The analytic obstacle and the geometric transfer

The computation starts with finite loop diagrams on a cylinder. Pairing states at a horizontal cut give a finite transfer matrix, and diagram slides determine its vacuum vectors by polynomial interpolation. Two scalar polynomials, a Pfaffian \(Q\) and a determinant \(R\), encode the normalizations. Their definitions and the exact physical gluing formulas appear in Section 2.

A formal asymptotic exponent would be insufficient: the coefficient of its leading term could vanish. Sections 3–5 address this issue directly. An oscillator trace is evaluated in two Gaussian channels; coercivity of the short-range interactions justifies the change of expansion. A long-circle argument produces expansions with controlled remainders. Positive Fourier measures then express confluent \(Q\) and \(R\) in terms of polynomial norms and force their leading coefficients to be nonzero. One physical consequence is that the mass of winding polygons, counted modulo row translation on the regular \(n\)-column cylinder, satisfies \[J_n\sim\frac{\sqrt3}{6n}.\] Section 6 strengthens this information to an exponential tail in the long direction. Section 7 gives a separate open-strip calculation of \(B_H\); its boundary vectors and determinant are proved in their own geometry.

The next question is how cylinder observables count long planar paths. An enclosing-loop gas with loop fugacity \(2\) converts, by a local flux reversal, to the mass of boundary chords visiting a given vertex (Section 8). A two-port identity (Section 9) counts two points on one polygon. Its oblique asymptotic in Section 10 is uniform even when the two intervals between the marks have different scales. The exponent controlling the scale ratio is independent of the requested expansion accuracy. This uniformity permits calibration at a smaller physical scale and yields the polygon second-moment estimate in Theorem 2.

Sections 11 and 12 complete the positive geometric argument. Strip crossing mass supplies confined paths and caps at specified gaps. The enclosing gas gives a lower first moment, while the polygon second moment controls the loss from pruning long paths. Paths of excessive aspect ratio are excluded by the cylinder pressure tail. The surviving paths are either bridges already, or arches to which we attach two disjoint exterior connectors. A bound on the number of double cuts controls recovery multiplicity. Summing the resulting bridges over the stated range of heights proves Theorem 1.

Earlier work and the role of exact normalization

Nienhuis’s analysis of the dilute \(O(n)\) model predicts the self-avoiding-walk spatial exponent \(3/4\) at \(n=0\) [12]. Lawler, Schramm and Werner formulated the \(\mathrm{SLE}_{8/3}\) scaling-limit and boundary-weight \(5/8\) predictions [10]. Duminil-Copin and Smirnov proved the exact honeycomb connective constant using a parafermionic observable whose local cancellation becomes a positive boundary identity after summation [2]. They also recorded the predicted boundary powers \(A_g\asymp g^{-5/4}\) and \(B_H\asymp H^{-1/4}\) in their discussion of conformal invariance [2]. The present finite estimates establish these bounded-factor powers in the port conventions above; the height-averaged window then locates critical mass near the corresponding length scale.

The decay of the strip crossing mass has been studied directly. Beaton, Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann proved \(B_H\to0\) in their study of surface adsorption [1]. Glazman and Manolescu gave a shorter proof and a logarithmic subsequence bound [5]. Krachun and Panagiotis proved a polynomial upper bound for \(B_H\) and quantitative sub-ballisticity for uniform honeycomb walks [9]. Their use of bridge renewal levels to recover sewn walks is also a precedent for the positive geometric counting below. Our constructions keep the cuts recoverable while imposing the length and confinement bounds needed for the window.

The rhombic weights and their Yang–Baxter relations belong to the integrable dilute-loop setting. Ikhlef and Cardy related the local parafermionic equations to the integrable weights [7]; Glazman established their criticality for self-avoiding walks on rhombic tilings [4]. Glazman and Manolescu proved boundary two-point invariance for columnwise rhombic half-plane tilings with angles in \([\pi/3,2\pi/3]\) [5]. The periodic spectral and seam formulation has precedents in the dilute-model analysis of Zhou and Batchelor [15]. We derive the finite identities in our normalization and then prove the nonzero amplitudes, uniform errors and planar conversions used here.

The oscillator fields, cocycles and root-mode identities in the change of expansion come from lattice vertex-operator algebra [3]. For the present indefinite bilinear space, we establish finite coefficient support before taking traces, then prove absolute summability and continuation. The pressure argument also uses classical total positivity and variation diminution [13], followed by the connected-cluster sign structure of a repulsive gas [14]. The local proofs supply the strictness and bounds uniform in the cylinder circumference that these applications require.

The irreducible-bridge construction goes back to Kesten and is developed systematically by Madras and Slade [8, 11]. The appendix of [1] already gives its critical normalization and independent concatenation in the honeycomb mid-edge convention. Section 11.1 derives additional quantitative height and diameter bounds from our finite boundary estimates. This is a separate renewal consequence: the proof of Theorem 1 uses finite weighted counts and recoverable sewing directly.

Diagram identities

Interpret the same cell as a two-line swap \(B(s)\), with \(L,B,R,T\) interpreted respectively as old first, old second, new second, new first ports (’first’ on the left in an ordered sequence of lines). Its argument when exchanging rapidities is right divided by left (before the swap). Thus a row is an auxiliary line initially on the left successively exchanged through all columns, then closed round to itself. In the diagram calculus compose linearly by gluing, dropping any term with an occupation mismatch or with a closed cycle formed; these matching identities can then be applied also to the vacuum pairing states. The triangle map \(P\) replaces one line by two neighboring lines and has the empty diagram and the three single arcs of weight \(X\). The cap map \(C\) replaces no lines by two with either no arc or a cup of weight 1. A transposed map reverses inputs and outputs. At adjacent rapidities \(q m,m/q\) (in that order) the lower vacuum will use \(P\) on a line of rapidity \(m\). At \(q^3 l,l\) it will use \(C\). Indeed \(B(1)=1,\ B(q^2)=P P^t,\ B(q^3)=CC^t\). One can pass these maps through a line of arbitrary rapidity (sliding the triangle/cap), going between two swaps and one/zero on the fused/removed side. Also swaps satisfy inversion and the ordinary braid relation with rapidity ratios.

Figure 1 records the local states and port conventions.

Here is a verification. A square root is continued multiplicatively in each identity (using \(q^{1/2}=e^{\pi i/8}\)); different choices just conjugate on occupied ports since only the single corners change sign. For inversion \(B(s)B(s^{-1})=1\), write \((a,b,c,d,e)\), \((A,B',C',D',E)\) for weights \((u,U,v,w,W)\) of the two factors from bottom to top. The matching coefficients use \(B'+D'b=b+B'd=A c+C'a=B'b+D'e+E d=0\), \(A a+C'c=D'd=1\). A quarter turn changes \(s\) to \(q^3/s\). Thus bending a leg in inversion gives cap sliding, e.g. \(B_{23}(s)B_{12}(q^3s)(1\otimes C)=C\otimes1\). For triangle sliding the transpose version is \[(P^t\otimes1)B_{23}(qs) B_{12}(s/q)=B(s)(1\otimes P^t).\] For clarity the scalar check below labels exterior ports \(0,1,2,3,4\) (inputs \(0,1,2\), outputs \(3\) right and \(4\) left); juxtaposed pairs mean both arcs. Now lower-case \(a,\ldots,e\) denote weights at \(s/q\); write capital \(A,B,C,D,E\) at \(qs\) just in this table. Entries of the two sides (with the empty coefficient 1 omitted) are the equal expressions \[\begin{array}{c|l|l@{\qquad}c|l|l} 23&A+EXb&Xu&12&Ba+C Xc&X\\ 13&B Xc+C a&Xu&01&A Xd+b&XU\\ 02&B c+C Xa&XU&03&B Xa+C c&v\\ 34&B X+C Xb&U&24&B Xb+C X&Xv\\ 14&A Xa+X c&Xv&04&A Xc+X a&u\\ 01,23&A b+D Xd+E Xe&0&14,23&A Xc+D Xa&0\\ 12,34&E Xa&XU&01,34&B Xb+C Xe&XW\\ 01,24&B Xe+C Xb&0&02,34&E Xc&XW\\ 04,23&A Xa+D Xc&Xw&04,12&B Xd&Xu\\ 04,13&C Xd&Xw&03,12&E Xd&Xv \end{array}\] Here entries on the left of each equality just split by the connections at the triangle (closed loops dropped). These scalar equalities and the inversion coefficients follow by substitution; e.g. use \([n]=\sin(\eta+n\pi/8)\) and substitute \((\sin(10\pi/8)[5],\sin(10\pi/8)[0],-[5][0],[5][-2],-[-1][0])/([10][3])\), shifting brackets by \(\pm1\) for upper/lower-case, and \([n][j]=(\cos((n-j)\pi/8)-\cos(2\eta+(n+j)\pi/8))/2\). Transpose and reflection give, in particular, \(B_{23}(s/q)B_{12}(qs)(1\otimes P)=(P\otimes1)B(s)\). Finally \(B_{12}(s)B_{23}(sr)B_{12}(r)=B_{23}(r)B_{12}(sr)B_{23}(s)\): as functions of \(r\) clear the four linear pole factors from \(B(r),B(sr)\) (a cell has poles only at arguments \(q^5,q^6\), and bounded weights at 0 and \(\infty\)); matching coefficients have degree \(\le4\) and powers integer, or all half-integer between 0 and 4 by port parity (a square-root flip on a rapidity multiplies by \(-1\) per occupied end carrying it). Check \(r=1,q^2,q^3\) by identity or triangle/cap sliding of both halves of \(B(r)\) across the third strand. Check \(rs=q^2,q^3\) the same way (move the invertible \(B(s)\)’s across the equality). This proves the braid relation by interpolation. All these relations apply to a row and carry through closure with seam twist. Swaps on spatial neighbors intertwine row addition; on a cylinder moving the seam past a position conjugates by the sign \(\sigma\) on that occupied position (even parity at each cell). Carrying it past a whole pattern gives no sign.

Polynomial denominators

Both \(A_\delta=Q\) for \(\delta=1\) and \(A_\delta=R\) for \(\delta=2\) are symmetric polynomials of degree \(\le\delta(N-1)\) per variable: poles at coincident points cancel by equal rows (twice for the symmetric determinant), and cell-kernel denominators cancel from \(F^\delta\). Their recursions are \[ \begin{array}{l|l|l} &Q&R\\ \hline (l,q^3 l,Z)/ (Z) &(1+q^3)l\prod d(z,l) &\kappa^2[(1+q^3)l\prod d(z,l)]^2\\ (qm,m/q,Z)/(m,Z)&\sqrt2\,m\prod(z+m)&k\sqrt2\,m^2\prod(z+m)^2\\ (0,Z)/(Z)&\prod z &\prod z^2 \end{array} \tag{5}\] At infinity the leading coefficient is the smaller polynomial. The first line is pole-pair elimination. To check the second put \(f_m(s)=(m^4+s^4)/(m^4-s^4)\), \(A(s,t)=(s^2-t^2)/h(s,t)\). The Schur complement kernel after pivoting on \((b,c)=(q m,m/q)\), by substitution, is \[\begin{aligned} &A(s,t)+[A(s,c)A(t,b)-A(s,b)A(t,c)]/A(b,c)\\ &\qquad=f_m(s)f_m(t)[A(s,t)-A(s,m)+A(t,m)] \end{aligned}\] with the extra column becoming \(-f_m(s) A(s,m)\); for \(M\) the complement is \(f_m(s) f_m(t)[M(s,t)-M(s,m)M(m,t)]\). The skew brackets give the smaller Pfaffian including \(m\) (eliminate its pairing with the augmenting column, or if using the transformed extra column turn it into \(m\) by row/column additions). Meanwhile \(F\) contributes \(h(b,c)/(b-c)\prod(z+m)/f_m(z)\) relative to the fused set. These identities give the table. At \(0,\infty\) the row of \(M\) just decouples; for the Pfaffian the row supplies the column augmentation or eliminates it. Inversion of all variables transforms \(A_\delta\) by the monomial factor \(\prod z^{-\delta(N-1)}\).

Gluing rules

Glue a lower and an upper vacuum through the marked row opened at the seam, asking for both open ends occupied and connected, no other loops. Upper vacua are given by up/down reflection (\(s\mapsto q^3/s\), i.e. take spatial rapidities \(z_j^{-1}\) and row parameter \(1/(q^3t)\)); in particular the same \(A_\delta(Z)\) up to monomials serves as denominator, and they, like the lower vacua, are independent of \(t\). Matchings alone suffice in this gluing. For \(E_+\) take both vacua with positive twist. For \(J\) take \(\sigma=-1\) below and \(+1\) above, without any extra factor at the mark.

Here is why the mixed gluing works. Orient each rooted loop in the direction from last to first column at the mark (positive seam crossing), and temporarily multiply its weight by \[-i\exp(i\,{\rm turn}/4+i\pi h_0/2),\] where \(h_0\) is the signed total seam crossing number. For this argument use a flat square-cell cylinder with simple curves normal to and smooth across crossed sides. A simple winding loop has \(h_0=\pm1\), turn 0, and a contractible one has \(h_0=0\), turn \(\pm2\pi\). Indeed on the twice-punctured sphere successive crossings met along the seam alternate signs by separation. In the zero-translation case we can lift the loop to a planar simple closed curve; in the other case lift arbitrarily many periods starting at a highest point of the loop and close above, using simplicity and bounded added turn to get zero turn per period. The displayed factor changes sign on reversal, and in the winding case equals \(h_0\); summing it over the marked crossings of a given orbit under row translations thus gives 1 if winding and 0 otherwise. Rearrangement here is absolutely convergent.

For a fixed matched diagram in a half-cylinder we can represent its pairing canonically without crossing the seam (noncrossing along the ordered boundary). Replacing both half diagrams so, keeping the opened row, the factor is 1 (only the mark crosses). For a pair \(i<j\) below the row, traversed from \(i\) to \(j\), the actual lift has horizontal index displacement \(j-i\) or \(j-i-N\); any further shift would interlace a translate, impossible. Its contribution to turn, from incoming to outgoing normal, is \(+\pi\) or \(-\pi\) respectively (close along the boundary), and to seam number 0 or \(-1\). Thus the factor relative to canonical is \(1\) or \(-1\), exactly the lower seam twist, also for reversed traversal. Above the row the turn signs switch and the relative factor is always 1. This proves the mixed rule.

Consequently these sums are symmetric in \(Z\), obey both removal rules and both pair recursions without multipliers (just reducing to the same quantity at fused/remaining rapidities), and have a polynomial numerator on multiplying by \(A_{\delta_{\rm below}} A_{\delta_{\rm above}}\prod d(z,t)\) of degree at most \((\delta_{\rm below}+\delta_{\rm above})(N-1)+2\) per variable \(z_i\). Indeed all neighbor exchanges within the linear sequence of columns and the reduction surgery there avoid the opened seam, so cycles closed off during manipulations can always be killed. For a high-then-low adjacent reduction, slide \(P\) or \(C\) from below to above. It contracts there to the smaller vacuum: on the reversed rapidity relation above, exchange of the pair uses \(PP^t\) or \(CC^t\), while the exchanged vacuum is \(P\) or \(C\) times the smaller one. The maps \(P,C\) are injective on pairing states (read off components with both new sites vacant or with a single prescribed one occupied). Branches for reflected cells are obtained by angle complementation. The same slide with an ordinary invertible swap proves symmetry using the two vacuum exchange relations (upper exchange by the inverse): \(B(s)^t=B(s)\) on diagrams (reflection exchanging \(L,T\) and \(B,R\)), and upper rapidity ratios are the reciprocals of lower ones. Thus a swap left over at the upper boundary after interchanging two adjacent columns is absorbed by upper exchange covariance. Only adjacencies not across the mark are needed for this symmetry. At a variable extreme both vacua decouple on an empty port, and the horizontal-only propagation at that cell has limiting weight 1. Polynomiality and bounds follow from the vacuum bounds, the cell denominators and square-root parity. Formally at zero size \(E_+=J=1\) (straight auxiliary identity).

There is also a mark constraint. Take the marked first column of parameter \(x\): at \(t=x\) the mark port is tied to \(T\) there if either is occupied, at \(t=x/q^3\) to \(B\). Thus in the closed-row picture we can equivalently require a single loop through that slice port. Propagating the lower or upper state respectively through the row by invariance now reduces to gluing both vacua at a slice (with that one-cycle, first-port-occupied condition); cycles closed off within the propagated half cannot be the target loop so are correctly killed. One just corrects for the twist factor now included at the mark by the propagated row. Thus \(E_+\) at both parameters equals that pure slice gluing of positive vacua, and \(J(t=x)=-J(t=x/q^3)\) (negative twist from below only). Symmetry gives the analogous row-parameter constraints for any \(x\in Z\). In particular \(E_+(Z;t=x)\) for \(x=z_1\) gives the actual polygon mass through a first column top port in the honeycomb evaluation of the vacua, where applicable (in particular at regular homogeneity).

Now formula (2) for \(E_+\) has the stated degree, both pair recursions by the \(Q\)-table, and both removal values. This suffices for uniqueness when \(N\ge2\); at \(N=1\) add the evaluation \(t=x\) (value 0 on both sides). For \(YJ\), \(Y\) preserves pair recursions and multiplies removal values by \(\kappa\); allow denominator \(R^2\prod d\) and degree \(4(N-1)+2\) throughout. The proposed wave formula satisfies the pair recursions: the joint projection residual kernel of \(M\) transforms by \(f_m(s)f_m(u)\) relative to the fused value by the Schur identity, or trivially under removal of a pole pair \(l,q^3 l\) by Schur elimination. Reading the first orders as the external variable \(u\to\infty,0\) gives \(-f_m(s)p^+(s)\), \(+f_m(s)p^-(s)\) relative to the fused waves, and \(f_m(q^3t)=1/f_m(t)\). Removing instead a position by sending it to infinity multiplies \(p^+\) by \(1-k\) and preserves \(p^-\) (one more Schur complement, using \(M(s,u)\sim k s/u\)); at 0 their roles interchange. Thus \(YJ\)’s proposed expression has the right removal values. It obeys the denominator and degree bounds: each projection has denominator cleared by \(R\) and external factors \(h(s,z)\), by the bordered determinant (poles of \(F^2\) there cancel as for \(R\)). In a wave product unwanted poles in \(z\) at \(t,q^3t\) cancel because the partner wave vanishes at a generic interpolation node \(s=z\). Bounds at the extremes then suffice. At \(t=x\) the zero is multiplied by a pole at \(q^3t=q^3x\); at \(t=x/q^3\) by a pole on the opposite shift. The coefficients of these two poles in multiplicative coordinates are opposite (the row \(M(s,Z)\) is the only source, \(M(s,x)=M(x,s)\)); pairing the summands by symmetry of the two-wave expression gives the negative-value constraint exactly.

For later use the marked-row evaluations just proved may be recorded as \[ E_+(Z;x)=E_+(Z;x/q^3),\qquad J(Z;x)=-J(Z;x/q^3) \quad(x\in Z), \tag{6}\] with the common first value given by the occupied-port slice gluing.

More explicitly, on induction all pair nodes and 0 and infinity match, so the remaining ambiguity for \(YJ\) is only a scalar multiple of \[\mathcal W_N(Z;t)=\frac{\prod_{i<j}K(z_i,z_j)}{R(Z)^2} \prod_i \frac{z_i t}{d(z_i,t)}.\] This follows by divisibility and degree variable by variable and common scaling invariance. Its two shifted values do not sum to zero identically (the single marked-site factors agree and are nonzero, while for each other independent variable the ratio of the two factors is nonconstant). Thus the constraint proves equality starting with the displayed zero-size value.

Finally consider the color sum, with normalization \[L(Z)=R(Z)^2/\prod_{i<j}K(z_i,z_j),\qquad O_Z(r)=L(Z)^{-1}\sum_\alpha\prod_i b_{\alpha_i}\prod_{i<j}P_{\alpha_i\alpha_j}(z_j/z_i).\] It is symmetric. For a pair \(x,y\) (labels in that order), in the raw sum:

• At \(y=q^2x\) the contributing residue labels fuse as \(+0\mapsto+\), \(0-\mapsto-\), \(+-\mapsto0\) at \(qx\). At \(y=q^3x\) only \(+-\) contributes, going to vacant.

• The simple extraneous pole at \(y=qx\) cancels between \(+-\) and \(00\); at \(y=x\) all simple poles cancel by symmetry.

To verify residues, the pair factors involving a spectator \(u\) in the first rules multiply to the reduced ones times \(S(u/(qx))\) (fusion), or times \(S(u/(qx))S(u/(q^2x))\) (gap three), precisely the relative spectator factors of \(L\) by the table. For the canceling pole the spectator products agree. These identities follow immediately by inserting the entries of \(P_{ab}\). With coordinate \(y/y_0-1\), let \(D_0=k/(q-q^{-1})\) be the pole coefficient of \(S\) at \(q\). The remaining pair multiplier (for the residue relative to the reduced term) for each fusion choice is \(k D_0\) (use \(S(1)=2k\)), and for gap three \((k/2)S(q^2)D_0\). These are exactly the relative pair pole coefficients of \(L\): use \(K(1,y/x)\)’s linear coefficients \(-2i\sqrt2,-2\) at the two positive gaps in these coordinates, and the \(R\)-ratios above. At \(qx\) the two coefficients are \(-b_+b_- S(1)D_0,\ b_0^2D_0\). Symmetry covers the reversed locations.

Hence \(O_Z(r)\) obeys both pair recursions with no multipliers, and \(R(Z)^2 O_Z(r)\) is polynomial of degree \(\le4(N-1)\) per variable (all factors bounded at extremes). Removal at 0 or infinity multiplies \(O\) by \(r\) since all added pair factors tend to 1 and \(L\) drops to the smaller value. The pair recursions with a single endpoint value thus determine it uniquely at all sizes from \(O_\emptyset(r)=1\). For the three choices of \(r\) the claimed powers of \(Y\) satisfy the same checks by the denominator table, as required.

Charge space and the formal trace

Use real vectors \(D_j\), \(j\bmod8\), with one relation \(\sum D_j=0\); take the symmetric bilinear product \[D_0 D_j=(2,-1,1,-2,2,-2,1,-1)_j,\qquad O D_j=D_{j+1}\] translated cyclically. It is nondegenerate on this quotient (e.g. cyclic Fourier transform), but not positive. Define \(a\) by \((1-O+O^2)a=D_0\); its row \(aD_j\) is \((1,0,0,-2,0,0,1,0)\). Put \(\Lambda=\sum\mathbb Z D_j,\ S=(1-O)^{-1}\). The color vectors are \[a,\quad a-D_0=(O-O^2)a,\quad a-D_0-D_1=-O^3a ,\] all of square \(8/3\). Further put \[\delta=D_7,\quad R_\delta y=y-\delta(\delta y),\quad M_b=O R_\delta,\quad h=-OS\delta=-S D_0,\quad H_b=h^2=9/8 .\] The rows \(hD_j=(-1,0,-1,1,-1,1,0,1),\ Sa\cdot D_j=(1,1,1,-1,-1,-1,0,0)\). We have \(S+S^t=1\) (metric transpose), \(S\Lambda\cdot\Lambda\subset\mathbb Z\), \(M_b h=h,\ \delta h=1,\ ah=0\), and \[\det(I-yM_b)=(1-y^6)(1-y^3)/(1-y^2)\] on the quotient. These follow by multiplication (one can use \(S=-\sum_{j=0}^7 j O^j/8\)); for the determinant multiply \(\det(I-yO)\) by \(1+\sum_{n\ge1}y^n\delta O^n\delta\) by the rank-one change. In particular the fixed axis of \(M_b\) splits off.

Use parameters, initially \(L>0\), \[p=e^{-L/8},\qquad \mathfrak q=e^{-4\pi^2/L},\qquad s=e^{-2\pi i u/L},\qquad u=x+2\pi i d\] with angles \(d\) in an open interval of length 1 about 0. Operators in products act in order of increasing \(d\). The sources (variables \(z=e^{u/8}\)) near 1 have angles close to 0.

The coefficient algebra uses the lattice fields and root-mode relations of vertex-operator constructions [3]. We give the formulas and verify their use in this indefinite bilinear space, where convergence is a separate issue. At momentum \(P\), take polynomials freely generated from \(|P\rangle\) by oscillators \(l_{-m}(y)\), \(m>0\), linear in a vector \(y\) in the above space (complexified); let \([l_n(y),l_m(z)]=n\mathbf1_{m=-n} y z\), positive modes killing the constant, \(l_0(y)=yP\). Degree \(D\) counts oscillator degree (index \(m\) counted \(m\)) plus \(P^2/2\). Traces are coefficient traces on this polynomial space over momentum translates. Use the field \[V_\beta(s)= \exp\Big(\sum_{m>0} l_{-m}(\beta)s^m/m\Big) \exp\Big(-\sum_{m>0} l_m(\beta)s^{-m}/m\Big) T_\beta\, e^{\pi i S\beta\cdot P}s^{\beta P+\beta^2/2},\] where \(T_\beta|P\rangle=|P+\beta\rangle\) preserving oscillators. Mode powers count actual changes of \(D\); each specified coefficient has finite output on a polynomial. Momenta at the trace cut range over \({\cal C}=OS(Na)+S\Lambda\). Insert \(\mathfrak q^D e^{-2\pi iD}\widehat O\) at the left, where \(\widehat O\) transforms both momenta and oscillators by \(O\); the source charge shift is \(Na\) up to roots in \(\Lambda\), so the product maps back to this coset. Intermediate momenta always pair integrally with \(\Lambda\).

Gaussian change of channel

We specify the scalar evaluation for a fixed list of fields of charges \(\beta_i\) with \(\Delta_{\beta_i}=\beta_i^2/2\) in increasing angular order, either with \(U=O\), or in the transformed trace with \(U=M_b\) (including any root fields in the list). In the latter case it vanishes unless \(\sum h\beta_i=0\). Set \[b=\sum\beta_i,\quad P_b=OS b,\quad T_U=(1-U^{-1})^{-1}\] where \(T_U=1/2\) on a fixed axis. Define \(K_U(Y)\), \(\Re Y>0\), by applying to \(U\) the matrix function with values \[\sum_{m>0,\ m\in\mathbb Z-\nu}\frac{e^{-mY}}m , \qquad y=e^{2\pi i\nu}.\] At nonunit-value eigenvalues (that is, \(y\ne1\)) take a real lift \(\nu\) modulo 1 continued locally analytically in \(y\), differentiating on Jordan blocks. At 1 simply use positive integers (no Jordan block). Put \[C_U=\lim_{Y\downarrow0}(K_U(Y)+\log Y\,I),\qquad d_U(\beta)=e^{-\beta C_U\beta/2}.\] The limits (on the positive axis) exist by comparison with \(\sum_{m\in\mathbb Z_{>0}}e^{-mY}/m\): the harmonic coefficient differences are summable and changing a bounded shift in the exponential therein costs \(O(Y(1+|\log Y|))\) locally uniformly in the chosen spectral chart (hence also usable for Jordan derivatives). All the eigenvalues needed here are roots of unity by the characteristic polynomials. The limits are real in the symmetric part, by conjugation and \(U^t=U^{-1}\). For instance if the sequence \(\beta U^j\beta\) has period \(l\) with only corresponding finite-order components, then \[d_U(\beta)=l^{-\Delta_\beta}\prod_{j=1}^{l-1}|1-e^{-2\pi i j/l}|^{(\beta U^j\beta)/2}.\]

Take real parts \(x_i\) of all representatives in \([-L/2,L/2]\) initially. Sum, for \(M_b\), over \(f\in-\phi/(2\pi)+\mathbb Z\); the \(O\) case has just one term with all fixed-axis contributions below omitted. The phase in units of \(\pi\) is \[\Phi=-\sum_i\Delta_{\beta_i}-\sum_{i<j}\beta_i T_O\beta_j -2 f\,\frac{hP_b}{H_b} -\sum_{i<j}{\rm sgn}(x_j-x_i)\beta_i T_U\beta_j.\] Indices here label angular, NOT \(x\)-order. Include a kinetic factor \[\exp\left[-\sum_{\text{gaps in }x\text{-order}}\frac{\Delta u}{2H_b} \left(f+\sum_{\text{before gap}}h\beta_i\right)^2\right],\] with cyclic gap lengths in \(u\), value \(f\) of the momentum outside the extremes. For each \(x_i<x_j\) the short-range cross factor has logarithm \[-\beta_i\sum_{n\ge0}\left(K_U(u_j-u_i+nL) +K_U(u_i-u_j+(n+1)L)^t\right)\beta_j.\] Each field has self multiplier \[(L/(2\pi))^{\Delta_{\beta_i}}d_U(\beta_i) \exp\left[-\beta_i\sum_{n\ge1}K_U(nL)\beta_i\right].\] In these formulas the transpose of a complex matrix is still the bilinear metric transpose. Finally multiply by \[Z_U=\prod_{l\ge1}\det(I-\mathfrak q^l U)^{-1},\] and, for \(M_b\), also \(\sqrt{L/(2\pi H_b)}\). Coefficients from group actions and their factorials are as above, and root integrals now along increasing real coordinate have measure \(du/L\), thus no overall \(L\)-power from root fields. Use limiting boundary values where needed.

We will use \[\frac{Z_{M_b}\sqrt{L/(2\pi H_b)}}{Z_O} = e^{7L/192}\frac{P_2 P_8}{P_6 P_3 P_1},\qquad P_j=\prod_{n\ge1}(1-e^{-nL/j}).\] Indeed if \(F(t)=\sum\log(1-e^{-nt})\), then \(F(t)=F(4\pi^2/t)+t/24-\pi^2/(6t)+\frac12\log(2\pi/t)\). For one quick proof apply ordinary Poisson summation to \(|x|/(e^{t|x|}-1)\). Its Fourier coefficients at integers are the samples of the transform \(t^{-2}(y^{-2}-\pi^2/\sinh^2(\pi y))\), \(y=2\pi l/t\), obtained by summing the exponential series and the elementary cotangent partial fractions. They are absolutely summable. This gives \(1/t+2F'(t)=\pi^2/(3t^2)+1/12-8\pi^2 F'(4\pi^2/t)/t^2\); integrate using \(t=2\pi\). The formula for the determinant ratio follows using the characteristic polynomials above and \(\det(I-yO)=(1-y^8)/(1-y)\).

Recovery of the finite sums

The purpose of this normalization is exact coefficient recovery. We first fix the source multiplicity \(N\) and the real period, and fuse the sources; Section 4 will separately control the fused charge \(Na\) as \(N\) grows. Take \([p^0]\) at zero (Taylor, or Cauchy average) of the following after confluence: \[\mathscr P_N= w^N \frac{\mathcal T}{Z_O\,e^{-\pi i\Delta_{Na}}d_O(Na)(L/(2\pi))^{\Delta_{Na}}}.\] This gives the raw color sum. To check, initially separate the \(N\) sources near 1 (distinct angular and real coordinates), and divide \(\mathcal T/Z_O\) instead by the leading uncolored radial prescription there at \(O\), i.e. without the finite-period images. The \(d_O\) ratios for the middle color and last color relative to the first are \(\sqrt{2k}\) and 1 by the self formula. Pair phase differences from uncolored are even integral (\(S\) pairings change by integers). For inner/outer fields \(y,z'\) in radial order, ratio of inner to outer source variable \(\zeta\), the direct short kernel exponentiates to \[B_{z'y}(\zeta)=\prod_{j=0}^7(1-q^{-j}\zeta)^{z'O^j y}.\] Its ratio for colors versus \(a,a\) is \(P_{\alpha_{\rm inner},\alpha_{\rm outer}}(1/\zeta)\) in conjugate convention by substitution. Thus the zero-image sum with the displayed multiplier has exactly the wanted color weights (including \(w,u_*\) and the self ratios). Positive images add analytic factors starting with 1 near \(p=0\), using arguments multiplied by \(p^n,\ n\ge1\).

For fixed real period the fusion limit of this separated ratio gives the displayed ratio. Indeed before moving the conjugation into colors (or on the transformed side with pure \(V_a\)’s, root contours away), the leading local angular contraction of the \(N\) uncolored fields combines them to \(V_{Na}\); its scalar, from the common powers, translation phases and direct pair factors, is the same as in the pure uncolored angular trace. All other contractions are regular there and merge multiplicatively. In passing from that pure angular trace to the leading radial denominator we only strip finite image factors, smooth and merging the same way. This argument can first use small \(\mathfrak q\) with the absolute bounds below: in the transformed angular contractions at separated sources, factor off the common direct uncolored collision powers before estimating. After this cancellation the remaining Taylor/image factors are regular with the same uniform bounds through coincidence (root layers strictly away and fixed); opening/closing the source separations only changes lower-order terms of the angular-momentum quadratic in root counts and the axis index. Then both the fused transformed radial expression and the separation coefficient of the color expression used next continue along real positive periods. Equivalently one can fuse in the continued radial expressions with root contours still away. To commute fusion with \([p^0]\), let the separated sources approach 1 along generic fixed log-slopes times a small scalar. The normalized \(O\)-color expression is meromorphic in that scalar with bounded pole orders and holomorphic image series in a common disk of \(p\). Its fusion value is its constant Laurent coefficient (negative coefficients vanish by the fusion limit on real periods). Thus the recovered term is exactly the previously established homogeneous rational color sum, not a relative approximation in \(p\).

Stability and analytic justification

We now justify summing the transformed scalar formula over all root insertions. There are two distinct tasks. At small \(\mathfrak q\) we compare convergent angular expansions with the coefficient identities already proved. We then use a bound on the radial interactions to continue that identity to every positive period and to the bounded complex shifts of large periods needed in Section 4. Source multiplicity is fixed throughout this continuation and through the confluence argument of Section 3.3; uniform estimates for the fused charge \(Na\) will be proved separately in the long-circle argument.

Positive norms and nonzero leading terms

Denote homogeneous \(Q,R,Y\) by subscripts \(n\) at width \(n\). Use \(c=3/4\) and the Fourier weights \[\mu(\nu)=\frac{\nu\cosh(\pi\nu/4)}{\sinh(\pi\nu)},\qquad w_b(\nu)=\frac{k}{\sqrt2}\,\frac{\sinh(\pi\nu/4)}{\sinh(\pi\nu)} .\] Indeed \(M(e^s,1)=\int e^{i\nu s}w_b(\nu)d\nu\), and \(A(e^s,1)= -i\,\mathrm{PV}\int e^{i\nu s}\mu(\nu)d\nu/\nu\), \(A(x,y)=(x^2-y^2)/h(x,y)\). The transforms follow, for example, by summing the residues at \(\nu=i j\) for \(s>0\), including the half term at 0 in the principal value (use horizontal tops at half-integer heights, then symmetry/analyticity). Write \(h_j(b')\) for squared monic polynomial norms, after orthogonalization in a weight \(b'\). Confluence in log coordinates gives \[\frac{Q_{n+2}}{Q_n}=\frac{k^{2n+1}}{n!(n+1)!} h_n(\mu),\qquad R_n=k^{n(n-1)}\prod_{j=0}^{n-1}\frac{h_j(w_b)}{j!^2}.\] For the first identity take derivatives in the Pfaffian, ordered starting at degree 0. Opposite-parity entries between degrees \(a',b'\) are \(-i\cdot i^{a'}(-i)^{b'}\int\nu^{a'+b'-1}\mu\), and same parity gives zero. In even size the resulting determinant block (evens vs odds) is the Gram of successive even monomials up to row signs; in odd size the augment first eliminates degree zero, leaving the Gram of successive odd monomials. The alternating-order Pfaffian is the determinant in odd/even positions; the entry signs give overall \((-1)^{\lfloor n/2\rfloor}\), canceled by the reversed Vandermonde in \(F\). This proves the formulas and positivity.

Compare with \(\mu_0(\nu)=\nu/[2\sinh(c\pi\nu)]\le\mu(\nu)\), \(w_0(\nu)=k/[2\sqrt2\cosh(c\pi\nu)]\asymp w_b(\nu)\). Their Laplace transforms are respectively \((2c)^{-2}\sec^2(s/(2c))\), \(k/(2c\sqrt2)\sec(s/(2c))\), by the same residue calculation. Thus \(h_n(\mu_0)=n!(n+1)!/(2c)^{2n+2}\). Indeed expand the two-variable Gram kernel at \(s+s'\) by the addition formula: for \(\sec^2\) it uses features \(\sqrt{n+1}\sec^2(\tau)\tan^n(\tau)\), \(\tau=s/(2c)\), times the square-root mass. Their leading Taylor coefficients give the squared monic norms by triangularity.

From \(O_Z(r)\) in the finite calculation, the unprobed raw sums at the two twists are \(R_n^2/(2k)^{n(n-1)/2}\) and \(\kappa^{2n}Q_n^4/(2k)^{n(n-1)/2}\). Hence the long-circle normalizations of \(Q_n,R_n\) themselves correspond respectively to \[B_Q(n)n^{5/48},\quad B_R(n)n^{-7/24},\qquad B_Q(n)=(k/(2c))^{n^2/2}k^{-n/2},\quad B_R(n)=(k/(2c))^{n^2}2^{-n/2}.\]

A polynomial Laplace estimate

Let \(p_j\) be the positive-leading orthonormal polynomials for \(w_b\), write \(h_j=h_j(w_b)\) for its squared monic norms, and set \[f_j(\theta)=\int p_j(\nu)e^{\theta\nu}w_b(\nu)\,d\nu.\] The following estimate retains its first correction because the leading terms in the winding calculation will cancel after taking real parts.

Winding and rooted polygon masses

We now apply the normalized polynomial estimates to the physical gluing formulas. Throughout, \(J_n\) counts winding polygons modulo whole-row translations on the regular \(n\)-column cylinder, as in Proposition 3.

We next evaluate \(J_n\) at the regular honeycomb point, in original conventions \(t=q^{-1}\). Confluence in the wave formula yields \[J_n=\frac{(-1)^{n+1}\sqrt2}{Y_n}\Re[P_n(\pi/4)P_n(\pi/2)],\quad P_n(\theta)=e^{-i\theta}-\sum_{j<n}p_j(-i) f_j(\theta).\] This just uses projection onto polynomial degrees \(<n\) in the Fourier kernel followed by evaluation at \(-i\), with its conjugate or reflection for the other waves.

Set \(A_n=|p_n(i)|=\sqrt{(Y_n+Y_{n+1})/k}\); \(p_n(-i)=(-i)^n A_n\) by the positive-off-diagonal recurrence. For \(r=\tan(\theta/(2c))\) at each of our two angles, \[P_n(\theta)=(-i)^n A_n f_n(\theta) \left[\frac1{1+i r}+\frac1{4n}\frac{-i r}{(1+i r)^2}+o(n^{-1})\right].\] For \(r<1\) this sums the tail starting at \(n\); the full series reproduces \(e^{-i\theta}\) by convergence and Taylor projection (absolute on smaller disks by the positive recurrence and estimates above). For \(r>1\) sum the negative prefix backwards, dropping the exponentially negligible free term. In both directions use \(A_{n+j}/A_n=1+j/(4n)+o(n^{-1})\) for fixed \(j\) by smooth expansion; \(f_{n+j}/(r^j f_n)=1+o(1/n)\) by the preceding estimate. The sums with first-order errors are dominated by geometric series times polynomials in \(|j|\), e.g. restrict indices first to \([n/2,2n]\) using the smooth estimates there and then exponential tails.

The two \(r\)’s are reciprocal. The product of the leading brackets is purely imaginary; the real correction is \(-2/(4n(r+r^{-1})^2)\). Using \(A_n^2/Y_n\to 2/k\) and the first ratio limit above gives \[ J_n\sim \frac{2(1/4)}{c(r+r^{-1})n}=\frac{\sqrt3}{6n}. \tag{11}\]

At the alignment \(t=1\), \(E_+\) is the total polygon mass through a top port in the physical chain by the mark reduction (both vacuum halves independent of row parameter). In original conventions the pole-pair recursion gives \[ E_+=1+\kappa\left[\frac{1+q^3}{d(1,1)}\frac{Q_{n+1}Q_{n-1}}{Q_n^2}-1\right] = (2-\sqrt2)/3+O(n^{-2}) \tag{12}\] by the smooth expansion. In particular the plane total polygon mass through a fixed port (hence also through a fixed center) is finite: any finite selection projects injectively for sufficiently large width, and we have symmetry of the edge directions.

The continued transfer eigenvalue

We first identify the continued eigenvalue by a scalar interpolation problem. Keep the pairing states with their annular homotopy, because winding loops now survive with fugacity \(a\). The state space is still finite: each cap has only two possible directions around the smaller boundary (displacement less than a full circumference by interlacing of translates). All allowed states can be built from vacant rows at the positive honeycomb point: close an innermost cap first, across its specified interval on the disk side containing no ends, propagating the others as in the finite proof, and repeat. Powers on the nonempty block at zero fugacity are therefore transient by the convergent vacuum bound. Thus the continued eigenvalue and its vector, normalized on the empty component, are analytic locally. Use at first also \(Z\) close to \((1,\ldots,1)\). All rows (spectral parameters \(t\)) commute, by inserting an auxiliary swap and its inverse between the two auxiliary lines at the cut, sliding the swap past each column by braid and canceling around the cylinder; the diagram identities just keep the pairings with homotopy when used in the annulus, since the changed patches preserve connections up to isotopy and their internal loops are contractible. Write \(\Lambda(t)\) for the eigenvalue on that same line. It is rational with denominator at most \(D_5 D_6\), \(D_r(t)=\prod_{z\in Z}(z-q^r t)\), using row parity of single corners and the empty row-output evaluation. At \(0,\infty\) a vacant output requires a vacant input and sees the free horizontal straight line or empty, so \(\Lambda(0)=\Lambda(\infty)=1+a\). At zero fugacity \(\Lambda\equiv1\).

For each \(x\in Z\) \[\Lambda(x)\Lambda(q^{-2}x)=\Lambda(q^{-1}x),\qquad \Lambda(x)\Lambda(q^{-3}x)=1 .\] Indeed in the lower row at \(t=x\) the site is identity, turning its horizontal incoming line up and bottom leg to the right. Thus \(PP^t\) at site \(x\) of the upper row \(t=q^{-2}x\) has a triangle collapsing the two incoming horizontals. Slide this collapse backwards through all other columns (relation \((1\otimes P^t)B_{12}(qs)B_{23}(s/q)=B(s)(P^t\otimes1)\), upper left input numbered first). On reaching the outgoing side of site \(x\), the two triangles there assemble into its cell \(B(q)\) on the fused row. The cap slide at the other specialization works identically, yielding just vertical continuation. This uses the coherent auxiliary half-powers. For generic \(Z\) these equations and endpoint data determine the analytic germ: at linearization at 1 the rational difference with zero value at 0 has basis \(t/(t-q^k z)\), \(k=2,3\). The two rows of linearized constraints at each site have full total rank, as seen for widely separated ordered positive \(z\)’s, where the matrix tends to block triangular with diagonal blocks with entries \(c_k+c_{k+2}-c_{k+1},c_k+c_{k+3}\) in the respective rows, \(c_j=(1-q^j)^{-1}\). Each block determinant is \(((2-\sqrt2)/2)^2\). This proves uniqueness successively on Taylor coefficients.

The following three-term expression is the finite spectral form we will verify; the periodic seam formulation has precedents in [15]. Use a degree \(n\) polynomial \(U_b(t)\), with roots \(u_j=q^{-1}e^{\theta_j}\). Put \(P(t)=U_b(t)D_4D_5D_6(t)\), using subscripts for shifts of either polynomial at \(q^r t\). The candidate is \[\Lambda= d P_{-2}/P_0+P_{-1}P_2/(P_0P_1)+d^{-1}P_3/P_1.\] The two node relations follow immediately using \(P_2=P_3=P_4=0\) at \(t=x\), generically; equivalently the three site-dependent factors apart from \(U_b\) and \(d\) are \(D_2D_3,D_0D_3,D_0D_1\) respectively with common denominator \(D_5D_6\). Endpoints are correct. Cancellation of unwanted root poles (including in \(P_1\)) holds if at \(u_j\) \[D_0 U_{b,-1}U_{b,2}/(D_2 U_{b,-2} U_{b,1})=-d .\] We use simple roots with no mutual shifted coincidences.

At \(d=i\), impose that the grade 3 modulo 4 in \(P\) vanish identically. For distinct positive \(Z\) such a solution is given in variable \(s=qt\) by \[\frac{P(t)}{\prod_{x\in Z}(x^4+s^4)}= \frac{p_Z^+(-s)}{-s}=1+\sum_x \frac{b_x}{s^2-\sqrt2 x s+x^2},\] with \(p_Z^+\) the wave defined earlier (the Gram kernel strictly positive by Fourier). Indeed the quadratic terms on clearing have no forbidden grade, give zeros at \(s=-q^{\pm1}x\), and the interpolation gives the further zero \(s=-x\). Thus the required divisibility holds. Then the candidate is identically 1: after clearing \(P_0P_1\) check at \(0\) and all \(q^j x\). On the orbit starting at \(x\), \(P_2=P_3=P_4=0,\ P_0=iP_6,\ P_7=i(P_5-P_1)\) by grade elimination, giving equality throughout. Confluence is allowed since the wave exists there by the confluent positive Gram; the leading coefficient in \(s\) of \(P\) stays 1.

The remaining \(n\) zeros in \(s\) are positive simple at distinct positive sites. Here are details by variation. This is a total-positivity and variation-diminishing argument in the framework of Schoenberg [13]; we prove the strictness and decay needed for the present kernel. In log coordinates, convolution against the kernel with Fourier multiplier \(\sinh(\pi\nu/4)/\sinh(3\pi\nu/4)\) takes \(-p_Z^+(-e^y)\) to \(p_Z^+(e^y)\). Indeed it acts identically on the exponential (contour shift in Fourier inversion gives exponential decay with rate \(>1\), so use the multiplier at imaginary frequency) and takes \(-M(-e^y,x)\) to \(M(e^y,x)\), by the Fourier transform of \(\sin b/(\cosh y-\cos b)\), which is \(2\pi\sinh((\pi-b)\nu)/\sinh\pi\nu\) by residues. This kernel is strictly totally positive. To recall a proof, by the sine product (equivalently integrate the integer-pole cotangent partial fractions for its log derivative) it is proportional to the convolution of normalized double exponentials of rates \(4k/3, k\ge1, 3\nmid k\) (the independent centered series converges by variances). Each factor has nonnegative minors on ordered tuples: decompose into two one-sided exponential kernels, whose minors reduce to step matrices, and integrate determinants. Infinite tails inherit nonnegativity by convergent convolution, for example by dominated Fourier inversion retaining several factors. Strict positivity at order \(l\) follows by convolving sufficiently many initial factors: the double exponential has strict determinant between aligned ordered tuples differing by motion of one point without passing neighbors (in composing step matrices there is then an open set of strict interlacings). Any two tuples can be joined by at most \(2l\) such steps via intermediate points far off, with strictness on open sets. The final tail determinant into a given tuple is positive somewhere (linear independence of translates by nonzero Fourier transform). The determinant integrations here are just antisymmetrization on products of ordered lines. This proves the assertion. If the input of our convolution had fewer than \(n\) sign changes, a linear combination of translates at some of the \(n\) output zeros \(\log x\) could match its signs: use the determinant of that many translates (one more than the number of changes) evaluated at the change points and the running point. Strictness and ordering give the signs, contradicting zero integral (the kernel decays faster than required for the exponential). Thus \(-p_Z^+(-s)\) has at least \(n\) sign changes, proving the root claim.

Counting phases and sparse end roots

We detail the analytic control of the root equations. The target is an inverse bound uniform in the circumference; in particular the first and last roots cannot be treated by a bulk Riemann-sum approximation. For positive sites put \[A_b(x)=-\arg(1-e^{ib}e^x)/\pi, \quad A=A_{\pi/4},\quad K=A-A_{\pi/2},\quad k_b=K'.\] Arguments are continued from 0 at the left end. Thus \(2\pi k_b(x)=t_0/(\cosh x-t_0)-1/\cosh x,\ t_0=2^{-1/2}\). At \(d=i\) order the real roots increasingly; their pole equations are \[F(\theta_j)=j-1+3/4,\qquad F(x)=\sum_{z\in Z}A(x-\log z)+\sum_j K(x-\theta_j),\quad 1\le j\le n .\] Indeed the fractions give these phases modulo 1 and the labels are constant by continuity on ordered distinct sites. At widely separated sites the wave formula has one root \(e^{\theta}\sim(1+\sqrt2)z\) at each scale: in the displayed quadratic fractions the zero equations at \(s=-z\) give a triangular limiting system on \(b_x/x^2\) with diagonal \(1/(2+\sqrt2)\) and entries 1 for larger columns, hence nonzero solutions there and roots on bounded site ratios by substitution. Then \(A(\log(1+\sqrt2))+K(0)=3/4\), previous sites contribute 1 each. These equations persist at homogeneous confluence (bounded coefficients with fixed leading term by the Gram; no roots at 0 in the limit since the constant fraction there is the homogeneous wave limit \(p_Z^+(-s)/(-s)|_{s=0}=(-1)^n Y_n\ne0\) proved above). Different labels imply simplicity there too. For nearby \(d\) the targets are shifted uniformly and analytically by the phase increment divided by \(2\pi\).

Hereafter at homogeneous confluence the continuum comparison has cumulative \(H(-\infty)=0\), \[h_b(x)=H'(x)=\frac{4}{\sqrt3\pi}\frac{\sinh(8x/3)}{\sinh(4x)},\qquad H=A+k_b*H\] (by the same Fourier formula, multiplier for \(h_b\) is \((2\cosh(\pi\nu/4)-1)^{-1}\)). Define \[E(x)=\#\{j:\theta_j\le x\}-n H(x)+1/3,\qquad B_j=n h_b(\theta_j).\] At jumps \(E\) takes one-sided values \(\pm1/2+(k_b*E)(\theta_j)\), since \(F=nH+k_b*(E-1/3)\) and \(\int k_b=1/4\). Monotonicity between then gives \(\|E\|_\infty\le (1/2)/(1-\|k_b\|_1)<.711\).

The linearization of the root equations is the map \[(\mathcal L_n\eta)_j =F'(\theta_j)\eta_j-\sum_l k_b(\theta_j-\theta_l)\eta_l.\] Its natural displacement norm is \(\|\eta\|_B=\max_j B_j|\eta_j|\): a root in a sparse tail is allowed a larger displacement than one in the bulk.

For fixed large \(n\) this solution germ also exists at sites sufficiently near homogeneity; at \(d=i\) it coincides with the grade-elimination solution above by continuity and local uniqueness. Thus by the scalar uniqueness for generic sites and then continuity, the continued eigenvalue is given by our candidate. Root poles cancel as polynomial factors before evaluation (also at an accidental specialization of a node), so evaluations away from \(D_5D_6=0\) are analytic on the small homogeneous root disk.

The pressure error at homogeneous sites

The uniform inverse has constructed the eigenvalue on a fixed parameter neighborhood. We now need its logarithm to be of order the reciprocal circumference. We sharpen the estimate at homogeneous sites to order \(1/n\). We need only real phases \(d=e^{i\phi}\) close to \(i\), so the deformed \(\theta_j\) are real by the construction. Now let \(N(x)=\#\{\theta_j\le x\}\) for these roots, and \(F\) also denote the deformed counting phase. Put \(\ell=\phi/(2\pi)-1/2\), so \(F(\theta_j)=j+\ell\). The shift estimate and exponential density bounds imply \(N-nH\) remains uniformly bounded. With \(C_\phi=-(4/3)\phi/(2\pi)\), \[F=nH+k_b*(N-nH),\qquad (1-k_b*)(N-nH-C_\phi)=N-F+\ell+1/2=:r .\] We will estimate the change of logarithmic root products by integrating against this discrepancy. The needed cancellation is stronger than its uniform boundedness:

Connected clusters and exponential height tails

We explain the geometric consequence. The pressure coefficients count connected families of winding polygons; their common sign will turn the analytic bound into positive height moments. On a finite long strip of this cylinder with vacant boundary input and output the partition function is a gas of pairwise disjoint winding polygons. Its log per row coefficients tend to \([a^p]L_n\) by simplicity/dominance locally at zero fugacity. The connected-cluster formula at order \(p\) sums products of the \(X^{\rm length}\) weights divided by \(p!\), with multiplier the alternating subgraph sum over connected spanning subgraphs of the overlap graph (also conflicts for repeated polygons). This follows by expanding disjointness pair factors and taking the formal log. This is the Mayer connected-graph expansion; see [14] for the expansion and its sign in a repulsive gas. Nonzero multipliers have common sign \((-1)^{p-1}\) and modulus at least 1, for example by deletion-contraction for the linear coefficient of the graph coloring polynomial. Thus we may lower bound absolute coefficients even in infinite length by restricting to stars where all other polygons meet the first (first counted per unit row, by the finite-strip limit).

The passage from a real interval to a coefficient bound is the positive-coefficient principle used in [14]. This sign property also completes the coefficient estimate: the power series for \(-L_n(-a)\) has nonnegative coefficients. Its radius must reach across our short interval by analyticity along that interval (otherwise at the first positive convergence boundary all derivative series still sum to the derivatives by monotonicity, and Taylor expansion there gives positive convergence farther out). Evaluating inside then bounds each fixed coefficient by \(O_p(1/n)\).

We next turn a coefficient into a height moment. Given a winding polygon \(P\), the weighted choices of winding loops meeting it satisfy \[ \sum_{Q:\,Q\cap P\ne\varnothing}X^{|Q|} \ge (h(P)-C)_+J_n, \tag{14}\] where \(Q\) here runs over actual row positions. To prove this, choose vertices \(p_+,p_-\) on the same bipartite sublattice, within bounded distance of the highest and lowest levels of \(P\). Their row-height difference is an integer \(D\ge h(P)-C\).

For a second winding polygon \(Q\), write \(Q_{b,r}\) for its translate by column phase \(b\in\mathbb Z/n\mathbb Z\) and row shift \(r\in\mathbb Z\). Define the signed mean height \(m(Q)\) by integrating the difference between the indicator of its lower domain and that of the lower half-cylinder, divided by one period’s area per row. Column translation preserves \(m(Q)\), and \(m(Q_{b,r})=m(Q)+r\). For disjoint essential curves, mean-height order agrees with their nesting order, since the area between them is positive.

Put \(t=m(P)-m(Q)\). If \(r>t\) and \(Q_{b,r}\) is disjoint from \(P\), then \(p_+\) belongs to its lower domain. If \(r-D<t\) and \(Q_{b,r-D}\) is disjoint from \(P\), then \(p_-\) belongs to its upper domain. For each integer \(t<r<t+D\), average these two necessary conditions over column phases. The relative query points have the same distribution: their row coordinates agree after subtracting \(D\), and their common bipartite type makes the remaining displacement an integer column shift. The upper and lower open domains of \(Q\) are disjoint. Hence the two probabilities of meeting \(P\) sum to at least one.

There are \(D-O(1)\) such integers \(r\). Their upper shifts \(r\) and lower shifts \(r-D\) form disjoint intervals, so summing over all shifts gives at least \(D-O(1)\) intersecting translates in the phase average. Finally sum \(Q\) modulo row translation with weight \(X^{|Q|}\). Column translation permutes this weighted family, so its phase average is the original sum, with no additional factor of \(n\). This proves (14).

Let \(c_{n,p}=|[a^p]L_n(a)|\). Evaluation of the nonnegative series at a fixed positive argument \(a_*\) inside the interval gives \(c_{n,p}\le C a_*^{-p}/n\) for every \(p\ge1\). Restrict the ordered connected-cluster sum to tuples whose first polygon meets every other one. Their overlap graphs are connected, and the multiplier has modulus at least one. For \(m\ge0\), the factor \(1/(m+1)!\) in that expansion gives \[(m+1)!c_{n,m+1} \ge\sum_PX^{|P|} \left(\sum_{Q:\,Q\cap P\ne\varnothing}X^{|Q|}\right)^m \ge J_n^m\sum_PX^{|P|}(h(P)-C)_+^m.\] Since \(nJ_n\) is bounded below, choose \(\varepsilon>0\) so that \(\varepsilon/(a_*nJ_n)\) is uniformly less than one. Divide the last inequality by \(J_n^m\), multiply by \((\varepsilon/n)^m/m!\), and sum. The exponential moment is bounded by \[\frac{C}{a_*n}\sum_{m\ge0}(m+1) \left(\frac{\varepsilon}{a_*nJ_n}\right)^m=O(n^{-1}).\] The harmless factor \(e^{\varepsilon C/n}\) restores \(h(P)\) in place of \((h(P)-C)_+\). This proves the winding estimate in (13). It also gives suppression beyond \(n^{1+\epsilon}\) faster than every fixed power, even with fixed length powers: deterministically \(|P|=O(n(h(P)+1))\).

The contractible height tail follows by a local joining operation. Join any such polygon to an arbitrary winding one by shifting the latter to put a bottom extremity just above a top extremity of the contractible one. More explicitly, using horizontal triangle-side levels, a highest triangle-center vertex must be down-pointing, using both lower-slant connections. Take the hexagonal face for which it is the lower left vertex just below mid-level. Intersection with the lower boundary of this face is then precisely a single arc of one or two edges (if the other extremity-level vertex on it is used, it too must use the shared lower neighbor). Align the winding polygon by lattice translations to have a lowest vertex (up-pointing) on the corresponding upper left of that face, giving the symmetric upper arc. They are otherwise disjoint, so flipping the two arcs to the complementary face edges gives a single winding loop with bounded length change and retaining the height up to bounded loss. Reversing from an output costs at most order its length in face choices and an additional factor \(O(n)\) to restore the original horizontal phase. Thus multiplying the contractible tail mass by \(J_n\) bounds it by a polynomially weighted winding tail, sufficient again with length powers allowed.

To obtain the second estimate of (13), apply the winding exponential moment with a smaller exponent, absorbing the polynomial recovery factors and the deterministic bound \(|P|=O(n(h(P)+1))\). Division by \(J_n\asymp n^{-1}\) introduces only another polynomial in \(n\). The bounded change of height in the join changes the constant. This completes the proof of Proposition 10.

Half-strip polynomials

The next proposition supplies the denominator and degree bounds needed when the two half strips are glued. Its proof constructs the polynomial vectors before identifying them with the physical sums.

A crossing determinant

To form \(a_n\), glue \(A_L E\) in the fixed source row to the upper reflected \(O\), matching occupations and requiring precisely the single path, no loops. This proves rational continuation. Sliding spatial swaps gives symmetry (opposite ratios in the reflected vector; diagram transpose of a swap has the same weights, then use inversion). The same gluing reduces at a gap-three pair to the remaining quantity without multiplier: carry the cap from below through the source row, then contract by \(C^t\) above. This gives the smaller reflected vacuum there since its exchange on the reversed rapidity relation uses \(CC^t\), the exchanged value itself uses \(C\), and \(C\) is injective. At \(z_n=p\) lower and row carry a last vacancy; the vacant-last component above also drops by right reflection followed by right-wall removal. Finally \(a_n\) is invariant under \(z_n\mapsto1/z_n\) by the full-strip arc sum: all occupations of that last column are consecutive paired contacts from the left, with product weights involving \(uU,v\) only. As for edge reflection before, both sides converge on a common open set (upper halves handled by vertical reflection). Symmetry gives separate inversions everywhere.

Consequently \[P_n={\cal B}_n Q_o^2\prod_j(x_j+b)\] is a symmetric polynomial in \(x\) of degrees \(\le2n-1\). The half-strip polynomial bounds and the inserted row denominator make it Laurent in \(z\) (half-powers match at occupied ports); inversion then gives polynomiality in \(x\). Before rescaling the gluing stays bounded at single extremes: if the upper odd occupation grows it faces either a lower even occupied end decaying or a decaying single corner. Both the wall specialization \(x=a\) and pair specialization \(H(x_i,x_j)=0\) reduce \({\cal B}\) without multipliers.

Define \[\begin{aligned} \Delta(x,y)&=2x+\sqrt2 y+s_0,\\ N(x,y)&=(a+x+\sqrt2 y)\Delta(x,y)-2H(x,y)\\ &=\sqrt2 xy+(2a+s_0)x+\sqrt2(a+s_0)y+\tfrac32 ,\\ M_o(x,y)&=\frac{x-a}{x+b}\frac{N(x,y)}{H(x,y)}. \end{aligned}\] Then \[{\cal B}_n=\frac{F_o^2}{Q_o^2}\det M_o(x_i,x_j).\] Here and initially in what follows statements using kernels are rational before specialization. Indeed the proposed numerator \(P_n\) is polynomial with the indicated degree: \(H\) poles cancel, including on the diagonal where \(M_o(x,x)=1-\tau^2 v\) by substitution, coincident positions cancel twice by rows and columns, and at a single infinity the site’s two off-diagonal directions decouple and its diagonal tends to \(\gamma\). Thus the leading \(x^{2n-1}\) coefficient is \(\gamma\) times the smaller proposed numerator. At \(a\) the off-diagonal row vanishes and the diagonal is 1, giving wall reduction. At \(H(x,y)=0\), the double-pole determinant term gives pair reduction by \[-(x-a)(y-a) N(x,y)N(y,x)=(x+b)(y+b)(x-y)^2(x+y+d)^2 .\] Indeed modulo \(H\), \((x-a)\Delta=(x-y)(x+y+d)\) and \((a+x+\sqrt2 y)(a+y+\sqrt2 x)=(x+b)(y+b)\). Thus by induction from size zero the only ambiguity at each size in \(P_n\) is a constant multiple of \(\prod_{i<j}H(x_i,x_j)\prod_i(x_i-a)\).

We fix it at infinity in the last column with all other columns regular honeycomb. Vary \(v\) there from its honeycomb value towards 1. Every visit to that column gives consecutive arches from its left edge of weights \(K_\ell=uU v^{\ell-1}=\tau(1-v)v^\ell\) (\(\ell\ge1\)), total \(\tau v\). The excess left-return mass above \(a_{n-1}\) consists of trains of \(k'\ge1\) disjoint ordered upward jumps with these weights. Inside the old strip they use one initial bridge from the source to the interface, one ending bridge to the left boundary, and \(k'-1\) intermediate upward interface-to-interface excursions, all mutually avoiding. Indeed for \(2k'\) successive interface ends, the outer arches pair \((1,2),(3,4),\ldots\). The two inner bridges have terminals in ascending order by planarity; no inner pair spans either terminal. Any nonempty prefix/suffix outside them would thus close into loops, so they end at \(1,2k'\). Then 2 must pair inside to 3 (a later partner again traps loops), and so on. Conversely these pieces in order and without conflicts form precisely such arcs if the last bridge ends above the source. For width zero of the old strip use the empty bridges only. By half-turn symmetry, translation and reversal the final bridge and intervening excursion masses per start are \({\cal B}_{n-1}\) and \(a_{n-1}\).

Hence \[0\le \Delta a_n\le \frac{{\cal B}_{n-1}^2\tau v}{1-\tau v a_{n-1}}, \qquad \lim_{v\uparrow1}\Delta a_n=\tau {\cal B}_{n-1}.\] Here \(\tau a_{n-1}<1\). For equality in the limit, at fixed train length normalize each jump by its total: the gaps diverge in probability. For any fixed shapes of the finite interior pieces this eventually removes all avoidance and end constraints with probability tending to 1. Dominated convergence and the geometric train tail suffice. This calculation defines an analytic summed arc mass along the interval \(v<1\) in question (locally, \(\sum |K_\ell|a_{n-1}<1\)) and agrees with the actual gluing near honeycomb. Thus the polynomial identity defining \(P_n\) using that mass continues along it even with fixed coincident other parameters or a zero of \(Q_o\). Since \({\cal B}_n\to\gamma{\cal B}_{n-1}\), the leading coefficient there agrees with the determinant’s. The possible ambiguity factor has nonzero leading coefficient there (\(x_j=b\) for others), proving the formula.

It suffices to estimate even sizes at \(x_j=1\) (square rhombi), by \[{\cal B}_{2n}(b,\ldots,b)\ \le\ {\cal B}_n(1,\ldots,1)\ \le\ {\cal B}_n(b,\ldots,b).\] Here are convergence and comparison details. In mixtures of the two angles \(\pi/3,\pi/2\) all weights are nonnegative with single routes strictly positive and doubles bounded by the product of the two single weights (\(w\le u^2,\ W\le U^2\); use the explicit weights/identities). One last honeycomb column is arch-by-arch dominated by one square since \(0<v_H<v_S<1/2\) in \(K_\ell\). Conversely two honeycomb columns dominate one square there: for each arch interval restrict to paths going up each row carrying one occupied vertical channel, giving mass at least \[uU(1,v) \begin{pmatrix}v&u^2\\ U^2&v\end{pmatrix}^{\ell-1} \binom{1}{v}\] with honeycomb \(u=X,\ U=v=X^2\); the two off-diagonals switch channels via two corners, and the end factors allow starting/ending via a horizontal propagation. The matrix sends the indicated column vector to at least \(v(1+u^2)\) times itself, \(>v_S\) in multiplier, and \(uU(1+v^2)>u_S U_S\). For example \(X>.541,\ .2928<v<.294,\ .320<v_S<.322,\ 2a<.766\) by the radicals above, \(u_SU_S=2a(1-v_S)v_S\). Distinct intervals use disjoint rows and have the required connectivity.

Induct on the number of squares simultaneously over widths to get transience and convergence. Spectra on the nonempty even and odd blocks are unchanged by permuting (nonsingular swaps at these ratios), also for vertically reflected angles. With a square placed last the two-column domination bounds \(a_n\) by a finite one by induction. This also bounds all lower and upper arc components: drop mutual conflicts of separate arcs using the product bound (self double visits keep their prescribed weights). For a single lower arc between fixed top ports append disjoint fixed tails in fresh rows to two left ports (route the nearer one first horizontally to the wall continuing the farther vertically, then the other); for a variable left-to-top arc append one such fixed tail. The resulting masses are bounded by full left-return sums by translation and reversal. Reflection interchanges the two directions of return and gives the upper version. Accessibility gives transience as before, hence also in every permutation, where gluing again gives the actual finite mass. We may thus use swap symmetry and repeatedly move a column to the last wall to apply either comparison, proving the displayed bounds. The honeycomb value is also nonincreasing in width by return inclusion.

Square-point asymptotic

We now estimate the crossing determinant at \(x_1=\cdots=x_n=1\), for even \(n\). The positive column comparison proved above will then transfer the estimate to every honeycomb width. There are two reductions. First we express the large determinant quotient as a quotient of residual Hermite evaluation determinants of sizes three and one. We then evaluate the cancellation in the three-column determinant; its first nonzero normalized term has order \(n^{-3}\).

Doubled change of expansion

The constraint \(\sum_i\sigma_i t_{a_i}=0\) in (16) introduces a fixed axis in the original color channel. After conjugation, the root channel has a different fixed axis. The construction below identifies both axes and the momentum sums they permit. Its convergence proof will also differ from Section 3.4: short-root interactions alone are not the object bounded there; part of the kinetic propagation must be included. Use \(n=m\), the conjugate convention, and \((\phi_+,\phi_-)=(0,0)\) or \((\pi/4,\pi/4)\). The following data replace the one-family vectors of the scalar change of channel above. Use \(D_j^\sigma\), \(j\bmod8\), modulo the radical, rotated by \(O\), with rows \[D_0^\sigma D_j^\sigma=(2,-1,1,0,0,0,1,-1),\qquad D_0^\sigma D_j^{-\sigma}=(0,0,0,-2,2,-2,0,0).\] Put \(D_0^\sigma=(1-O+O^2)a^\sigma,\ a_s=a^++a^-,\ \delta^\sigma=D_7^\sigma,\ d_*=\delta^++\delta^-\), and \(M_*=O R_{\delta^+}R_{\delta^-}\). Use \(S=(1-O)^{-1}\) with value \(1/2\) on the fixed space. The old and new fixed axes and squares are \[h_o=\tfrac14\sum_j(D_j^+-D_j^-),\ H_o=4;\qquad h=-OS d_*,\ H=h^2=9/4.\] Color vectors in each family are \(a^\sigma,\ a^\sigma-D_0^\sigma,\ -O^3 a^\sigma=a^\sigma-D_0^\sigma-D_1^\sigma\), paired with \(h_o\) as \(\sigma(1,0,-1)\). All intermediate momenta are taken in the full lattice dual to the root span \(\sum\mathbb ZD_j^\sigma\).

Here the translation cocycle in a field \(V_\beta(s)\) (including source and root fields) uses \(e^{\pi i \mathcal B(\beta,P)}\) in place of \(e^{\pi i S\beta\cdot P}\). The lift \(\widehat O\) has also a momentum phase \(e^{\pi i\chi(P)}\) on input. Use \[\begin{aligned} v O^j a^\sigma &=\sigma(-1,0,1,2,3,4,1,-2)_j,\qquad v'=(O^{-1}v-v-h_o)/4,\\ \mathcal B(b,P)&=S b\cdot P+[(h_o b)(vP)-(vb)(h_o P)]/4,\qquad \chi(P)=(h_oP)(v' P). \end{aligned}\] Some useful checks: radicals are the common sum and difference alternating sum; on the sum/difference sectors the metric matrices for contractions are the circulants adding/subtracting the two displayed rows. The \(a^\sigma\) rows on \(D_j^\sigma,D_j^{-\sigma}\) are \((1,0,0,0,0,0,1,0),(0,0,0,-2,0,0,0,0)\) respectively. We have \(hD_j^\sigma=(-1,0,-1,1,-1,1,0,1)\), \(ha^\sigma=0\). The determinants on the quotient are \[\det(I-yO)=(1-y^8)^2/(1-y^2),\quad \det(I-yM_*)=(1-y^6)^4(1-y)^2/[(1-y^3)^2(1-y^2)^3].\] Indeed before quotienting the common and difference \(M_*\) determinants are \((y-1)^2(y^2-y+1)(y^2+y+1)^2\), \((y+1)^2(y^2-y+1)^3\). In particular no Jordan derivative on a fixed axis is needed. \(\mathcal B+\mathcal B^t\) is the vector metric, and \(\mathcal B\) of any root or color against a root in either order is integral. For instance \(vD_j^\sigma=\sigma(0,1,2,3,0,1,2,-1)_j\); the two rows of \(\mathcal B(a^\sigma,D)\) are \((1,1,1,1,0,0,1,0),(0,0,0,-2,-1,-1,-1,0)\). These checks follow by multiplying the circulants, using \(S=-\sum jO^j/8+4\,{\rm proj}_{\rm fix}\). For \(v'\beta=0\), in particular roots of indices \(7,0,1\), \(\widehat O V_\beta\widehat O^{-1}=V_{O\beta}\).

Use the oscillators, degree and torus coordinates \(p=e^{-L/8},\ \mathfrak q=e^{-4\pi^2/L},\ s=e^{-2\pi i u/L}\) as before. Temporarily separate the sources (\(z_i=e^{u_i/8}\) near 1, vector \(a^{\sigma_i}\)). In the angular trace with \(\mathfrak q^D e^{-2\pi iD}\widehat O\) conjugate each source field by the product over types of \[g_\sigma=\exp(\xi_\sigma F_1^\sigma)\exp(\xi_\sigma F_0^\sigma), \quad F_j^\sigma=f_{D_j^\sigma},\quad \xi_\sigma=e^{-i\phi_\sigma}.\] As before \(f_b=[V_{-b}]_0,\ e_b=-[V_b]_0\) use zero modes. Their residue commutators and Weyl action hold with \(S b\cdot P\) replaced by \(\mathcal B(b,P)\) (the required integral exchange signs follow from the preceding identities). On a separated source this gives the three color fields with coefficients \(1,\xi_\sigma,\xi_\sigma^2\); the two successive residue cocycles use \(\mathcal B(D_0,a)=\mathcal B(D_1,a-D_0)=0\) within that family, and the opposite family commutes here.

Equivalently remove the conjugation, replacing the background at the cut by \(\widehat M_*\) and root zero-mode exponentials distributed around the trace. Here \(\widehat M_*=\widehat O W_+W_-\) uses the same root Weyl matrices as above with parameters \(\xi_\sigma\); on momentum \(P\), each \(W_\sigma\) uses multiplier \[(-\xi_\sigma)^{\delta^\sigma P} e^{-\pi i(\delta^\sigma P)\mathcal B(\delta^\sigma,P)}\] and reflects vectors and oscillators in that root. (The two cross cocycles are zero.) More precisely, suppress both family superscripts and multiply the indicated factors over families; write \(\widetilde B=\exp(-\xi F_0)\exp(\xi F_1)\exp(\xi F_0)\), \(B=\widehat O\widetilde B\widehat O^{-1}\), \(X_-=\exp(\xi f_\delta), X_+=\exp(-e_\delta/\xi)\). Cycling the conjugators gives background \(\widehat O\exp(-\xi f_\delta)\widetilde B\). Commute \(\widetilde B\) left through the \(f_\delta\)’s (it uses \(-D_1,-D_0-D_1\), products nonnegative), then cycle as \(B\) and commute through the sources. Now use \(\widehat O\exp(-\xi f_\delta)=\widehat M_* X_- X_+\). Move \(X_+\) across \(B\) by conjugating \(B\), then through the sources. Thus in order of action the root circles have the following vectors in each family and reduced angular coordinates (\(u=x+2\pi i\,d_i\)): \[ \begin{array}{c|ccccc} r_i&\delta&-D_2&-D_1-D_2&\delta-D_1-D_2&-\delta\\ \hline d_i+d_0&0&1/2&1/2&1/2&d_f\\ h r_i&1&1&1&2&-1 \end{array} \tag{17}\] Here take \(d_0=1/4,\ d_f<1\) sufficiently close to 1; sources lie just after the first layer. Indeed \(B\) uses two commuting roots \(-D_2,-D_1-D_2\) by the single-pole bracket; conjugating it by \(X_+\) adds only the displayed composite (product \(-1\) with the second in the same family), with no further brackets. Products within each simultaneous layer are nonnegative. All exponential coefficients are fixed constants from these group actions (their precise bracket signs are immaterial).

We describe scalar evaluations, using the short kernels \(K_U\), self constants \(d_U\), \(T_U=(1-U^{-1})^{-1}\) with value \(1/2\) on invariants, and oscillator determinants \(Z_U\) from the Gaussian change of channel above. For a list of charges \(\beta_i\) in angular order, \(b=\sum\beta_i\), take \(U=O\) (color evaluation) or \(M_*\) (root-circle evaluation), fixed vector \(h_U=h_o\) or \(h\), square \(H_U\). Require \(h_U b=0\); there can also be a parity restriction as below. Use exactly the short cross, self/image, and kinetic factors previously given with \(h_U,H_U\); in particular the kinetic exponent in \(x\)-order is \[-\sum_{\rm gaps}\Delta u\, E_U(f+\sum_{\rm before} h_U\beta_i),\qquad E_U(y)=y^2/(2H_U).\] Use the prefactor \(Z_U\sqrt{L/(2\pi H_U)}\). The allowed \(f\) run over \(\mathbb Z\) or \(f_0+\mathbb Z,\ f_0=-(\phi_++\phi_-)/(2\pi)\), respectively. Phases include the same radial-order phase \(-\sum_{i<j}\operatorname{sgn}(x_j-x_i)\beta_i T_U\beta_j\) in units of \(\pi\), and \(-2 f h_U P_0/H_U\) where \(P_0\) is a representative of angular cut momenta. Other phases are the position-independent angular ones evaluated there, including translations between fields. Indeed the free Gaussian transformation only uses \(P=-T_U b\) on the perpendicular complement, the axis Gaussian summation with unit steps in \(h_U\), and the geometric oscillator traces. We give the phase and lattice checks to justify this use.

For \(O\) the cut condition \(P=O(P+b)\) gives \(P=OS b+\gamma h_o,\ \gamma=n'-vb/4\), exactly \(n'\in\mathbb Z\) by \(\mathcal B(b,D)\) integral. Then \[-P^2+\mathcal B(b,P)+\chi(P+b)=-b^2/2-8\gamma n' .\] Thus the angular phase including field translations is \(-\sum\Delta_{\beta_i}-\sum_{i<j}\mathcal B(\beta_i,\beta_j)\) modulo 2 (\(vb\) integral). For \(M_*\), write \(n_\sigma=\delta^\sigma(P+b),\ k'=\sum n_\sigma\delta^\sigma\), \(e'=b-k'\). We must have \(P=O(P+e')\), \(h_o e'=0\), so \(n_+-n_-=h_o b\) and \(P=OS e'+\gamma h_o,\ \gamma=l-v e'/4,\ l\in\mathbb Z\). Self-consistency on the two \(\delta\)’s now says \(h b=0,\ 2l=-\mathcal B(b,\delta^+-\delta^-)\), by \(S d_*=d_*-h\). Thus either no terms occur or the \(n_\sigma\)’s step together freely with \(P\) stepping by \(h\). The phase part \[-P^2+\mathcal B(b,P)-\mathcal B(k',P+b)+\chi(P+b-k') \equiv -{e'}^2/2-\mathcal B(k',b)\pmod 2\] as in the \(O\) case. With \((-1)^{\sum n_\sigma}\) included this gives \(-b^2/2+\mathcal B(b,k')\), unchanged under the step since \(\mathcal B(b,d_*)=hb\). Thus only \((\xi_+\xi_-)^{\rm step}\) modulates the summation. Ordinary Poisson transformation gives the specified indices. All residual phases, for the two parameter choices, depend on integral charge multiplicities modulo fixed integers (and on the two orders), using rational bilinear data.

Let \(\mathcal T\) denote the resulting fused trace at \(u=0\), vector \(\alpha=m a_s\) before group action. Recovery at confluence uses \[ \mathcal H(1^m;1^m;\lambda) =[p^0]\, \frac{\rho^m e^{im(\phi_++\phi_-)}\mathcal T} {Z_O\sqrt{L/(2\pi H_o)}\ e^{-\pi i\Delta_\alpha} d_O(\alpha)(L/(2\pi))^{\Delta_\alpha}} . \tag{18}\] To check, for separated sources divide by the leading uncolored radial prescription at \(O\), i.e. with \(f=0\) and no finite-period images. Choose angular and real orders identical. The ratios of \(d_O\) on the three colors are \(1,1/\sqrt\rho,1\), and dimensions agree. The relative pair phase at \(f=0\) is \((\mathcal B(\beta_i,\beta_j)-S\beta_i\cdot\beta_j)-(\mathcal B(a^{\sigma_i},a^{\sigma_j})-S a^{\sigma_i}\cdot a^{\sigma_j})\) modulo 2 by integrality of the differences. Since \(v\beta_i=\sigma_i(-1,-1,-2)_{a_i}\), this gives \(q^{-\sigma_i\sigma_j b_{a_i a_j}}\), \(b_{+0}=b_{0-}=1,\ b_{+-}=3\), completed antisymmetrically. The relative kinetic multiplier is \((z_j/z_i)^{\sigma_i\sigma_j(t_{a_i}t_{a_j}-1)}\) per pair by neutrality. Including the direct kernels with factors \((1-q^{-j'} z_i/z_j)^{\beta_j O^{j'}\beta_i}\) relative to the uncolored ones gives precisely \(P^{\sigma_i\sigma_j}\) in conjugate convention by substitution. The normalization therefore yields (16). All other terms use image factors regular at \(p=0\) and/or \(p^{f^2},\ f\ne0\). Confluence gives (18) exactly as for the earlier torus formula: the leading local product factor of uncolored fields is common upstairs and in the leading denominator by vertex contraction (conjugation preserves it). In the transformed expression root layers stay clear of the sources. In the color ratio one can take the constant coefficient in the separation, with uniform Laurent bounds near \(p=0\) since direct ratios are meromorphic with bounded pole orders and all image series are regular. Thus that coefficient commutes with \([p^0]\).

Convergence check

The estimate needed for summing the doubled trace is the following. Let \(\mathscr W_{\rm root}\) denote the product of the root short-range factors and root self constants/images in the \(M_*\) prescription. Omit the root field powers of \(L/(2\pi)\), which cancel against zero-mode measures, and omit all source contractions.

Leading projection

Cauchy average (18) with \(\Re L=2\ell,\ \ell=6\log m,\ |\Im L|\le8\pi\). The fused source self ratio of \(d_{M_*}\) to \(d_O\) is \(\mathcal R^{2m^2}\), \(\mathcal R=(2k/3)^2/\sqrt{2k}\), since the source is common (same contractions as in the earlier unprobed calculation, doubled). Moreover \(a_s(M_*^2-M_*+I)=0\) as a contraction row. The determinant and Poisson prefactor ratio is \(e^{L/32}(1+O(e^{-\ell/4}))\): in angular product notation the oscillator quotient alone is \(F_8^2F_3^2F_2^2/(F_6^4F_1^2)\), \(F_j=\prod_{n'\ge1}(1-\mathfrak q^{j n'})\), to which we apply the elementary product transformation above; the leading constant \(3/4\) cancels against \(\sqrt{H_o/H}\).

Put \(E(y)=y^2/(2H),\ E_0=E(f_0)\le1/72\). We claim that after division by \[\rho^m\mathcal R^{2m^2} m^{3/8-12E_0}\] the result has the form \(c_{\mathfrak c}+O(e^{-\ell/24})\) with fixed constants on finitely many size congruence classes. We give truncation details. Work first before Cauchy averaging. Use real coordinates \(x\in[-\ell,\ell]\) and a common bend as above, zero except in fixed large seam widths, taking values \(\pm\Im L/2\) at \(x=\pm\ell\), with shape in seam coordinates independent of \(\ell\). Let \(s'=\ell-|x|\) denote distance from the seam.

The source profiles on the five root circles are multiples of \((0,-1,-1,0,1,1)\) for \(a_s M_*^j r_i,\ j=0,\ldots,5\); multipliers are respectively \(1,0,-1,0,-1\). Nonzero profiles thus contribute in modulus at most \(C\exp(-c m e^{-|x|/6})=C\exp(-c\exp(s'/6))\) including images. Indeed off the bend the direct product log with one unit of profile uses exponents \((0,-1,-1,0,1,1)\) on \(\left|1-e^{(-|x|+2\pi i(d_i-j))/6}\right|\). Comparing numerator and denominator gives the strict bound for unit profile at \(d_i<0\) or opposite profile at \(d_i>0\), since \(\cos(\pi(d_i-1)/3)+\cos(\pi(d_i-2)/3)\) has the sign of \(d_i\) here. In bounded seam widths and for the more distant images costs are bounded per particle. All remaining (uncoupled) particles have strictly positive integral \(h\)-charge. In particular \(K\) is at most twice the number of negative particles by neutrality, and all negatives are localized. The source self images cost \(O(1)\) in the log.

We can discard \(K>\ell^{2/3}\) and configurations with any coupled particle at \(s'>\ell^{1/4}\) with arbitrarily high exponential accuracy in \(\ell\). Indeed the bin estimate and negative-particle localization supply \(\exp(-c K^{2-1/4})\) (either a negative lies beyond distance \(K^{1/4}\), or use the bin squares up to there), retaining a fixed fraction of localization. Volumes and multiplicities cost at worst \((C\ell)^K\), as zero-mode self powers cancel length denominators in the measures. The \(f\)-sum costs \(O(1+K)\) with Gaussian tails by the reserved kinetic bound, and the divided-out empty propagation with energy \(E_0\) costs only \(e^{2\ell E_0}\).

For the retained counts the full kinetic rate on macroscopic empty gaps can be used, not just the small reserved fraction. Shorten all cyclic gaps between sorted roots longer than \(s_0=C_1\log\ell\) to \(s_0\), for estimating the flat real root modulus. The resulting period is between \(s_0\) and \(O((1+K)s_0)=o(\ell)\) (shorten the empty circle too). Actual short-root logs and self images differ from their shortened flat values by \(o(1)\) for sufficiently large \(C_1\), before the bend: any modified image path crosses a long gap in both versions so has exponential decay, even with Jordan factors. The bend error is absorbed against the shortened bin penalty, since near pairs remain near after shortening. Apply (19) there, keeping the order and the value of \(f\) across the cut. Thus any interval of length \(R'\) with kinetic momentum different from \(f_0\) costs \[\exp(-(R'-o(\ell))/9)\] up to subexponential factors: on the deleted portions we have the actual propagation with \[\min_{z\in(f_0+\mathbb Z)\setminus\{f_0\}} (E(z)-E_0)\ge1/9.\] The baseline \(E_0\) on the shortened part costs only \(e^{o(\ell)}\). This bound includes summation/integration since \(K\le\ell^{2/3}\), with the reserved shortened kinetic again available at \(|f|\gg K+1\). Outside the localized widths the momentum increases by positive units only. Consequently any root at \(s'\ge\ell/2\) forces such an interval of length \(\ell/2-o(\ell)\) towards one of the two sides. Likewise if the entire middle is empty but its momentum is not \(f_0\). These cases are therefore errors smaller than \(e^{-\ell/24}\).

It remains to put the retained particles in infinite seam coordinates \(\hat x=x\pm\ell\) about 0 (plus on \(x<0\)), complex positions \(\hat u=u\pm L/2\). Shapes of these paths are fixed, with \(|\hat x|=s'\). Middle momentum is exactly \(f_0\), determining \(f\) by seam charges. After removal of \(e^{-E_0 L}\), the kinetic expression now uses \(E(y)-E_0\) on this line, \(y=f_0\) at both infinities, no period dependence. Keep the direct short-root interactions in these coordinates and drop the rest (self images also); the latter paths have real lengths at least \(\ell\), so log errors \(K^2 O(e^{-(1/6-o(1))\ell})\) since minimum positive frequency for \(M_*\) is \(1/6\). For source factors keep just the first harmonic (\(1/6\)) at \(L/2\pm\hat u\) for a root, and at \(L\) for self images. These give fixed exponents since \(m e^{-L/12}=e^{-i\Im L/12}\). All other harmonics/images in these logs are exponentially smaller (error \(O(e^{-\ell/4})\) suffices), using \(|\hat x|\le\ell^{1/4}\) for coupled particles and no Jordan terms on source contractions. The retained exponents themselves obey the same localization. Thus replacement by these fixed prescriptions costs an admissible error (absolute normalized volume bounds \(e^{o(\ell)}\) here follow by the shortening argument). All remaining constants/phases depend only on seam data, \(\Im L\) and possibly the size class, including order signs via the two sides of the seam.

These fixed integrals can now be extended to the whole line and all counts with the same error. To see absolute tail bounds directly, treat dyadic blocks of furthest distance \(R\) to \(2R\) at large \(R\), estimating the direct kernels by stability on an auxiliary period, say \(8R\), with \(f_0\) outside the particles. Extra distant paths cost \(K^2 O(e^{-cR})\), absorbed by bins. The fixed bend is still allowed and any differing imaginary gap phases only use real energies in modulus. Large counts \(K>R^{2/3}\) or coupled particles beyond \(R^{1/4}\) again cost arbitrarily high exponential order. At smaller counts repeat shortening: a furthest positive particle forces an interval of incorrect momentum of length at least \(R-o(R)\) since negatives are all within \(R^{1/4}\) and the two asymptotic momenta are \(f_0\). Hence such blocks cost \(e^{-R/9+o(R)}\), summable starting at \(R=\ell/2\) with the required accuracy. The other extensions with furthest distance below \(\ell/2\) use the auxiliary period on scale \(\ell\) for the count and localization bounds. This proves convergent constant amplitudes with exponential accuracy, uniformly in the imaginary shifts, and establishes the claim by Cauchy averaging.

For \(\lambda=0\), \(\mathcal H=Q_{2m}^2/(2k)^{m(m-1)}\). The positive Fourier norm comparison in the calibration gives \(Q_{2m}\ge (2k/3)^{(2m)^2/2} k^{-m}\) directly by products, hence \(\mathcal H\ge \rho^m\mathcal R^{2m^2}\). The predicted leading power here is \(5/24<6/24\), so none of its leading constants can vanish. For \(\lambda=2\) the result for \(\mathcal H\) is at least as large as for \(\lambda=0\) by positivity of the gas; the power is \(3/8\), a difference \(1/6<6/24\). Again leading constants cannot vanish. Dividing now gives (15).

Plane version and flux reversal

Write \(\mathcal Z_R(o)\) for the weight of sets of disjoint honeycomb polygons within distance \(R\) of \(o\), each strictly enclosing \(o\), with fugacity 2, including the empty set. For face centers, (15) gives the upper bound \(C R^{1/12}\) by fitting two independent such balls at the two marks of a cylinder of even circumference comparable to \(R\). Conversely, winding separators in (15) cost at most a bounded gas multiplier even discarding all avoidance: they must intersect the slice and their single-loop sum is bounded by \(\sum (h(P)+C)X^{|P|}\) modulo row translations, \(O(1)\) by the pressure estimate. Contractible separators lift uniquely around a prescribed representative of the puncture on their disk side. Those violating lift diameter \(N^{1+2\epsilon}\) at circumference \(N=2m\) cost superpolynomially small single-loop mass. Indeed for heights \(>N^{1+\epsilon}\) this is the cylinder height tail (count slice placements using the height). At smaller heights project the same lifted polygons to a cylinder in another lattice direction of period \(\lceil N^{1+3\epsilon/2}\rceil\); no collision is possible by the height restriction, while their transverse height now exceeds this new period by a power. The same tail bound applies there, since recovery of anchored lifts from loops modulo row shifts costs at most polynomially in length and period (the fixed enclosed point lies within a length of the polygon). Multiplying this small mass by twice the full gas partition bounds the loss with any such bad loop marked. The remaining contractibles lift into two independent plane gas spaces of radius \(C N^{1+2\epsilon}\) as an upper bound, by splitting according to enclosed puncture. Taking \(\epsilon\downarrow0\) gives \(\mathcal Z_R(o)=R^{1/12+o(1)}\). This also holds for \(o\) a honeycomb vertex (triangle center): the segment to a neighboring hexagon center avoids polygon crossings except at the vertex; at most one loop through the vertex can occur in the face gas, with bounded total weight by the plane rooted estimate. Discarding that loop thus costs only a bounded factor, besides bounded changes in radii.

There is a constant-factor comparison for a finite convex domain of triangles with a triangle at center \(v\) strictly inside (all neighbors in the domain). Let \(\mathcal Z_D(v)\) restrict the enclosing gas to this domain, and let chordal arcs in \(D\) have both ends free (sum over ordered boundary midports, weights \(X\) per center). Then \[ \sum_{\omega\ {\rm visiting}\ v} X^{|\omega|}\ \asymp\ \mathcal Z_D(v) \tag{20}\] with uniform constants. Here are details. Put \(\tau=3/8\); the complex extension weights \(X e^{\pm i\tau\pi/3}\) sum to 1. Start at an outer boundary port, keep extending from mid-edge to mid-edge unless at the boundary (after nonempty travel) or advancing toward an already visited center. The terminal collision at such a center decomposes into a prefix up to just before that visit and a detour around a simple polygon returning towards the center; reverse the detour as alternative. The arrival leg on the earlier visit is outside the polygon, and the two added turns including that visit are \(\pm4\pi/3\), so these dead contributions cancel. Summing terminal real flux gives exactly 1 from all boundary exits, using \(\cos(\tau\,{\rm turn})\); in the full convex domain this cosine is positive and uniformly bounded below by closing the simple arc along the boundary as before.

Remove now the triangle at \(v\), exposing three inner ports. From the same outer sources the identity still holds, with extra exits at these ports. Subtracting and summing over the outer sources gives \[T=\sum_{\omega\ {\rm visiting}\ v} X^{|\omega|}\cos(\tau\,\operatorname{turn}(\omega)),\] where \(T\) is the total real outer-to-inner flux in the punctured domain. The sum on the right is over ordered boundary chords of the full domain. Its cosine factors lie between \(\cos(3\pi/8)\) and \(1\). Reversing the outer-to-inner paths preserves their real weights, so \(T\) is also the inner-to-outer flux summed over the three inner sources. For an inner source the extension identity works with the entire enclosing gas in the punctured domain included (path and gas disjoint). Indeed a collision detour reached from outside still cancels; reached from inside it encloses \(v\) and its added turns are \(\pm8\pi/3\), giving factor \(-2\) after summing orientations. To see the turns, the positive circuit omits at closure the bend at that center (\(+\pi/3\) from the convex side with unused leg outside, \(-\pi/3\) with unused leg inside), substituting its opposite at first visit instead. In the inside case this dead term now cancels exactly against the prefix colliding with a separate selected polygon of fugacity 2 there; these are all collisions with gas, and other gas loops remain unchanged. Thus total real exit flux from each such source is \(\mathcal Z_D(v)\). Paths reaching the outer side prohibit any enclosing loop, so contribute exactly the needed cross sum. Inner-to-inner exits have strictly negative cosine, since completing with the missing center gives a polygon with total turn \(\pm2\pi\), adding only \(\pm\pi/3\). Let \(U\ge0\) be the absolute value of this inner-to-inner flux, summed over the three sources and including the disjoint gas. The three extension identities therefore give \[T-U=3\mathcal Z_D(v).\] Completing an inner-to-inner arc through the missing center multiplies its weight by \(X\) and gives a polygon through \(v\). Each such polygon has two orientations. Dropping its avoidance with the remaining gas therefore bounds \[U\le 2X^{-1}\mathcal Z_D(v) \sum_{P\ni v}X^{|P|} \le C\mathcal Z_D(v),\] by the finite rooted plane polygon mass. Hence \(3\mathcal Z_D(v)\le T\le(3+C)\mathcal Z_D(v)\). Comparing \(T\) with the positive weighted chord sum proves (20).

Trace and bulk normalization

Use all the doubled data of the enclosing-loop calculation. Introduce one charge \(H_\diamond\) in the common sector of that lattice by \[H_\diamond D_j^\sigma=\mathbf1_{j=6}-\mathbf1_{j=3}.\] For each mark use a field of charge \(Y=H_\diamond\) at the mark, and place one counter-field (for both) of charge \(-2H_\diamond\) near the outer seam, say at \(u=L/2-c_s+2\pi i d_c\) for large period, \(c_s>0\) fixed small and \(0<d_c<1\) generic. Only use these extra fields in the marked case. On probe fields use translation exponent \(\mathcal B(Y,P)-2SY\cdot P\); on other fields use \(\mathcal B\) as before. The probe modification has zero total coefficient on the initial momentum (leaving only inter-field translation phases). The vector \(H_\diamond\) has \(h_o H_\diamond=v H_\diamond=0,\ hH_\diamond=3/4\). Its modified exchange sign with a root is ordinary (by \(S+S^t=1\)); and products with \(D_j^\sigma\) of indices \(7,0,1,2\) vanish. Thus probes are unaffected by the color conjugators and commute with all root operations used here.

Temporarily separate all sites \(z=e^{u/8}\) with \(u=x+2\pi i d\) near the indicated angles \(d=0,1\), in increasing angular order within a range strictly less than 1; take regular fields in increasing real order also, and all probes later than them in real order. Insert \(a^\sigma\) at regular sites, conjugated by the enclosing-loop \(g\)’s. Divide the resulting trace by the following uncolored prescription: take the \(O\) scalar channel on uncolored \(a^\sigma\) and all \(Y\), with its prefactor, self constants including field powers, and direct short factors only, omitting also the direct factors to the counter-field. Use the phase (units of \(\pi\)) \[-\sum_i\Delta_{\beta_i}-\sum_{i<j}[\mathcal B(\beta_i,\beta_j)+2SY_j\cdot\beta_i] -\sum_{i<j}{\rm sgn}(x_j-x_i)\beta_i T_O\beta_j\] in angular order, \(Y_j=0\) at regular sites; use for the kinetic factor \(\prod_{x_i<x_j}\exp((h_o\beta_i)(h_o\beta_j)(u_j-u_i)/(2H_o))\). This is a formal normalization, not requiring \(h_o\)-neutrality. Finally multiply by \[\prod_{{\rm reg}\ i<j}D_{\sigma_i\sigma_j} \prod_{{\rm reg}\ i}(\sqrt\rho\,e^{i\phi_{\sigma_i}})\] and, if marked, also by \(\rho ab\prod_{z\,{\rm reg}} f_+'(a,z) f_+'(b,z)\), with \(f'\) the mark polynomial in conjugate convention.

The period constant coefficient (\(p=e^{-L/8}\), allowing half powers) of this prescription is \(F_Z^{(p_*)}\). Indeed expansion in original colors gives exactly the phase step computation at \(O\) from (16): only neutral colors occur and at \(f=0\) the kinetic term is the pair expression above on them by neutrality. Thus regular-regular ratios, self factors and color coefficients give the same pair and site factors as before. For a color \(b'\) and outer probe \(Y\), the direct relative factor against the uncolored site is \[\prod_j(1-q^{-j}z/z_{\rm probe})^{YO^j(b'-a^\sigma)}.\] Mixed phase from the indicated pair equals \(-2b'Y\) if the color is earlier in angle, and zero otherwise, hence no relative phase. Our row on the two color changes \(-D_0,-D_0-D_1\) thus supplies \(f_c'/f_+'\) at each mark, and all counter-field direct factors tend to 1. Image factors are regular additional series and nonzero \(f\)’s have \(p^{f^2}\).

Removing the conjugators gives the \(M_*\) channel with precisely the same root species (17) and coefficients as before. All root-circle fields there commute through uncolored regular fields (orthogonal products), so can be interspersed with them keeping the indicated order among the circles. The total charge in either channel is unchanged by probes, and the only new translation effects besides the ordinary \(\mathcal B\) formula are \(-2\sum_{i<j}SY_j\cdot\beta_i\). Thus all the previous cut/phase tests work (integral root products on the dual lattice); at \(M_*\) require \(\sum h\beta=0\), with the same possible parity obstruction and step \(h\). Use \(f\in f_0+\mathbb Z,\ f_0=1/4\). All residual phases from the cut, translations and order signs have rational data, depending as regards regular multiplicities only on congruence classes. Formal properness, oscillator contraction and continuation at finite real period are justified exactly by the doubled estimates: external charges have fixed bounded multiplicities during those identities, trace reordering uses probes commuting as above, cut support and quadratic time-average tests change only by bounded charge translates, and root counts themselves have neutral total \(h\)-charge. The bounded axis shifts from probes can be included using a portion of kinetic margin in the bin estimate, at cost exponential in total real length. Separated root and external angles then give absolute summability. For reaching a limiting counter position one can first continue with it strictly inside the interval then move it at large period at its nonsingular angle. Near the seam bend use a constant bend at that position.

Here are contraction data. For regular \(a^\sigma\) on \(M_*\) we have \(a^\sigma(M_*^2-M_*+I)=0\) as contractions. Its row \(a^\sigma M_*^j a^\tau\), or \(a^\sigma M_*^j H_\diamond\), is \((4,2,-2,-4,-2,2)/3\). On the five root columns in order (17), \(a^\sigma M_*^j r_i^\tau\) vanishes for opposite type and for the same type is \((1,0,-1,0,-1)_i\) times \[c_j'=(0,-1,-1,0,1,1)_j .\] On \(O\) the rows against \(a^\sigma,a^{-\sigma},H_\diamond\) respectively are \[(4,2,1,-1,-2,-1,1,2)/3,\quad (4,2,-2,-4,-2,-4,-2,2)/3,\quad (4,2,1,-1,-2,-4,-2,2)/3 .\] Also \(H_\diamond M_*^{-1}\delta^\sigma=1\). For example to check by multiplication, on sum/difference use cyclic coefficient columns with shift \(V\), metric circulant \(C\) the sum/difference of the two rows given before, regular column \((I-V+V^2)^{-1}e_0\). Multiply by \(V(I-e_7 e_7^{\mathsf T}C)\) for \(M_*\); the common coefficient column of \(H_\diamond\) (same in both families) is \((7,-7,-5,-19,-17,17,19,5)^{\mathsf T}/48\).

At confluence (including angular boundary contact), divide each direct central pair factor in the \(M_*\) expression by its uncolored \(O\) counterpart. Indeed \(K_U(Y'+2\pi i d)=U^dK_U(Y')\), \(d=0,\pm1\), and the singular exponent \(\gamma_i U^d\gamma_j\) agrees on both sides for uncolored central fields (all orthogonal to both \(\delta\)’s). Thus there is a finite constant ratio in increasing real order. When rows \(\gamma_i U^j\gamma_j=A(j)\) have finite period \(l\), the modulus of the constant of that individual short factor after its power is removed is \[T(A,d)=l^{-A(d)}\prod_{j=1}^{l-1}|2\sin(\pi j/l)|^{A(j+d)},\] and its phase rational here by the directed log formula. With either two regular sites or regular then mark, multiplication of \(T(M_*,d)/T(O,d)\), \(d=k_j-k_i\), by respectively \(|D_{\sigma_i\sigma_j}(z_i,z_j)|\) or \(|f_+'(z_j,z_i)|\) gives precisely \(b_d^2\). These identities follow by inserting the displayed 6- and 8-rows for \(d=-1,0,1\), with \(|D_+|=|(1+q^{2d})(1+q^{2d}+\sqrt2 q^d)|,\ |D_-|=1,\ |f_+'|=|(q^d-q^2)(q^d-q^3)|\). Self ratios similarly give \(\sqrt\rho\, d_{M_*}(a^\sigma)/d_O(a^\sigma)=1/(2c)\). Mark-only constants (also the counter self ratio) are bounded independently of size. Kinetic phases from the formal normalization at confluence are unit with rational data. Hence the whole extensive constant has modulus \({\cal D}(Z)^2\) up to fixed factors, and after dividing this out the dependence on regular multiplicities there is at most by congruence class. Field period powers cancel (root powers against measures). The determinant/prefactor ratio gives \(e^{L/32}\) times a convergent series at large \(\Re L\) as calculated earlier.

Initially choose first root layer at small \(d_s>0\), others at \(d_s+1/2,d_s+d_f<1\) (notation (17)). They stay clear of confluences. Absolute bin control permits taking confluence in the transformed ratio. On the old color side take just the constant coefficient in separation along generic fixed slopes: direct color ratios are meromorphic with bounded pole orders, other image series regular there and at \(p^{1/2}=0\), including the omitted counter factors as regular series. Thus this limit gives the claimed coefficient rule at confluence as well.

Circle expansion at two scales

Physical interpolation

First in the unmarked case the comparison \(|Q|\ge c_1{\cal D}\) forces nonvanishing of the leading coefficient, locally even at nonrational limiting compositions. Indeed fix any possible residue class of the family/angle counts occurring along an infinite sequence. To approach an admissible limit on it one can take the counts within bounded error of the exact scaled composition, preserving residues (and orientation at a tie if needed). One can use either nearly equal splitting since the identity computes the symmetric \(Q^2\) regardless of the chosen split. If the coefficient there vanished, holomorphy gives error \(O(1/|Z|)\) in evaluating it along this approximation, contradicting the comparison since \(5/24=3/8-1/6<1/2\). Compactness thus gives \[|Q(Z)|^2\asymp {\cal D}(Z)^2 |Z|^{5/24}.\] Also the mass \(C_a\) of loops visiting just a prescribed mark (no constraint at other sites) is bounded above by a constant at any such physical list. Use the one-port mark reduction to \(E_+(Z;t)\) at \(t=a\) (take the seam there). By its formula and the cap reduction for \(Q(Z,q^3a)\) the nonconstant term is a bounded prefactor times \(Q(Z,a)Q(Z\setminus a)/Q(Z)^2\). The just-proved estimate gives the bound since the products in \({\cal D}\) cancel up to fixed factors. In particular \(0\le C_{ab}\le C_a\le C\).

For bounded \(K_0\), equation (23) now gives (22) directly. For large integral \(K_0\), put \(m_0=K_0^{1/4}\). We calibrate the marked expansion using physical values near this smaller scale. Choose one sufficiently large accuracy \(J>4C_2+1\) and keep its finite rate and degree list fixed. At \(m_0\) the remainder satisfies \[K_0^{C_2}m_0^{-J}=o(K_0^{-5/24}) =o((K_0m_0)^{-1/6}).\] The strict power margin also absorbs any fixed logarithmic factor.

The polygon second moment

We now convert the two-port estimate into a bound on all ordered vertex pairs of a polygon. Good displacements are counted directly; polygons concentrated near a lattice side line require a separate width argument. For plane honeycomb polygons \(P\), modulo primitive lattice translations, the resulting estimate is \[ \sum_{R\le{\rm diam}\,P<2R} X^{|P|}|P|^2\le C R^{2/3}(1+\log\log R). \tag{24}\]

Critical irreducible bridges

As a separate consequence of the finite bridge bounds, we obtain the classical irreducible-bridge law. The following result is not needed for the logarithmic window proved in the next section. We use the classical construction [8, 11]. Its critical normalization and independent concatenation in the honeycomb mid-edge convention are also given in [1]; the finite bridge bounds here yield the additional quantitative tails below. Cutting a strict bridge at all its intermediate single-crossing levels decomposes it uniquely into strict bridges with no intermediate single-crossing level; these are the irreducible bridges. The cuts are ordered by height. Each portion lies strictly between its two cut lines: a return past a line would cross it again. Conversely, irreducible pieces concatenate freely, translating the next starting port to the previous terminal port. Their center sets lie in disjoint open slabs, so the concatenation is self-avoiding and its critical weight is the product of the weights.

Aspect cutoff

Let \(x,y\) be normal level coordinates in two distinct lattice-side directions. Write \(K_D(h)\) for the mass of strict \(x\)-bridges of height \(D\) from a fixed source whose absolute \(y\)-deviation from that source is at most \(h\). We first prove \[ K_D(h)\le C D^{-1/4}e^{-cD/h}\qquad(D\ge h\gg1). \tag{25}\] The cylinder tail will supply the exponential factor. To use it, we join two long, narrow bridges into a polygon, retaining enough possible cut heights that the construction can be repeated at constant cost.

Fix \(D\ge h\), and choose an integral gap \(g_0\) comparable to \(10h\) on an \(x\)-wall. A step along this wall changes \(y\) by one. Start two bridges on that wall with gap in \([g_0-2h,g_0+2h]\). For each \(d\in[D,2D]\), the two translated bridge increments have product mass \(K_d(h)^2\). Interchanging the increments reverses the change in the terminal gap. At least one of these two choices leaves that gap in the same interval, so the retained product mass is at least \(K_d(h)^2/2\). The two paths are disjoint because their starting gap exceeds \(2h\).

Extract the first simultaneous qualifying \(x\)-cut in \([D,2D]\) whose prefixes have the required confinement and terminal gap, allowing the terminal wall as a cut. The terminal wall qualifies. From a selected cut, the two discarded suffixes are strict bridges of the same height. Summing over possible terminal walls bounds their total product mass by \(\sum_{j\le D}B_j^2\le C\sqrt D\), including the empty suffix at \(j=0\). Thus, for every admissible starting gap, the family of distinct prefix pairs has mass at least the common quantity \[W:=cD^{-1/2}\sum_{d=D}^{2D}K_d(h)^2.\] The first-cut convention recovers this pair after any continuation in the next \(x\)-slab.

Repeat the construction for \(k\) stages. At each stage the starting gap is admissible, so the same lower bound \(W\) applies. Close the two initial ends and the two terminal ends by exterior caps, each of mass at least \(ch^{-5/4}\) and diameter \(O(h)\). The resulting polygon has \(x\)-extent between \(kD-O(h)\) and \(2kD+O(h)\). Although its lateral center may drift between stages, its spread in \(y\) at any fixed \(x\)-level is \(O(h)\): the stages occupy disjoint \(x\)-slabs and the terminal gap is always comparable to \(h\). It therefore projects without collision to the regular cylinder of circumference \(C'h\), for a sufficiently large fixed \(C'\).

For fixed \(h,D\), the number of inputs producing a cylinder polygon is polynomial in \(k\). Choose a starting port on its lift and the terminal level at which to remove the two caps; there are polynomially many choices in \(k,h,D\). The successive pair cuts are then read by the first-cut rule. The cylinder height tail, with the deterministic length bound used to absorb these recovery choices, consequently gives \[c h^{-5/2}W^k \le \operatorname{poly}(k,h,D)\,h^C e^{-ckD/h}.\] Taking \(k\)th roots and then \(k\to\infty\) yields \(W\le C e^{-cD/h}\), and hence \[\sum_{d=D}^{2D}K_d(h)^2\le C\sqrt D\,e^{-cD/h}.\] To obtain one prescribed height, enclose the paths counted by \(K_D(h)\) in their parallelogram and clip at an \(x\)-level \(d\) in a proportional bin below \(D\). Positive boundary flux gives \(K_D(h)\le C K_d(Ch)\) at each such level. Sum its square over the \(\asymp D\) available levels and use the preceding bin bound. This proves (25) when \(D/h\) is large; the ordinary bound \(K_D(h)\le B_D\le CD^{-1/4}\) covers bounded ratios.

It follows that arches and bridges in a \(y\)-strip of height comparable to \(h\), from one fixed wall source, satisfy \[\mu\{\operatorname{diam}\omega>s\} \le C h^{-1/4}e^{-cs/h}\qquad(s\ge C_0h).\] Indeed clip by two \(x\)-walls at separation a small fixed multiple of \(s\), so every such path is lost. Average over \(\asymp s\) central wall sources and reverse the new exits. There are \(O(h)\) ports on the new walls, and reaching those central sources costs at most \(C K_{\lfloor c's\rfloor}(Ch)\) per reversed source. The preceding estimate gives \(C(h/s)s^{-1/4}e^{-cs/h}\), which is at most the claimed bound. Exhaustion by larger parallelograms justifies the argument in an infinite strip.

Length moments

For diameter at most \(s\), let \(F_{A,p}(s,g)\) be the length-\(p\) mass of arches from a fixed source with gap in \([g,2g)\), and let \(F_{B,p}(s,d)\) be the length-\(p\) mass of bridges from a fixed source at exact height \(d\). With \(b(s)=1+\log\log(e^e+s)\), we prove, for \(p\in\{1,3/2\}\) and \(1\le g\lesssim s\), \[ F_{A,p}(s,g)\le Cb(s)s^{p/3}g^{p-1/4},\qquad \sum_{1\le d\le s}d^{-1/4}F_{B,p}(s,d) \le Cb(s)s^{1/2+4p/3}. \tag{26}\] Fixed changes between diameter and level units only change the constants. Both estimates come from sewing a path into a polygon. The main issue is how many cuts of the resulting polygon can recover an input path.

A simple double cut is a lattice-side grid line meeting a polygon at exactly two crossing ports. For fixed comparability constants, let \(M_g(P)\) count such cuts in the required orientation, with crossing gap comparable to \(g\) and at distance at most \(Cg\) above the polygon’s bottom. For every fixed integer \(k\ge1\), \[ \sum_P\mu(P)M_g(P)^k\le C_k g^{-2+k/2}, \tag{27}\] where polygons are counted modulo primitive translations. To see this, first select \(r\) distinct cuts in increasing height order. Translation fixes one crossing of the first cut, and its partner has \(O(g)\) choices. The lower and upper outer arcs have specified ends at gaps comparable to \(g\), so each costs at most \(Cg^{-5/4}\). Between two consecutive cuts, polygon connectivity forces two strict bridges; if their height difference is \(j\), their product mass is at most \(B_j^2\). Summing each positive gap up to \(Cg\) costs \(O(\sqrt g)\). The total is therefore \[C_r g\,g^{-5/2}(\sqrt g)^{r-1}=C_r g^{-2+r/2}.\] Expanding \(M_g^k\) by its number \(r\le k\) of distinct levels proves (27).

For an arch at gap comparable to \(g\), attach below its wall a cap of mass at least \(cg^{-5/4}\) and diameter \(O(g)\). Its original wall is one of the cuts counted by \(M_g\). Choosing that cut and one crossing recovers the arch and cap up to a fixed number of orientations. Consequently \[c g^{-5/4}F_{A,p}(s,g) \le C\sum_{\operatorname{diam}P\le Cs}\mu(P)|P|^p M_g(P).\] Let \(Q_2(s)=\sum_{\operatorname{diam}P\le Cs}\mu(P)|P|^2\). Summing (24) over dyadic scales gives \(Q_2(s)\le Cb(s)s^{2/3}\). Hölder’s inequality with \(k=2/(2-p)\) bounds the last display by \[C Q_2(s)^{p/2} \left(\sum_P\mu(P)M_g(P)^k\right)^{1/k}.\] After multiplying by \(g^{5/4}\), the power of \(g\) is \(5/4+(-2+k/2)/k=p-1/4\), giving the first estimate.

For a bridge of height \(d\le s\), choose a second bridge of the same height, of diameter \(O(d)\) and mass at least \(cd^{-1/4}\) by transverse tightness. Translate it by one of \(\asymp s\) shifts at distance comparable to a sufficiently large fixed multiple of \(s\). The two bridges are disjoint; caps of gaps comparable to \(s\) close them at both walls. The two caps contribute \(cs^{-5/2}\), and the shift choices contribute \(cs\), for a net factor \(cs^{-3/2}\) besides the second-bridge mass. The output has diameter \(O(s)\). Its two joining walls are counted by \(M_s\), so the recovery multiplicity is \(O(M_s^2)\). Thus \[c s^{-3/2}\sum_{1\le d\le s}d^{-1/4}F_{B,p}(s,d) \le C\sum_{\operatorname{diam}P\le Cs}\mu(P)|P|^pM_s(P)^2.\] Use Hölder with \(k=4/(2-p)\) and power \(2/k\) on the cut moment. The resulting power of \(s\) is \(3/2+p/3+(-2+k/2)2/k=1/2+4p/3\). Since \(b(s)^{p/2}\le b(s)\), this proves the second estimate.

Proof of the logarithmic window