta_1,\qquad \mu_P^H=\delta_0.\] The separate coefficient sign in the table belongs to the local factor and does not multiply these measures. The fused measures are the union of the \(U,L\) measures at zero displacement: \[\mu_{F_{\rm pair}}^s=-2\delta_0,\qquad \mu_{F_{\rm pair}}^o=\delta_{1/2}-\delta_{-1/2},\qquad \mu_{F_{\rm pair}}^H=-\delta_{3/2}-\delta_{-3/2}.\] In the order \(P,M,O_+,O_-,U,L,F_{\rm pair}\), the total of both the short measure and the \(H\)-measure is \[\bar q_i=(1,-1,0,0,-1,-1,-2)_i\] in that species. Thus the physical charge vector is \(q_{(\alpha,i)}=\bar q_i e_\alpha\) for species \(\alpha\). For a label \(\iota=(\alpha,i)\), put \[c_\iota=\frac{\bar q_i}{\sqrt2},\qquad \boldsymbol c=(c_\iota)_\iota,\qquad a_c=\frac89.\] The jump of the common charge at label \(\iota\) is \(-c_\iota\).

For \(k>0\), the short and outer height kernels are \[\begin{aligned} p(a)&=\frac{\cosh((1/2-|a|)k)}{2\sinh(k/2)} &&(|a|\le1,\ \text{period }1),\\ o(a)&=\frac{\sinh((1-|a|)k)}{2\cosh k} &&(|a|\le2,\ \text{antiperiod }2). \end{aligned}\] They are positive covariance kernels. More explicitly, their Green series are \[p(a)=\sum_{n\in\mathbb Z}\frac{k e^{2\pi ina}}{k^2+(2\pi n)^2}, \qquad o(a)=\frac12\sum_{n\in\mathbb Z} \frac{k e^{\pi i(n+1/2)a}}{k^2+\pi^2(n+1/2)^2}.\] Set \[C_i(k)=\int\cosh(ka)\,d\mu_i^H(a),\qquad S_i(k)=\int\sinh(ka)\,d\mu_i^H(a).\] Define the dimensionless matrix \(\mathcal H\) by \[ \begin{aligned} \mathcal H_{\alpha i,\beta j}(k) ={}&\delta_{\alpha\beta} \left[ \iint p(a-b)\,d\mu_i^s(a)d\mu_j^s(b) +\iint o(a-b)\,d\mu_i^o(a)d\mu_j^o(b) \right]\\ &+\sum_{\sigma=\pm}(P_\sigma)_{\alpha\beta}\Phi^\sigma_{ij}(k), \end{aligned} \tag{62}\] where the two uses have \[\begin{aligned} \Phi^{\sigma,{\rm pre}}_{ij} &=-\mathsf h_\sigma C_iC_j,\\ \Phi^{\sigma,{\rm post}}_{ij} &=-\mathsf h_\sigma(C_iC_j-S_iS_j) +4\mathbf1_{\{\sigma=-\}}\iint \sinh\!\big(k(|a-b|-3)_+\big)\,d\mu_i^H(a)d\mu_j^H(b). \end{aligned}\] The first formula is the Gaussian absolute moment: reality of \(\xi\) on the real line gives \(\Re\xi(x+ia)=(\xi(x+ia)+\xi(x-ia))/2\). The second uses the original charges and \(\cosh(k(a-b))=\cosh(ka)\cosh(kb)-\sinh(ka)\sinh(kb)\).

We verify the continuation term, since it controls the collision directions. The only singular part of \(K_0\) below height \(4\) is \(K_{\rm lead}(s)=-4P_-\log(1+s^2/9)\). For \(k>0\), the transform of its negative real part at height \(a\) is \[\frac{k}{2\pi}\widehat{-\Re K_{\rm lead}(x+ia)} =P_-\left[-4e^{-3k}\cosh(ak) +4\sinh\!\big((|a|-3)_+k\big)\right].\] Indeed the two real logarithms involve \(x^2+(3-a)^2\) and \(x^2+(3+a)^2\); their transforms use \(e^{-|3-a|k}\) and \(e^{-|3+a|k}\). Crossing height \(3\) therefore replaces the growing factor \(e^{-(3-|a|)k}\) by \(e^{-|3-|a||k}\), giving the displayed correction. In the pre-collision absolute moment all extreme height differences are less than \(3\), so that correction is absent.

Let \(V^r_{\iota\jmath}(x)\) be the even extension of the negative log magnitude of one forward residual pair, for \(x\ne0\). In the pre-collision use it denotes the analogous real symmetrized pair. It is integrable: possible singularities at zero are logarithmic, and the residual factors decay exponentially at infinity. Expanding the short logarithm gives transform \((2\pi/k)(p(a)-1/k)\); the outer logarithm gives \((2\pi/k)o(a)\). The relation \(K^r=K_0+\pi|x|\mathcal S\) adds \((2\pi/k)\mathcal S/k\) to the \(H\) residual transform. Since both charge totals are \(\bar q_i\), removing the full-factor linear terms leaves \[ \widehat V^r(k)=\frac{2\pi}{k} \left(\mathcal H(k)-\frac{a_c}{k}\boldsymbol c\boldsymbol c^t\right), \qquad k>0. \tag{63}\] There is no constant ambiguity here because the residual kernel decays at infinity.

Coercivity at a fixed displacement

The useful positive part of \(\mathcal H\) is most easily found after normalizing the fused coordinate. In this calculation only, put \(F=F_{\rm pair}/2\), halving all three signed measures. It can be included as an auxiliary coordinate in the pre-collision calculation and removed afterward. Let \[D_0=\operatorname{diag}(1,1,1,1,1,1,2)\] on \(P,M,O_+,O_-,U,L,F\), and let \(D_F=\operatorname{diag}(D_0,D_0)\) on the two species. Then \(\mathcal H_{\rm phys}=D_F\mathcal H_{\rm norm}D_F\). A coefficient vector \(v_{\rm phys}\) has normalized coordinates \(D_Fv_{\rm phys}\), so physical particle count is represented there by \[\mathfrak n_{\rm norm}\big|_{\text{one species}} =D_0^{-1}\mathbf1_7=(1,1,1,1,1,1,\tfrac12).\] The normalized charge row is \((1,-1,0,0,-1,-1,-1)\). In particular \(F\) has count \(1/2\) and charge \(-1\), whereas \(R_d=(U+L)/2\) has count \(1\) and charge \(-1\).

This distinction explains the fixed-displacement hypothesis. At \(d=0\), the height measures of \(R_d\) and \(F\) agree but their count values do not. Thus the all-ones count bound below cannot be uniform through \(d=0\). The same-species \(U,L\) factor also has the simple collision pole from Section 5, with magnitude proportional to \(1/|x\pm2id|\) near collision. The pointwise estimate consequently needs constants depending on \(d\). The contour argument sums the series at fixed endpoints on the two sides of the collision.

In normalized coordinates subtract from \(\mathcal H\) the species-off-diagonal matrix \[K_{\rm norm}(k)= \begin{pmatrix}0&\lambda VV^t\\ \lambda VV^t&0\end{pmatrix}, \qquad \lambda=\tfrac38\tanh(2k),\] where \[V_P=0,\quad V_M=1,\quad V_{O_\pm}=\tfrac12,\quad V_U=V_L=e^{-|d|k},\quad V_F=1.\] Write \(\mathcal H'=\mathcal H-K\), using congruence by \(D_F\) for both matrices when returning to the physical basis. Reflection of heights, together with negation of outer signs, preserves these forms; the outer at height \(1\) is unchanged by antiperiodicity. Thus \(P,M,F\) are even, while \(O_+\) and \(O_-\), and \(U\) and \(L\), are exchanged. The exact measure identity \[O_++O_-=P+M\] also holds for \(V\). Its two species copies generate a fixed nullspace \(\mathcal N\). Both physical count and common charge vanish on \(\mathcal N\).

The Fourier energy and box confinement

We now prove Theorem 15. Split the full matrix as \[\mathcal H(k) =Q_\infty\tanh(ek) +\bigl(K(k)-K(\infty)\tanh(ek)\bigr)+J(k).\] After multiplication by \(2\pi/k\), each of the first two matrices represents an entrywise nonnegative pair potential. To see this, put \[\phi_L(x)=-2\log|\tanh(\pi x/(4L))|.\] It is positive and increasing in \(L\), and its transform is \(2\pi\tanh(Lk)/k\). The first matrix has nonnegative coefficients by the lemma. Each nonzero entry of \(K\) is a positive multiple of \(\tanh(2k)e^{-rk}\), with \(r\ge0\); here \(r>0\) exactly when an entry involves \(U\) or \(L\). If \(r=0\), the difference above gives a positive multiple of \(\phi_2-\phi_e\). If \(r>0\), \(K(\infty)\) vanishes in that entry and the potential is the convolution of \(\phi_2\) with the positive Poisson kernel of parameter \(r\). These pair potentials can therefore be dropped from a lower bound on the negative log weight, on every forward pair and self-image.

For the remaining regular part, use the even symbol \[\widehat{\mathcal R}(k) =\frac{2\pi}{|k|} \left(J(|k|)-\frac{a_c}{|k|}\boldsymbol c\boldsymbol c^t\right), \qquad k\ne0,\] with its continuous value at \(k=0\). The low expansions above show bounded one-sided derivatives there; terms in \(|k|\) are allowed. At infinity the symbol is integrable, with tail \(O(k^{-2})\), and its piecewise second derivatives are integrable. Fourier inversion and two integrations by parts therefore show that \(\mathcal R(x)\) is continuous, bounded, and \(O((1+x^2)^{-1})\). Its periodization \[\mathcal R_T(x)=\sum_{r\in\mathbb Z}\mathcal R(x+rT)\] converges absolutely.

Index the physical particles by \(\alpha=1,\ldots,N\), with labels \(\iota_\alpha\) and positions \(x_\alpha\) in one period, and define the vector of point measures \[\rho_\iota=\sum_{\alpha:\,\iota_\alpha=\iota}\delta_{x_\alpha}, \qquad \widehat\rho(k)=\int_T e^{-ikx}\rho(dx),\qquad k\in(2\pi/T)\mathbb Z.\] The forward regular energy, including nonzero self-images, is \[\begin{aligned} E_{\mathcal R} &:=\sum_{\alpha,\beta}\ \sum_{\substack{r\in\mathbb Z\\x_\beta+rT>x_\alpha}} \mathcal R_{\iota_\alpha\iota_\beta}(x_\beta-x_\alpha+rT)\\ &=\frac12\sum_{\alpha,\beta} (\mathcal R_T)_{\iota_\alpha\iota_\beta}(x_\beta-x_\alpha) -\frac12\sum_\alpha\mathcal R_{\iota_\alpha\iota_\alpha}(0)\\ &=\frac1{2T}\sum_{k\in(2\pi/T)\mathbb Z} \widehat\rho(k)^*\widehat{\mathcal R}(k)\widehat\rho(k) -\frac12\sum_\alpha\mathcal R_{\iota_\alpha\iota_\alpha}(0). \end{aligned}\] Indeed the half-double sum pairs the forward image \((\alpha,\beta,r)\) with its reverse \((\beta,\alpha,-r)\). Only \((\alpha,\alpha,0)\) is excluded from the gas. The subtracted diagonal is therefore the unperiodized value \(\mathcal R_{\iota_\alpha\iota_\alpha}(0)\); every nonzero self-image remains.

Closure of the state path gives \[\boldsymbol c^t\widehat\rho(0)=0,\qquad p_c'=-\boldsymbol c^t\rho,\qquad ik\widehat p_c(k)=-\boldsymbol c^t\widehat\rho(k)\quad(k\ne0).\] The common component \(p_c=(p_1+p_2)/\sqrt2\) is periodic because the total charge is in \(8\mathbb Z(1,-1)\). Put \(\eta=\theta/(\sqrt2\pi)\) and \(\bar p_c=T^{-1}\int_Tp_c\). Parseval’s identity now gives the exact cancellation \[ \begin{aligned} E_{\mathcal R} +\pi a_c\Re\int_T(p_c+\eta)^2\,dx ={}&\frac{\pi}{T}\sum_{k\ne0} \widehat\rho(k)^*\frac{J(|k|)}{|k|}\widehat\rho(k) +\frac1{2T}\widehat\rho(0)^t\widehat{\mathcal R}(0)\widehat\rho(0)\\ &+\pi a_cT\bigl[(\bar p_c+\Re\eta)^2-(\Im\eta)^2\bigr] -\frac12\sum_\alpha\mathcal R_{\iota_\alpha\iota_\alpha}(0). \end{aligned} \tag{67}\] Here and below the sum uses \(k\in(2\pi/T)\mathbb Z\). At each nonzero frequency the negative term \(-\pi a_c|\boldsymbol c^t\widehat\rho(k)|^2/(Tk^2)\) in the pair energy is canceled by the state square. At zero, \(\widehat\rho(0)\) is perpendicular to \(\boldsymbol c\), so the limit of (64) gives \[\frac1{2T}\widehat\rho(0)^t\widehat{\mathcal R}(0)\widehat\rho(0) \ge\frac{\pi\gamma}{T} \left|\mathfrak n^t\widehat\rho(0)\right|^2.\] No positivity of \(\widehat{\mathcal R}(0)\) on other vectors is required. The mean square in (67) bounds a positive multiple of \(T\bar p_c^2\) up to \(O_\Theta(T)\). The diagonal correction costs \(O_d(N)\), independently of \(T\).

Let \(g_T\) be the normalized periodic Gaussian whose Fourier multiplier is \(e^{-k^2/2}\), and let \(\nu=\mathfrak n^t\rho=\sum_\alpha\delta_{x_\alpha}\). Combining (64) with (67), and including the local factors, gives for every nonzero post-collision weight \[-\log|W_{d,\theta}| \ge c_2\left(\|g_T*\nu\|_{L^2(T)}^2+\|g_T*p_c\|_{L^2(T)}^2\right) -C_2(N+T).\] Here \(c_2,C_2\) depend only on \(d,\Theta\). The local constants have logarithmic cost \(O_d(N)\). The Berry factors containing real state charges are phases; their \(\theta\)-dependent parts cost \(O_\Theta(N)\). For the pre-collision product (61), use the same identity with the real angle \(\vartheta=\Re\theta\). Its symmetrized local constants are finite for fixed \(d<0\), so they cost \(O_d(N)\), and the same bound holds with \(-\log W^{\rm pre}_{d,\Re\theta}\) on the left. The explicit heat and deterministic costs outside that product are absorbed afterward as described following the theorem. Zero weights already satisfy the desired inequality.

The first norm controls the box counts. In fact, with \(h_T=g_T*g_T\), \[\|g_T*\nu\|_2^2=\sum_{\alpha,\beta}h_T(x_\alpha-x_\beta) \ge c_1\sum_{j=0}^{T-1}n_j^2,\] because \(h_T\) is positive and has a uniform positive lower bound for pairs in the same unit box. Each jump of \(p_c\) has magnitude at most \(\sqrt2\). For \(x\) in box \(j\), express \(p_c(x)-(g_T*p_c)(x)\) by integrating its increments against the Gaussian. A jump in a box \(r\) boxes away is then multiplied by a Gaussian tail bounded by \(C e^{-c(r-2)_+^2}\). The resulting sequence of bounds is summable uniformly after periodization. Discrete Young’s inequality therefore gives \[\|p_c-g_T*p_c\|_2^2\le C_1\sum_j n_j^2.\] Within box \(j\), the same jump bound gives \[\sum_j p_{c,j}^2\le 2\|p_c\|_2^2+C_2\sum_j n_j^2.\] Using a smaller positive constant, the two smoothed norms consequently control \(\sum_j(n_j^2+p_{c,j}^2)\). Finally \(\sum_j n_j\le\varepsilon\sum_jn_j^2+C_\varepsilon T\) absorbs the linear costs. This proves Theorem 15.

The pointwise bound gives the required absolute summability. For a fixed common integer \(p_1+p_2\), there are eight classes in \(\mathbb Z^2/8\mathbb Z(1,-1)\); the Gaussian bound on the common charge sums over that integer. In each box, integration and the ordinary particle factorials reduce to a convergent sum of the form \(\sum_{n\ge0}A^n e^{-cn^2}/n!\). Thus both the pre-collision Gaussian integration and the post-collision gas series are absolutely summable at the fixed displacements used in Section 5.

The local state and its gauge

Fix a sufficiently small positive post-collision displacement \(d\), as in Section 6. Take \(T=bn\), where the integer block size \(b\) will be chosen sufficiently large and then held fixed. In block \(j\) write \(x=jb+t\), \(0<t<b\). Positions at ties or block boundaries form null sets and are discarded.

A block configuration \(Y\) is a finite ordered list of positions \(t\) and particle labels. It is measured by Lebesgue measure on each ordered simplex and counting measure on labels; the empty configuration is an atom. This is equivalent to the factorial convention for indistinguishable particles. Write \(q(Y)\) for its total charge vector. The state before a block is \[S=(Y_{\rm prev},p),\qquad p\in\mathbb Z^2/8\mathbb Z(1,-1),\] where \(Y_{\rm prev}\) records the preceding block in translated coordinates. Its successor after inserting \(Y\) is \[ S'=(Y,p-q(Y)). \tag{68}\] We use the indicated charge jumps to choose one consistent lift within each block. State measure is the product of the block-configuration measure and counting measure on \(p\). Particle types with zero local factor may be omitted. In particular \(P,M\) in each species have nonzero local factors, so their charges can be used to connect states.

Separate the forward residual interactions into pairs in the same block, pairs in neighboring blocks, and pairs at least two blocks apart. Let \(w_\theta(S,S')\) be the product of all constants, phases, and the integral-square weight in \(Y\), starting at \(p\), together with residual factors within \(Y\) and from \(Y_{\rm prev}\) to \(Y\). The site factors involving \(B_T\) are omitted for now. The function \(w_\theta\) is continuous and nonzero on each allowed open position simplex.

For a closed sequence of \(n\) blocks, the sum of the absolute values of the omitted tail log-magnitudes is at most \[C e^{-cb}\sum_j |Y_j|^2.\] Indeed an interaction across \(l\ge2\) blocks has horizontal separation at least \((l-1)b\), and each residual log decays exponentially there. This also applies to short periodic sequences, with their images included. For large \(b\), stability therefore bounds the sum of the local costs by \[ \sum_j-\log|w_0(S_j,S_{j+1})| \ge \frac{c'}b\sum_j\bigl(|Y_j|^2+p_{c,j}^2\bigr)-C_b n, \qquad p_{c,j}=\frac{p_{1,j}+p_{2,j}}{\sqrt2}. \tag{69}\] Fix an open parameter neighborhood \(\mathcal U_\theta\) whose closure \(D\) is compact in the complex \(\theta\) plane. The same bound, with adjusted constants, holds for the sum of \(-\log\sup_{\theta\in D}|w_\theta|\). The change from \(\theta=0\) costs at most a constant times \(b(1+|p_c|+|Y|)\) per block, which is absorbed by the quadratic terms. Constants in this section may depend on \(b,d,D\).

There is consequently a real measurable function \(\Phi\) on states such that \[ \left|e^{\Phi(S')-\Phi(S)}w_\theta(S,S')\right| \le \exp\!\left(C_b-\delta_b (|Y_{\rm prev}|^2+|Y|^2+p_c^2)\right), \qquad \delta_b=c''/b>0, \tag{70}\] for \(\theta\in D\) and allowed successors. To construct it, subtract a smaller quadratic cost from the edge costs used in (69), and add \(C_b\), so that every finite directed cycle has nonnegative modified cost. Let \(\Phi(S)\) be the infimum of modified costs of paths from a reference state to \(S\). Every state can be reached and returned from: insert \(P,M\) charges as needed at distinct positions, and then insert the required final block. A fixed return path bounds the infimum below by the negative of its cost, so the infimum is finite. The directed triangle inequality gives (70). For measurability, the intermediate positions in the infimum may be restricted to rational translated coordinates, using continuity on the open simplices; the terminal state \(S\) remains arbitrary and fixed. The discrete constraints depend only on the labels. On each label stratum this is a countable infimum of functions continuous in the endpoint, hence is measurable.

Exact propagation of the interaction tail

Choose the logarithm of each far residual factor using the expansion of \(K^r\) and the convergent logarithmic expansions of the short and outer factors. For an earlier label \(u\), a later label \(v\), and sufficiently large positive separation \(s\), that log has a representation \[ g_v e^{-D_0s}h_u \tag{71}\] on a complex sequence space, with bilinear row-column pairing. Here \(D_0\) is a direct sum of finite Jordan arrays whose diagonal values are \(\pi l/6\), \(l\ge1\). Their sizes and the entries of their nilpotent parts are uniformly bounded; those parts generate the polynomial factors below. The components of \(g_v,h_u\) have at most polynomial growth. To obtain this representation, write the residual log as \[\sum_{l\ge1}e^{-\pi l s/6}P_{l;uv}(s).\] The polynomials have uniformly bounded degree and polynomially growing coefficients, also after the fixed imaginary height shifts. A nilpotent chain generates each power of \(s\); separate coordinates can be used for each ordered label pair and degree. Any positive intervening delay then puts the resulting vectors in \(\ell^2\).

Set \[ \begin{split} A_0&=e^{-bD_0},\\ G(Y)&=\sum_{(t,v)\in Y}g_v e^{-D_0(t+b/2)},\\ H_1(Y_{\rm prev})&=\sum_{(t,u)\in Y_{\rm prev}} e^{-D_0(3b/2-t)}h_u. \end{split} \tag{72}\] Thus \(\|G(Y)\|\le Ce^{-cb}|Y|\) and \(\|H_1(Y_{\rm prev})\|\le Ce^{-cb}|Y_{\rm prev}|\), while the trace norm of \(A_0\) tends to zero as \(b\) increases. The offsets in (72) put exactly two blocks between a source and its first use. For example, a particle \((t_u,u)\) in \(Y_{j-2}\) enters the field used by \((t_v,v)\) in \(Y_j\) with exponent \[g_v e^{-D_0(t_v+b/2)}e^{-D_0(3b/2-t_u)}h_u =g_v e^{-D_0(2b+t_v-t_u)}h_u,\] which is (71) at their actual separation. Each additional factor \(A_0\) adds one more block of delay.

On the holomorphic Bargmann space over this sequence space, with orthonormal monomials \(z^\alpha/\sqrt{\alpha!}\), define the edge operator \[ \mathcal K(S,S')f(z)=e^{G(Y)z}f(A_0z+H_1(Y_{\rm prev})). \tag{73}\] Finite-mode Bose and Fermi trace–determinant formulas are derived in [12]. Here the affine terms and infinite-mode limit are also needed.

For large \(b\), the trace norm of (73) is at most \[ C\exp\!\left(C(\|G(Y)\|^2+\|H_1(Y_{\rm prev})\|^2)\right). \tag{74}\] Indeed linear composition \(C_Bf(z)=f(Bz)\) acts by symmetric tensor powers and has trace norm at most \((1-\|B\|_1)^{-1}\) when \(\|B\|_1<1\). Factor (73) through \(C_{4A_0}\), with outside linear dampings \(z\mapsto z/2\) carrying the exponential multiplier and the translation. Such damped operators are bounded with the estimate in (74): a map \(e^{gz}f(dz+h)\), \(d=1/2\), is a scalar times \(W_vC_dW_w\), where \[W_vf(z)=e^{-\|v\|^2/2+\bar v^tz}f(z-v), \qquad v+dw=\bar g,\quad dv+w=-h.\] The operators \(W_v\) are unitary and strongly continuous, by Gaussian translation in finite dimensions or directly on total exponential vectors.

For a closed sequence, multiply the edge operators in increasing block order. Composition and trace give \[ \begin{gathered} \operatorname{tr}\prod_{j=0}^{n-1}\mathcal K(S_j,S_{j+1}) =\det(I-A_0^n)^{-1}\exp\!\left(\sum_jG(Y_j)z_j\right),\\ z_{j+1}=A_0z_j+H_1(Y_{j-1}),\qquad z_n=z_0. \end{gathered} \tag{75}\] In finite one-particle dimension, the trace of \(e^{gz}f(Bz+h)\) for small \(\|B\|\) is \(\det(I-B)^{-1}\exp(g(I-B)^{-1}h)\). Integrating its coherent diagonal \(e^{gz+\bar z^t(Bz+h)}\) against normalized complex Gaussian gives this formula: Gaussian monomial orthogonality gives the trace resolution, and square completion works first for Hermitian \(B\) and then analytically in its entries. Composition proves (75) in finite dimension. Increasing Jordan-block truncations converge in trace norm by the preceding factorization: \(C_{4A_0}\) converges in trace norm by its tensor series, and the outside factors and their adjoints converge strongly. The periodic solution \(z_j\) sums the sources in blocks up to \(j-2\) in the periodic past. The explicit delay calculation above shows that the exponential in (75) is precisely the entire omitted tail, including its images.

The marked profile comparisons will apply this construction after establishing their local bounds and suppression estimates.

Empty-block traces and spectral consequences

Let \(E_0\) be the homogeneous operator with every site multiplier equal to one. Let \(I_0\) be the operator retaining only the labels \(M,F_{\rm pair}\). Both depend on \(\theta\). The exact traces are \[ \begin{split} \operatorname{tr}E_0^n &=\det(I-A_0^n)^{-1}/c(bn),\\ \operatorname{tr}I_0^n &=8\det(I-A_0^n)^{-1}\sum_{k\in\mathbb Z}u_k^n, \end{split} \qquad n\ge2. \tag{78}\] Here \[u_k=\exp\!\left[-\frac{4\pi b}{9}(k+\theta/\pi)^2\right].\] The first equality follows from the empty contour at \(m=0\) and the gas identity. In the second, every retained particle has strictly negative common charge. A closed sequence can therefore contain no particles, and the eight representatives at each \(k=p_1+p_2\) give the displayed empty-block sum. The same reasoning gives the trace formula with the sum restricted to a consecutive range of \(k\) for a triangular diagonal compression to that range. This full formula will also be used for the marked intervals.

We recall the spectral facts needed to read these traces. For a compact Hilbert–Schmidt operator, the limsup of the absolute \(n\)th roots of its power traces is its spectral radius, and those traces determine every nonzero eigenvalue with its algebraic multiplicity. To see this, approximate the operator by finite rank with norm error less than \(\epsilon\). Outside radius \(\epsilon\), the error resolvent exists and finite-dimensional inversion makes the full resolvent meromorphic with finite-rank pole parts. Resolvent contour integration isolates the Riesz blocks. The resolvent identity makes each Riesz projection idempotent and commuting, and Laurent comparison identifies the higher pole parts with powers of the nilpotent part on that block. The complementary powers are exponentially bounded at any radius containing the remaining spectrum; their traces have the same rate by retaining two Hilbert–Schmidt factors. On an isolated spectral radius the trace is therefore a sum of algebraic multiplicities times eigenvalue powers. Cesàro extraction of distinct unit-circle frequencies shows that this sum cannot tend to zero unless all its coefficients vanish. Applying this successively off zero proves the assertions. Separating resolvent contours give locally uniform bounds for norm-continuous families.

It follows from (78) and \(\log c(T)/T\to\pi/16\) that all nonzero eigenvalues of \(E_0\) are independent of \(\theta\), and \[ r(E_0)=r_E=e^{-b\pi/16}. \tag{79}\] For \(|\Re\theta|<\pi/2\), the unique leading eigenvalue of \(I_0\) is \(u=u_0\), with algebraic multiplicity eight. It is semisimple in this strip. Along successor edges a negative particle charge increases \(k\); the operator acts from an argument state to an output state, so in that operator direction \(k\) can only stay or decrease. Split into the three triangular diagonal compressions with negative, zero, and positive \(k\). Their traces are (78) restricted to the corresponding ranges, so the two side blocks are spectrally separated from \(u\). In the zero block, a successor state must have empty \(Y\). Splitting off these empty fibers leaves, per representative, the diagonal operator \(uC_{A_0}\) on an empty fiber and zero on the other diagonal block. The constants give a simple semisimple eigenvalue \(u\) for \(uC_{A_0}\); \(C_{A_0}\) is strictly contracting on higher Bargmann degrees.

For completeness, semisimplicity passes through each of these finite triangular decompositions as follows. A generalized \(u\)-eigenvector has zero component on the incoming nonresonant side by invertibility there. Its component on the resonant block is already a genuine eigenvector. Applying the operator minus \(u\) leaves a generalized eigenvector on the outgoing nonresonant side, which must also be zero. Thus, locally uniformly on compact neighborhoods in the strip, \[ (I_0/u)^s=P_I+O(e^{-cs}) \tag{80}\] in operator norm, where \(P_I\) is its Riesz projection. For \(E'=E_0/r_E\), peripheral Jordan blocks are retained: locally uniformly, \(\|(E')^s\|\le C(1+s)^r\), and its powers equal their finite peripheral Riesz expansions up to exponentially small error. The uniformity follows from fixed separating contours and the fixed finite ranks of the Riesz blocks.

The logarithmic interval and its exterior words

We now return to the balanced \(2m\)-site problem, with \(m\) even and sufficiently large at fixed \(b\). Put \[\kappa_1=\pi/3,\qquad L=\left\lfloor\frac{\log m}{\kappa_1b}\right\rfloor, \qquad a_m=me^{-\kappa_1bL}\in[1,e^{\kappa_1b}).\] For the labels \(P,O_\pm,U,L\), the infinite-volume factors before taking the \(m\)th power are, respectively, \[B(x),\qquad B(x\mp i),\qquad B(x+i(3/2+d))^{-1},\qquad B(x-i(3/2+d))^{-1}.\] Call these functions \(\omega_v(x)\); the labels \(M,F_{\rm pair}\) have factor one. The explicit formula for \(B\) gives \(|\omega_v(x)|<1\) and \[ \omega_v(x)=1-\Lambda_{v,\pm}e^{-\kappa_1|x|} +O(e^{-2\kappa_1|x|})\quad(x\to\pm\infty), \qquad \Re\Lambda_{v,\pm}>0. \tag{81}\] In particular \[|\omega_v(x)^m|\le e^{-cm e^{-\kappa_1|x|}},\qquad |\omega_v(x)^m-1|\le Cm e^{-\kappa_1|x|}.\] Inside \(|x|\lesssim bL\), these factors suppress all labels except \(M,F_{\rm pair}\); outside they approach one. This is the logarithmic interval over which the interior and exterior propagation rates differ.

Let \(T_{m,j}\) be the one-block kernel of Proposition 17 with the factor \(\prod_{(t,v)\in Y}\omega_v(jb+t)^m\). Define the central product and the exterior differences by \[ \begin{split} \widetilde C_m&=\prod_{j=-L}^{L-1}(T_{m,j}/u),\\ D_{+,k}&=T_{m,L+k}/r_E-E',\\ D_{-,k}&=T_{m,-L-1-k}/r_E-E',\qquad k\ge0, \end{split} \tag{82}\] where products in the spatial coordinate are in increasing order. A finite subset of exterior indices on a side has length \(l_\pm\) equal to one more than its maximum, and length zero when it is empty. Its word \(W_+\) consists of the factors on indices \(0,\ldots,l_+-1\), with \(D_{+,k}\) at chosen indices and \(E'\) otherwise. Define \(W_-\) in the same way in reverse order. The empty subset gives the identity operator. The occupied index interval of a nonempty word ends at its largest chosen index, with no factors beyond that index.

For a centered interval of \(n\ge2L\) blocks containing the central range, expand the exterior factors and rotate the trace to begin with \(C'_m=(u/r_E)^{2L}\widetilde C_m\). The term for two compatible exterior words has the form \[ \operatorname{tr}\bigl(C'_m W_+(E')^{s-l_+-l_-}W_-\bigr), \qquad s=n-2L. \tag{83}\] The next lemma identifies what is needed to estimate this trace: a trace-norm bound for the central product, exponential moments of the word lengths, and a limiting difference from a bare power trace. It allows all peripheral nilpotent terms of \(E'\).

A coefficient lemma for signed traces

The preceding decomposition leads to the following abstract statement. The operator \(E\) describes the exterior, \(C\) contains the central interval, and the two word families collect the perturbations at its ends. Their lengths include the unperturbed blocks between chosen perturbations. The lemma extracts the coefficient of each peripheral eigenvalue from a difference of traces.

Bounds for the central product and the words

Fix \(b,d\) and a compact parameter neighborhood \(D\) inside \(|\Re\theta|<\pi/2\). The gauged kernel bound and (81) give \[ \|D_{\pm,k}\|_2\le Ce^{-\kappa_1bk}. \tag{86}\] For \(-L\le j<L\), the Hilbert–Schmidt norm of \(T_{m,j}/u-I_0/u\) is at most \(C\exp(-c e^{\kappa_1bk})\), where \(k\) is the number of full blocks between \(j\) and the nearer edge. These estimates follow by telescoping the products of the specified scalar multipliers. A telescoping term costs at most a particle count times the displayed scalar difference; the quadratic gauge bound absorbs that count. In the central estimate, any configuration omitted by \(I_0\) contains at least one suppressed particle.

Expand the central differences against the bounded powers (80). Their norms are summable from both ends. Keeping a Hilbert–Schmidt factor at each endpoint gives \[ \|\widetilde C_m\|_1\le C. \tag{87}\] Likewise the peripheral projections and nilpotent powers of \(E'\) are uniformly bounded on \(D\) by separating contours, with orders bounded by their fixed finite ranks. Thus \(\|(E')^k\|\le A(1+k)^{r_J}\) there, for a fixed nonnegative integer \(r_J\). For a nonempty exterior word, pair each intervening power with the farther chosen perturbation at index \(k\). There is no unrestricted gap, so \[\|W_\pm\|\le \prod_{k\text{ chosen}}A_1(1+k)^{r_J} e^{-\kappa_1bk}.\] Its length is at most the sum of \(k+1\) over its chosen indices. Hence for \(0<\epsilon<\kappa_1b\), \[ \sum_W e^{\epsilon l(W)}\|W\| \le\prod_{k\ge0} \left(1+A_1(1+k)^{r_J} e^{-\kappa_1bk+\epsilon(k+1)}\right)<\infty, \tag{88}\] uniformly in \(m\). Equations (87) and (88) give the uniform central and word bounds required by Lemma 18.

We also need the dependence on the position of each edge within a block. At the right edge, with \(s=x-bL\), replace the suppressed powers by \(\exp(-a\Lambda_{v,+}e^{-\kappa_1s})\). At the left edge, with \(s=x+bL\), use \(\exp(-a\Lambda_{v,-}e^{\kappa_1s})\). These profiles define kernels analytic in \(a\) near the closed interval \([1,e^{\kappa_1b}]\). Choose this neighborhood so that \(\Re(a\Lambda_{v,\pm})\) remains positive for every suppressed label; the same domination and summable bounds then apply. The exterior differences in (82) converge to their profile differences at \(a=a_m\) with error \(Ce^{-cL}e^{-ck}\). In the first \(\lfloor L/4\rfloor\) central blocks from each edge the analogous normalized error is \(Ce^{-cL}\). Indeed the error from the two-term expansion of \(\log\omega_v\), after multiplication by \(m\), is \(O(me^{-2\kappa_1|x|})\) in these regions. Deeper central blocks differ from \(I_0/u\) by a superexponentially small amount.

It follows that there is an analytic trace-class family \(C(a,\theta)\) such that \[ \widetilde C_m=C(a_m,\theta)+O(e^{-cL}) \quad\hbox{in trace norm}. \tag{89}\] Construct it by taking the ordered left interior profile from its edge inwards, then \(P_I\), then the right interior profile from inside to its edge, and taking limits in the two lengths. Inserting an additional block next to \(P_I\) changes the product by a summable superexponential error, since \(P_I\) is fixed by \(I_0/u\) and all partial products are bounded. For comparison with \(\widetilde C_m\), the middle portion after the first interior blocks from both edges is \(P_I+O(e^{-cL})\) in operator norm by (80) and the same expansion. The bracketing products have bounded Hilbert–Schmidt norms. Replacing their edge portions by the profile kernels then gives (89) by telescoping.

The finite-site limit and coefficient extraction

For fixed \(m\), define the normalized physical quantity \[ R_m(\theta)=e^{-m^2\kappa_\infty}\frac{N(X)^2}{P_0} G_{A,B}(h+h^{-1}),\qquad \kappa_\infty=\lim_{T\to\infty}\kappa_T, \tag{90}\] with homogeneous \(X\) and \(h=e^{-2i\theta}\). The fixed-site contour limit, the gas identity, and the \(m=0\) trace in (78) identify \(R_m\) with the \(n\to\infty\) limit of the ratio of the trace with periodized site factors to \(\operatorname{tr}E_0^n\). Use blocks \(-\lfloor n/2\rfloor\le j<\lceil n/2\rceil\).

In the numerator normalized by \(r_E^n\), the periodized site factors can be replaced by the actual single-profile factors \(T_{m,j}\), not by their edge limits, with an error tending to zero. At fixed \(m\) each block difference costs \(O_m(e^{-cn})\) in the kernel estimate. To control its surrounding products, expand the unperiodized exterior differences by subsets in increasing or decreasing exterior index. Pair each power gap between successive perturbations with the farther index; the polynomial power bound for \(E'\) is then absorbed by the exponential decay in (86). Only one unrestricted power gap remains, giving a polynomial bound in \(n\) for any exterior half-segment. A whole subinterval has at most two such portions and a fixed central part. Expanding the additional periodization differences retains these polynomial bounds, while their exponential smallness makes the total error tend to zero. The same argument applies to traces by keeping two Hilbert–Schmidt factors.

The bare normalized traces \(\operatorname{tr}(E')^n\) are bounded: their peripheral expansions contain only multiplicity times eigenvalue powers, since positive nilpotent powers have trace zero. Multiplying the known ratio limit by these bounded traces and using the replacement just proved therefore gives \[ \operatorname{tr}\prod_j(T_{m,j}/r_E) -R_m\operatorname{tr}(E')^n\longrightarrow0 \tag{91}\] at fixed \(m\). No lower bound on the bare trace is used.

We now apply Lemma 18 to (83). Each actual exterior half has length \(s/2+O(1)\), rather than allowing every pair with \(l_++l_-\le s\). Extending to that latter set adds only pairs with total word length at least \(s/2-O(1)\). By (88) and the polynomial power bound, their contribution is \(O_m((1+s)^r e^{-cs})\). It is negligible at fixed \(m\). Thus (91) is the lemma’s difference hypothesis with \[E=E',\qquad C=C'_m=(u/r_E)^{2L}\widetilde C_m, \qquad a=2L.\]

For a peripheral eigenvalue \(\zeta\) of \(E'\), write \(P_\zeta\) for its Riesz projection, \(d_\zeta=\operatorname{rank}P_\zeta\), and \(J_\zeta=(E'-\zeta)P_\zeta\). The coefficient polynomial supplied by the lemma is, in the present notation, \[ \mathcal P_{\zeta,m}(s)= \sum_{W_+,W_-}\zeta^{-l_+-l_-} \operatorname{tr}\!\left( C'_mW_+\left[\sum_{t\ge0} \binom{s-l_+-l_-}{t}\zeta^{-t}J_\zeta^tP_\zeta\right]W_-\right). \tag{92}\] The nilpotent sum is finite. The word sum is coefficientwise absolutely convergent by (88). The lemma gives directly \[ R_m=\frac{\mathcal P_{\zeta,m}(0)}{d_\zeta\zeta^{2L}}, \qquad |R_m|\le C|u/r_E|^{2L}. \tag{93}\] The constant is uniform in \(m\) on the fixed parameter neighborhood: the word moments, central trace norm, and peripheral nilpotent bounds are uniform, and \(|\zeta^{2L}|=1\) makes the changing offset harmless.

The analytic dependence on the edge position is also retained. Replace \(\widetilde C_m\) in (92) by \(C(a,\theta)\) and each exterior difference by its profile limit. Their exponentially small weighted errors telescope in the absolutely summable word series. Hence, locally uniformly in \(|\Re\theta|<\pi/2\), \[ R_m(u/r_E)^{-2L}d_\zeta\zeta^{2L} =\mathcal F_\zeta(a_m,\theta)+O(e^{-cL}), \tag{94}\] where \(\mathcal F_\zeta\) is analytic in \(a\) near \([1,e^{\kappa_1b}]\) and locally analytic in \(\theta\). Fixed separating contours give the stated uniform spectral bounds. Equations (93)–(94) are the operator outputs used after the remaining homogeneous scalar normalization.

Confluence and the even cross Gram matrix

Put \(\eta=\pi/4\) and \(x_j=e^{\eta w_j}\). The kernel of \(N\) is \[B_0(w)=\frac{\sinh(\eta w)}{\cosh(\eta w)+\cos(\pi/4)}.\] Compare it with \(C(w)=\tanh(\pi w/6)\). Schur’s Pfaffian identity [11] gives \[\operatorname{Pf}_{j,k}\frac{y_j-y_k}{y_j+y_k} =\prod_{j<k}\frac{y_j-y_k}{y_j+y_k}.\] For clarity, multiplying the Pfaffian by \(\prod_{j<k}(y_j+y_k)\) produces an alternating polynomial of Vandermonde degree, hence a constant times \(\prod_{j<k}(y_j-y_k)\). The constant is one by taking positive \(y_1\gg y_2\gg\cdots\): the limiting skew matrix with all upper entries one has Pfaffian one by first-row induction. Taking \(y_j=e^{\pi w_j/3}\) gives the pair product for \(C\). As the \(w_j\) tend to zero, \[\frac{d(x_j,x_k)}{x_j-x_k}C(w_j-w_k) \longrightarrow p_0\frac{\pi/6}{\eta}=\frac{2p_0}{3}.\] Thus \(\mathcal P(X)\operatorname{Pf}C(w_j-w_k)\) tends to \((2p_0/3)^{m(2m-1)}\).

We next make the ratio of the two confluent Pfaffians explicit. For either analytic odd kernel \(B=B_0,C\), apply the same divided-difference transformations to its rows and columns. At confluence the entry of orders \(r,s\) is \[\frac{(-1)^s}{r!\,s!}B^{(r+s)}(0),\qquad 0\le r,s<2m.\] Entries with \(r,s\) of the same parity vanish. After grouping the even and odd orders, the Pfaffian is therefore, up to common signs, the determinant of the cross matrix with \(r=2a\), \(s=2b+1\). With the Fourier convention \(\widehat h(k)=\int_{\mathbb R}e^{-ikw}h(w)\,dw\), this cross entry is \[-\frac{(-1)^{a+b}}{(2a)!(2b+1)!} \frac1{2\pi}\int_{\mathbb R}k^{2(a+b)}\widehat{B'}(k)\,dk.\] The signs and factorials cancel in the ratio for \(B_0\) and \(C\). Consequently that ratio is the ratio of Gram determinants of \(1,k^2,\ldots,k^{2m-2}\) for the weights \(\widehat{B_0'}(k)\,dk\) and \(\widehat{C'}(k)\,dk\).

Both weights are strictly positive and have all moments finite. To compute them, put \(\widetilde B_\beta(t)=\sinh t/(\cosh t+\cos\beta)\), so \[B_0(w)=\widetilde B_{\pi/4}(\pi w/4),\qquad C(w)=\widetilde B_{\pi/2}(\pi w/6).\] Applying \(\cos\beta+\sin\beta\,\partial_\beta\) to the transform of \((\cosh t+\cos\beta)^{-1}\) from Lemma 5 gives \(\widehat{\widetilde B_\beta'}(\xi) =2\pi\xi\cosh(\beta\xi)/\sinh(\pi\xi)\). Scaling the two identities above gives \[\widehat{B_0'}(k)=\frac{8k\cosh k}{\sinh4k},\qquad \widehat{C'}(k)=\frac{6k}{\sinh3k}.\] At zero these expressions use their continuous limits. For \(t=e^{-|k|}\) their ratio is \[\frac{\widehat{B_0'}(k)}{\widehat{C'}(k)} =\frac23 e^{\phi(k)},\qquad \phi(k)=\log\frac{(1+t^2)(1-t^6)}{1-t^8}.\] Extend \(\phi\) continuously by \(\phi(0)=\log(3/2)\). Set \(u=3k/\pi\) and \[\mu(u)=\frac{2u}{\sinh\pi u},\qquad f(u)=\phi(\pi u/3).\] Use \(\mu(0)=2/\pi\). The measure \(\mu(u)\,du\) is a probability measure, and \(f\) is real, continuous, even, and exponentially decaying. Let \(P_m\) be the orthogonal projection in \(L^2(\mu(u)\,du)\) onto \(\operatorname{span}\{1,u^2,\ldots,u^{2m-2}\}\). Changing variables in the Gram determinants gives the exact identity \[ N(1,\ldots,1) =\left(\frac{2p_0}{3}\right)^{m(2m-1)} \left(\frac23\right)^m \det(P_m e^fP_m|_{\operatorname{ran}P_m}). \tag{96}\] Here \(e^f\) denotes multiplication by \(e^{f(u)}\). We now evaluate this last determinant at logarithmic order.

A logarithmic statistic for even tests

For the function in (96), \[\frac1{2\pi}\int_{\mathbb R}f(u)\,du =\frac3{\pi^2}\int_0^\infty\phi(k)\,dk.\] Expanding the three logarithms into geometric series gives \[\int_0^\infty\phi(k)\,dk =\frac{\pi^2}{24}-\frac{\pi^2}{36}+\frac{\pi^2}{48} =\frac{5\pi^2}{144}.\] Lemma 19 therefore makes the determinant in (96) equal to \(m^{5/48+o(1)}\). Substitution proves (95).

The fused cylinder increment

On a finite cylinder of circumference \(M\) with closed rims, let \(Z\) be the partition of exactly one unoriented noncontractible polygon. As in Section 2, it is the coefficient of \(n\) in the loop sum with contractible-loop weight zero and noncontractible-loop weight \(n=p+p^{-1}\). Write \(\mathcal T_p(\phi)\) for the oriented row transfer matrix with row parameter \(\phi\), and put \(\alpha=\pi/8\).

The row matrices with the same twist commute. Indeed, multiply the local exchange identity across the sites and trace over the two auxiliary spaces. Their twist \(p^{s_1+s_2}\) commutes with the exchanging matrix. This proves commutation where that matrix is invertible, and then identically by continuation. In particular, a stack matrix element between empty rims, and its coefficient \(Z\), is symmetric in its rows.

We will also need the order of the error in fusing two rows. Let \(U=\operatorname{im}J\subset V_1\otimes V_2\) for the embedding at \(2\alpha\) in Section 2, and at site \(j\) set \[X_j(\phi)=R_{2j}(\phi)R_{1j}(\phi+2\alpha).\] The local intertwining preserves \(U\otimes V_j\) and, through the isomorphism \(J:V\to U\), induces \(R(\phi+\alpha)\). At \(\phi=0\), \[X_j(0)=P_{2j}R_{1j}(2\alpha),\qquad \operatorname{im}X_j(0)\subset U\otimes V_j,\] where \(P_{2j}=R_{2j}(0)\) is the swap. The inclusion follows because \(R_{1j}(2\alpha)=J_{1j}J_{1j}^*\) and the swap carries the embedded spaces \(1,j\) to \(1,2\). Thus the operator induced by \(X_j(\phi)\) on the quotient by \(U\otimes V_j\) is \(O(\phi)\). Its product over the \(M\) sites is \(O(\phi^M)\).

The twist preserves this invariant subspace and satisfies \(p^{s_1+s_2}J=Jp^s\). Taking the auxiliary trace on the invariant subspace is therefore the trace of the fused row; the similarity through \(J\) introduces no factor involving \(J^*J\). Taking the trace on the quotient gives \[ \mathcal T_p(\phi)\mathcal T_p(\phi+2\alpha) =\mathcal T_p(\phi+\alpha)+O(\phi^M). \tag{107}\] This is a locally holomorphic statement in \((\phi,p)\) near \((0,i)\). After taking an empty-rim stack matrix element, the loop sum is a polynomial in \(n=p+p^{-1}\). Since \(dn/dp=2\) at \(p=i\), extracting its coefficient of \(n\) preserves the order of vanishing.

Put one new outer row on \(L\) honeycomb rows, whose parameters are all \(\alpha\), and subtract the partition without the new row. We next describe the height limit of this increment. A loop wholly in the new row gives \(v^M\). Otherwise the lift of the loop uses \(2k>0\) ports into the base strip per period. No exterior port is used. The new row therefore pairs consecutive ports in \(k\) occupied intervals, each with one turn of weight \(u\), one of weight \(d\), and straight tiles of weight \(v\).

Here the topology forces a particularly simple matching in the base. Number the lifted ports in increasing order so that the outer intervals are \((2j-1,2j)\). In a noncrossing lower perfect matching, the interior of a pair is matched internally, so the pair’s endpoints have opposite parity. If its left endpoint is odd, the block including the endpoints is closed under both matchings. If its left endpoint is even but the right endpoint is not the next port, the interior is closed under both matchings. Either case creates a finite component. The lift of one noncontractible polygon is an infinite component, so the lower pairs are exactly \((2j,2j+1)\).

For \(1\le d<M\), let \(H_M(d)\) be the weight of half-plane arches from a fixed port to the port \(d\) steps to its right that are disjoint from all their nontrivial horizontal translates by \(M\). More generally the coefficient for \(k\) lower arches requires the arches and all their translates to be mutually disjoint. At fixed \(M\) these coefficient sums converge as \(L\to\infty\): dropping the mutual disjointness conditions bounds each by a product of the finitely many \(H(d)\) involved. The coarse summability proved in Section 2 suffices for this limit.

The tile weights satisfy \[ud=c_\alpha v(1-v),\qquad c_\alpha=2\sin\alpha.\] There are at most \(M-2k\) straight tiles in a \(k\)-interval term. Hence the limiting increment is a polynomial \(F(v)\) of degree at most \(M\). For one lower gap of length \(d\), the upper interval has \(M-d-1\) straight tiles and \(M\) possible positions in a labeled period. These positions count each unoriented polygon once. Isolating these terms gives \[ \begin{split} F(v)&=v^M+ Mc_\alpha v(1-v)\sum_{d=1}^{M-1}v^{M-d-1}H_M(d)\\ &\qquad +(1-v)^2Q_M(v),\\ F(1)&=1,\qquad F'_v(1)=M\left(1-c_\alpha\sum_{d=1}^{M-1}H_M(d)\right), \end{split} \tag{108}\] for a polynomial \(Q_M\) (zero when no term with two intervals is possible).

We now pass (107) to these increments. Commute its two added rows to opposite closed rims of the base stack. Inclusion–exclusion says that the difference between the two-row increment and the sum of the one-row increments consists precisely of loops touching both added rows. On a closed rim no tile can use a double turn. On a fixed complex parameter neighborhood each remaining outer weight is bounded in absolute value by a constant times the corresponding positive honeycomb weight. There are only \(2M\) outer tiles. In the arch decomposition from the top, at least one base arch of a loop touching both rims reaches depth comparable to \(L\). Its weight tends to zero by monotone convergence in the finite sums \(H(d)\), after dropping the other disjointness conditions. The same domination makes this convergence locally uniform in the two outer parameters.

The one-row increments converge coefficientwise as polynomials in \(v\), and therefore locally uniformly in their parameters. Cauchy’s formula now passes every vanishing derivative of order below \(M\) in (107) to the limiting increments. With \[y=i(2\phi-3\alpha),\qquad v=\frac{\cosh y-\sin\alpha}{\cosh y+\cos\alpha},\qquad f(y)=1-F(v),\] the resulting identity near \(y=-3i\alpha\) is \[ f(y)+f(y+4i\alpha)-f(y+2i\alpha) =1+O((y+3i\alpha)^M). \tag{109}\]

A positive moment representation

For \(k>0\), introduce column vectors indexed by \(j=0,\ldots,M-1\) and positive scalar weights \[\begin{gathered} A_j(k)=\partial_\alpha^j\sinh(\alpha k),\qquad B_j(k)=\partial_\alpha^j\cosh(\alpha k),\\ w(k)=\frac{2(1+4\sinh^2(\alpha k))}{\sinh(\pi k)},\qquad w_0(k)=k w(k). \end{gathered}\] Since \(F(1)=1\), the function \(f(y)\) is a linear combination of \((\cosh y+\cos\alpha)^{-r}\), \(1\le r\le M\). The first \(M\) parameter derivatives of \[\frac{\sin\alpha}{\cosh y+\cos\alpha} =\int_{\mathbb R}e^{iky}\frac{\sinh(\alpha k)}{\sinh(\pi k)}\,dk\] span these powers: the derivative of order \(j\) has a nonzero coefficient of the denominator power \(j+1\). Lemma 5 therefore gives a vector \(\mathbf c\) such that \[f(y)=\int_{\mathbb R}e^{iky} \frac{A(k)^T\mathbf c}{\sinh(\pi k)}\,dk.\] The integral and its required derivatives converge locally uniformly in \(|\Im y|<\pi-\alpha\).

At \(y=-3i\alpha\), the three arguments in (109) are \(-3i\alpha,i\alpha,-i\alpha\), all in this strip. Split the Fourier integral into positive and negative \(k\). For derivative order \(j\), this gives a factor \(2i^jk^j\) times \(\cosh(3\alpha k)\) if \(j\) is even, and times \(\sinh(3\alpha k)-2\sinh(\alpha k)\) if \(j\) is odd. Since \(B_j(k)\) is \(k^j\cosh(\alpha k)\) or \(k^j\sinh(\alpha k)\) in the two cases, the identities \[\frac{\cosh(3z)}{\cosh z} =\frac{\sinh(3z)-2\sinh z}{\sinh z} =1+4\sinh^2z\] then turn the \(M\) derivative equations into \[ D\mathbf c=B(0),\qquad D=\int_0^\infty B(k)A(k)^T w(k)\,dk. \tag{110}\]

Write \(\Delta(z)=\prod_{i<j}(z_j-z_i)\) for the Vandermonde. The matrix \(D\) is nonsingular. Indeed, the ordered determinant expansion gives \[\begin{aligned} \det D={}&\int_{0<k_1<\cdots<k_M} \det[B(k_1),\ldots,B(k_M)]\\ &\quad\cdot\det[A(k_1),\ldots,A(k_M)] \prod_iw(k_i)\,dk_i. \end{aligned}\] Both determinants in the integrand are strictly positive. To see this, expand \(\sinh(uk)\) or \(\cosh(uk)\) in its positive power series. For each increasing list of powers \(p_\ell\), the coefficient minor after taking successive derivatives at \(u=\alpha\) is a positive factor times \(\det[(p_\ell)_{\underline j}]=\Delta(p)\). The other minor is a generalized Vandermonde in the ordered positive \(k_i\), also positive. Its nonvanishing follows by the usual zero bound obtained from Rolle’s theorem after successively dividing out the lowest power; its sign follows by separating the positive \(k_i\) in scale. The positive series expansion now proves the claim.

Set \[K(z)=A(z)^TD^{-1}B(0).\] For real \(y\to+\infty\), the residue at \(k=i\) in the Fourier formula for \(f\) gives \(-2iK(i)e^{-y}\) as its first term. On the polynomial side, \[1-v=2(\cos\alpha+\sin\alpha)e^{-y}+O(e^{-2y}).\] Equating the coefficients proves \[ F'_v(1)=\frac{-iK(i)}{\cos\alpha+\sin\alpha}. \tag{111}\] We next express the quantity on the right by a positive expectation.

Give \(0<k_1<\cdots<k_M\) the probability density proportional to \[ \begin{gathered} \det[A(k_1),\ldots,A(k_M)]\\ {}\cdot\det[B(k_1),\ldots,B(k_M)]\prod_iw(k_i). \end{gathered} \tag{112}\] This law has a useful orbital form. For \[D_k=\operatorname{diag}(k_1,\ldots,k_M,-k_1,\ldots,-k_M), \qquad Q=\mathcal U\operatorname{diag}(\alpha I_M,-\alpha I_M)\mathcal U^*,\] where \(\mathcal U\) has normalized Haar measure on \(U(2M)\), put \[I(k)=\mathbb E\exp(\operatorname{tr}(D_kQ)).\] Then (112) is proportional to \[ \Delta(k^2)^2\prod_iw_0(k_i)\,I(k). \tag{113}\] In particular it is a positive probability density.

For completeness, we derive the paired confluent form used here from the Harish-Chandra–Itzykson–Zuber integral [19]. For distinct real lists \(z,u\) of size \(d\), put \(D_z=\operatorname{diag}(z)\) and \(D_u=\operatorname{diag}(u)\). Normalized unitary integration gives \[\int_{U(d)}e^{\operatorname{tr}(D_z\mathcal U D_u\mathcal U^*)}\,d\mathcal U =C_d\frac{\det[e^{z_i u_j}]}{\Delta(z)\Delta(u)}, \qquad C_d>0.\] One derivation uses the isotropic compression mechanism of [1]. Compress \(\mathcal U D_u\mathcal U^*\) to a leading \((d-1)\)-dimensional minor. Its ordered eigenvalues \(l_i\) interlace the ordered \(u_i\), and the last resolvent entry is \[\sum_{j=1}^d\frac{\omega_j}{t-u_j} =\frac{\prod_{j<d}(t-l_j)}{\prod_{j\le d}(t-u_j)}.\] Here \(\omega\) is uniform on the simplex, since its entries are squared moduli of a uniform complex unit vector. The residues give a bijection between the simplex interior and the interlacing region. Its absolute Jacobian is \(\Delta(l)/\Delta(u)\), as follows by differentiating the residues, or by passing through the coefficients of the monic numerator. The compressed matrix, conditional on its eigenvalues, is again unitarily invariant. Induction with test parameters \(z_i-z_d\) integrates each exponential column over \([u_j,u_{j+1}]\); column subtraction and the factor \(e^{z_d\sum u_j}\) give the displayed determinant formula. The simplex normalization changes only \(C_d\).

Apply this formula to the lists \(\pm k_i,\pm u_i\). The absolute paired Vandermonde is \[|\Delta(k_1,\ldots,k_M,-k_1,\ldots,-k_M)| =2^M\prod_i k_i\,\Delta(k^2)^2,\] and similarly for \(u\). Splitting the exponential determinant into its sinh and cosh blocks therefore gives a positive constant times \[\frac{\det[\sinh(k_i u_j)]\det[\cosh(k_i u_j)]} {\Delta(k^2)^2\Delta(u^2)^2\prod_i k_i u_i}.\] Divide both numerator determinants by \(\Delta(u)\) and let \(u_i\to\alpha\) by divided differences. The remaining \(u\) factors are constants, and the numerator becomes the two derivative determinants in (112). This proves (113). Also \[I(k)\le e^{2\alpha\sum_i k_i},\qquad w_0(k)\sim4k e^{-6\alpha k}\quad(k\to\infty),\qquad w_0(0)=2/\pi,\] so the density is integrable and extends continuously when one point approaches zero.

Orbital derivatives and stochastic comparison

We will apply the same estimates to the unpinned and pinned densities. For \(e=0\), write \(\mathbb O_N^{(0)}\) for the unpinned orbital law of size \(N\). For \(e=4\), write \(\mathbb O_N^{(4)}\) for the law obtained by pinning one point of the size-\(N+1\) orbital law. Their densities have the form \[\Delta(k^2)^2\prod_{i=1}^N k_i^e w_0(k_i)\,I^{(e)}(k),\] where \(I^{(4)}\) retains its zero pair. All coordinates below are sorted and positive.

For a positive one-point weight \(w_*\), let \(\mathbb P_N[w_*]\) denote the pure law with density \(\Delta(k^2)^2\prod_iw_*(k_i)\), whenever integrable. Write \(\mu\preceq_{\rm st}\nu\) when the expectation of every bounded coordinatewise increasing function under \(\mu\) is at most its expectation under \(\nu\).

Every pure law used here is positively associated: the covariance of two bounded increasing functions is nonnegative. Here is an induction that also covers any fixed upper endpoint. The size-one assertion is the one-dimensional covariance inequality. Given the last point \(t\), the earlier points have the same pure form with weight \(w_*(k)(t^2-k^2)^2\) below \(t\). If \(s<t\), the likelihood ratio for conditioning at \(s\) relative to conditioning at \(t\) is proportional to \[{\bf1}_{k_{N-1}<s}\prod_{i<N} \left(\frac{s^2-k_i^2}{t^2-k_i^2}\right)^2.\] It decreases in every coordinate, including across its cutoff. Association at size \(N-1\) therefore makes the earlier points stochastically increasing in the last point. Conditional covariance and the one-dimensional covariance inequality then prove association at size \(N\). In particular an increasing (respectively decreasing) likelihood ratio tilting a pure law moves the points upward (respectively downward) in stochastic order. Truncation extends this rule to the unbounded nonnegative tests used below.

The logarithmic derivative of \(w_0\) is \[D_0(k)=\frac1k+3\alpha\tanh(3\alpha k) -\alpha\tanh(\alpha k)-8\alpha\coth(8\alpha k).\] It satisfies \[ \begin{gathered} -6\alpha<D_0(k)<0,\qquad D_0(k)\longrightarrow-6\alpha,\\ D_0(k)=-\tfrac{40}{3}\alpha^2 k+O(k^3)\quad(k\downarrow0). \end{gathered} \tag{114}\] For the upper bound put \(u=\alpha k\) and use \(8\coth(8u)>1/u+4\tanh(4u)\). For the lower bound, it suffices to use \[\frac{16u}{e^{16u}-1}+\frac{6u}{e^{6u}+1}<1,\] which follows from \(e^{16u}\ge1+16u+(16u)^2/2\) and \(e^{6u}\ge1+6u\). The limits follow by expansion.

By Lemma 22, the logarithmic derivative of the orbital density relative to the pure density with weight \(k^e w_0(k)e^{2\alpha k}\) is \(2q_i-2\alpha\le0\). Hence \[ \mathbb O_N^{(e)} \preceq_{\rm st}\mathbb P_N[k^e w_0(k)e^{2\alpha k}], \qquad e=0,4. \tag{115}\] For this pure law its first \(j\) points are in turn stochastically below \(\mathbb P_j[k^e w_0(k)e^{2\alpha k}]\). Indeed, conditioning on the later points multiplies the latter density by \(\prod_{i\le j<r}(k_r^2-k_i^2)^2\) and an upper cutoff, both decreasing in the first \(j\) coordinates.

The largest point of this standalone size-\(j\) law is at most \(Cj\) with probability \(1-O(e^{-cj})\), with constants valid for \(e=0,4\). To verify the needed uniformity, condition on all but one of its unsorted points. The remaining density is proportional to \(p(k)^2W(k)\), where \(\deg p\le2j\) and \(W(k)=k^e w_0(k)e^{2\alpha k}\). On \([j,2j]\), \(W\ge c e^{-C_0j}\), while \(W\) is bounded above by a fixed polynomial times an exponentially decaying function. Choose \(2j+1\) interpolation points in uniformly spaced small subintervals of \([j,2j]\), with widths and intervening gaps bounded below, at which \(|p|\le C\|p\|_{L^2[j,2j]}\). Lagrange interpolation, using denominators bounded below by \(c^{2j}l!(2j-l)!\), gives for \(k>2j\) \[|p(k)|\le\|p\|_{L^2[j,2j]} [C_1(k+2j)/j]^{2j}.\] After weakening the exponential decay to absorb the fixed polynomial, this implies, for a sufficiently large fixed \(A\), \[\begin{aligned} \frac{\int_{Aj}^\infty p(k)^2W(k)\,dk} {\int_0^\infty p(k)^2W(k)\,dk} &\le C e^{C_0j}\int_{Aj}^\infty [C_1(k+2j)/j]^{4j}e^{-c_0k}\,dk \\ &\le C e^{-c_1j}. \end{aligned}\] The last inequality follows after \(k=jt\) because the linear negative term in \(t\) eventually dominates \(4\log(C_1(t+2))\). A union bound proves the maximum estimate.

Fix \(0<\epsilon<1\) and let \(m=\lfloor N^{1-\epsilon}\rfloor\). Applying the preceding bounds to the first \(j\) points and summing over \(j\ge m\) shows that under either orbital law the event \[E=\{k_j\le C\max(m,j)\text{ for every }j\}\] has probability \(1-O(e^{-c m})\). On \(E\), for \(i\le m\), \[\sum_{\ell=1}^N(k_i+k_\ell)^{-1} \ge c\sum_{\ell=m}^N(m+\ell)^{-1} \ge c'\log(N/m).\] Lemma 22 therefore gives \[ \sup_{\substack{i\le m\\ k\in E}}q_i \le\frac{C}{\epsilon\log N}=o(1) \tag{116}\] for each fixed \(\epsilon\).

Gamma comparison and evaluation

For \(\ell,\beta>0\) define \[W_{\ell,\beta}(k)=|\Gamma(\ell+i\beta k)|^2,\qquad b=\pi\beta.\] These weights will bound the points in the direction needed for the decreasing product in Lemma 21. For the event \(E\) above, write \(k_{\le m}=(k_1,\ldots,k_m)\) and \(k_{>m}=(k_{m+1},\ldots,k_N)\).

We next compute the particular pure statistic appearing here. Throughout this calculation \(\ell,\beta>0\) are fixed. Normalize \(|\Gamma(\ell+iu)|^2\,du\) to a probability measure on \(\mathbb R\). Its Fourier transform is \(\cosh(t/2)^{-2\ell}\): writing \[\Gamma(\ell+iu)=\int_{\mathbb R}e^{(\ell+iu)y-e^y}\,dy\] reduces the transform of its squared modulus to the autocorrelation of \(e^{\ell y-e^y}\), which is evaluated by scaling \(e^y\). Shifting this Fourier contour by a small positive or negative imaginary amount also gives exponential decay of the weight. Thus the generating function \[(1-iz)^{-\ell+iu}(1+iz)^{-\ell-iu} =\sum_{n\ge0}P_n(u)z^n\] gives the Meixner–Pollaczek polynomials \(P_n^{(\ell)}(u;\pi/2)\) [16]. Integrating the product of two generating functions near zero gives \((1-zz')^{-2\ell}\). Consequently the \(P_n\) are real, have degree \(n\) with leading coefficient \(2^n/n!\), have parity \(n\), and have squared norms \[c_n=\frac{(2\ell)_n}{n!}\sim\frac{n^{2\ell-1}}{\Gamma(2\ell)}.\] Their evaluations needed below satisfy \[\begin{aligned} (-1)^jP_{2j}(0)&=\frac{(\ell)_j}{j!} \sim\frac{j^{\ell-1}}{\Gamma(\ell)},\\ (-1)^jP_{2j}(i\beta)& \sim\frac{2^{2\beta-1}j^{\ell+\beta-1}}{\Gamma(\ell+\beta)}. \end{aligned}\] For the second equivalent, take the even coefficients of \((1-t)^{-\ell-\beta}(1+t)^{-\ell+\beta}\). If \(\ell\ge\beta\), factor it as \((1-t^2)^{-\ell+\beta}(1-t)^{-2\beta}\) and use the positive binomial convolution. For \(\ell>\beta\), its binomial equivalents, with the summation index divided by \(j\), give the leading constant \[\frac{2^{2\beta-1}}{\Gamma(\ell-\beta)\Gamma(2\beta)} \int_0^1v^{\ell-\beta-1}(1-v)^{2\beta-1}\,dv =\frac{2^{2\beta-1}}{\Gamma(\ell+\beta)};\] the case \(\ell=\beta\) is immediate. If \(\ell<\beta\), the coefficients of \((1+t)^{\beta-\ell}\) are \(O((1+n)^{\ell-\beta-1})\) and are absolutely summable. Convolution with the first factor is asymptotic to its coefficient times their sum \(2^{\beta-\ell}\); the indices of the summable factor above \(n/2\) contribute only \(O(n^{2\ell-1})\), smaller than the main term. This proves the displayed equivalents.

Use the even polynomials \(P_{2j}(\beta k)\) on the positive axis. Their evaluation kernel through \(j=L-1\) is the kernel of \(\mathbb P_L[W_{\ell,\beta}]\), up to its fixed one-point normalization. The determinant expansion for its intensity at zero therefore gives \[\rho_{\Gamma,L}(0)\asymp_{\ell,\beta} \sum_{j<L}\frac{P_{2j}(0)^2}{c_{2j}} \asymp_{\ell,\beta}1+\log L.\] Pinning one point of this law gives \(\mathbb P_{L-1}[k^4W_{\ell,\beta}]\); denote its expectation by \(\mathbb E_{\Gamma,L}^{\rm pin}\). Replacing evaluation at zero in one determinant by evaluation at \(i\) changes its Vandermonde by exactly \(\prod_j(1+k_j^{-2})\). Hence \[ \begin{split} \mathbb E_{\Gamma,L}^{\rm pin}\prod_j(1+k_j^{-2}) &=\frac{\sum_{j<L}P_{2j}(0)P_{2j}(i\beta)/c_{2j}} {\sum_{j<L}P_{2j}(0)^2/c_{2j}}\\ &\asymp_{\ell,\beta}\frac{L^\beta}{1+\log L} =L^{\beta+o(1)}. \end{split} \tag{117}\] Indeed the denominator summands are asymptotic to a positive constant times \(j^{-1}\), and the numerator summands to a positive constant times \(j^{\beta-1}\). In particular the exponent statement is for fixed \(\ell,\beta\).

We can now finish the estimate of (111). For the unpinned orbital law, its lower stochastic bound in Lemma 23 bounds its decreasing minimum-tail event from above by the Gamma minimum tail. Conversely, (115) and the comparison of the first point to the standalone one-point pure law bound that event from below by the one-point law with weight \(w_0e^{2\alpha k}\). Taking \(t\downarrow0\) for each \(M\) gives \[c\le\rho(0)\le C(1+\log M).\] The constants may depend on the fixed comparison parameters.

For the pinned law, the product in Lemma 21 is decreasing. The first Gamma comparison therefore bounds its expectation above by (117) with \(L=N+1=M\), since the pinned law has \(N=M-1\) moving points. For a lower bound, restrict to \(E\) and discard all factors after \(m\), which are at least one. Conditional on the later points, the second Gamma comparison bounds the remaining decreasing product below by the pure size-\(m\) expectation, namely (117) with \(L=m+1\). Its bound is multiplied by \(\mathbb P(E)=1-O(e^{-cm})\).

To record the order of limits, first fix \(b_+>6\alpha\) for the upper bound, and fix \(b_-<6\alpha\) and \(\epsilon>0\) for the lower bound. Choose their \(\ell\) parameters as in the comparison lemma, and only then let \(M\to\infty\). Put \(a_M=\log F'_v(1)/\log M\). The pin and residue formulas give \[(1-\epsilon)\frac{b_-}{\pi} \le\liminf_{M\to\infty}a_M \le\limsup_{M\to\infty}a_M \le\frac{b_+}{\pi}.\] Letting these fixed parameters approach \(6\alpha\) and then \(\epsilon\downarrow0\) proves \[ F'_v(1)=M^{3/4+o(1)}. \tag{118}\]

Wedge comparison

We return to the physical arches. Define the positive deficit \[\delta_M=\frac1{c_\alpha}-\sum_{d=1}^{M-1}H_M(d) =\frac{F'_v(1)}{c_\alpha M} =M^{-1/4+o(1)}.\] The coarse bound \(c_\alpha\sum_{d>0}H(d)\le1\) from Section 2, together with \(H_M(d)\le H(d)\) and \(\delta_M\to0\), gives the saturation \[\sum_{d>0}H(d)=\frac1{c_\alpha}.\] Consequently, with \(A(n)=\sum_{d\ge n}H(d)\), \[A(M)\le\delta_M.\] Every arch contained within distance strictly less than \(M/2\) of its source is disjoint from its nontrivial translates by \(M\). Thus the arches counted in the deficit all reach distance at least \(M/2\), giving \[\delta_M\le W_{M/2}.\] Choosing \(M\) just above \(2R\) proves the required lower exponent for \(W_R\) and the stated upper exponent for \(A(n)\).

For the upper radius bound, intersect the upper half-plane with the wedge between rays of angles \(0\) and \(\theta=\pi/3\). A source on its horizontal ray has distance \(a\in\{1/2,3/2,\ldots\}\) from the vertex. The signed balance holds on the infinite wedge. To justify this, first exhaust the half-plane by convex tessellation polygons with extending horizontal bases. The real contribution of their same-side arches tends to one by saturation. Every other real boundary coefficient is at least \(\sin\alpha\), so the total raw weight of remote exits tends to zero. Remote exits in the wedge intersections are a subset of these exits. The fixed-ray exit sums converge absolutely by the same positive bound, and the finite signed identities therefore pass to the wedge.

Put \(s=3/8\). For same-side walks the phases are \(e^{is\pi}\) to the left and \(e^{-is\pi}\) to the right, while exits to the other ray have phase \(e^{is\theta}\). Let \(L_<(a)\) and \(L_>(a)\) be the lost left and right same-side weights on restriction from the half-plane to the wedge, including endpoints removed by the restriction. Subtract the balances and project perpendicular to \(e^{is\theta}\). This gives \[\sin(s(\pi-\theta))L_<(a) -\sin(s(\pi+\theta))L_>(a)=0, \qquad L_<(a)=rL_>(a),\quad r=\sqrt2.\]

For positive half-integer ports \(a,b\), let \(D(a,b)\) be the lost same-side weight with those fixed endpoints. Reversal gives \(D(a,b)=D(b,a)\). The missing negative ports to the left contribute \(A(\lceil a\rceil)\), so \[L_<(a)=A(\lceil a\rceil)+\sum_{0<b<a}D(a,b), \qquad L_>(a)=\sum_{b>a}D(a,b).\] Summing \(L_<=rL_>\) over \(0<a\le T\) gives the positive identity \[\begin{aligned} (r-1)\sum_{0<a<b\le T}D(a,b) +r\sum_{0<a\le T<b}D(a,b) &=\sum_{0<a\le T}A(\lceil a\rceil)\\ &\le T^{3/4+o(1)}. \end{aligned}\] Because \(r>1\), this bounds \(\sum_{0<a\le T}(L_<(a)+L_>(a))\) by the same order. After translating the source to a fixed port, the wedges increase with \(a\); their total loss therefore decreases. The summed bound implies \[L_<(a)+L_>(a)\le a^{-1/4+o(1)}.\] Intersect two reflected wedges whose vertices are horizontal distance \(a\) on opposite sides of the source. Their intersection is a triangle of diameter \(O(a)\). An arch reaching distance \(Ca\) for sufficiently large fixed \(C\) leaves at least one wedge, so its weight is bounded by the sum of these two losses. This proves the upper exponent for \(W_R\) and completes Proposition 20.

A bound for segments between height extrema

We record a segment bound used later in the marked-bond summability argument. Measure vertex height in units of one third of the spacing between horizontal tessellation lines. Heights are integers and change by one or two on an edge.

For the later use, split an \(n\)-edge walk at a highest vertex and treat the two halves from that vertex. Within either half, go to the last occurrence of its lowest remaining level, then to the last occurrence of its highest remaining level, and continue alternately. These are segments of Lemma 24. Their positive integer spans decrease strictly after a possible equality of the first two, since each chosen extremum was the last occurrence of its level. The sum of the spans is at most \(2n\), so there are \(O(\sqrt n)\) segments and at most \((2n+1)^{O(\sqrt n)}\) span lists. For a prescribed list, successive use of the lemma bounds the endpoint-summed product of segment weights by a constant to the number of segments. Recovering the weight of the joined walk costs \(x^{-1}\) per shared endpoint. These are the subexponential concatenation bounds needed in Section 12.

They also recover the reciprocal exponential growth rate of full-plane SAWs. Let \(c_n\) count \(n\)-edge SAWs from a fixed vertex. A highest vertex has only \(O(n^2)\) possible positions, and the split has at most \(n+1\) positions. Summing the span lists and the preceding product bounds gives \(c_nx^{n+1}\le\exp(O(\sqrt n\log n))\), hence \(\limsup c_n^{1/n}\le x^{-1}\). Lattice symmetry gives \(c_{n+m}\le c_nc_m\), so the limit \(\mu=\lim c_n^{1/n}\) exists. If \(\mu<x^{-1}\), then \(c_nx^n\le Cr^n\) for some \(r<1\). An arch reaching radius \(R\) has at least a constant times \(R\) vertices, so summing this bound over its possible lengths would make \(W_R\) exponentially small, contrary to Proposition 20. Thus \(\mu=x^{-1}\). [3]

Boundary chords and interior visits

We first convert the arch estimate into the partition of macroscopic chords. The following consequence of the signed balance makes the comparison precise.

Recall the regular tessellation hexagon \(D_R\), the ordered first-exit chords \(\mathcal C_R\) of diameter at least \(R/10\), and their raw partition \(Z_R^{\rm chord}\). For an upper bound at a source \(a\), choose a small similarly aligned tessellation hexagon \(P\) contained within distance \(R/25\) of \(a\), with \(\operatorname{dist}(a,\partial P)\ge cR\). Such a choice is possible with fixed constants for large \(R\) by taking its center within a bounded lattice distance of \(a\). A chord of diameter at least \(R/10\) reaches distance at least \(R/20\) from \(a\), and hence is lost on restriction to \(D_R\cap P\). Lemma 25 bounds its weight by a constant times the artificial exits of that intersection. These exits are a subset of the artificial exits of \(P\) intersected with the supporting tessellation half-plane at \(a\). Applying the lemma to this half-plane cap bounds them by \(CW_{cR}\). Thus the macroscopic weight at every source is at most \(R^{-1/4+o(1)}\).

For a lower bound choose the ports in a fixed middle portion of one side, at distance at least \(cR>R/10\) from every other side. A half-plane arch from such a source reaching radius \(R\) is either a macroscopic chord contained in \(D_R\) or is lost on restriction to \(D_R\). By Lemma 25, all lost arch weight is bounded by a constant times the exits of \(D_R\) on the other sides, and these are macroscopic as well. Thus the macroscopic weight at each of these sources is at least a constant times \(W_R=R^{-1/4+o(1)}\). There are \(O(R)\) boundary ports in all and order \(R\) chosen ports. We have proved \[ Z_R^{\rm chord}=R^{3/4+o(1)}. \tag{119}\]

Let \(\mathcal Z(R)\) be the \(+2\) partition around the central face center in \(D_R\). For a honeycomb vertex \(v\) in \(D_R\), write \[\mathcal V_R(v)=\sum_{\substack{\gamma\in\mathcal C_R\\v\in\gamma}} x^{N(\gamma)}.\] This is a raw mass under the finite chord ensemble, so \[\mathbb E_R^{\rm chord}N =\frac1{Z_R^{\rm chord}}\sum_{v\in D_R}\mathcal V_R(v).\] For \(v\) at distance at most \(R/10\) from the center, every first-exit boundary chord visiting \(v\) is macroscopic for large \(R\). A translated hexagon of side \(\lfloor R/2\rfloor\) centered at each incident face lies inside \(D_R\). Positivity under domain inclusion and Proposition 26 therefore give \[\mathcal V_R(v)\ge c\mathcal Z(\lfloor R/2\rfloor) \quad\text{uniformly for these bulk vertices}.\] For every vertex not adjacent to a boundary port, a translated hexagon of side \(3R\) centered at an incident face contains \(D_R\). The same proposition gives \(\mathcal V_R(v)\le C\mathcal Z(3R)\) uniformly. There are order \(R^2\) bulk vertices and at most order \(R\) vertices adjacent to boundary ports. Each of the latter contributes at most one to the normalized mean, since a chord visits a vertex at most once. Hence \[\begin{aligned} \frac{cR^2\mathcal Z(\lfloor R/2\rfloor)}{Z_R^{\rm chord}} &\le\mathbb E_R^{\rm chord}N\\ &\le\frac{CR^2\mathcal Z(3R)}{Z_R^{\rm chord}}+CR. \end{aligned}\] Using (119) and \(\mathcal Z\ge1\) yields \[\begin{aligned} R^{5/4+o(1)}\mathcal Z(\lfloor R/2\rfloor) &\lesssim \mathbb E_R^{\rm chord}N\\ &\lesssim R^{5/4+o(1)}\mathcal Z(3R). \end{aligned}\] In particular \(\mathbb E_R^{\rm chord}N\ge R^{5/4+o(1)}\). Section 11 will supply \(\mathcal Z(R)=R^{1/12+o(1)}\), completing the mean exponent \(4/3\) for this finite chord law.

The physical sum and the estimate to prove

Put \(\tau=e^{i\pi/3}\). After a lattice rotation or reflection, a permitted cut is a periodic path of tessellation sides whose steps are \(1\) and \(\tau\). Translating it by the time step \(\tau^2\) gives rows of rhombi: the spatial projection transverse to \(\tau^2\) is strictly increasing, so these rows tile the plane. A period has \(M\) sides and a nonzero lattice translation \(P\). We call a site horizontal when its cut step is \(1\), and mark the honeycomb bond crossing each of two horizontal sites \(i,j\). Choose the seam just before \(i\). The period then consists of consecutive nonempty lists \(A,B\), with \(i\) first in \(A\) and \(j\) first in \(B\); write \(A'=A\setminus\{i\}\) and \(B'=B\setminus\{j\}\).

On the cylinder modulo \(P\), let \(H^{\rm poly}\) be the increasing closed-rim height limit of \[\sum_{\substack{\lambda\ {\rm a\ single\ simple\ unoriented\ cycle}\\ \lambda\ {\rm uses\ both\ marked\ bonds}}} x^{|\lambda|}.\] The sum includes contractible and noncontractible cycles. The proposition below proves that the limit is finite.

The associated physical site variables have shifted-to-horizontal ratio \(q=e^{i\pi/4}\). Indeed the angles of the two rhombi relative to the time step are \(\pi/3\) and \(2\pi/3\); at one common physical row their spectral differences are \(\alpha\) and \(2\alpha\) for the shifted and horizontal sites, respectively. At both differences the scalar weights are exactly the honeycomb weights after splitting each rhombus into two equilateral triangles. At \(2\alpha\) the weights \(u,d\) and \(a,b\) are interchanged relative to \(\alpha\).

For a generic nonzero site list \(X\) on the same two lists, define a separate algebraic quantity \[ H^{\rm flip}(X)= L_- f_i\, i^{\sum_{a\in A'}s_a^{\rm phys}} e_j R_+,\qquad f_i=|-\rangle_i\langle+|,\quad e_j=|+\rangle_j\langle-|. \tag{122}\] Here \(s_a^{\rm phys}\) is the diagonal site-spin operator, and \(R_+=R_i\), \(L_-=L_{-i}\) are the selected vacuum vectors of Lemma 8. Their generic normalization is the one fixed in Section 3.

The physical identity has the broader domain: its proof uses the selected vacuum vectors, a finite nonvanishing result for their scalar normalizer, summability on a fixed cylinder, and a two-arc phase calculation. The count restrictions in Theorem 29 enter later, when Sections 13–15 estimate the horizontal representation. We keep the algebraic and positive sums separate until Proposition 28 is proved.

The generic flip formula

Put \(U=x_i\), \(V=x_j\), and retain arbitrary nonzero values of these variables. For the unmarked lists \((A',B')\), let \(C_s\) denote the summand in (6) for \(N^2\mathscr G\) at \(h=1\), including its phases. Thus the labels satisfy \(Q=\sum_{A'}s_a=\sum_{B'}s_a\). For a pair \((u,v)\) write \(e_k(u,v)=v-q^k u\).

Finite nonvanishing at the physical sites

The physical vacuum argument needs only finite nonvanishing. The growth of the same scalar with the number of sites will enter later in Section 13.

Summability on a fixed cylinder

The phase of two marked arcs

We calculate phases on the reference flat cylinder with horizontal cut and angle \(\pi/3\), keeping the scalar weights of the physical tilted rows. The local pictures and phases after the telescoping removal in Section 2 are unchanged by this choice. Place the seam at zero and write the endpoint abscissae as \(0<i<j<M\).

Consider a smooth regular embedded lift of one directed arc \(X:[0,1]\to\mathbb R^2\) from \((i,0)\) to \((j+kM,0)\), \(k\in\mathbb Z\), disjoint from all of its nontrivial horizontal \(M\)-translates. Its endpoint tangents are vertical with signs \(\zeta_i,\zeta_j\in\{-1,1\}\), upward positive. Use a small perturbation to make its interior cut and seam-ray crossings transverse, retaining the absence of interior cut crossings at seam or endpoint abscissae and its prescribed tangent turns. The actual reference lattice arcs have this absence before smoothing. Let \(x_z\) and \(\epsilon_z\) be the abscissae and signed directions of the interior cut crossings, and define \[\mu=\sum_{\text{interior crossings }z}\epsilon_z\delta_{x_z} +\frac{\zeta_i}{2}\delta_i +\frac{\zeta_j}{2}\delta_{j+kM},\qquad C(a)=\mu((-\infty,a))+\frac12\mu(\{a\}).\] The total mass is zero, since the interior signed crossing total is \(-(\zeta_i+\zeta_j)/2\). In particular \(C\) has compact support. Define the finite sums \[C_0=\sum_{\ell\in\mathbb Z}C(\ell M),\qquad C_{\rm per}(a)=\sum_{\ell\in\mathbb Z}C(a+\ell M)-C_0.\] There is no interior crossing at a seam or an endpoint translate. At an endpoint, \(C\) includes half of an atom already of mass \(\zeta/2\), hence the contribution \(\zeta/4\).

Identification of the physical matrix element

At a physical list we may henceforth write \(H_{\rm lg}=H^{\rm poly}=H^{\rm flip}\). At nearby generic lists only \(H^{\rm flip}\) is used. In particular any use of (123) at a nongeneric list means the limit of its combined expression. The next section estimates that algebraic quantity before taking the physical limit.

The endpoint source and the finite-list regulator

Take the combined limit of (123) at \(U=V=e^{\eta w_e}\). This is the only specialization of its two endpoint variables. Write \(W_\sigma=(w_{\sigma,1},\ldots,w_{\sigma,n_\sigma})\) for the unmarked coordinates in species \(\sigma=1,2\), with \(n_\sigma\ge0\), and set \(x_{\sigma,a}=e^{\eta w_{\sigma,a}}\). The full ordered list is \((U,e^{\eta W_1},U,e^{\eta W_2})\) and has size \(M=n_1+n_2+2\). Let \(P_0(W_1,W_2)\) be the all-zero term of the finite spin sum on the unmarked lists. Empty products have value one.

The pair factors and sign cancellation for a defect \(\delta=-s=\pm1\) at centered coordinate \(v=w+i\delta/2\) are unchanged. Before Gaussian splitting, its potentials against a zero site in the same and opposite species are respectively \[V_{d,-}(w)=q^{-1}\frac{e_1e_7}{e_0e_6},\qquad V_{c,-}(w)=q\frac{e_3^2}{e_4^2},\qquad e_k=1-q^ke^{\eta w},\] and \(V_{\rho,+}(v-i/2)=V_{\rho,-}(v+i/2)^{-1}\). The local factor after sign cancellation is \(\sqrt c\), before these zero-site interactions. The extra \(p_0/4\) in the matched prescription came from the opposite-species zero at the same coordinate.

The phase \(q^{-6Q}\) and the two endpoint ratios in (123) contribute \(\exp(i\delta D_e(v-w_e))\) per defect, where the odd branch \(D_e\), real on the real line, is determined by \[e^{-iD_e(v)} =q^{-3}\left(\frac{e^{\eta v}-q^{11/2}} {e^{\eta v}-q^{5/2}}\right)^2.\] It is analytic on \(|\Im v|<5/2\) and tends exponentially to \(\pm\nu\) at the two real infinities, where \(\nu=3\pi/4\). Indeed \[D_e'(v)=\frac{2\eta\sin(5\pi/8)} {\cosh(\eta v)-\cos(5\pi/8)},\qquad \widehat{D_e'}(k)=4\pi\frac{\sinh(3k/2)}{\sinh4k}.\]

Fix \(0<r<1/8\). Start with distinct real unmarked coordinates and a real endpoint coordinate, all of absolute value less than \(r/8\), and take \(T>8r\). Use the Gaussian and Wick data of Section 4. In the current (8) of species \(\sigma\), put \(z=1\), use the product of \(U_{\delta,T}\) translated to its own list \(W_\sigma\), and use the circle \(\gamma_r\) enclosing that list. Replace \(\varphi_\sigma(v)\) by \(\varphi_\sigma(v)+d_T(v)\), where \[ d_T(v)=\sum_{\ell\in\mathbb Z} \bigl[D_e(v-w_e+\ell T)-D_e(v-w_e-T/2+\ell T)\bigr]-\nu. \tag{126}\] This function has zero mean, changes sign on translation by \(T/2\), and tends to \(D_e(v-w_e)\) on compact sets as \(T\to\infty\). Call the resulting Wick current \(\mathcal C^L_{\sigma,T}\). With the zero-mean source \[A_\sigma(x)=\sum_{w\in W_\sigma}a_T(x-w),\qquad A=(A_1,A_2)^t,\] define the normalized regulator \[ \mathfrak C_T^L= \frac{\mathbb E\!\left[ e^{i\int_T(DA)\cdot\chi} e^{-2\pi\sum_\sigma n_\sigma\ell_\sigma/T} \mathcal C^L_{1,T}\mathcal C^L_{2,T}\right]} {\mathbb E e^{i\int_T(DA)\cdot\chi}}. \tag{127}\] The source denominator is a nonzero Gaussian exponential.

The changed gas data

Write \[g_+=g_d+g_c,\qquad l(k)=\frac{kE(k)}{1+g_+(k)E(k)},\qquad L_*:=l(0)=\frac89.\] Let \(y_T=\partial^{-1}d_T\) have zero mean. Define \(J_T^{\rm loc}\) by applying the Fourier multiplier \((l(k)-L_*)/k^2\) to \(d_T'\), using its regular value at zero. This is a periodized difference of translates to \(w_e\) and \(w_e+T/2\) of a fixed exponentially decaying function. It is analytic at position heights of absolute value less than \(5/2\) relative to either center. These facts follow by multiplying \(\widehat{D_e'}\) by the displayed symbol: the symbol is regular at zero and meromorphic with polynomial growth on a narrow strip about the real frequency axis.

Use the six complete particle lists (32), the fused union list (50), and the state gas (57), now at the fixed post-collision displacement \(d=1/8\). The residual pair factors, local activities, state space, and charge jumps are those of that gas at this displacement. Its changes are the following.

1. In each \(H\) dress of species \(\sigma\), replace \(B_T(v)^m\) by \(\prod_{w\in W_\sigma}B_T(v-w)\). These products use only that species’ unmarked list.

2. In the integral square of (55), replace \(\theta\) by \(d_T(x)\); in the local Berry phase of a particle at \(x_j\), replace it by \(d_T(x_j)\).

3. For an \(H\) entry at height \(a'\) and center \(x\), with signed power \(r\), add to its dress logarithm \[ r\left[J_T^{\rm loc}(x+ia') -L_*\bigl(y_T(x+ia')-y_T(x)-ia'd_T(x)\bigr)\right]. \tag{129}\]

The last logarithms are bounded and exponentially localized near the two kinks, for \(|\Im w_e|\le0.6\) and the listed heights. They are evaluated on every \(H\) entry, preserving the listed internal order and the analytic fused union value. In particular the fused particle’s kink multiplier is generally not one.

For later boundary evaluation, the site part of item 1 must first be simplified by (51). For a particle in species \(\sigma\) it is \[ \begin{array}{c|l} P&\displaystyle\prod_{w\in W_\sigma}B_T(x-w)\\ M,\ F_{\rm pair}&1\\ O_+&\displaystyle\prod_{w\in W_\sigma}B_T(x-w-i)\\ O_-&\displaystyle\prod_{w\in W_\sigma}B_T(x-w+i)\\ U&\displaystyle\prod_{w\in W_\sigma}B_T(x-w+i(3/2+d))^{-1}\\ L&\displaystyle\prod_{w\in W_\sigma}B_T(x-w-i(3/2+d))^{-1}. \end{array} \tag{130}\] The value one in the \(F_{\rm pair}\) row concerns only these site products, not item 3. Denote the resulting gas by \(Z_T^L\).

We will also use the following precise variant. Omit \(d_T\), put \(z=e^{i\theta}\) in the currents, and use the constant real angle \(\theta\) in the gas. Equation (131) then holds with the two \(d_T\) terms absent, at every supported period and every finite pair of initial real lists in \(|w|<r/8\), including coalescing lists by continuity, with bounds for fixed lists and \(\theta\) in a fixed compact set. Bounded functions of the winding may be inserted and carried to functions of the total species charges. In Section 15, the homogeneous lists \((0,m_*)\) in this variant are taken to \(m_*\to\infty\) at one fixed supported \(T\); their post-collision site factors have modulus at most one, which supplies the needed domination in that distinct auxiliary limit.

Continuation and the fixed-list period limit

At each fixed supported period, continue the finite lists through generic coordinates whose imaginary parts, including the endpoint coordinate, have total spread less than one. We use \(|\Im w_e|\le0.6\) and \(|\Im w_{\sigma,a}|\le1\). For the final application, take \(w_e=-if\) and \(w_{\sigma,a}=ih_{\sigma,a}\), where \(0\le f\le0.6\) and the \(h_{\sigma,a}\) lie in the single neighborhood of \(-f,1-f\) supplied by Proposition 38, still with total spread strictly less than one.

This continuation acts first on the finite residue expression for (127). Its Gaussian source-to-defect contractions use the rapidly converging high-frequency symbol \(\widehat G D\widehat a\); its low frequencies in the period limit use the same odd step primitive. Defect-to-defect contractions remain in their analytic strips, and Wick poles between generic sites are avoided at spread less than one. The \(U\) factors are meromorphic by their defining products. Thus the finite meromorphic identity continues from the initial real set. For each fixed pair of counts, its finite residue expression converges locally uniformly as \(T\to\infty\) on compact subsets of the specified generic domain of spread less than one that avoid its poles. Indeed such a compact set has a common high-frequency decay margin and uniform convergence of the symmetric low-frequency sums, and there are only finitely many residue terms. Holomorphic uniqueness therefore identifies this limit with the continued right side of (128).

On the gas side, continue the already contracted Fourier scalars, not an uncontracted translate of \(a_T\) outside its strip. The contracted scalars need only the stated site differences or the wider kink strip, so their period limits converge. Use the simplified site factors (130) before evaluating any boundary value. They have no poles for \(|\Im w_{\sigma,a}|\le1\) on the post-collision real centers: the inverse factors may equivalently be shifted by \(3i\) using (51). At fixed period and fixed lists these factors are locally bounded per particle. The stability bound therefore gives locally absolute summability along the continuation. This proves both (131) and (128) for the generic continued lists just specified. It asserts no bound uniform in the list sizes from this continuation alone.

The growing scalar normalizer

The exact orbital ratio in Lemma 31 now supplies the power of \(M\) needed after scalar cancellation.

Cancellation and the interface for propagation

We finish the scalar calculation at the fixed continued lists. For an unmarked pair, denote its factors in \(P_0/\mathcal S(X_{\rm un})^2\) by \(S_d,S_c\) for the same and opposite species, where \(X_{\rm un}=(e^{\eta W_1},e^{\eta W_2})\). Here \(\mathcal S\) is the site prefactor of Lemma 31. Writing \(u,v\) for the site exponential variables and \(C\) for its comparison kernel at their log difference gives \[S_d=\frac{(v-u)^2(v^2+u^2)} {(v^2-\sqrt2uv+u^2)(v^2+\sqrt2uv+u^2)C^2},\qquad S_c=\frac{(v-u)^2(v+u)^2} {(v^2+\sqrt2uv+u^2)^2C^2}.\] Each unmarked site versus one endpoint similarly gives, using \(F_0\), \[S_e=\frac{(v-u)^2}{(v^2+\sqrt2uv+u^2)C^2}.\] The distinct-site quadratic contractions in (131) converge to \(-\log S_d\) and \(-\log S_c\). Their self exponentials are \(S_d(0)^{-1/2}=2/3\) per unmarked site. The contraction \(-i\int_T A^tD\mathbf1\,d_T\) converges to \(-2\log S_e\) per unmarked site.

Here are the transform checks for these cancellations. In units \(2\pi/k\), the transforms of \(\log S_\rho\) are \[-\left[\frac{P_\rho}{\sinh4k} -\frac{\cosh3k-1}{\sinh3k}\right],\] where \[\begin{aligned} P_d&=\cosh4k+\cosh2k-\cosh3k-\cosh k,\\ P_c&=\cosh4k+1-2\cosh k,\\ P_e&=\cosh4k-\cosh k. \end{aligned}\] For \(d,c\) these full transforms equal \(\widehat\lambda_0(k)\widehat{\mathfrak a}_\rho(k)\). Their negatives use \(\widehat\lambda_0(-k)\) instead, exactly as in the quadratic \(A\) contraction. For \(e\) the bracket is \(\tanh(3k/2)\sinh k/\sinh4k\), and \[-2\widehat{\log S_e}(k) =\widehat\lambda_0(k)\frac{\widehat{D_e'}(k)}{k}.\] The kink dipole inserts the additional factor \(1-e^{-ikT/2}\); its second term averages to zero in the periodic Riemann sum by regularity at zero and high-frequency decay. These identities first hold for decaying logs on the real difference line, then continue to the stated fixed-list domain.

Insert \(\mathcal S(X)^2\) into (128) and use (131). All distinct unmarked pairs and unmarked-to-endpoint pairs cancel by the preceding identities. The \(M-2\) self factors \(2/3\) cancel \((2/3)^{-M}\) from the reciprocal square of (132), up to a constant. The reciprocal squared Schur factor of the two endpoints, multiplied by \(cU^2\), is bounded at \(U=V\) by its nonzero confluent value. The limit of \(R_T^d\) is bounded for \(|\Im w_e|\le0.6\).

Consequently, fix any supported integer block length \(b\). For every fixed pair of generic imaginary lists in the common neighborhood above, with total spread less than one including \(w_e=-if\), the limit below exists along \(T=4Nb\). For every \(\varepsilon>0\) and all sufficiently large even \(M\), \[ |H^{\rm flip}(X)| \le C_\varepsilon M^{-5/24+\varepsilon} \left|\lim_{N\to\infty}c(4Nb)Z_{4Nb}^L\right|. \tag{133}\] The constant and threshold are uniform in \(0\le f\le0.6\), the list orders, and their finite counts. Existence of the limit comes from the finite-list regulator limit and the convergence of the contracted scalars, with the lists held fixed; the uniform prefactor comes from (132) and the exact cancellations. No uniform operator estimate is being inferred from this scalar argument.

Sections 14 and 15 now estimate the right side uniformly for lists with the proportions in Theorem 29. Once that bound is available, approach the physical spread-one heights separately for each fixed pair of lists. Proposition 28 supplies the regular limit of \(H^{\rm flip}\) and its identification with \(H^{\rm poly}\) there; no period limit at the boundary is needed.

Suppression for translated site lists

For one species, a site at imaginary height \(h\) contributes to a suppressed particle \(\tau\) the factor \[\omega_\tau(x;h)=B(x+i(a_\tau-h))^{\epsilon_\tau}, \qquad \begin{array}{c|ccccc} \tau&P&O_+&O_-&U&L\\\hline a_\tau&0&-1&1&13/8&-13/8\\ \epsilon_\tau&1&1&1&-1&-1 . \end{array}\] The \(M\) and \(F_{\rm pair}\) site factors are exactly one after the per-site meromorphic cancellations supplied by the identities for \(B\). These are statements about the \(B\)-products; the additional marked-source dresses are separate.

The mixed spectral radius

Fix the post-collision displacement \(d=1/8\) and a sufficiently large integer block size \(b\), using Proposition 37. On any fixed compact set of real angles, choose a common gauge with quadratic slack. For a constant real angle \(\theta\), let \(E_\theta\) be the homogeneous operator allowing all particle types, and let \(I_\theta\) allow only \(M,F_{\rm pair}\) in both species. Let \(X_\theta\) allow all types in species 1 and only \(M,F_{\rm pair}\) in species 2. These are the Hilbert–Schmidt operators constructed from the common gauge and Bargmann tail in Section 7. The empty-contour power traces give \[r(E_\theta)=r_E=e^{-\pi b/16}.\] The angles used below are \(\theta=\pm\nu\), where \(\nu=3\pi/4\).

A coefficient lemma for two exterior gaps

The fixed-list limit in (133) controls a trace with two long exterior portions, one at each limiting angle. The following version of Lemma 18 records the frequency collision that occurs when both portions have length \(2N-L\).

Proof of the unequal-interval estimate

From selected bond pairs to planar polygons

We finish the length-square estimate for planar polygons modulo lattice translations: \[ \sum_{\lambda:\,\operatorname{diam}\lambda\le H} x^{|\lambda|}|\lambda|^2\le H^{2/3+o(1)}. \tag{145}\] The argument first removes polygons with too few locally available bond directions. For the remaining polygons it selects many bond pairs and constructs, for each pair, a cylinder satisfying every hypothesis of Theorem 29.

Fix a large integer \(K\). A vertex of a polygon is called good if the cyclic window of \(2K\) path edges centered at that vertex uses bonds crossing tessellation sides of all three orientations. Write \(N=|\lambda|\) for the polygon length.

Choose \(K\) as in Lemma 41. For each prescribed exponent slack \(\epsilon>0\), choose a fixed distance cutoff large enough both for Lemma 42 and for its resulting \(m\ge m_\epsilon\) in Theorem 29. In a polygon with at least \(N/2\) good vertices, lattice packing excludes only \(O(N)\) ordered pairs within this fixed cutoff. Thus for sufficiently large \(N\) the retained pairs number at least \(cN^2\). Choosing a used bond in each window loses only a bounded multiplicity: each selected bond can be assigned by only \(O(K)\) vertices along the polygon. The finitely many shorter lengths contribute a bounded total modulo translations.

Fix the first bond at one translation representative of each bond type. In a dyadic displacement annulus of radius \(d\), there are \(O(d^2)\) possible second bonds. For each selected pair among these possibilities, Lemma 42 embeds the diameter-truncated plane polygons injectively into its positive cylinder sum. Its parameters satisfy \(m\asymp d\) and \(n\asymp H\), with the discrepancy bound \(0.6\). Theorem 29 therefore bounds that sum by \(C_\epsilon H^\epsilon d^{-4/3}(H/d)^{1/3}\). Summing gives \[\begin{aligned} \sum_{d\ \mathrm{dyadic},\ d\lesssim H} C_\epsilon d^2H^\epsilon d^{-4/3}(H/d)^{1/3} &\le C_\epsilon H^{1/3+\epsilon} \sum_{d\ \mathrm{dyadic},\ d\lesssim H}d^{1/3} \\&\le C_\epsilon H^{2/3+\epsilon}. \end{aligned}\] Combine this with the summable family from Lemma 41 and the bounded short lengths. Since \(\epsilon>0\) was arbitrary, this proves (145).