Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice

1 Introduction

A critical self-avoiding walk crossing a strip of height h is expected to have length of order h^{4/3}. A comparison of total mass and first length moment suggests this scale, but cannot locate a substantial mass of paths there: a small collection of very long paths could carry the entire first moment. This paper supplies a quantitative window by combining an exactly normalized cylinder calculation with a positive second-moment and sewing argument.

We realize the honeycomb lattice as the centers of the unit equilateral triangles in the triangular tiling. A port is the midpoint of a triangle side. A path between ports visits a self-avoiding sequence of triangle centers and has critical weight

where |\omega| counts the visited centers. Gluing two paths at a port multiplies their weights. Lengths of vertex-rooted walks counted by edges have a different endpoint convention; all statements here use port paths and center length.

Fix one lattice-side direction, and number its parallel grid lines by integer height, with one interline distance as unit. A strict bridge of height H\ge1 starts at a fixed port on line 0, ends at any port on line H, and has all centers strictly between the two lines. Let \mathcal B_H denote these bridges. The start is fixed separately in each sum; the terminal lateral position is free. Translations and lattice symmetries identify the possible starts. Write

The averaging in height is part of the conclusion. A particular height, lateral endpoint, or exact center length is not selected by this theorem. The window is useful for subsequent renewal conditioning because its length scale is explicit and its loss is only logarithmic. Passing from such a window to one prescribed length requires a further local renewal argument.

Several finite estimates have independent uses. For an arch in a half-plane, let A_g be its critical mass from one fixed boundary port to the boundary port at positive gap g, in lattice-side units. The path has no intermediate boundary contact. For a plane polygon P, length again counts its vertices; unrooted sums below are modulo primitive lattice translations.

Here and throughout, f\asymp g means two inequalities with positive constants independent of the size parameter. Constants may depend on explicitly fixed confinement or angular parameters. The first-moment bound in Theorem 1.2 has fixed transverse confinement; later height-averaged moment bounds allow larger diameter cutoffs with their stated losses.

1.1 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

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 1.2.

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 1.1.

1.2 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, Sections 3.3–3.4 and 4.1]. 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, Section 4]. 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, Theorem 10]. Glazman and Manolescu gave a shorter proof and a logarithmic subsequence bound [5, Proposition 1.1]. Krachun and Panagiotis proved a polynomial upper bound for B_H and quantitative sub-ballisticity for uniform honeycomb walks [9, Theorems 1–2]. 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, Section 3]; Glazman established their criticality for self-avoiding walks on rhombic tilings [4, Theorems 1.1–1.2]. Glazman and Manolescu proved boundary two-point invariance for columnwise rhombic half-plane tilings with angles in [\pi/3,2\pi/3] [5, Theorem 1]. The periodic spectral and seam formulation has precedents in the dilute-model analysis of Zhou and Batchelor [15, Section 2]. 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, Section 2.1]. 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, Sections 2.2–2.3]. 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 1.1 uses finite weighted counts and recoverable sewing directly.

2 A periodic loop calculation at finite circumference

Set

When half-powers of q occur we use this particular angle. Consider an infinite combinatorial cylinder of rhombi (a square grid suffices topologically), N per horizontal row. There are ports (side midpoints) L,B,R,T (left, bottom, right, top) on each cell in that cyclic order, common to neighboring cells. A diagram in the cell is a noncrossing matching of a subset of these ports. In gluing diagrams the two occupations of each common port must agree; matchings then join across it, weights multiplying. Use configurations with finite support. For a ratio s=z_j/t=e^{3ig/4} in column j, put \eta=3g/8 (so e^{i\eta}=z_j^{1/2}/t^{1/2}), D=\sin(5\pi/4+\eta)\sin(5\pi/8-\eta). Weights are 1 for vacant,

for a single arc (LT or BR; LB or TR; opposite sides respectively), and

for LT,BR and LB,TR, respectively (the Yang–Baxter rhombus weights; the identities needed here are checked below). At s=q (the indicated half-power branch) this is the honeycomb SAW weight: split the rhombus into unit equilateral triangles, one with sides L,T, one with B,R, each supporting a single arc between two distinct side midpoints via the center with weight X (or vacant with weight 1). Thus the walk goes between centers, with weight X per visited vertex. At s=q^2 the reflected picture holds. The regular cylinder quotient in either case is by translation along a horizontal row of triangle sides. [5]

Mark the auxiliary port L of column 1 (the seam), in one row. Consider configurations consisting of precisely one loop, with weights given by the indicated cells, homogeneous up and down the cylinder. Let E_+(Z;t) denote their mass when required to contain the mark. Here Z=(z_1,\ldots,z_N). Let J(Z;t) denote instead the mass of single winding loops modulo translations by whole rows (no mark condition, and quotient by vertical/row translations only, NOT cyclic shifts of columns). Initially these sums can be taken at the regular honeycomb point and in a neighborhood; elsewhere we will use rational continuation.

Use the following notation:

Augment the Pfaffian by a last column 1 (with the skew row) in odd size only. Determinantal quotients etc. mean rational functions, so limits or continuation from distinct parameters are understood.

For the color sum choose constants b_0=k, b_+b_-=k/2 and b_++b_-+k=r, and define

Set

All quotients are interpreted as rational functions and then continued to coincident parameters.

The relation between parafermionic cancellation, rhombic geometry and integrable loop weights was developed by Ikhlef and Cardy [7]. Glazman’s treatment gives the corresponding weighted-walk criticality and corrected general loop-weight formulas [4, Theorems 1.1–1.2 and Section 5]. We verify the particular normalizations and all diagram identities used here.

We prove Proposition 2.1 in three steps: construct the actual convergent vacua, determine their polynomial numerators, and glue them with the required mark or twist. The boundary estimate needed for the first step is proved next. At a horizontal slice the lower vacuum V^\sigma has one component (weighted sum) for each pairing of a subset of the N ports, noncrossing on the disk. It accumulates diagrams closed below the slice except for precisely these ends (no closed loops), with an extra factor \sigma for every occupied horizontal seam-crossing port, \sigma=\pm1. We forget homotopy beyond this sign. Noncrossing is necessary also on the half-cylinder (cap off below a finite diagram). In particular V^\sigma_\emptyset=1. Row addition T_\sigma(Z;t) acts linearly, killing any resulting cycle, and including the extra seam factor also in the added row. That is, each column of this finite matrix attaches one row in all possible ways just above the specified input pairing; what is needed from the input is only which ends are joined, since cycles of either topology are discarded. The empty input vector refers to the unit coordinate for the all-vacant state.

Here are convergence/uniqueness details for this construction. A single arc in such a half-cylinder at the honeycomb point lifts to a half-plane SAW between boundary mid-edges (touching that boundary only at its two ends). The total X-mass of such SAWs from a fixed port is finite, even summed over all other boundary ports. To see this, exhaust by convex polygons made of equilateral triangles on that side with the source in the interior of a bottom side. For a midpoint s use the sum f(s) of walks terminating there in the polygon, weighted by X^{\#\text{vertices}}\exp(-5i\,{\rm turn}/8) (’turn’ is total signed turn of travel), starting at the boundary source in the inward direction and not touching the boundary intermediately; include the empty walk with weight 1. At a triangle center v_0, \sum_{s\sim v_0}(s-v_0)f(s)=0. Indeed contributions for a walk approaching an unvisited v_0, and its two extensions to exit midpoints there (turns \pm\pi/3), cancel by 1+2X\cos(7\pi/8)=0. A walk stopping while approaching v_0 which was already visited pairs with the reversal of the detour since that visit (thus using the other exit from v_0); the incoming leg at the earlier visit is on the outside of the simple polygon closed by the detour (it is reached from the boundary without crossings). The added turns starting from this leg are 4\pi/3,-4\pi/3 (the positively oriented polygon contributes 2\pi-\pi/3 without its closure bend, and the entry bend is -\pi/3; likewise for negative orientation); for terminal rays from the center at angles 0 and -2\pi/3 respectively the coefficients cancel by e^{-5i\pi/6}+e^{-2i\pi/3+5i\pi/6}=0, and rotations/reflection give the other cases. Thus summing over centers leaves only boundary terms; divided by the inward source direction (times the common half-edge length), this gives that the sum of X^{\#\text{vertices}}\exp(3i\,{\rm turn}/8) over nonempty exits is 1. Closing a nonempty simple arc counterclockwise along the convex boundary shows the turn from inward to outward normal has absolute value at most \pi (total 2\pi, with two extra \pi/2 corners and boundary contribution between 0 and 2\pi). Taking real parts gives total exit mass at most 1/\cos(3\pi/8). This proves the bound by exhaustion. This is the boundary parafermionic cancellation at the honeycomb critical activity [2]. The local cancellation and the boundary flux equality need no convexity provided the detour polygon cannot enclose the source (as for an outer boundary source with all detour edges interior); convexity here is for uniform positivity.

Dropping mutual avoidance now bounds all components of V^+ (the number of ends at most N). They are limits of T_+^m on the empty input. At s=q, each pairing is accessible with positive weight from that input: working down from the slice, join adjacent paired ends (after previous joins) in a fresh row with corners and horizontal propagation, propagating all other ends vertically, iterating. This uses noncrossing and single arcs only. Since the empty row-output functional only sees the empty input, convergence of the monotone series for every state and this accessibility imply summability of powers of the nonempty block, so its spectral radius is <1 (the same reflected argument at s=q^2). Absolute domination handles the twist and a small complex neighborhood by continuity, including convergence of marked sums via gluing two half-cylinders. Thus V there is characterized by T V=V,\ V_\emptyset=1.

2.1 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

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

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.

2.2 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

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

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)}.

2.3 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

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 (2.2) 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

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

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

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.

3 Change of expansion for the color sums

The purpose of this section is to represent the finite color sums of Section 2 by a convergent Gaussian expansion on a circle. Proposition 3.1 gives the scalar change of channel for a fixed list of charges; Section 3.3 then recovers the raw finite sum exactly by a constant coefficient. This representation is the input to the long-circle estimate in Section 4. We first set up the charge space and justify the formal trace operations, then prove the scalar identity, and finally establish the stability and convergence needed to sum over all root insertions.

Use the conjugate convention q\mapsto q^{-1} in the preceding factors P_{ab}, and write

We use \phi=-7\pi/8,-\pi/8, giving r=1,\kappa^2 respectively. By a raw sum we mean without the multiplier \prod K/R^2; the variables of interest are Z=1^N.

3.1 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

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

all of square 8/3. Further put

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

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,

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, Section 2.1, equations (2.2)–(2.5)]. 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

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.

Color shifts and the change of twist

Zero modes of roots are constant coefficients in s, equivalently contour integrals with ds/(2\pi i s). For a norm-two root b' used below, set e_{b'}=-[V_{b'}]_0,\ f_{b'}=[V_{-b'}]_0.

Write F_j=f_{D_j}, u_*=\exp(-i\phi), and g_b=\exp(u_*F_1)\exp(u_*F_0). Our trace \mathcal T inserts g_b V_{Na}(1) g_b^{-1} at the fused source, or, for temporarily separated sources, inserts g_b V_a(s_i)g_b^{-1} at each of them. The commutators below show that the latter field is the sum of the three color fields with coefficients 1,u_*,u_*^2; the two addition phases are 1 by the rows above.

Our immediate task is to move the color shifts from the source fields to the background twist, leaving pure sources. We first verify the root relations needed for that conjugation, then justify every trace reordering at finite coefficient support. Analytic summability will be proved in Section 3.4.

The modes e_{b'},f_{b'}, together with l_0(b'), obey the 2-by-2 traceless-matrix commutation relations (convention e upper triangular). To check, commuting annihilators to the right in V_\beta(s)V_\gamma(s') gives (1-s'/s)^{\beta\gamma}. Including translations, the ordered product has factor e^{\pi i S\beta\cdot\gamma}(s-s')^{\beta\gamma} times creation and annihilation exponentials separately combined and the one-site momentum powers on the original momentum (with the combined translation). For a root and a root or source charge the phase commutator supplies (-1)^{\beta\gamma}, so moving the root coefficient contour gives ordinary residues. At product -1 the commutator with [V_\beta]_0 simply adds \beta to the other field’s charge with coefficient e^{\pi i S\beta\cdot\gamma}, and at nonnegative product it vanishes. Opposite roots have, in -V_{b'}(s)V_{-b'}(s'), the kernel s s'/(s-s')^2 times the expansion starting with 1+(s-s')\sum_n l_n(b')s'^{-n-1}, giving [e_{b'},V_{-b'}(s')]=\sum_n l_n(b')s'^{-n}. The reversed-root computation is analogous. These coefficient statements follow also directly from the binomial expansions on the two sides of the pole. Moreover [l_n(y),V_\beta(s)]=y\beta\,s^n V_\beta(s). Thus the triples using the shifted field modes at power -n and l_n(b') transform under the adjoint zero-mode action just like at n=0. Both root zero modes are locally nilpotent by degree conservation (shifting repeatedly by the same root eventually raises the momentum degree too far).

For a second expression let \widetilde B=\exp(-u_*F_0)\exp(u_*F_1)\exp(u_*F_0), B=\widehat O\widetilde B\widehat O^{-1}. Thus B is an exponential of u_* F_2+c u_*^2[V_{-D_1-D_2}]_0 with commuting summands (the constant c from the bracket will not be needed). The factor \widetilde B commutes with f_\delta, and B with e_\delta, by nonnegative products. Shifting the root indices through \widehat O gives

Here \widehat M_b=\widehat O W, where

This is the root SL_2 matrix with rows (0,1/u_*),(-u_*,0). It takes oscillators to their transforms by R_\delta (by the adjoint triples), and on constants

Indeed for n=\delta P\ge0, |P\rangle is a highest vector of weight n by degree, mapped by this matrix to (-u_*)^n f_\delta^n/n! acting on it. This is the elementary degree-n string action (on f^j/j! the raising coefficients are n-j+1). The translation cocycles in the constant component of f_\delta^n multiply the Laurent constant of \prod_j s_j^{1-n}\prod_{i<j}(s_i-s_j)^2, namely (-1)^{n(n-1)/2} n! by the two alternating determinants, whose sign cancels the cross cocycles. Negative n uses the analogous lowest vector and u_*^n e_\delta^{-n}/(-n)!.

Conjugate the whole trace by g_b B, preserving the degree insertions. Since B,X_+ commute through the uncolored sources, the result has background \widehat M_b, pure source field(s), and root zero-mode exponentials from X_+,B,X_-. Give them ordered angles \epsilon\eta_i, respectively for

(with the sources between the first and second). Coefficients and factorials remain those of the indicated exponentials.

Finite support at the trace cut

All trace reorderings here are first at fixed cut degree (degree just before the rightmost operator acts, with the initial state identified with the final one) and fixed powers selected from the source fields, i.e. source mode degrees; in the one-source case only its degree-conserving power enters. They are proper despite the indefinite form. Any cut momentum with a contributing diagonal term, including in cycling the conjugators, must lie in a translate of OS\,\operatorname{span}(D_7,D_0,D_1,D_2), by the equation after \widehat O (move left-root shifts inside by O^{-1}, and include the shift by \delta when using W). The form on that span after applying OS is positive: the four-column Gram has diagonal 9/8, off-diagonal 1/8 except between first and last, -7/8. Thus cut degree bounds the cut momentum in its discrete coset and leaves only finitely many oscillator indices. Successive zero-mode exponentials on each input vector are finite, and fields with prescribed mode degrees have finite outputs. This justifies coefficient composition and cyclicity; in cycling a product past the cut one just matches finite closed matrix paths at the specified degrees (cut degree unchanged since the cycled operators are of degree zero). The root shifts preserve all the relevant lattice translates, as \Lambda\subset S\Lambda. Analytic summability is discussed below.

3.2 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

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

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

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

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

Indices here label angular, NOT x-order. Include a kinetic factor

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

Each field has self multiplier

In these formulas the transpose of a complex matrix is still the bilinear metric transpose. Finally multiply by

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

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).

3.3 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:

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

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.

3.4 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.

A positive covariance for the four root layers

The full boson form is indefinite. The positivity we need concerns only the real part of the short-range covariance on the four root layers, excluding the order phases and kinetic factors. Write their charges as

Fix \eta_1<0<\eta_2<\eta_3<\eta_4, and put the roots on layers d_i=\epsilon\eta_i. We will choose one sufficiently small \epsilon>0 and keep it fixed. All constants below may depend on these layers; no estimate uniform as \epsilon\downarrow0 is required.

For a real period L, set t'=4\pi^2|l|/L and y=e^{-t'}. The Fourier matrix of the unsigned covariance at a nonzero frequency is 2\pi^2/L times A_\epsilon(t')/t', where

and P is made symmetric. Here \mathsf T is ordinary transpose on the four-component column array. The two angular oscillator terms give P: their coefficients are real, so their real parts contribute cosines. The fixed-axis Fourier comparison in the scalar-channel proof subtracts 2vv^{\mathsf T}/(t'H_b). At frequency zero we use the continuous limit of A_\epsilon(t')/t'. This is also the correct zero Fourier coefficient: the line kernel has an integrable local logarithmic singularity and exponential tails, and the nonzero sample formula holds for arbitrary real periods, hence identifies its continuous line Fourier transform.

We claim that, after choosing \epsilon sufficiently small, there is c_\epsilon>0 such that

The verification has three frequency regimes. First set \epsilon=0 and retain only the three distinct rows. Put

Direct substitution gives the three leading principal minors of P and the contraction needed for the rank-one subtraction:

Indeed the derivative of -\log y-z_y, multiplied by 9y(y+1)^2e_y^2, is minus (1-y)^2 times the palindromic polynomial of degree eight with first five coefficients 81,390,1145,2042,2476; the difference vanishes at y=1. Thus the three-row subtracted matrix is strictly positive for t'>0. Writing A_0^{(3)} for this three-row principal block, its removable small-frequency limit is

The leading principal minors of this limit are 44/27, 244/243 and 272/243. On restoring the fourth row, the sole additional zero direction at \epsilon=0 is the duplicate difference. Since \partial_dP_{14}|_{d=0}=-2t', separating its two layers lifts this direction positively to first order in \epsilon. After division by t' these expansions are removable at zero. They establish (3.2) on each fixed bounded frequency interval, including zero, for sufficiently small fixed \epsilon.

A separate argument supplies uniformity at high frequency. At \epsilon=0 the matrix approaches the positive semidefinite Gram matrix C=(r_ir_j)_{ij}; on its first three rows the upper triangle is 2,-2,-1;2,1;2. The vector v is orthogonal to \ker C. In the scale s'=\epsilon t', the layer perturbation has derivative (-C_{ij}|\eta_j-\eta_i|). Represent C by Euclidean vectors c_i. For z\in\ker C,

To make this lift uniform in the crossover, fix a large T independently of \epsilon. For t'\ge T, the positive semidefinite matrix A_0(t') has two eigenvalues bounded below by a fixed a>0; its complementary two-dimensional eigenspace E_{t'} tends to \ker C. In the variable s' we have, uniformly for t'\ge T,

Consequently H(t') is bounded below by a fixed b>0 on E_{t'} when T is large. Decompose a vector into p\in E_{t'}^\perp and q\in E_{t'}. The unperturbed quadratic form is at least a|p|^2 and is nonnegative on q. Absorb the possible O(s')|p|^2 term and the mixed s'H(t') term in half of a|p|^2, at cost O(s'^2)|q|^2; for sufficiently small fixed s_0>0, this cost and the Taylor remainder are absorbed in b s'|q|^2 whenever 0<s'\le s_0. This gives a lower bound c(|p|^2+s'|q|^2) uniformly in this regime. In particular, no comparison between e^{-t'} and s' is needed: the full positive semidefinite A_0(t') was retained.

For s'\ge s_0, the entrywise product C_{ij}e^{-s'|\eta_j-\eta_i|} is strictly positive definite, by the positive Fourier transform of the exponential covariance, and tends to 2I at infinity. Its least eigenvalue is uniformly positive on this range. The rank-one subtraction is O(1/t'), and the remaining errors from y decay exponentially, uniformly when \epsilon|\eta_j-\eta_i|<1/2. Since t'\ge s_0/\epsilon, both errors are uniformly small after decreasing \epsilon. Thus the order of choice is T, then s_0, then a sufficiently small \epsilon. These high-frequency bounds overlap the bounded-frequency estimate and prove (3.2) for all t'>0 with one fixed \epsilon.

From the covariance to an unsigned root-count bound

Partition the real circle into successive bins of lengths between 1/2 and 2. Let K_{I,i} count roots of species i in bin I, and put K_I=\sum_iK_{I,i} and K=\sum_IK_I. We claim that the product of root short-range factors, including finite self images and the constants d_U(\gamma_i), satisfies

The constants are uniform for all sufficiently large real periods. The product here excludes the factors L/(2\pi) from root self multipliers and the coordinate measures; those pair to give du/(2\pi) per root in the integrated formula.

To obtain a bounded kernel from the logarithmic covariance, split the diagonal scalar symbol 2/(t'+a_*)I from A_\epsilon(t')/t', with a_* sufficiently large. At infinity the latter symbol is 2I/t'+O(t'^{-2}), and away from infinity use (3.2). The remainder is therefore an integrable symbol with a bounded continuous spatial kernel, and it still dominates c/(1+t'^2)I. The removed scalar kernel has nonnegative spatial values away from the diagonal: use 2/(t'+a_*)=2\int_0^\infty e^{-s(t'+a_*)}\,ds. It consequently causes no growth in pairs of distinct points of the same species. We may drop those factors when proving an upper bound.

The charge signs do not change the lower comparison. If \rho_i=\sum_{\text{species }i}\delta_x and D_\sigma=\operatorname{diag}(1,-1,-1,-1), the covariance is conjugated by D_\sigma, while the scalar diagonal comparison is unchanged. Its quadratic energy is bounded below by

where G_L is the positive periodized exponential kernel corresponding to 1/(1+t'^2). The first inequality uses its uniformly positive value on distances within one bin. Insert and then remove the diagonal terms of the bounded remainder at a cost O(K). The singular scalar diagonal was never inserted. Finite self images and the constants d_U also cost O(K). This proves (3.3), including for the signed charges. In particular coincident same-species logarithmic singularities cause no absolute-value problem. On compact subintervals of L>0, use bins scaled to the period if necessary; the same bound holds with locally adjusted constants.

Equality at small \mathfrak q and continuation in the period

We first compare the two analytic expressions at small \mathfrak q>0, retaining the distinct angular layers and, initially, separated sources. At their radii |s_i|=\mathfrak q^{-d_i}, contract the free oscillators for each fixed list of fields. For K roots and axis index n, the angular zero momenta give a power of \mathfrak q equal to the time-average of the momentum norms divided by two. Its homogeneous quadratic part in the counts and n, subject to hb=0, is at \epsilon=0 the cut norm divided by two on the positive space OS\operatorname{span}(D_7,D_0,D_1,D_2) used above. The norms before and after the collapsed root product agree by twist invariance. Its only kernel in the root-count cone consists of equal numbers of \delta and -\delta, with n=0: a vanishing cut momentum forces the root sum to be n\delta, and neutrality then forces that sum to vanish. Opening the interval between these two layers lifts the remaining direction positively to first order, since \delta^2=2. With our fixed sufficiently small \epsilon, the exponent is therefore

where constants may depend on the fixed external fields.

For each list and momentum, the formal oscillator coefficients can now be combined by the contraction formula. Their absolute Laurent-series norms on the layer contours cost at most \exp(O(K^2+1)). Direct ratios are strictly inside their annuli except for equal-layer contacts of identical roots, whose factors are integer-power polynomials. Image series converge geometrically even with the Jordan factors, and Z_U has a convergent series as well. These bounds also justify extracting the specified root coefficients on the contours. The quadratic \mathfrak q decay dominates these costs when \mathfrak q is small. The endpoint O-formula converges with separated sources. Together with the earlier finite coefficient-support argument, this proves equality of the analytic expressions there by grouping coefficients.

For continuation along the positive L axis, use (3.3). At fixed L and fixed source multiplicity, separated external fields cost only \exp(O(K)) in the root bound, and the momentum sum has Gaussian tails for |f|\gg K+1. Factorials from the root exponentials then give absolute summability. Since root layers remain separated from the sources, source confluence at fixed L is harmless after removing the common uncolored collision factors as in Section 3.3. This continues the equality of the two normalized trace expressions to every positive real period. The raw finite sum is then recovered by [p^0] as in Section 3.3; neither this continuation nor the fixed-multiplicity fusion asserts uniformity in the source multiplicity.

We finally establish the complex-period bounds used by Cauchy averaging. After fixing the layers and their stability margin, choose a common bend

whose Lipschitz constant is sufficiently small relative to c_0. For any prescribed bounded range of \Im L, the bend may be confined to a sufficiently large fixed width at the seam when \Re L is large. For small real separation y', it changes the imaginary displacement by O(\|\alpha'\|_\infty|y'|), whereas the kernel derivative is O(1/|y'|). At large distance the derivative decays exponentially, including its polynomial Jordan factors. Summing the changes over bins therefore costs at most a small multiple of K+\sum_IK_I^2 in the real logarithm, which is absorbed in (3.3). The modulus of the kinetic factor still uses the real gap lengths.

For holomorphy, fix L_0>0 and parameterize all contours by

where \theta is real and piecewise smooth, \theta(x+L_0)=\theta(x)+1, \theta(0)=0, and \theta(\pm L_0/2)=\pm1/2. For an arbitrary fixed L_0, restrict to a sufficiently small complex neighborhood so that the derivative times the imaginary shift is small. For large L_0, a large fixed seam width instead accommodates any prescribed bounded imaginary shift range with the same small slope. For real L near L_0 this monotonically reparameterizes straight contours. For the nearby complex periods under consideration, signs of real-coordinate order are still determined by x. On each fixed sorted chamber the integrands, with their continued branches, are holomorphic almost everywhere in L and are locally dominated by the summable bounds just proved, comparing with the curves at the real-part period. Integration thus gives the claimed continuation, moving external positions with the contours when necessary. There is no need to refold by the order of \Im(u/L) after bending. Source coincidences use exactly the regular fixed-multiplicity fusion limits of Section 3.3.

4 Long-circle expansion

5 Calibration and positive masses

The long-circle expansion permits a leading term whose coefficient could vanish. Positive polynomial norms will first rule this out for Q and R. We then extract the Laplace estimate needed both for winding loops here and for open-strip crossings in Section 7. The physical winding amplitude will be J_n\sim\sqrt3/(6n); the rooted polygon mass will remain uniformly bounded.

5.1 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

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

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

5.2 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

The following estimate retains its first correction because the leading terms in the winding calculation will cancel after taking real parts.

5.3 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 2.1.

We next evaluate J_n at the regular honeycomb point, in original conventions t=q^{-1}. Confluence in the wave formula yields

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,

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

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

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.

6 Winding pressure and long-direction tails

The calibrated winding mass J_n\asymp n^{-1} controls the number of winding loops per row. We now control how far those loops, and contractible loops, extend along the cylinder. All polygons in this section are counted modulo whole-row translations, with the column labels fixed. Write h(P) for vertical extent in row units.

We obtain the first estimate from the pressure of a gas of disjoint winding polygons, then join a contractible polygon to a winding one to obtain the second. Give winding loops fugacity a=d+d^{-1} near a=0,d=i, continuing to kill contractible loops. Let L_n(a) be the logarithm of the transfer eigenvalue continuing the empty-row-sector eigenvalue 1 at a=0, evaluated at the physical row parameter. The analytic part of the proof continues this eigenvalue analytically along a real interval about zero independent of n and proves |L_n(a)|\le C/n there. The coefficients of -L_n(-a) are nonnegative; this converts the real-interval bound into

with fixed A. The factorial in the connected-cluster expansion then turns these coefficient bounds into the exponential height moment.

6.1 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

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, Section 2]. 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

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

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

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, Theorem 1 and p. 17]; 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.

6.2 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

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

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,

(by the same Fourier formula, multiplier for h_b is (2\cosh(\pi\nu/4)-1)^{-1}). Define

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

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.

6.3 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),

We will estimate the change of logarithmic root products by integrating against this discrepancy. The needed cancellation is stronger than its uniform boundedness:

Dominant spectral terms. Use spectral parameter qt=e^z, and call the three summands of the candidate T_1,T_2,T_3 respectively. Their consecutive ratios are \exp(2\pi i F(z))/d and \exp(2\pi i F(z+i\pi/4))/d by the phase formulas (here arguments are in log-coordinates). On the line z=b+i\pi/8, b real, terms 2 and 3 relative to 1 are bounded by

for fixed c'>0. On z=b-i\pi/8 terms 1 and 3 relative to 2 satisfy the same bound. To verify this, take log moduli and replace root sums by n h_b integrals (here h_b still denotes the continuum kernel above, independent of the horizontal shift b). Errors are O(1) by discrepancy, kernels differentiated along the horizontal having uniform exponential decay about b. At argument w=b\pm i\pi/8, the continuum phase per column is H(w)=A(w)+\int K(w-y)h_b(y)dy; its imaginary part has the desired signed lower bound. Indeed it vanishes at spatial infinities, and the horizontal derivative has imaginary part that of h_b(b\pm i\pi/8), equal to \mp(2/(\sqrt3\pi))\sinh(8b/3)/\cosh(4b). For T_3/T_1 on the upper line one adds the phase at w+i\pi/4. The summed continuum phase still has imaginary part tending to zero at the ends, but now derivative h_b(w+i\pi/4). Indeed continue the differentiated identity by lifting the convolution contour upward to keep the first pole at y=(w+i\pi/4)-i\pi/4=w below it (lift by less than \pi/4; isolated meromorphic poles are harmless). Returning to the real contour adds the residue correction -h_b(w) to this continued formula since the residue in y of K'(w+i\pi/4-y) is -1/(2\pi i). The resulting imaginary derivative of the sum is -(4/(\sqrt3\pi))\sinh(8b/3)/\cosh(4b), again sufficient.

The dominant summand (1 above, 2 below) is itself bounded uniformly. Indeed the same integral substitution with O(1) error applies to its log modulus; the root factors per root are smooth log ratios of e^y-e^{b+i\alpha}, with angles a fixed distance from zero modulo 2\pi. The resulting thermodynamic term per site must vanish identically: at every fixed b use d=i, where the eigenvalue sum equals 1 and the relative bounds just proved tend exponentially to zero.

The pressure change. Compare that dominant term to itself at d=i as the roots deform, writing g(y) for the complex log of its root factor tending to zero at +\infty. It is a sum of one or two log differences as above, with angle differences (angles taken in (0,2\pi)) as follows:

• •

  Upper: numerator from z-i\pi/2, denominator from z, giving difference 3\pi/2.

• •

  Lower: numerators from z-i\pi/4,z+i\pi/2, respective denominators from z+i\pi/4,z, differences 3\pi/2,-3\pi/2.

The dressed derivative p=(1-k_b*)^{-1}g' then satisfies the quadrature hypothesis uniformly. Indeed in a Fourier convention with e^{i\nu(y-b)}, a single difference’s transformed derivative is 2\pi i(e^{-\nu\alpha}-e^{-\nu\beta})/(1-e^{-2\pi\nu}) by shifting up a period. Resolving the convolution replaces the denominator’s \sinh(\pi\nu) by \sinh(3\pi\nu/4)(2\cosh(\pi\nu/4)-1). The double zeros at \pm4i/3 then yield only simple poles, by the angle differences. Exponential horizontal decay and contour shift beyond these first poles prove the claim, including derivatives. Thus against N-nH-C_\phi the integral of g' equals that of p against r by even convolution. The change of log root products is minus the integral of g' against the change of N. Its constant contribution g(-\infty)(C_\phi-C_{\pi/2}) exactly cancels the explicit phase change in T_j (g(-\infty) is 3\pi i/2 above, zero below). The remaining log change costs C e^{4|b|/3}/n.

Altogether, using the bounded dominant term directly if this last bound is large, we have |\Lambda(t)-1|\le C |\cosh(4z/3)|/n on the two lines. The same holds between by maximum modulus on division: the rational function in e^z has bounded endpoints and no poles there. Consequently L_n(a) continues analytically along a fixed short real interval about zero, with modulus O(1/n) there (the simple real root construction extends locally analytically even when an evaluation aligns with a removable pole).

6.4 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, Propositions 2.1, 2.6 and 2.8] 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, Section 2.3]. 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

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 (6.2).

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

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

The harmless factor e^{\varepsilon C/n} restores h(P) in place of (h(P)-C)_+. This proves the winding estimate in (6.1). 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 (6.1), 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 6.1.

7 Strip crossing

We calculate the total critical mass of bridges across a honeycomb strip. Number the parallel triangle-side grid lines in any fixed lattice direction at unit spacing. A strict bridge of height H\ge1 joins a fixed midport on level 0 to a midport on level H, with every visited triangle center strictly between the two lines. Its terminal port is otherwise unrestricted. Write B_H for the sum of X^{|\omega|} over these bridges, where |\omega| counts visited centers, and put B_0=1.

The proof expresses lower half-strip states by polynomials and extracts a determinant for the crossing mass. Positive column comparison relates the homogeneous square-rhombus point to physical honeycomb width. We then evaluate the determinant at that square point by a positive Gram matrix and a three-column Hermite remainder. Positivity and strip exhaustion identify the finite determinant identities with the infinite-strip mass.

Keep the cells, ports, weights, swaps B(s), triangle/cap maps P,C and loop-killing diagram calculus of the periodic finite calculation above, but do not identify the two side boundaries. Use n columns extending along all integer rows. In this section recenter the rapidities: column j has row-cell argument s=p z_j,\ p=q^{3/2}=e^{3\pi i/8}; spatial swaps still use the right z divided by the left. Put

Thus d=\sqrt2 a,\ s_0=\sqrt2 b,\ ab=\sqrt2/4. Honeycomb (g=\pi/3 in the cell parametrization) is z=q^{-1/2},x=b, and square rhombi give z=x=1. Vertical reflection changes z to z^{-1}, swapping u,U and w,W. Useful weight identities (with coherent half-powers) are

Indeed with \eta as in the cell definition, the sine denominator is -(x+b)/2, and g_z=\sin(5\pi/8+\eta)/b,\ h_z=\sin\eta/b, linear combinations of z^{\pm1/2}; the identities follow by the addition formulas.

At a slice (e.g. just over row 0), let E be the vector of lower half-strip arc sums with no side contacts and ends on the slice, and O the same with exactly one end on the left side (summed over rows up to 0). No loops. States are noncrossing pairings of subsets of the ordered ports, with in the odd case a single occupied unpaired port attached to the left side, called the defect (no pair spans it). Row addition with sides vacant is denoted T, on either block; adding a row with a left source only, from even to odd, is A_L. Thus

These sums converge, and the systems with this normalization are nonsingular, at either homogeneous honeycomb angle (s=q or q^2) and locally around each. Indeed by the boundary flux identity proved above on convex polygons of equilateral triangles, each boundary source of a finite parallelogram of these cells has bounded total positive X-weight of simple exit walks (first exit, at mid-edges), uniformly in the dimensions: boundary coefficients are \cos(3\,{\rm turn}/8) with |{\rm turn}|\le\pi. Thus on exhaustion individual arc sums in the lower half-strip are bounded (also from a fixed top port to a variable left port), and one may drop mutual avoidance. Every allowed state is accessible from empty even by rows, using once a source row for odd: working downwards, join a pair consecutive among occupied sites in a fresh row using corners and horizontals, continuing others vertically, repeat and finally route the single defect to the left. Single arc weights are strictly positive. Since the empty output on the source-free even block sees only empty input, convergence of the positive series for each nonempty even and each odd state and this accessibility imply summability of powers on the nonempty even block and on the odd block. Both spectral radii are <1. This yields also local absolute convergence and rational continuation (in chosen square roots). Braid intertwining with T,A_L and inversion, using row addition from left to right, imply the rational covariance of the actual E,O: exchanging adjacent parameters acts by the swap on either vector (empty even value preserved).

Write a_n for the mass of arcs from a fixed left mid-edge back to the left strictly above, before any further side contact, in the infinite strip, and

Below gives the same mass by reversal and translation. At homogeneous honeycomb {\cal B}_n is exactly the positive crossing mass from that port to the right boundary. Indeed the same convex flux identity on truncated rows gives coefficient \cos(3\pi/8) on returns to either part of the left side and 1 on crossings (turns \pm\pi,0 by closing along the convex boundary). Remote top exits go to zero: they are the single-top-defect outputs after empty lower rows, one A_L, then a growing number of upper source-free rows in the odd block, which is transient. Reflection bounds bottom exits. Pass to the infinite strip on left/right by positivity; \tau=2\cos(3\pi/8). We take a_0=0,{\cal B}_0=1. These strips in the physical tiling are precisely bands between triangle-side grid lines along one lattice direction (n equal spacings); the other such directions follow by symmetry.

We will need the edge reflection rule on the lower vectors. At the last column reflection z\mapsto1/z simply multiplies each component by (U/u)^{o} with o=0,1 its port occupation: successive contacts from the left are linked there in consecutive pairs with weights uU v^{\ell-1} (\ell\ge1 their row gaps), except possibly a last contact linked to the top with factor u. At the first column it multiplies an even component by (u/U)^o, and sends O to the following vector componentwise:

Here D_* adds a first-site defect if that port was vacant, or else deletes the first-site end, turning its partner into the defect.

For details, fix the rest of a lower-half diagram and list the first column’s occupied R’s at t_1<\cdots<t_l\le0, followed in this list by the final T if occupied. In the even calculation pair consecutively using the single vertical channel, with factor U for a final R-to-top link. In the odd calculation the left source attaches to an odd-index member j of this list, with consecutive pairings before and after. If it attaches to R_j and is bounded above by the next R-time t_+, sum over source positions relative to the unmodified weights of all other pairs. The factor is

where t_- is the previous time or -\infty. Indeed from below/above one jumps along a free vertical segment, with a double cell at a coincident neighboring time. Both correction coefficients are -\gamma v, so this respects the multiplicative reflection. For R_j last in the list, omit the final neighbor term and subtract U^2 v^{-t_j}/(1-v) (cutoff at row 0); the reflection difference is then the claimed multiple of the even R_j-to-top weight U v^{-t_j}. Finally for attachment to the top itself the analogous factor is u/(1-v)+(w-u^2/(1-v))v^{-t_l}/u (omit the last term when there is no previous contact). Its reflected value minus u/U times the old gives the same stated coefficient times the vacant-top value. In the difference terms substituting the end only reroutes the defect, so loop exclusion upon gluing the rest agrees with D_*. These local geometric sums prove the identities at convergent parameters and thus rationally. To justify a common open set one may start at homogeneous honeycomb. The same local formulas for a single reflected edge, summing over finite fixed rest diagrams, bound its positive sums by the original finite E,O; accessibility as above gives transience also there, hence a common complex neighborhood by continuation. The one-column rules are interpreted with no intervening strip and work identically.

Use a different Pfaffian here than in the periodic problem:

The augmentation is again a final column 1 in odd size. This is a symmetric polynomial of degrees at most n-1, by cancellation at equal rows. At x_i=a it drops that site multiplying by \prod_{j\ne i}(x_j+b); at x_i\to\infty its leading coefficient drops the site. Indeed the skew column against other sites is then constant, supplying or eliminating the augment, using H(y,a)=(y-a)(y+b). There are recursions (remaining sites denoted by S in x coordinates):

The first is skew pole elimination (if j=[q^3 l],k'=[l], then R=(y-[q^6 l])(y-[q^{-3}l])). For the second induct in the size, interpolating in any spectator. At internal pole pairs H(j,k')=0 among spectators the recursions agree by R([q m])R([m/q])=(h+j)(h+k')R(h) using the factors just given. At spectator value -h the result is zero by the first rule (simultaneous poles paired with the two sites); at a it agrees by ([q m]+b)([m/q]+b)=(h+a)(h+b); at infinity the leading coefficients agree. These 2(n-3)+2 finite tests plus the leading term suffice for n\ge3, and size two is immediate.

7.1 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.

7.2 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

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

Then

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

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

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

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

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.

7.3 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}.

From the crossing kernel to interpolation remainders. For even n, adding the all-ones matrix to the skew kernel defining Q_o does not change its determinant: the adjugate of an even-size skew matrix is skew, and its quadratic form on the all-ones vector is zero. The new numerator is

Consequently the crossing determinant gives the rational identity

Here K(x,y)=1/H(x,y), and K_S(x,y) is K(x,y) minus its polynomial interpolant in y of degree less than |S| at the Hermite nodes

The repeated node in S_3 prescribes both value and first derivative. To verify the second equality, use the polynomial congruences

The second follows from N\equiv(a+x+\sqrt2y)\Delta and (y-b)(y+a)\equiv(x+a)(a-x-\sqrt2y) modulo H. Multiplying either numerator by its node polynomial and dividing by the indicated row factor produces K plus a polynomial of the required interpolation degree. Its Hermite data vanish, so it is exactly K_S. The row and column factors cancel to give the displayed quotient. All these identities are rational identities before confluence; we will prove that the denominator is nonzero at the square point.

Confluence as a positive Gram matrix. Use the positive measure and its orthonormal polynomials from Section 5:

The leading coefficient of each p_j is positive. The calibrated Jacobi coefficients and Lemma 5.2 give

for fixed 0<\theta<c\pi, with C(\theta)>0; the Jacobi diagonal is zero. The convergent product S_j in the earlier Laplace estimate has been absorbed into this first-order expansion.

The useful Gram representation is

Indeed,

and the formula follows by subtracting the Fourier transforms at s-t and s+t. Convergence and analytic continuation cover the arguments used below. Put

These polynomials have odd degrees 1,3,\ldots,2n-1 with nonzero leading coefficients. Thus their span is the full odd polynomial space of degree at most 2n-1, and

is positive definite. Dividing the determinant of K(x_i,x_j) by its two Vandermonde factors and sending all x_i to 1 gives \det G. The same confluence applied to K_S is a finite-rank perturbation of G.

Here is that perturbation explicitly. Write d_S=|S|, and let \lambda_\ell, 1\le \ell\le d_S, be the Hermite functionals in the y variable: evaluation at b for S_1, and evaluation at b, first derivative at b, and evaluation at -a for S_3. Use the polynomial interpolation basis

The polynomial e_m has degree m-1 and nonzero leading coefficient, so the matrix E_{\ell m}=\lambda_\ell(e_m) is invertible. Define also

The jet matrix of the subtracted interpolant is CE^{-\mathsf T}B^{\mathsf T}. The finite-rank determinant identity therefore gives

This displays the promised reduction from size n to size d_S. For a function f with finite polynomial moments, define its moment projection by

This agrees with orthogonal projection whenever f belongs to L^2(w_b), but needs only the displayed moments. The small residual matrix has entries

The moment integrals are finite at all the Hermite data used here. Only polynomial evaluation is used at im; no square-integrable test vector at that argument is required.

Set

For y=\cos\theta, the projection in the last display becomes

Multiplying the value rows by \sin\theta and combining the derivative row with the value row at \alpha converts the three Hermite rows into value at \alpha, \theta derivative at \alpha, and value at \beta. These operations and \det E involve only fixed nonzero constants. Thus it remains to estimate

Imaginary evaluations and the cancellation scale. Put

We will normalize each residual by i^L A_L(m)f_L(\theta). All three normalized columns have the same leading term. Their first independent differences have orders L^{-1} and L^{-2}, producing the determinant factor L^{-3}. For that reason we need two finite differences of A_j(m), rather than just its leading power.

The Jacobi recurrence gives

Over two steps the positive pair (A_j,A_{j+1}) is multiplied by

Its sum therefore grows as j^{h_m} with a positive limiting coefficient: after removing this power the relative errors are summable. The difference divided by the sum contracts by 1-4cm/j+O(j^{-2}), with O(j^{-2}) forcing, and tends to zero. This also identifies the same leading coefficient in the two parities. In particular,

For \Delta_2 A_j=A_{j+2}-A_j, write the exact recurrence as

It follows that j\Delta_2A_j/A_j\to2h_m. Differencing this exact expression once more, using j^2\Delta_2 e_j\to2 and j^2\Delta_2t_j\to-4cm, gives

These limits also give uniform first- and second-difference bounds at all sufficiently large indices.

Define

For \theta=\alpha, where \rho<1, the full odd polynomial series reproduces i\sin(m\theta), so this residual is the forward tail j=L,L+2,\ldots. To justify reproduction, compare Taylor coefficients: each odd monomial has its exact finite polynomial expansion. The estimates above give local convergence for \rho<1, and the nonnegative Taylor coefficients of f_j supplied by the positive Jacobi recurrence justify absolute Taylor interchange. For \theta=\beta, where \rho>1, use the negative finite prefix backwards; the free term is exponentially negligible after normalization. Thus U_m is the signed sum of

with coefficient (-1)^{d/2} for even d\ge0 at \alpha, and -(-1)^{d/2} for even d<0 at \beta, apart from that negligible term. The forward and negative-backward geometric sums both give

The normalization of the derivative row is legitimate: subtract f_L'(\alpha)/f_L(\alpha) times the first row before dividing it by f_L(\alpha). The normalized determinant therefore has rows U_m(\alpha),U_m'(\alpha),U_m(\beta). Set

Since L\ell_m\to h_m and the h_m are distinct, this is defined for all sufficiently large L. Keep the first column, replace the second by L(U_2-U_1), and replace the third by

The first two terms in the discrete Taylor expansion

are annihilated exactly in the third column. The remaining coefficient has a nonzero limit because h(h-1) is quadratic and h_1,h_2,h_3 are distinct. The second column has a nonzero linear coefficient. Hence, up to fixed nonzero column factors, the limits of the three columns are the same Hermite rows of

The operator D inserts the exponent d in either geometric sum.

We include the bounds needed to pass to these limits. After subtraction of the constant and linear terms, the discrete Taylor remainder multiplied by L^2 is bounded by C(1+|d|)^2 when L/2\le L+d\le2L, by the second-difference bound. For fixed d, f_{L+d}/f_L\to\rho^d. Throughout that central range the same ratios are bounded by geometric factors times a polynomial, and the remaining indices contribute exponentially small tails, even after multiplication by the displayed powers of L. The derivative row has the same control. Indeed the Jacobi recurrence and the first-order Laplace expansion give

Subtracting the corresponding expression at L proves convergence of the derivative of f_{L+d}/f_L to the derivative of \rho^d for fixed d, and bounds it by the same geometric factor times C(1+|d|) in the central range. Polynomial factors do not affect the exponential tail bounds. This proves convergence of all three Hermite rows.

Nonvanishing and the physical exponent. The three limiting functions span S times polynomials of degree at most two in

Their value and first derivative at \alpha, together with their value at the distinct point \beta, form a nonsingular Hermite matrix; the change from \theta to \mu has nonzero derivative. Undoing the column operations shows that the normalized three-column determinant has magnitude comparable to L^{-3}. The normalized scalar denominator tends to S(\rho(\alpha))\ne0.

In particular the confluent denominator \det K_{S_1} does not vanish at large even sizes. The preceding row and column identities then also give Q_o(1,\ldots,1)\ne0, so the rational crossing formula may indeed be specialized there. Undoing the row and column normalizations and canceling the scalar denominator yields

Here positivity comes from the convergent strip comparison, \rho(\alpha)\rho(\beta)=1, and the remaining power is -3+h_2+h_3=-3+1+7/4=-1/4. Finally the positive column inequalities and monotonicity in honeycomb width extend the two-sided bound from even square sizes to all integer honeycomb widths: {\cal B}_n\asymp n^{-1/4}. This proves Proposition 7.1.

8 A divided slice and enclosing loops

This section has two outputs: a constant-factor cylinder estimate for an enclosing-loop gas, and a positive comparison between its plane version and the mass of all boundary chords visiting a specified interior vertex. The latter comparison, equation (8.6), turns the analytic partition function into a local length observable.

On the periodic cylinder split a slice of m+n ports at two prescribed interfaces between columns into a first interval \mathcal X of m and its complement \mathcal Y of n (the rapidities are denoted by the same intervals). At the honeycomb point we can mark two hexagon centers at the split, exactly on the slice. Let \mathcal F(\mathcal X;\mathcal Y;\lambda) count diagrams consisting of disjoint loops, giving \lambda to each loop separating the two marks, zero to every other loop. Separation is on the sphere obtained by capping both ends, so can just be tested by odd parity of occupied ports of that loop in \mathcal X. We include the empty diagram with weight 1. We will use

Here is a finite formula and proof. Give each site type \sigma_i=1 in \mathcal X, -1 in \mathcal Y. Define pair denominators and scalar functions (subscript given by the product \varepsilon=\sigma_i\sigma_j) by

Use pair factors P^\varepsilon_{ab}(z_j/z_i) of colors a,b\in\{+,0,-\} by the same rule as the periodic color sum but with S_\varepsilon: P_{++}=P_{--}=1,\ P_{00}=S_\varepsilon,\ P_{+0}=P_{0-}=S_\varepsilon(u/q),\ P_{+-}=S_\varepsilon(u/q)S_\varepsilon(u/q^2), completed by transposed inversion. We can also use the conjugate convention q\mapsto q^{-1} here. Write t_a=1,0,-1 on these colors, \rho=2-\sqrt2. Take w_0^\sigma=1,\ w_\pm^\sigma=\sqrt\rho\,e^{\pm i\phi_\sigma}, where 2\cos(\phi_++\phi_-)=\lambda. Then the raw sum is

Indeed one glues two periodic V^+’s, lower and reflected upper, at that slice (no eligible loop can be entirely in one of the halves). The parity test uses matchings only. Absolute convergence and rational continuation, hence polynomiality with Q^2 and degree \le2(m+n-1) per variable, follow from the periodic vacuum bounds. There is separate symmetry and both pair recursions of the periodic calculation hold within a type (reduce \mathcal F with no multiplier). This uses the same swap cancellation and sliding of P or C across the gluing, with contraction into the reflected vacuum by the P P^t,C C^t exchange relation. All small added maps are confined to the one interval, so loops closed off and killed on one side in doing this had even parity, and through parities are unchanged. Extreme-site removal again holds. One more condition when m=n is to set all x_i=q^3 y_i simultaneously, pairing \mathcal X,\mathcal Y in reverse orders. Lower cap reductions peel from the \mathcal X-to-\mathcal Y interface, reflected upper cap reductions from the other interface. They give at each matched pair either two vacant caps or two cups closing a loop of odd parity, independently (take compatible half-power branches). Thus \mathcal F=(1+\lambda)^m. These conditions determine Q^2\mathcal F inductively starting with size zero: within the larger group test any variable at the 4(\max(m,n)-1) gap-two/three nodes, 0 and infinity (leading coefficient). Only at m=n>0 is there a possible difference, divisible by \prod z_i\prod_{i<j,\ \sigma_i=\sigma_j}K(z_i,z_j) and hence a scalar multiple by degree, fixed by the cap condition.

To check (8.2), use symmetry and residues in u=z_j/z_i near same-type positive gaps. At q^2, labels +0,0-,+- fuse to +, -,0; at q^3, +- disappears. The condition on total charge is preserved with bijective reductions for each fixed choice. Spectator factors relative to the reduced ones are S_\varepsilon(z/(qz_i)) and S_\varepsilon(z/(qz_i))S_\varepsilon(z/(q^2z_i)) respectively, exactly those of Q^2/\prod D by the periodic table. The relative pair pole coefficients (coordinate u/u_0-1) agree also: 1/(q-q^{-1}) for fusion and \rho S_+(q^2)/(q-q^{-1}) for gap three. At q the 00,+- spectators agree for every color and type and residues cancel using \rho S_+(1)=1; at 1 use same-type transposition. No cross-type poles are needed. On multiplying the sum by \prod D, at z_i=0 each term has order t_{a_i}^2 by neutrality; at infinity the degree bound is similarly improved by t_{a_i}^2. For a_i=0 the removal multipliers are precisely \prod_{j\ne i}z_j^2 and 1 respectively (shifts contribute total power zero of q, by the same neutrality). This proves the polynomial bounds and removal rules. At paired specialization the cross zeros force t_{a(x_i)}\le t_{a(y_i)}, thus equality on every pair. Distinct pairs then decouple by S_+(v)^2 S_-(q^3v)S_-(q^{-3}v)=1; each pair contributes \rho(1+\lambda). Meanwhile Q^2/\prod D=\rho^m by the cap recursion (spectator factors cancel between pairs). This proves the formula. Conjugating q gives identical checks.

8.1 Doubled change of expansion

The constraint \sum_i\sigma_i t_{a_i}=0 in (8.2) 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

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

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

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

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

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

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):

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

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

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

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

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 (8.2). All other terms use image factors regular at p=0 and/or p^{f^2},\ f\ne0. Confluence gives (8.4) 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].

8.2 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.

Finite coefficient support and analytic continuation. We now apply the estimate to justify the formal conjugations and the summation of the trace. These are separate from the finite covariance calculation above.

For the formal trace step first fix cut degree and source mode degrees. Before the Weyl change, on cycling g^{-1} past the cut, all root shifts in the closure with O are negative counts from indices 7,0,1 of the two families. The cut then has proper support: on these six columns the Gram of OS projected off h_o is strictly copositive (diagonal 11/16; same-family off-diagonals -3/16 at distance one and 3/16 at distance two, opposite-family off-diagonals bounded below by -1/16). Thus degree bounds the counts, fixed-axis momentum and oscillator degree there. Cycling B after commuting as specified just rotates finite matrix paths of this kind. After that cycle the Bruhat and residue-commutation identities need only be used per fixed input vector, locally finite. The final expanded prescription is itself proper: its time-average angular momentum norm controls a strictly positive quadratic in the counts up to O_m(1), and also the square along the freely summable axis (around a count-dependent center). Indeed with homogeneous root data the solution of the jump profile having axis mean zero is the t\downarrow0 limit of solving (\partial_d+t)y=\sum r_i\delta_{d_i} with twist taking the end to the start by M_*. Integration by parts, or the two directed exponential propagators, gives \int y^2 as the count contraction against P(t)/(2t). Thus (8.5), with nonnegative entries added back, gives the assertion (external charges only translate by bounded profiles for fixed m). Angular degree between positions differs from cut degree by fixed source modes only. So along any formal coefficient path the time-average oscillator degree is nonnegative, which bounds the time-average momentum norm from above at fixed cut and source modes. Counts, free axis momentum and cut oscillator degree are therefore bounded there. This proves final support finiteness (together with the earlier cycling bound it suffices: once all cycling is done, equality of the remaining operators on each fixed input vector at fixed modes can be used before summing over that vector); zero-mode exponentials and Weyl operations per vector used here are the locally nilpotent/sliding identities of the torus lemma. It also proves absolute convergence after Wick contraction for small \mathfrak q>0: at radii prescribed by the angles the momentum power of \mathfrak q is exactly half that time average, and oscillator Taylor factors cost at worst \exp(C_m(1+K^2)) absolutely (direct factors within simultaneous layers are polynomials). Thus coefficients can be grouped to implement contour zero modes, using separated sources first.

After Gaussian change the bin estimate gives absolute summability for every positive real period, with Gaussian tails in |f|\gg 1+K; fixed external fields off the root circles pose no problem. We can therefore continue equality along this axis and to the bounded imaginary period shifts at large real period, as in the torus argument, then fuse sources by the local product rule. More explicitly, for continuation from a given real period parameterize integration positions by their real coordinates plus (L-\Re L_{\rm initial}) times a fixed real bend profile, plus 2\pi i d_i. Use a profile zero at the external positions and Lipschitz with increments of one per period (small derivative at large real period, or just take small enough increments of the parameter locally at bounded period to ensure small slopes after bending). Orders of real coordinates are preserved in a neighborhood in real part and convergence is locally uniform along the required imaginary changes, so the piecewise kernel integrals continue analytically. Simultaneous layer contacts have only nonnegative direct powers; no poles are moved through external fields. These arguments justify equality used in (8.4).

8.3 Leading projection

Cauchy average (8.4) 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

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 (8.5) 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

up to subexponential factors: on the deleted portions we have the actual propagation with

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 (8.1).

8.4 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, (8.1) 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 (8.1) 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

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

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

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

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 (8.6).

9 Two ports on a polygon

Glue both positive vacua at a slice with columns in cyclic order [a,\mathcal X,b,\mathcal Y], attributing weight only to a single loop containing both named ports a,b (at the interfaces in time of consecutive rows). Denote this sum by C_{ab}. The identity below expresses this observable in the divided-slice colors. Using m,n,\sigma,D,P,w for the regular variables (\mathcal X,\mathcal Y) exactly as in (8.2) of the divided-slice calculation (q=e^{\pi i/4},\ \phi_+=\phi_-=\pi/4), we have

10 Two-port bound in oblique sections

We give the power upper bound from (9.1). We need it with both types of triangular rhombi (q,q^2 as cell ratios), i.e., up to common rotation the spatial rapidities are 1,q. For positive integers m,K_0, take m unmarked sites between the two marks on the first side of the slice, and K_0m on the other. On both parts take the same proportions 1-\theta,\theta of the rapidities 1,q, 0\le\theta\le1/2, with actual integer counts. Mark rapidities a,b can each have either value. The bound we will show (large m\ge K_0^{1/4}) is

These are physical evaluations of the vacuum formulas. Take a row boundary of forward lattice side steps in two directions e_0,e_1 of consecutive angles 0,\pi/3 in the triangular lattice; successive rows translate by e_1-e_0. This places exactly the triangular splittings for q,q^2, respectively, in the rhombi. The horizontal period is the sum of the steps. Half-cylinder vacuum convergence still holds here. In fact the lifted region has the forward zigzag as boundary. Exhaust the region by a finite zigzag segment and its distant parallel translate, with straight connecting sides parallel to the row translation. For a midport source on the base the lift of counterclockwise tangent angle, starting there with angle \alpha_s, ranges in [0,7\pi/3] until returning to the source. The turn of any simple arc to another boundary port of lifted angle \alpha is \alpha-\alpha_s-\pi (close it along the boundary). Thus the complex boundary flux argument with turn multiplier 3/8, rotated by a fixed phase at the source, has strictly positive real coefficients bounded below. Indeed the outer-source identity sums X^{|\omega|}e^{i(3/8){\rm turn}} to 1 by the two extensions at each step and reversal of collision detours from outside. It gives a uniform bound on the mass to base ports. The same exhaustion bound works on the other side by symmetry. This suffices for the previous convergence and accessibility proof (all single routes positive, double weight the product of allowed separate triangle routes when nonzero). The rational normalized vacuum coincides with the convergent actual one by transfer invariance and uniqueness wherever its denominator does not vanish; nonvanishing of Q also follows from the next comparison.

For Z such a list define

We have |Q(Z)|\ge c_1{\cal D}(Z). Use the positive Fourier weight comparison \mu\ge\mu_0 of the calibration, now before confluence with distinct ordered real angles in a range slightly larger than [0,\pi/4]. By expanding exterior powers, the Fourier skew Pfaffian is (up to common phase and factorial) an integral over positive frequencies \nu_j with columns e^{-\nu_j\arg z},e^{\nu_j\arg z} in a determinant, and column 1 in augmented size, against \prod(\mu(\nu_j)/\nu_j)d\nu_j. Differences at zero are taken as usual (absolutely integrable). The determinants all have fixed sign after ordering frequencies: real exponentials with distinct real exponents on an ordered real set form a nonsingular matrix, by successive differentiation/Rolle after dividing by one exponential. Hence replacement by \mu_0 only decreases the modulus. The resulting skew kernel is (2c)^{-1}\tanh(\log(x/y)/(2c)), by the transform already used. Its Pfaffian gives a factor (2c)^{-\lfloor |Z|/2\rfloor} times the product of individual hyperbolic tangents, by the identity for (u_i-u_j)/(u_i+u_j) (compare residues by pair deletion, then the remaining constant by antisymmetry; use last u_j=0 for augmentation). Multiply by F and take confluence.

We will use the same expansion for the two finite observables

The unmarked expansion, together with the positive comparison just proved, will determine the size of |Q(Z)|. The marked expansion will then be calibrated by the elementary physical bound C_{ab}\le C_a, where C_a is the mass through one port. This separates the normalization from the decay between the marks.

Here are the size conventions used in the two calculations.

In the marked case |Z|=(K_0+1)m+2, and the later-angle fraction is \theta in both regular families. Each mark can have rapidity 1 or q. For the unmarked case use the fugacity-zero identity (8.2); partition each angle class as equally as possible between the families, with order at our disposal. Orient the two angles so that the total later fraction is at most 1/2, using inversion and common scaling if necessary. These operations preserve |Q| and {\cal D}.

Write p_*=0,2 for the number of marks. In the finite color and mark weights use the conjugate convention q\mapsto q^{-1} and \phi_+=\phi_-=-\pi/4; the physical sites still have later rapidity q. Indeed the glued mass is real on nearby unit-circle arrays and invariant, as a rational function, under simultaneous rapidity inversion by upper/lower reflection. Its coefficients can therefore be conjugated.

The composition parameters are the two angle fractions in each regular family in the marked case, and all separate regular multiplicities divided by s_1 in the unmarked case. We allow them to vary in fixed small complex neighborhoods of the indicated limiting compositions, including a zero later fraction. The final physical bound uses identical real later fractions in the two marked families. Angular boundary limits are taken by blocks, one for each angle and field kind, in a fixed order within an angle.

There is also a finite phase index. For each fixed root list, its orders and lattice indices, the part of the residual phase depending on regular multiplicities is a rational quadratic polynomial in those multiplicities. Here the physical source angles are 0,1; the generic root and counter-field angles and the bounded imaginary period shift remain contour data. Express the axis index as f=1/4+n with integer n and write any binomial terms in ordinary monomials. Before summing root occurrences, choose one integer D_{\rm ph} clearing the finitely many underlying rational charge and bilinear-form coefficients in these monomial expressions. Each root-list phase polynomial is an integer combination of those coefficients. Thus the same D_{\rm ph} works for every list and order, independently of K_0 and J. Fix the regular multiplicities modulo 2D_{\rm ph}. Changing them by this modulus changes each such polynomial by an even integer. Their phase dependence is therefore fixed on these classes. This does not freeze the phases depending on root lists, orders or continuous contour data.

We first construct the trace representing these finite quantities and remove its extensive scalar factors. The proof of the proposition then separates an edge of length 6\log K_0 from the free propagation length 6\log s_1. Only the latter enters the accuracy-dependent convolutions. This is what keeps the exponent C_2 fixed.

10.1 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

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)

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

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 (8.2): 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

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 (8.3) 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 (8.3), a^\sigma M_*^j r_i^\tau vanishes for opposite type and for the same type is (1,0,-1,0,-1)_i times

On O the rows against a^\sigma,a^{-\sigma},H_\diamond respectively are

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

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 (8.3)). 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.

10.2 Circle expansion at two scales

10.3 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

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 (10.2) now gives (10.1) 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

The strict power margin also absorbs any fixed logarithmic factor.

Interpolating the angle fraction. Fix a target phase class and common physical fraction \theta. At each of finitely many sizes m' near fixed multiples of m_0, choose an integer m' within bounded error preserving the size class. Use equally spaced neighboring admissible later counts j' of the prescribed class in the first family, and exactly K_0j' in the second. Choose them within bounded distance of m'\theta and within the physical range; one-sided choices suffice at an endpoint. The bound C_{ab}\le C and the unmarked Q estimate imply

at these physical points. Formula (10.2) gives the same bound for its truncated expression there.

Interpolate that expression in the common angle parameter. A fixed sufficiently large number of the neighboring grid points has bounded interpolation weights. Holomorphy and the coefficient bound give a derivative remainder at most

for any prescribed fixed interpolation order A. Choose A large enough that this and the formula errors are negligible. We have therefore bounded the truncated expression at the exact target fraction \theta, without a divisibility requirement on m'\theta.

Interpolating the size. Write m=m_0e^u. The truncated expression is a linear combination of the finitely many functions e^{-\gamma u}u^i in its fixed rate and degree list. Choose fixed evaluation values of u for which their matrix is invertible; linear independence guarantees such a choice. The bounded rounding of each m' perturbs these values by O(m_0^{-1}), so the inverses stay bounded. Thus every coefficient in this shifted basis, including its factor m_0^{-\gamma}, is O((K_0m_0)^{-1/6}) at the exact target composition.

For u\ge0, the higher-rate polynomial terms are bounded by a constant times e^{-3u/2}. At the rate 3/2 there is no polynomial factor. Consequently

Undo the normalization, using |Z|\asymp K_0m and the unmarked Q estimate. The principal term gives

The remainder relative to this target is at most C_JK_0^{C_2+5/24-J/4} for J\ge3/2, so the same choice of J absorbs it. All choices can be made on fixed local neighborhoods; a finite cover of the composition interval makes the constant uniform. This proves (10.1).

10.4 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

Pairs away from lattice side lines. Call ordered vertex pairs good for displacements of length at least R^{1/3} and at Euclidean distance greater than a sufficiently large fixed constant from all lattice side lines through zero. To count their weighted mass fix the first polygon vertex (either type) and sum over displacements to the other. A good displacement is well inside some rotated e_0,e_1 cone. At each visited vertex the polygon uses two of three midports and therefore at least one oriented-side choice in these two directions. Sum over both choices there. For any prescribed such pair, one forward zigzag can include those two sides: between the end of the first and start of the second is a nonnegative integral combination of e_0,e_1. If it takes m steps, m is comparable to the distance. On the rest of the zigzag period after the second mark take K_0 copies of the intermediate step counts, K_0\asymp R/m integral and large enough that the period vector has length >4R. Reflection exchanging the side directions allows the later proportion \le1/2. Projection now embeds all the required polygons injectively at the marks (lift at the first mark), preserving physical weights. Thus (10.1) applies since m\gtrsim R^{1/3}. At dyadic displacement scale r this costs R^{1/3} r^{1/3} up to constants. Hence (10.3) holds with the good-pair count in place of the squared length.

Polygons concentrated near a side line. We can discard with superpolynomial length-weighted accuracy polygons of transverse width below R/\log^2 R relative to any lattice side direction. Project these contractibly to a cylinder along a different (nonparallel) lattice side with period \lceil C R/\log^2 R\rceil, C large fixed; translates do not meet by the width bound, whereas the height there is of order at least R. The exponential pressure height estimate applies, also with length powers, and a plane polygon modulo translations is recovered by lifting.

For length l\gtrsim R, short displacements account for O(lR^{2/3}) pairs. If there is not already a fixed positive fraction good, some constant-width band through one vertex parallel to a lattice side then contains a positive fraction of all vertices by pigeonhole, implying l=O(R). Group these exceptional polygons by that side direction and dyadic transverse width w\gtrsim R/\log^2R relative to it. At least c w vertices lie at distance \gtrsim w from the band, by the width and bounded step size. For each, the other-direction side-line tubes through it meet the band in bounded regions, and a sufficiently small constant times R as minimum displacement length still leaves a positive fraction of the band vertices giving good partners (the band has constant width). Thus at least a fraction c' w/R of all ordered pairs are good with displacement of order R, transverse component at most Cw. There are only O(wR) such displacement vectors and each costs O(R^{-4/3}) by the argument above. The good-pair mass in this bin is therefore at most

Each such polygon has at least c(w/R)|P|^2 good pairs, so its squared-length mass, summed over the bin, is at most

Only O(1+\log\log R) width bins lie between R/\log^2R and O(R). Together with the ordinary good-pair estimate and the negligible thin polygons, this proves (10.3).

11 Positive bridges, caps, and growth

We now turn the partition estimates into positive families of open paths. The immediate goals are transverse tightness, length-controlled half-plane growth with recoverable cuts, and caps at prescribed boundary gaps. These are the geometric inputs needed to convert a long arch into a bridge without losing more than logarithmic factors. Use triangle-center walks with boundary endpoints at midports, weight X^{|\omega|} per walk (|\omega| counts centers), so gluing at a midport multiplies weights. Triangle-side levels in any fixed lattice direction are numbered at unit spacing (one interline distance as unit). A strict bridge of height h\ge1 goes from a fixed port on level 0 to level h, with all its centers strictly between. By Proposition 7.1 its total mass is B_h\asymp h^{-1/4}; include B_0=1. Reflection/translation give the same conventions in any of the side directions. The arch on a half plane is a nonempty return to the source side before any other contact, and A_g denotes its mass to a port at positive horizontal gap g (one of the two displacements), integer in lattice side lengths.

We recall the useful clipping rule: for a boundary port of a finite convex domain of triangles with straight sides, exit flux with coefficient \cos((3/8)\,{\rm turn}) is 1 by the extension identity. Coefficients are fixed by boundary tangents and uniformly positive. Cutting such a domain by a lattice side line (keeping the source) thus bounds the mass of all lost exits, including excursions of arcs to retained endpoints past the cut, by a constant times new exits at the cut; conversely positive lost terms give lower bounds on these new exits. Indeed coefficients to shared endpoints agree. More generally on a simple polygonal domain made of triangles with the source on its outer (counterclockwise) boundary, the same complex identity from the finite proof holds: collision detours are approached from outside so still cancel. If \alpha is the tangent angle at an exit port lifted by following this boundary from the source (angle measured from the starting boundary tangent), the path turn to that port is \alpha-\pi, by closing counterclockwise from there to the source (the two extra corners are +\pi/2 each). In the straight strip the identity 1=\cos(3\pi/8) A^{[h]}+B_h holds as in the strip calculation, A^{[h]} summing both directions of return. Thus total half-plane arch mass is 1/\cos(3\pi/8), and the arch height tail has order B_h.

Here are two tightness consequences with fixed constants.

1. 1.

   Arch mass of diameter >r is \le C r^{-1/4}. Take an equilateral triangle of side order r, slightly smaller than r, with horizontal base. For a source in the central third of the base compare to the half-plane identity. All arches of diameter >r are lost, hence bounded by exits on the other sides. Average over base sources there and reverse from those sides: for each other-side port, arcs ending on the central base must penetrate to normal distance c r or larger from that port’s side. Clipping by a parallel line at such distance thus bounds their sum by C B_{\lfloor c'r\rfloor}. There are O(r) ports on the other sides, proving the bound.

2. 2.

   Bridges of height d remaining at transverse displacements at most Td from the source’s normal have mass \ge c d^{-1/4} uniformly for sufficiently large fixed T, even specifying either weak sign for the endpoint displacement from that normal. Indeed clip the strip at lateral distance of order S\gg d by two slant lattice side lines. Mass lost is bounded by lateral exits by the strip identity. Those exits are included in the analogous half-plane clip with same lateral lines and a distant parallel top at height \asymp S. There the half-plane identity bounds their sum by lost arches, all of diameter \gtrsim S. This proves tightness by the preceding estimate; for a sign use reflection symmetry in the source’s normal.

The enclosing-loop estimate supplies length control for the growth construction below. We record its finite-domain consequence before choosing the growth prefixes, so that length can be included in their cut tests.

A cut is read at a port on a lattice-side grid line. In the forward paths considered below, each candidate u-level lies between the starting and terminal levels. An interior candidate is a single-crossing cut if the path crosses it at exactly one port. A terminal candidate is treated as a cut when all centers lie on its starting side; it becomes single crossing after an extension strictly beyond that wall. The first qualifying cut is the lowest candidate with this cut property whose prefix satisfies the specified diameter, confinement, terminal-position, and, when present, length conditions. These auxiliary conditions concern the prefix alone. If u=t is selected, every center of the suffix lies in u>t, or the suffix is empty at a terminal cut. Removing that suffix or replacing it by another extension in u>t changes neither the crossing counts at earlier candidates nor their prefix conditions, and the selected level remains eligible as a terminal or single-crossing cut. Thus the first-cut rule recovers the same cut.

The length restrictions will be needed for the final exterior connectors. The cap construction below uses only the mass, geometry, and recovery conclusions; retaining the length restrictions costs only fixed constants.

We next construct caps at prescribed gaps. The first step gives a lower bound at some gap on each scale. A one-cell-thick boundary deformation then compares any smaller prescribed gap to that one.

The same comparison gives the pointwise upper bound. For large g, each of order g integer gaps j\in[g/2,g) satisfies A_j\ge cA_g. Every arch at such a gap has diameter at least g/2, so the diameter tail proved above gives

Thus A_g\le C(1+g)^{-5/4}, with bounded gaps absorbed by changing C.

11.1 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, Appendix]; 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.

12 From moments to logarithmic length windows

We prove Theorem 1.1. The first step is to retain a substantial length-weighted mass after cutting off exceptional diameters and lengths. The second step uses the length-controlled growth just proved to attach two disjoint connectors to the surviving arches and controls the number of ways to recover the input from an output bridge. We keep \mu(\omega)=X^{|\omega|}; counts without a spatial root on plane polygons are modulo the primitive translations.

We use the constant-factor cylinder estimate (8.1) and its plane ball upper bound C r^{1/12} at diameter scale r, together with the following strip consequence. In an infinite strip between horizontal lattice-side walls at distance h, for a point v in the middle third (vertex or face center) the enclosing gas satisfies

Indeed use (8.1) on the regular cylinder with even period n along the walls of order h/(C_1\log h), C_1 sufficiently large. Winding separators can be dropped at cost a constant factor as before. The single-loop weight of contractible separators of height >h/6 is negligible by the exponential pressure bound (multiply by height plus a constant to include placements at the slice). Removing all configurations containing one thus costs a negligible fraction of the contractible gas. All others lift around the corresponding two marks inside infinite strips of the required height, regardless of their width, splitting into two plane gas spaces as before. This and the vertex transfer (bounded mass of polygons through the vertex) give the bound.

12.1 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

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

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

Taking kth roots and then k\to\infty yields W\le C e^{-cD/h}, and hence

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 (12.1) 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

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.

12.2 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,

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,

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

Expanding M_g^k by its number r\le k of distinct levels proves (12.3).

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

Let Q_2(s)=\sum_{\operatorname{diam}P\le Cs}\mu(P)|P|^2. Summing (10.3) 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

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

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.

12.3 Proof of the logarithmic window