Crossing and inverse transport

The two special ranks of \(B\) are the reason for introducing \(A\) and \(C\). Direct substitution gives \(B(1)=I,\ B(Q_\lambda^3)\propto C C^t,\ B(Q_\lambda^2)\propto A A^t\) (both proportionalities are equalities at \(d_*\)). Crossing reads \[g(3\lambda-x)B_{23}(Q_\lambda^3/r)(C\otimes I) =g(x) B_{12}(r)(I\otimes C).\] Indeed at numerator level \(u\leftrightarrow U,\ w\leftrightarrow W\), while \(t,v\) stay. In entries one multiplies \(B(r)_{b,-k;-s,a}\) by \(\chi^{a-b}g(x)/g(3\lambda-x)\) to get \(B(Q_\lambda^3/r)_{s,b;a,k}\). This follows from the two blocks (e.g. a doubly occupied no-swap output with charges \(p,-p\) interchanges the numerators of \(w+W\chi^{2p}\) and \(W+w\chi^{2p}\); like charges interchange \(w\) and \(W\)).

Also \(B(r)B(r^{-1})=I\): the single block uses \(a_j b_j=s_j^2-a_0^2\), and \(w(r)w(r^{-1})=1\). On the cup/vacuum sector \(g(x)(w+nW)=p_0-a_0 a_3\) is the opposite \(t\) numerator since \(b_2b_3=p_0-a_0 b_5,\ b_5+n b_1=a_3\); the \(U\) numerators reverse sign. Here \(c^tc=n\), and \((p_0+a_0b_3)(p_0-a_0a_3)-n s_2^2a_0^2=g(x)g(-x)\) by \(\sin(6\lambda)=(1-n)s_2\). For \(n\ne0\) split off the complementary doubly occupied space orthogonal to \(c\), then continue to \(n=0\). In particular \[\begin{gathered} B_{12}(r)B_{23}(Q_\lambda^3 r)(C\otimes I)=D(x)(I\otimes C),\\ B_{23}(r)B_{12}(Q_\lambda^3 r)(I\otimes C)=D(x)(C\otimes I),\\ D(x)=g(-x)/g(3\lambda+x). \end{gathered}\]

Exchanging three adjacent columns

The next identity ensures that moving a column past two others gives a consistent result. For arbitrary spectral parameters \(r,z\), as an identity of meromorphic matrices on \(\mathcal H^{\otimes3}\), it reads \[B_{12}(z) B_{23}(rz) B_{12}(r)= B_{23}(r)B_{12}(rz)B_{23}(z).\] Equality at \(z=1,1/r\) follows by identity and unitarity. At \(z=Q_\lambda^3\) it follows by the two displayed \(C\)-transports and transpose, and at \(z=Q_\lambda^3/r\) by crossing on the range and on the transposed range of the middle rank-one insertion (the crossing scalars agree on the two sides). In more detail at the latter point the products up to common scalar read \(B_{12}(Q_\lambda^3/r)(I\otimes C)(I\otimes C^t)B_{12}(r)\) and \(B_{23}(r)(C\otimes I)(C^t\otimes I)B_{23}(Q_\lambda^3/r)\). There is also equality at \(z\to0\). Indeed after a common scalar \(-e^{8i\lambda}\) the limiting \(B(z)\) is \[|p,k\rangle\ \mapsto\ \xi^{pk}|k,p\rangle+(\xi-\xi^{-3})[p=1,\ k=-1]|1,-1\rangle,\qquad \xi=-e^{2i\lambda}\] by substitution. Two such leading transports move the initial third spin \(k\) to the first slot, with factor \(\xi^{k(S_1+S_2)}\) on the passed pair (\(S|p\rangle=p|p\rangle\)), plus for \(k=-1\) a term moving \(+1\) there instead and applying \((\xi-\xi^{-3})(L\otimes \xi^{-S}+\xi^S\otimes L)\), where \(L|1\rangle=|-1\rangle\), \(L=0\) on the other basis elements. These pair operators commute with \(B(r)\): the second is symmetric about vacancies, preserves occupancy and kills \(c,c^t\) on contraction by \(\chi\xi+(\chi\xi)^{-1}=0\). This verifies the claimed equality at zero. (At \(z\to\infty\) there is analogously a scalar \(-e^{-8i\lambda}\), weighted swap with \(\xi^{-1}\), and the opposite one-way flip.)

Now multiply the braid relation by \(z^4 g(y)g(x+y)\), \(z=e^{iy}\). Entries become polynomial of degree at most eight, of fixed parity by the sign rule (track the initial third site ending first). For generic \(r,\lambda\) the four finite nonzero test points and their negatives, together with zero, suffice, proving the identity.

Fusion to a single spin

The cup \(C\) removes a pair of sites. The map \(A\) instead replaces two sites by one. The precise transport identities for this second operation are \[\begin{aligned} B_{12}(r)B_{23}(Q_\lambda^2 r)(A\otimes I)&=d_f(x)(I\otimes A)B(Q_\lambda r),\\ B_{23}(r)B_{12}(Q_\lambda^2 r)(I\otimes A)&=d_f(x)(A\otimes I)B(Q_\lambda r),\qquad d_f(x)=b_3/a_5. \end{aligned}\] Here \(D(x)=d_f(x)d_f(x+\lambda)\), both transport scalars being 1 at \(d_*\). The second equality follows from the first by unitarity (\(d_f(-x-2\lambda)=1/d_f(x)\)).

To verify the first, the braid identity places the output already in range \(I\otimes A\), using range \(B(Q_\lambda^2)={\rm im}\, A\). Read the components on slots \((s,0,b)\) there. Write primes and tildes on normalized weights for shifts \(+2\lambda,+\lambda\), respectively, \(X=X_\lambda\), \(d_f=d_f(x)\). The required scalar equalities are \[\begin{aligned} d_f\tilde t&=t t'+n X U u',\\ d_f\tilde u&=X(u t'+v u')=t u'/X+U(n w'+W'),\\ d_f\tilde U&=u v'+v U'/X=X(t U'+U v'),\\ d_f\tilde v&=v v'+X u U'=t v'+U U',\\ d_f\tilde w&=u u'+v w',\qquad d_f\tilde W=v W'. \end{aligned}\] Indeed the last applied \(B\) contributes on rows \((0,0)\) as \(t\) or \(U\chi^p\) from \((p,-p)\), and on \((s,0)\), \(s\ne0\), as \(u,v\) from a single \(s\). Thus vacuum input, expanded first as \(|000\rangle+X\sum_p\chi^p|p,-p,0\rangle\), gives \(tt'+n X Uu'\) and the cup term with \(u v'+v U'/X\) after extracting the output \(A\). Double input expanded from \((p,k)\) as \(X(|p,0,k\rangle+|0,p,k\rangle)\) gives \(X(t U'+U v')\) on cup annihilation and \(u u'+v w',v W'\) on double propagation. The two single inputs similarly give respectively \(X(u t'+v u'),t v'+U U'\) from spin on slot one and \(t u'/X+U(n w'+W'),v v'+X u U'\) from spin on slot two.

For substitution drop normalizing denominators, replacing \(d_f\) by \(a_2 b_3\). The numerator lists at prime and tilde arguments in order \(t,u,v,U,w,W\) are \[(p_0+a_2b_1,s_2b_1,a_2b_1,s_2a_2,-a_0b_1,a_2a_1),\quad (p_0+a_1b_2,s_2b_2,a_1b_2,s_2a_1,b_1b_2,a_1a_0).\] The \(U\) formulas use \(b_1+a_0/X=a_1\) and \(X[p_0+a_0(b_3+b_1)]=s_1s_3+a_0b_2=a_1b_3\); the \(v\) formulas use \(a_1b_2=s_1s_2+a_0 b_1\), \(s_3b_1+s_2 a_0=s_1 b_3\).

For \(u\) use \(X[p_0+(a_2+a_0)b_1]=a_2b_2\); the second expression with \(b_1/X=b_2-a_0\) uses \(-p_0-s_1s_2+b_2b_3-n a_0b_1+a_2a_1=0\). Finally for \(t\), use \(a_2 b_3=p_0+a_0 b_1\) and \(a_2 b_1=s_2s_1-a_0a_1\), reducing to \(s_3(b_3-a_1-b_1)=(1-n)s_1 b_1\). These all follow by elementary sine addition, and \(w,W\) immediately by \(a_2 b_2=s_2^2-a_0^2\). This finishes the tensor checks.

Rows, twists, and fusion

Take \(N\) spatial slots \(1,\ldots,N\) periodically, seam between last and first, twist \(\eta\ne 0\). With site parameters \(z=(z_1,\ldots,z_N)\), \(t_i=z_i^2\), define the open row \(M_{p,k}(z;u)\) by fixing \(s_0=p,s_N=k\) and summing over intermediate spins in \[(M_{p,k})_{b;a}=\sum \prod_{i=1}^N B(z_i/u)_{s_{i-1},b_i;a_i,s_i}.\] (The \(u\) here is a spectral parameter.) Put \(T_\eta(z;u)=\sum_s\eta^s M_{s,s}(z;u)\), also \(T_\eta(\varnothing;u)=1+\eta+\eta^{-1}\). These closed transfers operate on total spatial charge zero and are rational in \(y=u^2\) by parity. Ordinary adjacent exchanges of the site list \(z\) intertwine \(T_\eta\), even each \(M_{p,k}\), via \(B_{i,i+1}(z_i/z_{i+1})\) from old to new order, by the braid relation. Rotation of \(z\) one slot left likewise intertwines \(T_\eta\) by \(\Omega |a_1,\ldots,a_N\rangle=\eta^{a_1}|a_2,\ldots,a_N,a_1\rangle\) by charge conservation. \(T_\eta(z;u)\)’s commute as \(u\) varies. Indeed for two auxiliaries (lower row \(u_1\), upper \(u_2\)) the site factor from ordered slots (spatial input, two incoming auxiliaries) to (two outgoing auxiliaries, spatial output) is \(K_i=B_{23}(z_i/u_2)B_{12}(z_i/u_1)\). The braid identity interchanges the auxiliary parameters by \(B(u_1/u_2)\), generically invertible and commuting with their combined twist, so their combined twisted trace is unchanged.

There are product identities at any site \(j\), writing \(e^{ix_{ij}}=z_i/z_j\): \[\begin{aligned} T_\eta(z;z_j)T_\eta(z;Q_\lambda^{-2}z_j) &=\Big(\prod_i d_f(x_{ij})\Big)T_\eta(z;Q_\lambda^{-1}z_j),\\ T_\eta(z;z_j)T_\eta(z;Q_\lambda^{-3}z_j)&=\Big(\prod_i D(x_{ij})\Big) I . \end{aligned}\] Indeed in the first case each \(K_i\) transports the auxiliary subspace \({\rm im}\,A\) by the second \(A\)-identity above, and \(K_j\) has output only into this subspace. Thus the trace restricts to it, with exactly the fused twist by charge, giving the first formula. Similarly use \(C\) for the second. At \(\lambda=d_*\), crossing gives on total charge zero \[T_\eta(z;u)^t=T_{\eta^{-1}}(z^{-1};Q^{-3}/u)\] with site inverses in original order: cross at each \(B\), transpose the two-spin matrix (it is symmetric), and negate auxiliary spins; powers across sites cancel by total charge.

Noncrossing pairings and loop weights

The spin representation alone does not express self-avoidance. For that purpose we keep the pairing of the occupied boundary slots. A link state is a noncrossing partial matching drawn below the cut of the cylinder, with the winding class of every arc recorded. Slots not matched are vacant. The following description makes the resulting finite state space and its spin representation explicit. Lift the cylinder to its periodic covering strip. Each lifted arc cuts off an interval \([i,j]\) with \(i<j<i+N\) on the cut: any larger separation would interlace with a translate. All translates of all intervals have noninterlaced endpoints. Include unpaired slots as vacancies and include the empty state. This is a finite formal basis. It maps linearly into spin tensors of charge zero by giving each arc independently spins \(s,-s\), \(s=\pm1\), at the indicated ends, summing with factor \[\chi^s \eta^{k s},\qquad k=[\text{interval crosses the seam}].\] Call the range the link image (no injectivity assumed); the empty coefficient passes through unchanged. It transforms by \(\Omega\) for the above rotation. Ordinary projections testing vacancy or total occupancy at any given slots preserve the image.

The transfer rows \(T_\eta\) intertwine this map with spinless diagrammatic transfers, loop weights \(n\) for a contractible loop and \(\eta+\eta^{-1}\) for a winding one. To see the local choices, put \((a,k;s,b)\) at the (bottom,right;left,top) ports of a square. Positive arrows point upward on vertical edges and leftward on horizontal edges. The six unoriented local weights are shown in Figure 1. Empty has weight \(t\); a bottom-left or right-top connection weight \(u\); a vertical or horizontal straight connection \(v\); a bottom-right or left-top connection \(U\); double connections of the first turn type have weight \(w\), of the second \(W\). The matrix sums these with an extra oriented factor along each strand \(\chi^{-T/2+m_{\rm out}-m_{\rm in}}\), where \(T\) is signed turning in units of a quarter turn (round the diagrams), \(m_{\rm hor}-m_{\rm vert}=1/2\). Indeed bottom to left and right to top, and their reversals, have exponent zero, bottom to right and top to left exponent \(+1\), inverses \(-1\), as in the cup factors.

A lower boundary arc between vertical stubs has turn \(-2s\) by closing along its interval. Seam factors count signed left crossings. Both kinds of phases thus concatenate.

Contractible simple loops have turn \(\pm4\), essentials turn 0 and signed crossing number \(\pm1\). For the latter crossing count view the cylinder as a punctured plane and use the index of a Jordan loop; for turning concatenate arbitrarily many lifted periods starting at a highest point and close simply above their height by three segments, bounding the total turn independently of the number of periods. Summing free orientations gives the claimed loop weights, leaving again a link state.

Local adjacent \(B\) acting on spatial slots also preserves the link image by the same diagram calculus drawn now with two bottom, two top vertical stubs and all other stubs propagating. Here \(u\) propagates a single occupied straight, \(v\) slides it past a vacancy, \(w\) propagates both, while \(U,W\) are cup/cap choices with factors \(c,c^t\) by the signed half-turns. Similarly \(C,A\) in insertion order across no seam and their transposes into the shorter spaces preserve link images (short cups/caps or vacant and single-spin propagation). In particular at \(\lambda=d_*,\eta=\pm i\), all loops have zero spinless weight and an exchange \(B\) preserves the all-empty coefficient on this image (also \(t=1\)).

The honeycomb weights

We have identified the loop weights in a finite row. We next identify the positive physical specialization, where every occupied vertex has weight \(x_*\). At homogeneous \(z_i=1,\lambda=d_*,u=Q^{-1}\), diagram rows split each square into two trivalent vertices sharing an edge, carrying respectively the bottom-left and right-top ports. Each visited vertex contributes precisely \(x_*\) by the local weights, with at most one strand visiting it; these are honeycomb diagrams. More explicitly take \(h=(\sqrt3,0), v=(\sqrt3/2,3/2)\), vertices \[P_{ij}=ih+jv,\qquad R_{ij}=P_{ij}+(h+v)/3\] (modulo horizontal period \(Nh\) on the cylinder); \(P_{ij}\) neighbors \(R_{ij},R_{i-1,j},R_{i,j-1}\). At \(u=Q^{-2}\) instead group bottom-right and top-left; the bases are \(h,v'=v-h\), and the pair displacement \(P'\to R'\) now \((v'-h)/3\), so the same honeycomb cylinder with a different grouping. The line between rows is a horizontal cut through perpendicular edges at their midpoints. Also at general nearby \(\lambda,u=Q_\lambda^{-1},z_i=1\), divide each cell by its empty weight to give the first honeycomb geometry with \(X_\lambda\) per visited vertex.

A finite bound for boundary paths

To pass from a finite-height cylinder to its stationary state, we require an upper bound on the weight of each boundary arc. We derive it from the honeycomb parafermionic identity of Duminil-Copin and Smirnov [2], with spin \(5/8\). The same estimate also controls crossings of a strip.

Take a bounded convex domain cut out along row-cut lines of the lattice and their rotated counterparts (by honeycomb symmetries); the boundary stubs are perpendicular edges ending at midpoints, never at corners. Fix a starting boundary midpoint \(o\). Sum over walks to edge midpoints \(p\), each walk visiting interior vertices self-avoidingly, changing edge at every vertex, depositing \(x_*^{\#\mathrm{vertices}}\exp(-i\sigma\,{\rm wind})\), \(\sigma=5/8\), to \(F(p)\). Winding is the signed total turn in radians; include the trivial walk. At every vertex \(V\) inside, with neighboring midpoints as complex coordinates, we have \[\sum_p(p-V)F(p)=0.\] Indeed an arrival without prior \(V\) from the edge side opposite \(V\) (including the trivial arrival if appropriate) groups with its two extensions through \(V\), by \(1+2x_*\cos((2+\sigma)\pi/3)=0\). The remaining arrivals from outside \(V\) to \(p\) have passed via \(q\to V\to r\) for the other two midpoints; pair with using \(q\to V\to p\) and the reversed excursion now ending at \(r\). The loop via \(r\) returning \(p\) closed back at \(V\) is simple with \(q\) outside (prefix from the boundary). For instance arrange \(q,r,p\) round \(V\) at angles \(0,2\pi/3,-2\pi/3\): that loop is positive and winding after the prefix adds \(5\pi/3-\pi/3=4\pi/3\), versus its negative for the partner. Their contributions cancel since \(\cos((2+4\sigma)\pi/3)=0\); reflection is identical.

Summing leaves only the boundary. Each path to another boundary midpoint has \(|{\rm wind}|\le\pi\) by closing from exit to entrance along the positively directed convex boundary (corner turns \(+\pi/2\) at each endpoint and intermediate boundary turn in \([0,2\pi]\)). Relative to the initial inward step the real parts thus contribute at least \(\cos((1-\sigma)\pi)\) times masses after dividing by half-edge length, while the \(o\) term contributes \(-1\). Exhausting a half-plane or strip by such convex polygons now bounds, by an absolute constant, the total \(x_*\)-mass of boundary arches from a given boundary midpoint in that half-plane, or bridges from a boundary midpoint reaching the parallel other boundary in the strip.

The honeycomb connective-constant theorem [2] gives, for the number \(c_m\) of \(m\)-edge walks from a fixed vertex, \[\mu=\lim_{m\to\infty}c_m^{1/m}=\sqrt{2+\sqrt2},\qquad x_*=\mu^{-1}.\] Consequently \(x_*\) is the convergence radius for these edge-counted walks. The boundary estimate also yields a quantitative upper bound by the successive-extrema unfolding principle [9]; we give the argument, including its lattice endpoint conventions.

For completeness, the boundary estimate itself implies that \(x_*\) cannot exceed the plane convergence radius. The endpoint conversion is also needed in the quantitative estimate, so we give its details. Put \(Y=2\,({\rm vertical\ height})\); increments are nonzero integers of magnitude at most 2. For walks from a given vertex at one \(Y\)-extremum to their other extremum of specified \(Y\), confined between them (strictness not needed, span nonzero), the total \(x_*^{\#\mathrm{edges}}\) is uniformly bounded: extend to perpendicular cut midpoints just below and above the two extreme rows. At the minimum attach first a neighboring lower \(P\) if at an \(R\), and at the maximum a neighboring higher \(R\) if at a \(P\), by fixed conventions. Added vertices lie strictly outside the old heights, and the bridge bound applies with only bounded length adjustment. An arbitrary \(m\)-step walk splits at a lowest vertex. Along each arm going away, cut at the last highest vertex, then last lowest of the remainder, alternately until the end. The positive spans are decreasing strictly after a possible initial equality, and sum to at most \(2m\). Thus the original chronology is a concatenation of \(O(\sqrt{m})\) extremum-to-extremum bridges, some reversed back, with at most \((C m)^{O(\sqrt{m})}\) possible signed span lists. Multiplying the uniform bounds gives subexponential total \(m\)-step mass at \(x_*\). The matching opposite radius bound will also follow from the winding masses below.

In the notation just introduced, the quantitative conclusion is \[ c_mx_*^m\le \exp\!\bigl(C\sqrt m\log(m+1)\bigr) \qquad(m\ge1), \tag{2}\] for an absolute constant \(C\). For each extremum-to-extremum bridge in this decomposition, the extension adds \(a,b\in\{0,1\}\) vertices at its lower and upper ends. An \(l\)-edge bridge therefore has port weight \(x_*^{l+1+a+b}\); the exterior half-edges add no visited vertices, and inverse multiplicity is bounded. These conversions preserve the uniform piece bound used in the unfolding count.

Existence and uniqueness of the normalized stationary state

Return to a cylinder of fixed circumference \(N\). The boundary-path bound now proves convergence of the row iteration, rather than merely bounding its matrix entries. At either homogeneous physical transfer, \(\eta=\pm i\), consider the formal matching matrix at \(d_*\). It is nonnegative, loops killed. Its powers on the empty state increase coefficientwise since all-empty propagation is available. They are bounded: each remaining arc lifts to a planar arch in the lower half-plane from a given slot; a product of the just-proved bounds dominates each coefficient. Every nonempty matching \(J\) is reachable with positive coefficient: build one periodic arc at a time in decreasing interval length, extending already present ends vertically. No already occupied slot lies in the new intervals by noninterlacing. In one row a top-to-top arc can run horizontally between its two slots with positive allowed turns, simultaneously with the required vertical passes outside. Writing \(T\) for the matching transfer, for each \(J\ne\varnothing\) there are \(m\ge1\) and \(\epsilon>0\) such that \(T^m e_\varnothing\ge e_\varnothing+\epsilon e_J\) coefficientwise. Consequently \[0\le \epsilon T^n e_J \le T^{n+m}e_\varnothing-T^n e_\varnothing\longrightarrow0.\] Thus every nonempty basis vector is sent to zero in the limit. This gives a rank-one projection limit on the formal space (eigenvalue \(1\) simple, all others of modulus less than 1). Hence also on its spin image, with unique fixed vector normalized to empty coefficient 1 and obtained from vacuum by the transfer limit. At fixed \(\lambda=d_*,\eta=\pm i\), varying sites near homogeneous and fixing the auxiliary at a physical value, that simple normalized link eigenvector continues regularly with eigenvalue 1 (the empty coefficient of transfer is unchanged on this image). In particular uniqueness there also implies generic uniqueness if supplied with a rational normalized eigenvector.

Pair values and the two-site initial condition

The degree and endpoint requirements do not by themselves determine \(V_N\). We prescribe its values when two column parameters have the cup ratio. We impose pair prescriptions at \(z_j=Q^3 z_i\). For \(p=2\) prescribe one for every directed pair \(i\to j\) in the cyclic list. For \(p=1\) only require \(i<j\). First for an order \((\ldots,i,k_1,\ldots,k_r,j,\ldots)\), insert \(C\) at \(i,j\) made adjacent (just before the intermediate list) on \(V_{N-2}\) of the other sites, then transport \(j\) right to its specified position over the intermediate list by successive exchanges \(B(z_j/z_{k_l})\). Multiply by \[\Pi_{ij}=K_p u^p\!\!\prod_{k\ne i,j}[(t_k-q^{-3}u)(t_k-q^6 u)]^p,\qquad u=t_i.\] Call the result \(W_{ij}\). For a cyclic prescription in construction \(p=2\), rotate to this situation (cut outside the directed interval) and rotate the result back by the inverse powers of \(\Omega\). The choice of linearizing cut does not matter using cyclicity of the smaller tensor: a block rotated over the cut when comparing two choices either misses this operation or contains the entire affected block, which conserves the total spin of its spectators; also \(\Omega^N=I\) on charge zero. These prescriptions respect site parity under stripping (including simultaneous signs at paired endpoints); this follows by the sign rule for exchanges and equal starting occupancies at the two ends of \(C\).

At \(N=2\) write \(V_2=D|00\rangle+H c\). Use \[\begin{array}{c|cc} &D&H\\\hline p=2&t_1^2+(2+2\sqrt2)t_1t_2+t_2^2&P Q^3 z_1 z_2(t_1-t_2)/(1-q^3)\\ p=1&t_1+t_2&(Q^3+Q^{-3})z_1 z_2 . \end{array}\] Indeed \(c^tc=0\); exchange sends \(H\) to \(U D+wH\), equal to \(-H,H\) respectively. By the weights at \(z_1/z_2=e^{ix}\), the ratios needed are \(-U/(1+w)=-s_2\sin x/(\cos d_*+\cos(3d_*)\cos(2x))\) and \(U/(1-w)=s_2/(2s_3\cos x)\), giving the table. Rotation of \(c\) contributes scalar \(-1,1\) respectively. The prescribed adjacent pair values and other constraints now follow directly.

Compatibility of the pair prescriptions

We prove Proposition 3 by induction in \(N\), starting with the displayed initial vectors. The first step is to check that the prescribed values agree wherever two pair slices meet. All parameters not constrained by those slices are first taken generic. Meromorphic continuation will then cover every regular specialization. Each \(W_{ij}\) is covariant by ordinary adjacent swaps not swapping its two endpoints together. For endpoint \(j\) this follows by its transport definition and unitarity; for \(i\) moving past a slot \(k\) use \(B_{12}(z_k/z_i)(I\otimes C)=B_{23}(z_j/z_k)(C\otimes I)\), or reverse by unitarity. For spectator swaps use smaller-tensor exchanges and the braid relation if both spectators lie on the transport route. For a cyclic interval one can linearize across a cut outside the route not splitting the swap (moving a cut past a whole swapping block is compatible by charge); the prescriptions are also covariant by rotation for \(p=2\).

It follows that evaluations from two disjoint prescribed pairs at their intersection agree: place both pairs positive-adjacent without swapping the endpoints of either together, and apply smaller-tensor prescriptions and commuting insertions. In the cyclic case one may migrate an end around over other labels up to adjacency (the second pair can cross the whole first block this way) and rotate both away from the seam. The products of prefactors agree: for start parameters \(u,v\) the extra cross factor before power \(p\) is \(q^6\prod_{k=\pm2,\pm3}(v-q^k u)\), symmetric. All swaps of unrelated labels here are generically invertible.

At two pairs with common start (so equal ends), arrange the triple consecutively preserving the orientations; transporting one equal end past the other uses \(B(1)=I\). For common end, similarly use \(B_{23}(Q^3)(C\otimes I)=I\otimes C\); again values agree. At a chain intersection \(i\to j\to k\) both evaluations vanish: \(\Pi\) has an order-\(p\) zero at the third parameter, with at most one exchange pole there; when \(p=1\), \(i<j<k\) and the third parameter in either prescription is outside its transport route, so not a pole at all.

Interpolation and polynomial degree bounds

Assume that the smaller tensors have been constructed. We now construct \(V_N\), check that the apparent interpolation poles cancel, and prove its bounds in every parameter. For \(N\ge3\), interpolate after stripping in \(t_N\). Use nonzero nodes \[r_{i\epsilon}=q^{3\epsilon}t_i,\qquad i<N,\quad \epsilon\in J_p,\qquad J_1=\{1\},\ J_2=\{1,-1\}\] by those prescriptions (\(z_N=Q^{3\epsilon}z_i\)), and also zero with the required deletion input if \(s_N=0\). With \(\rho=[s_N=0]\), the Lagrange factors for nonzero nodes are \[L_{i\epsilon}=(t_N/r_{i\epsilon})^\rho \prod_{(a,\gamma)\ne(i,\epsilon)}\frac{t_N-r_{a\gamma}}{r_{i\epsilon}-r_{a\gamma}} .\] Use \(L_0=\prod_{a,\gamma}(1-t_N/r_{a\gamma})\) when needed. Homogeneity and charge zero hold, and the result unstripped stays in the link image since the extra scaling and interpolation weights depend only on occupancy at \(N\). Each prescribed input \(W\) is polynomial on the paired slice after stripping: every transport pole divisor involving the paired parameter and a spectator is canceled by \(\Pi\); at spectator zero the matrices stay bounded and parity suffices to strip, at simultaneous endpoint zero even the \(u^p\) supplies divisibility. Its stripped spectator degree has the desired bound since \(\Pi\) adds \(2p\) there, the raw smaller tensor grows by at most degree \(p(N-3)\), and bounded exchange and stripping then give the bound by parity (rounding down to integral degree).

Potential interpolation poles at coincident nonzero nodes from \(t_i=t_l\) cancel between the corresponding simple Lagrange poles by common start/end compatibility, and at \(t_i=q^{\pm6}t_l\) for \(p=2\) by chain vanishing. At \(t_l\to0,\infty\) for \(l<N\), the special terms \(i=l\), including stripping and Lagrange factor, grow respectively at worst as \[O(t_l^{\,1-(|s_l|+|s_N|)/2-\rho}),\qquad O(t_l^{\,p(N-1)-(|s_l|+|s_N|)/2-\rho});\] indeed the raw paired input has bounds from \(\Pi\) of powers \(p,\ p(2N-3)\), and the other factors of \(L_{l\epsilon}\) beyond \((t_N/r)^\rho\) contribute at worst powers \(-(p-1),-p(N-2)\). These bounds have no pole and give the required degree (round by parity), subleading at infinity for vacant \(l\). In other terms use the spectator bounds and the zero input factor \(t_l^p\) canceling the \(L_0\) pole. Thus all coefficients are polynomial after stripping with the claimed degrees.

Deleting a vacant spectator

We have a polynomial vector with the required degrees. To complete the induction, its deletion limits must agree with the smaller tensors in every variable, not just in the interpolation variable. Fix a spectator \(l<N\) with \(s_l=0\). At infinity, in nonspecial pairs only the smaller-tensor vacant \(l\) can contribute to the leading scale; if crossed it is passed exactly by the extreme swap. Thus these give \(t_l^{p(N-1)}\) times smaller interpolation terms (power \(2p\) from \(\Pi\) and \(p(N-3)\) from the smaller-tensor deletion, removed Lagrange factors tending to 1); the zero term likewise, special terms subleading as above.

At zero the nonspecial inputs scale to smaller inputs times \((\prod_{k\ne l,N}t_k^p)r_{i\epsilon}^p\), since the dropped spectator factors in \(\Pi\) give \(q^{3p}u^{2p}\); their extra Lagrange factors tend to \((t_N/r_{i\epsilon})^p\). These yield the nonzero-node terms needed after deletion. Special terms vanish if \(s_N\ne0\). Suppose \(s_N=0\) and write \(L'_0\) for the smaller zero basis. For \(p=2\) the special raw inputs before stripping at leading order are \[P q^{3\epsilon} t_l^2 \!\!\prod_{k\ne l,N}t_k^4\, V_{N-2},\] as both endpoints remain vacant and the traveling end swaps exactly at leading order. Rotations if needed here and for spectator deletion do not pick up a phase from crossed vacancies. Multiplication by \(L_{l\epsilon}\sim t_N^2L'_0/[q^{3\epsilon}(q^{3\epsilon}-q^{-3\epsilon})t_l^2]\) gives cancellation. The ordinary zero input then yields the required zero term using the smaller-tensor deletion \((L_0\sim (t_N/t_l)^2L'_0)\). For \(p=1\) the special raw input is similarly at leading order \((1+q^3)t_l\prod_{k\ne l,N}t_k^2 V_{N-2}\), with \(L_{l+}\sim t_N L'_0/(q^3 t_l)\), while the zero input has leading raw \(t_l\prod_{k\ne l,N}t_k^2 V_{N-2}\), with \(L_0\sim-t_N L'_0/(q^3t_l)\). These combine to exactly the desired zero term. Stripping spectator factors is common to all of these comparisons. This proves both deletion limits for \(l<N\); vacant \(t_N=0\) is already a node.

All pair values, exchanges, and rotation

The degree bounds and spectator deletions now let us test the remaining identities by interpolation. First impose every required pair \(i\to j\) not involving \(N\) by comparing as polynomials in spectator \(t_N\) on the slice. At the same nonzero interpolation nodes (still distinct generically), compatibility gives equality, and at zero if vacant use spectator deletion in \(W_{ij}\) by the same \(\Pi\), smaller-tensor and extreme-swap rules. Degree in \(t_N\) after stripping then proves the prescription.

For \(p=1\) deduce now the vacant infinity deletion at \(N\) as well: the difference in the leading coefficient from the required one, stripped, has all factors \(t_j-q^3t_i,\ i<j<N\), by the pair prescriptions and smaller-tensor infinity limit. Its total degree is at most \((N-1)(N-2)/2\), strictly less for any nonvacuum spin list. For the all-vacant component use also \(t_l=0\) for some \(l<N\) (the deletion limit there with smaller-tensor infinity gives agreement). Thus the difference vanishes.

Exchanges for either construction follow by testing a third label’s parameter (\(N\ge3\)). At all pair nodes involving that label the pair covariances apply (its order relative to the partner is unchanged if \(p=1\)); at zero for a vacant spin use deletion. These suffice by degree since the exchanged sites are external to the variable being tested. Similarly for \(p=2\) test any label’s parameter to show rotation using the cyclic covariances of the prescriptions and zero deletion. This also gives the remaining infinity deletion at \(N\) for \(p=2\).

Only rotation for \(p=1\) remains. Its pair prescriptions were imposed in linear order, so cyclic covariance is not already available. Consider the difference between \(V_N\) at the left-rotated list and \(\Omega V_N\). All ordered pair specializations among labels \(2,\ldots,N\) vanish by smaller-tensor cyclicity (the pair transport misses slot 1); the vacant deletion limits for these labels vanish too (difference at the specified leading scales). Thus any component vacant at one of \(2,\ldots,N\) vanishes identically using \(N-2\) pair roots there, zero and the infinity coefficient. The difference is supported on that tail all occupied, and exchanges by \(B\) on the tail by what was just proved. Projecting an exchanged adjacent pair to \(00\) forces \(c^t\) annihilation there; hence the exchange acts by scalar \(w(z_i/z_j)\) alone. Its poles now force both factors \(t_j-q^2t_i,\ t_j-q^3t_i\) in the stripped polynomial for every \(2\le i<j\le N\), by bringing these arguments adjacent (other generic swaps nonsingular); the \(w\) numerator is nonzero generically there. This exceeds the degree \(N-2\) per occupied variable since \(N\ge3\). Hence rotation holds. We have finished the induction for both constructions. Uniqueness follows from the same interpolation nodes and degree bounds at each induction step.

The empty component and the transfer fixed vector

It remains to normalize the constructed vector and identify it with the physical stationary state. Write \(D_p=V_N^{\rm empty}\), the coefficient of the all-vacant spin tensor. It is symmetric by exchanges and link loop killing. In fact with \(X=(t_i)\) \[\begin{gathered} D_2(X)=P^N\!\prod_{i<j}\left[\frac{p_K(t_i,t_j)}{t_i-t_j}\right]^2 \det[k(t_i,t_j)],\\ p_K(x,y)=x^2+\sqrt2xy+y^2,\quad k(x,y)=xy/p_K(x,y). \end{gathered}\] Indeed this expression is polynomial (all \(p_K\) poles canceled and a Vandermonde zero in each of rows and columns). It has both required endpoint limits by inspection (in particular correct maximum degree). At \(t_j=q^3u,\ u=t_i\), the double pole term \(-k(u,t_j)^2\det k_{\rm red}\) in the determinant gives exactly \(\Pi_{ij}\) times the smaller formula: \(-P^2 q^6/(1-q^3)^2=P q^3\) and the extra spectator factors reduce by factoring \(p_K\). These and the vacant zero input suffice by interpolation since the constructions of \(W\) preserve the inserted empty coefficient apart from \(\Pi\). The inversion formula \(D_2(X^{-1})=\prod_i t_i^{-2(N-1)} D_2(X)\) follows. We do not need an analogous determinant here for \(D_1\). Both denominators are generically nonzero, also on slices of a pair at ratios \(q^{\pm1,\pm2,\pm3}\) (meaning either sign in any of the three powers), by deletion down to the displayed two-site components.

We next prove the fixed-vector identity \(T_\eta(z;u)V_N(z)=V_N(z)\) for every auxiliary parameter \(u\), as a rational identity. This is the step connecting the interpolation construction to the transfer limit. Induct using one site variable \(t_l\); clear its two transfer pole factors and strip its output-spin parity. The difference has degree at most \(p(N-1)+2-|b_l|\) at output spin \(b_l\), by bounded extremes and rounding as before. It vanishes at the \(p(N-1)\) prescribed pair nodes involving \(l\), by \(W\), placing the pair adjacent and using \(C\)-transparency and smaller transfer (intertwining the swaps/rotations). At zero if \(b_l=0\) use deletion and extreme swap. It also vanishes at \(t_l=u^2\) and \(t_l=q^3u^2\) for auxiliary \(u\). Indeed put the site last in the first case \(z_l=u\): \(B(1)\) identifies \(s_{N-1}=a_N,\ s_0=s_N=b_N\) in row notation. The exchanges in the remaining row move the last argument left through all to the first position; the twist factor \(\eta^{b_N}\) undoes this rotation.

In the second case put the site first with \(z_1=Q^3u\); \(CC^t\) forces \(s_0=-b_1,\ s_1=-a_1\) with factor \(q^{a_1-b_1}\). Cross the remaining \(B(z_i/u)_{s_{i-1},b_i;a_i,s_i}=q^{a_i-b_i}B(z_1/z_i)_{b_i,-s_i;-s_{i-1},a_i}\). Phases cancel by charge, and this braid moves the first argument right to last while the twist again undoes rotation. The node count is sufficient (start the induction at \(N=0\)).

Set \(\Psi_\eta=V_N/D_p\); we call this normalized stationary vector the vacuum. These normalized rational vectors are regular by continuation at homogeneous sites and give exactly the physical transfer limits: even if a denominator representation vanishes there, generic uniqueness near those sites identifies the rational function locally with the regular link eigenvector already established. All exchanges/rotation, endpoint deletions with vacuum normalization (nonvacant outputs vanishing by the degree bounds), spin parities and transfer fixed identities thus hold in the stated limits or as rational identities. This completes the proof of Proposition 3.

Normalization and pair reductions

One has \(\langle I\rangle=1,\ \langle P_{l,+}+P_{l,-}\rangle=0\): gluing two link states, read the bra as arcs above with inverse twist (the arc turning and factors in its coefficients then have exactly the indicated signs). For a loop with signed left seam crossings \(N_L,N_U\) below and above respectively, the twist factor is \(\eta^{N_L-N_U}\), real (\(\pm1\)) if contractible and imaginary (\(\pm i\)) if essential, since \(N_L+N_U=0,\pm1\) respectively. The free orientations thus sum to zero by the turning phases. Occupancy restriction does not interfere, and the all-vacant coefficients are normalized.

By exchanges, transpose symmetry, unitarity and row intertwining, \(Z_\eta\) is symmetric in the site labels, rational in the \(t_i\) and \(y\) by the sign rule and site parity. In \(y\) poles can only come from \(\prod_i(y-q^2t_i)(y-q^3t_i)\), and both endpoints \(0,\infty\) give 1 since the weighted extreme swaps have total spatial charge zero, the scalar on each is 1 and one-way flips cannot restore the fixed auxiliary. Also sending a spectator \(t_k\) alone to either endpoint decouples that site, by the vacant leading limits of the vectors in both factors of the brackets and the extreme swap past a vacancy. At \(a=t_l\) one has the crossing value \[Z_\eta(a)=Z_\eta(q^{-3}a).\] Indeed put \(l\) last at \(y=a\); \(B(1)=I\) forces its output spin equal to the traced seam spin. Thus \(M_{-,-}\Psi_\eta=\eta P_{\rm last,-}\Psi_\eta\) by projecting \(T_\eta\Psi_\eta=\Psi_\eta\). At \(y=q^{-3}a\) put \(l\) first and \(B(Q^3)=CC^t\) similarly gives \(\eta P_{\rm first,+}\Psi_\eta\). Rotating both ket and bra from first to last multiplies the charged projector contractions by \(\eta^{2(\pm1)}=-1\); use the projector-sum vanishing.

Pair reductions hold rationally for otherwise generic variables, including generic \(y\): two site parameters \(t,q^2t\) replace by the single \(qt\), and \(t,q^3t\) delete. Arrange them ordered adjacent (across no seam) and use positive root ratios \(z_j/z_i=Q^2,Q^3\) respectively. The vacuum denominators in either order and either set of inverse arguments are nonzero generically on these slices by the decoupling limits down to the size-two vacua. In positive order the exchange \(B(Q^2)=A A^t\) or \(B(Q^3)=C C^t\) from reverse order places \(\Psi_\eta\) in the inserted image. Extract \(A\) or \(C\) by its left inverse. The result stays in the link image (\(A^t A\) diagonal \(1,2x_*^2,2x_*^2\), inverse just occupancy scaling, \(C^t C=1\)), has empty coefficient 1, and is a transfer fixed vector on the fused/deleted list by the spatial \(A,C\) transports, hence the corresponding shorter vacuum by generic uniqueness there.

In inverse-site arguments the order is opposite: exchange to positive order uses \(A A^t\) or \(C C^t\) on \(\Psi_\eta(z^{-1})\). By the result for positive order and injectivity, applying the transposed insertion alone thus yields exactly the shorter inverse vacuum. The same spatial transport identities hold in the open row, proving the pair reductions after contraction.

The sign of a marked polygon

We can now interpret the marked scalar at physical weights. The algebraic states are complex, but the \(-i\) marked observable will be a sum of positive polygon weights. At homogeneous sites interpret \(Z_\eta(q^{-k})\), \(k=1,2\), by surrounding the open row with the same physical rows (\(u=Q^{-k}\)) but twist \(\eta\) below and inverse twist above, vacuum on the two outer cuts, taking height limits. This gives the brackets by the transfer convergence and the crossed transpose identity. Draw first in oriented square topology as before, seam vertical. The right arrow fixed by the mark at its seam edge singles out a loop. All others sum to zero as before. Twist crossings of the marked loop now exclude the single marked crossing, and still use \(\eta^{N_L-N_U}\).

If contractible positive (counterclockwise), just before that right arrow on the seam upwards one is outside the loop, and just after one is inside; \(N_L=0,N_U=1\). For the negative loop \(N_L=1,N_U=0\). The two possible full phases are \(q^{-2}\eta^{-1}\) and \(q^2\eta\), both \(-1\) at \(\eta=i\), \(+1\) at \(-i\).

For net rightward essential, the point just before is in the bottom component and \(N_L=N_U=0\); for net left it is in the upper component and \(N_L=N_U=1\). Indeed crossings toggle sides, right arrows entering the left side of the oriented strand, and the net sign over the full seam records which component is which. Both essential phases are 1.

Similarly \(Z_\eta(1)=\eta\langle P_{\rm last,-}\rangle\) can use \(u=Q^{-1}\) on both sides, marking a downward arrow and a seam just to its right (the point on the cut there is thus on the left side of the arrow). For counterclockwise contractible \(N_L=-1,N_U=1\); for clockwise both 0, giving the same signs with the extra \(\eta\). For net right essential \(N_L=-1,N_U=0\); for net left \(N_L=0,N_U=1\), again total phase 1 including \(\eta\).

Consequently each marked value at minus twist counts both essential and contractible self-avoiding polygons through its edge with weight \(x_*^{|\gamma|}\) (\(|\gamma|\) the length), and at plus twist essential minus contractible. These statements pass to the limit since the minus sign choice of twist made both sums positive and their limiting bracket finite. The orientation of the marked loop is uniquely fixed. The three edges for \(1,q^{-1},q^{-2}\) sample exactly one of each honeycomb edge type up to lattice translations, on the same cylinder: the first two are across vertical and horizontal diagram ports of the first grouping (\(h,v\)); the third is across a horizontal port of the second (\(h,v-h\)), hence an internal pair edge of the first.

The sum of all polygon masses

We first prove the rational identity \[ S'_{-i,a}=c_*. \tag{3}\] This is true for \(N=1\), whose three terms are \(0,x_*^2,x_*^2\). Write \(D_-=D_1\) and \(D_+=D_2\) for the two vacuum polynomials (minus and plus twist). Multiply the inductive error by \((b+a)D_-(X)b^{N-1}D_-(X^{-1})\). This clears all non-monomial poles in \(b\) (at \(-a\) row poles before vacuum division are at most simple). Both endpoints of the error vanish by spectator decoupling and induction. Thus the product is polynomial of degree \(\le2N-2\) and has a zero at 0. Other zeros are at \(b=q^{\pm2}a\) and \(b=q^j a'\) for every other spectator \(a'\ne b\), \(j=\pm2,\pm3\), by pair reductions. For \(N\ge2\) these are \(1+2+4(N-2)=4N-5\) distinct zeros at generic parameters, more than the degree bound \(2N-2\). The polynomial therefore vanishes.

An unmarked determinant and a finite color sum

For the winding mass we need a second scalar representation. Set \[\begin{gathered} p_K(x,y)=x^2+dxy+y^2,\qquad \delta(x,y)=p_K(x,y)(x^2+y^2),\\ R_X=\frac{D_+(X)^2}{\prod_{x,y\ {\rm pair\ in}\ X}\delta(x,y)} =P^{2|X|}(\det k_X)^2\prod_{x,y\ {\rm pair\ in}\ X} f(x,y),\\ f(x,y)=\frac{p_K(x,y)^3}{(x-y)^4(x^2+y^2)} , \end{gathered}\] using unordered pairs and \(k=xy/p_K\). Define colors \(-,0,+\) with weights \(\ell_-=(-1-d+i)/2,\ \ell_0=P,\ \ell_+=q\ell_-\). Put \(G(r)=(1+r)^2/[(r-q)(r-q^{-1})]\), \(\beta=G(1)=2P\), so \(\sum\ell=1,\ \ell_-\ell_+=P^2/\beta\). For sites \(x,y\) of colors \(s,t\) multiply a pair factor \(g_{st}(y/x)\), with table \[g(r)=\begin{pmatrix} 1&G(r/q)&G(r/q)G(r/q^2)\\ G(rq)&G(r)&G(r/q)\\ G(rq)G(rq^2)&G(rq)&1 \end{pmatrix}.\] For a color assignment \(\sigma:\{1,\ldots,N\}\to\{-,0,+\}\) define its weight to be \[\prod_{j=1}^N\ell_{\sigma(j)} \prod_{1\le j<k\le N}g_{\sigma(j),\sigma(k)}(t_k/t_j).\] The relation \(g_{st}(r)=g_{ts}(1/r)\) makes the sum invariant under reordering the sites. Let \(\tau_X\) be the sum of these weights over all \(3^N\) assignments. Let \(F_X(y)\) be the same sum with the additional factor \(\prod_{j=1}^N L_{\sigma(j)}(t_j/y)\). To define these factors, put \(f_j(r)=1-q^j r\) and \[L_-=L=\frac{f_5 f_6 f_7}{f_2 f_3 f_4},\qquad L_0(r)=\frac{L(r/q)}{L(r/q^2)},\qquad L_+(r)=1/L(r/q^3).\] The sums do not depend on site order.

Residues and the determinant identity

We prove \(R_X=\tau_X\) by showing that both rational functions have the same residues and the same limits when one parameter tends to zero or infinity. The decorated sum has the same reduction rules, which we will need for the marked identity.

Here are the residues in either color sum between sites \(a',b\), using coordinate \(b/(q^j a')-1\). At \(b=q^2a'\) (\(j=2\)) one gets factor \(\alpha P\), \(\alpha=P/(q-q^{-1})\), times \(\prod_{z\ne b,a'} G(z/(q a'))\) times the same sum on the list fused to \(q a'\). Indeed pairs \((-,0),(0,+),(-,+)\) contribute, fusing to \(-,+,0\) respectively, by the simple \(G\)-residue \(\alpha\) and the color weights above. Their spectator table products with a spin \(s\) reduce by \(g_{i,s}(r)g_{j,s}(r/q^2)=G(r/q)g_{k,s}(r/q)\) for each fusion \(ij\to k\), \(r=z/a'\); \(L_i(t)L_j(q^2t)=L_k(qt)\). At \(b=q^3 a'\) only \((-,+)\) contributes, with factor \(\alpha G(q^2)P^2/\beta\), spectator factors \(G(r/q)G(r/q^2)\), and the pair deleted (decoration cancels). At \(b=q a'\), \((0,0)\) and \((-,+)\) residues cancel by the table and common decoration \(L(t)/L(q^{-2}t)\). Equal-site simple poles cancel by symmetry, and inverse ratios are treated by exchange. These checks for \(F_X\) hold at generic \(y\) and regularly for decorations away from their poles.

Then \(R_X=\tau_X\). Indeed \(R_X\) has only the simple pole loci above with \(j=\pm2,\pm3\). At \(b=q^3a'\), the empty-component pair prefactor in \(D_+\), squared over the \(\delta\) cross factors, yields exactly \(G(r/q)G(r/q^2)\) per spectator (remaining numerator roots \(q^5,q^6\) double, denominator roots \(1,q,q^2,q^3\)). At \(b=q^2a'\) use kernel Schur complements. For an ordered list \(Y\) for which the matrix \(k(Y,Y)\) is invertible, define \[k_Y^\perp(s,t)=k(s,t)-k(s,Y)k(Y,Y)^{-1}k(Y,t).\] The required identity, with \(Y=(a',q^2a')\) and \(Y'=(qa')\), is \[k_Y^\perp(s,t)=H(s/a')H(t/a') k_{Y'}^\perp(s,t),\qquad H(s)=(s^4-1)/(s^4+1).\] To see this scale \(a'=1\). The kernel \((1-H(s)H(t))k=2st(s^2-dst+t^2)/[(s^4+1)(t^4+1)]\) has rank three and agrees with \(k\) on the conditioning rows/columns at \(Y\) (the index matrix has diagonal \(1/P\), off diagonal \(1/d\)). Subtracting the \(Y\)-projector \(k(\cdot,Y)k(Y,Y)^{-1}k(Y,\cdot)\) leaves rank one in the same polynomial span vanishing at \(1,i\), hence proportional to \(H(s)k(s,q)H(t)k(t,q)\). At \(s=q\) comparing the \(H\)-pole gives multiplier \(-P\), as required.

Thus the squared determinant gains \(H(r)^4\) per spectator relative to fusion. Factoring the remaining pair terms gives \[H(r)^4\frac{f(r,1)f(r,q^2)}{f(r,q)}=G(r/q).\] Only the two residue constants remain. At \(N=2\) one has \[R_{(1,b)}=\frac{(v+2+2d)^2}{(v+d)v},\qquad v=b+b^{-1},\] whose residues are exactly \(\alpha P\) and \(\alpha G(q^2)P^2/\beta\). The cases \(N=0,1\) start the induction. For larger \(N\), the reduction formulas now show that all residues of \(R_X\) and \(\tau_X\) agree. Both functions have the same decoupling limits at zero and infinity, so their difference is zero.

Decorating the distinguished site

The three edges in the marked observable correspond to three equal specializations of the decorated color sum. More precisely, the quantities \(F_X(a),F_X(q^{-1}a),F_X(q^{-2}a)\) coincide; denote their common value by \(F_a(X)\). They all equal one half the color sum on spectators with extra factors \(H_s^v(z/a)\) given in order by \[L,\quad 1,\quad f_1 f_2 f_3/(f_4 f_5 f_6)=q^{-1}L(1/r).\] Here the last expression uses argument \(r\) (as do the \(f_j\)’s). Indeed the surviving color \(k\) at \(a\) for these arguments \(y\) is \(-,0,+\) respectively, with \(\ell_k L_k(a/y)=1/2\), and \(g_{k,s}(r)L_s(ra/y)=H_s^v(r)\) by substitution (use \(G=f_4^2/(f_1f_7)\)). Denote the spectator sum without half by \(\tau^v\).

The marked scalar identity

We now prove the final finite algebraic identity: \[ S_a R_X=4c_* F_a . \tag{4}\] The pair reductions determine all residues except three simple pole locations relative to the marked parameter. Symmetry then reduces the possible error to a small polynomial ambiguity, which is removed by the explicit one-, two-, and three-site calculations below. Write \(J_a\) for left minus right. Inductively this vanishes as any spectator \(b\to0,\infty\) by decoupling (at infinity \(\ell_- q+P+\ell_+q^{-1}=1\)).

Vacuum denominators beyond those of \(R_X\) cancel by the \(D_+\) inversion formula. Residues at \(b=q^j a'\), \(j=\pm2,\pm3\), for another spectator match by pair reductions and the common \(F_X,R_X\) factors, all decorations regular at generic such points taking \(y=a\).

At \(b=q^{\pm2}a\) residues match similarly by reduction to the fused mark; take the specialization \(y=a,q^{-2}a\) respectively for \(F_X\), decorations regular there and at the fused site. At \(b=q^{\pm3}a\), \(S_a\) is bounded as noted above and \(F_a\) has at most simple poles by \(H^v\).

Thus the only poles left for \(J_a\) can be simple at \(b/a=q^{\pm3},-1\). Also at \(b=q^{\pm1}a\), \(J_a=J_b\) by the crossing consequence above and taking a common specialization for \(F_X\) (all decorations regular in a neighborhood, the pair pole removable by the cancellation already shown). Hence for \(m=N-1\) spectator ratios \(r_j=t_j/a\), \(t_j\in X\setminus\{a\}\), by symmetry and homogeneity, \[J_a=\left(\prod_j B_0(r_j)\right)\sum_{j=0}^m c_j e_j(r),\qquad B_0(r)=\frac{r}{(r-q^3)(r+1)(r-q^{-3})},\] with \(e_j\) elementary symmetric: after factoring out the poles and endpoint zeros only degree \(\le1\) per ratio remains. The constants are label-independent. At \(b=q^\epsilon a\), \(\epsilon=\pm1\), a further ratio \(q^{3\epsilon}\) cannot be a pole, by the equality with \(J_b\), so \[c_j+\epsilon i d\,c_{j+1}-c_{j+2}=0\quad(0\le j\le m-2).\] Subtracting the two relations with \(\epsilon=\pm1\) gives \(c_1=\cdots=c_{m-1}=0\). If \(m\ge3\), either relation then also gives \(c_0=c_m=0\). For \(m=2\) remains \(c_0=c_2,c_1=0\); for \(m=1\) use \(B_0(1/r)=rB_0(r)\) in the same equality to give \(c_0=c_1\).

Winding and contractible polygon masses

The identities just proved give regular analytic continuations of the summed color expressions near homogeneous sites through \(R_X\) and \(S_a\). Let \(\tau_N\) be the homogeneous value of \(\tau_X\) with \(N\) sites. In the spectator sum \(\tau^v\), fix the distinguished parameter at \(a=1\) and let all \(N-1\) spectators tend to \(1\); write the resulting value as \(\tau^v_{N-1}\). Thus \(2F_a=\tau^v_{N-1}\) at homogeneous sites.

Let \(e_1,e_2,e_3\) be the three cylinder edges identified in Section 2.4, one of each honeycomb edge type. A polygon through more than one of them is counted once for each sampled edge it uses. Let \(E_N\) be the sum of \(x_*^{|\gamma|}\) over winding polygons through these three edges, with this multiplicity.

This identity supplies the finite quantity whose asymptotics we study next. The determinant positivity proved below shows that \(\tau_N\) is nonzero, so the identity then expresses \(E_N\) as a ratio of the two continued color sums.

The first response at the two edges.

We now sharpen the first coefficient, for which \(F_1=G_1\). In each edge coordinate \(u_e=N/z\) (right edge in \(x\)) or \(Nz\) (left edge), put \(b_N(u_e)=B_1(u_e/N)^N\to b_\infty(u_e)=e^{-b u_e}\), \(b=2\sqrt3\). Thus \(B_N(z)=b_N(u_e)\). Form \[g_N=(f_1-b_N)/u_e,\qquad h_N=(F_1/b_N-1)/u_e.\] They are analytic in the right half-plane with uniformly bounded boundary \(L^2(du_e)\) norms by (7) and the orthogonality estimates after change of variables. Also \(g_N/b_N\) is analytic in the left half-plane. For each right function its boundary values on \(u_e=i y'\) have Fourier inverse supported on \([0,\infty)\), and interior values are the Laplace transform of that density. Indeed one may close contours (the functions are \(O_N(|u_e|^{-1-1/\alpha})\) at infinity and \(O_N(|u_e|^{-1+1/\alpha})\) at zero), or use the Cauchy integral with denominator \(v'-u_e\) for evaluation at \(v'\) on the right, which by Plancherel picks out the nonnegative half-density. Similarly \(g_N/b_N\) has density supported on the negative half-line.

Take weak \(L^2\) subsequences for \(g_N,h_N\). The support conditions are weakly closed, and Cauchy evaluation is a bounded functional, so their interior values converge. Division by the inner phases \(b_N\to b_\infty\) converges strongly as a boundary multiplier. Consequently the limit \(g\) still has nonnegative density support, while \(g/b_\infty\) has nonpositive density support. Since division by \(e^{-bu_e}\) shifts the density left by \(b\), the density of \(g\) is supported in \([0,b]\).

To read the boundary data for the unscaled limits, use \[\frac{b_\infty(u_e)}{u_e} =\int_b^\infty e^{-u_e t}\,dt.\] The limits \(f_1\to b_\infty+u_e g\) and \(F_1\to b_\infty(1+u_e\lim h_N)\) can therefore be written \[f_{\rm edge}(u_e)=u_e\int_0^\infty e^{-u_e t}\varphi(t)dt,\qquad F_{\rm edge}(u_e)=u_e\int_0^\infty e^{-u_e t}\psi(t)dt ,\] with \(\varphi-1,\psi-1\in L^2(dt)\). The density of \(g\) is confined to \([0,b]\), whereas multiplication by \(b_\infty\) shifts the density of \(\lim h_N\) from \([0,\infty)\) to \([b,\infty)\). Hence, almost everywhere, \[\varphi(t)=1\quad(t>b),\qquad \psi(t)=0\quad(t<b).\]

Determining the edge limit.

Relation (6) passes to these limits on \(u_e>0\), since \(f_1\) in the edge coordinate is bounded by \(1+C\sqrt{u_e}\) and \(\mathsf T\) has enough exponential moments. The convolution dilates \(u_e\) by \(e^{\alpha s'}\), with \(s'\) integrated against \(\mathsf T(ds')\) (either edge by evenness). Laplace uniqueness gives \(\psi=\mathsf T*\varphi\) in the coordinate \(x'=\log(t/b)/\alpha\). These operations are bounded on constants plus \(L^2(dt)\): the density dilation \(\varphi(t)\mapsto\varphi(te^{-\alpha s'})\) has \(L^2(dt)\) norm multiplier \(e^{\alpha s'/2}\), covered by the stated exponential moments.

The factorization now joins the two prescribed half-lines. In \(x'\), set \[q_{\rm edge}=\mathsf T_+^{-1}*\psi=\mathsf T_-*\varphi.\] For \(x'<0\), the first expression vanishes by the supports of \(\mathsf T_+^{-1}\) and \(\psi\). For \(x'>0\), every argument sampled by the second expression lies where \(\varphi=1\), so it equals \(c'=\int\mathsf T_-(ds')>0\). Thus \(q_{\rm edge}=c'\mathbf1_{(0,\infty)}\), which determines the limit uniquely. Applying \(\mathsf T_+\) and integrating gives \[\int_0^\infty F_{\rm edge}(u_e)du_e=\int_b^\infty \psi(t)t^{-2}dt =\frac{c'}b\int e^{-\alpha s'}\mathsf T_+(ds')>0 .\] Absolute integrability follows from support and the stated \(L^2\) bounds; positivity follows from the real Laplace moments of the factors. Both edges have the same limit, without subsequence dependence. To return to the winding activity, use \(J_1(id)=\int L(id-x)F_1(x)dx\). Changing variables to \(u_e=N e^{-\alpha|x|}\) on the two halves, the bound \(|F_1(x)|\le C\exp(-c u_e)\) and the shifted \(L\)-asymptotic give by domination \[ N J_1(id)\ \longrightarrow 2 c_L \alpha^{-1}\cos(\alpha d)\int_0^\infty F_{\rm edge}(u_e)du_e>0 . \tag{8}\]

Projecting the logarithmic forcing.

Keeping \(b_N\) as before, set \[e_N=(f-g)/u_e,\qquad h_N=G/(u_e b_N),\qquad Q_N=(g-j_e)/(u_e b_N).\] The first two are analytic right with square-integrable boundary functions of positive Fourier-inverse support as above: for each \(N\), at zero \(f-g\) is \(O(|u_e|(1+|\log u_e|))\), \(G=O(|u_e|^{1/\alpha})\), and analogous inverse powers occur at infinity. Use \(P_+\) for the orthogonal projection on this boundary support \(t\ge0\). Then \[P_+(e_N/b_N)=-P_+ Q_N .\] Indeed the sum before projection is \((f-j_e)/(u_e b_N)\), rational and analytic in the left half-plane, regular at zero and decaying at infinity, so its Cauchy integral to the right vanishes. Moreover in unweighted boundary norm of \(u_e\), \[\Re\langle e_N,G/u_e\rangle\ge c_T\|e_N\|^2,\qquad \|h_N\|\le C\|e_N\|,\qquad \langle e_N,G/u_e\rangle=-\langle P_+Q_N,h_N\rangle .\] These follow by weighted (6) after dividing by \(u_e\) \((G=\mathsf T*(f-g))\), then by the boundary phase and support.

We check that \(P_+Q_N\) is uniformly bounded and converges weakly with right Laplace value at \(v'>0\) equal to \[\kappa\int_0^\infty \frac{e^{-v't}}{b+t}dt .\] Its Cauchy integral with denominator \(v'-u_e\), up the imaginary axis with factor \(1/(2\pi i)\), can be rotated to rays \(\arg u_e=\pm(\pi/2+\epsilon)\) for small fixed \(\epsilon>0\), oriented in and out respectively on lower and upper. There are no poles between and (9) justifies the endpoint passages. On those rays subtract \((\kappa\log N+k_e)e^{bu_e}\), which has zero integral by closing left. The resulting data before the Cauchy kernel are bounded uniformly in \(L^2\) there and converge weakly to \(-\kappa e^{bu_e}\log u_e\). Indeed for \(r=|u_e|\le N\) use (9) and \[|b_N^{-1}|\le e^{-cr},\qquad |b_N^{-1}-e^{bu_e}|\le C(r^2/N)e^{-c' r}\] on the rays (the expansion of \(B_1(u_e/N)^{-1}\) near zero, with strict modulus suppression along the left rays). For \(r\ge N\), \(Q_N=O((N/r)e^{-cN^2/r})\), and \[\int_N^\infty (N/r)^2e^{-2cN^2/r}\,dr =\int_0^N e^{-2c v}\,dv\le (2c)^{-1},\qquad v=N^2/r.\] Thus these tails are uniformly bounded in \(L^2\) and converge weakly to zero: their support leaves every compact interval, so one first tests against compactly supported \(L^2\) functions and then uses density.

The ray Cauchy evaluations are uniformly bounded in \(L^2\) up to the imaginary boundary from the right, by angular separation: the kernel bound \(C/(r+|\Im v'|)\) is bounded by the Schur test with power weight \(-1/2\). They therefore bound the boundary projections by Plancherel, and convergence of interior evaluations determines weak convergence. Rotating the limiting rays to the two banks of the negative real axis gives the log jump \(\kappa\int_0^\infty e^{-b s'}/(v'+s')\,ds'\). By Fubini this is \(\kappa\int_0^\infty e^{-v't}/(b+t)\,dt\), the claimed Laplace value.

The prescribed density tail and its sign.

Consequently \(e_N,h_N\) are uniformly bounded in \(L^2\). Taking weak subsequences as before, their limits give \[\lim e_N(u_e)=\int_0^\infty e^{-u_e t}\varphi(t)dt,\qquad G_{\rm edge}(u_e)=\lim u_e b_N h_N=u_e\int_0^\infty e^{-u_e t}\psi(t)dt\] on the right with \(\varphi,\psi\in L^2(dt)\) and \(\psi=0\) for \(t<b\). Dividing by the convergent boundary phases shifts the limiting density left by \(b\). The projection identity and the just-computed Laplace value therefore give \[\varphi(t)=-\kappa/t\qquad(t>b).\] This replaces the constant tail \(\varphi=1\) in the essential response. Again (6) dilates to \(\psi=\mathsf T*\varphi\) in \(x'=\log(t/b)/\alpha\); here the bound \(|u_e e_N(u_e)|\le C\sqrt{u_e}\) on positive \(u_e\) justifies the passage to the limit. The common intermediate function \[q_{\rm edge}=\mathsf T_+^{-1}*\psi=\mathsf T_-*\varphi =\begin{cases} 0,&x'<0,\\ -(\kappa/b)c_-e^{-\alpha x'},&x'>0, \end{cases} \qquad c_-=\int e^{\alpha s'}\mathsf T_-(ds')>0\] is determined by the same two support arguments. It determines both edge limits uniquely. Applying \(\mathsf T_+\) and integrating now gives \[\int_0^\infty G_{\rm edge}(u_e)du_e=\int_b^\infty \psi(t)t^{-2}dt =-\frac{\kappa c_-}{2 b^2}\int e^{-\alpha s'}\mathsf T_+(ds')<0 .\] We recover \(J(id)=(L*S)(id)+(L*G)(id)\); the first term is \(N\gamma_0\) for a fixed constant. In the second, on real \(x\) we have \(|G(x)|\le C\sqrt{u_e}\exp(-c u_e)\) for \(u_e=N e^{-\alpha|x|}\) by the \(h_N\) bound. Thus exactly the same kernel limit as in (8) applies to that term, giving \[ J(id)=N\gamma_0-\gamma_1/N+o(1/N),\qquad \gamma_1>0 . \tag{10}\]

Vacuum fusion and saturated components.

First note two polynomial consequences of the vacuum construction. One is the following \(D_1\) fusion, valid in either parity: \[ D_1(Y,q^{-1}x,qx)=\sqrt2 x\prod_{t\in Y}(t+x)\ D_1(Y,x). \tag{25}\] Indeed by symmetry and the \(q^3\)-recursion (place the pair first), at two spectators \(u,q^3u\) both sides reduce with matching factors since \[\frac{\prod_{\pm}(q^{\pm1}x-q^{-3}u)(q^{\pm1}x-q^{-2}u)} {(x-q^{-3}u)(x-q^{-2}u)}=(x+q^3u)(x+u).\] Induct in \(m=|Y|\), starting with \(D_1(a,b)=a+b\). Degree in a spectator is at most \(m+1\); zero and leading infinity match by deletion, and the \(2(m-1)\) pair nodes just found suffice when \(m\ge2\). For \(m=1\) test additionally at that spectator \(q^2x\): deletion of \(q^{-1}x,q^2x\) on the left gives \(2(1+q)(1+q^3)x^3=\sqrt2(1+q^2)^2x^3\).

The other consequence for size \(2k\) is a saturated component (\(s=\pm1\)): \[\Psi(z)_{s^k\,(-s)^k} =(d_0 q^s)^k \ \frac{\prod_{i<j,\ {\rm same\ spin}}S_2(x_i,x_j)S_3(x_i,x_j)} {\widetilde D(X)},\quad S_l(x,y)=\frac{z_y}{z_x}Q^{-l}-\frac{z_x}{z_y}Q^l=-iP_l(\tau).\] Indeed stripped \(V_{2k}\) has degree at most \(2k-2\) per occupied variable; same-spin swaps act by scalar \(w\), hence zeros at \(x_j=q^2x_i,q^3x_i\) for ordered pairs within either run (bring them adjacent within the run). The resulting product saturates the bound, giving the formula up to its stated constant. At \(x_k=x,\ z_{k+1}=Q^3 z_k\) adjacent deletion gives \(q^s\) times the smaller normalized vacuum component. Here \(\widetilde D(X)/\widetilde D(X_{\rm red})=d_0\prod_{t\in X_{\rm red}}S_{-2}(x,t)S_{-3}(x,t)\) directly by the \(D_1\) recursion (a sign per spectator, even number); the extra pair factors upstairs are precisely the displayed spectator product using \(S_{l+8}=-S_l\). This proves the constant by induction.

Resolving the two blocks.

On each block alone use the open-row operators \(M_{p',p'}\) with \(p'=-1\) on \(R\), \(p'=+1\) on \(L\), and generic auxiliary arguments. In block spin basis they are triangular: at the first change between input and output the output-minus-input spin equals the new intermediate auxiliary spin minus \(p'\), by charge conservation. Thus lex order can only increase for \(p'=-\), decrease for \(p'=+\). Diagonal eigenvalues assigned to a color list are products of \(w,v,W\), respectively for colors \(-,0,+\) meaning actual diagonal-list spins \(p',0,-p'\). These spectra are simple generically: local numerator zero exponents in the auxiliary square relative to the site square are \(\{-2,-3\},\{0,-3\},\{0,-1\}\), disjoint from other generic site orbits. Insert the identity in the form of the tensor product spectral resolution in \(\mathscr H\) between the vacuum components; the block charges commute and require precisely the color-sum constraint. The factor \(\eta^{S_L}\) is scalar at those charges.

Each term with a zero color \(x\) equals the corresponding term on a larger system replacing that site by \(q^{-1}x,qx\) in order (roots \(Q^{-1}z,Qz\)), colored \(-,+\). Indeed positive-order pair fusion inserts the map \(A\) on the lower vacuum, and \(A^t\) on the inverse-site vacuum gives the shorter inverse vacuum, by the same vacuum argument as in the marked-row reductions. The spatial transport fuses the open row on this subspace, and all block charges agree. The chosen spectral label lifts uniquely: zeros for the two local numerators now use one choice from \(\{\{-3,-4\},\{-1,-4\},\{-1,-2\}\}\) and one from \(\{\{-1,-2\},\{1,-2\},\{1,0\}\}\), with common denominator exponents \(1,2,3,4\); all nine multisets are distinct modulo 8. The required fused vacancy zeros thus match just \((-,+)\). So the spectral projections intertwine on the inserted subspace. These arguments apply also to several simultaneously split generic sites, by the same simple spectra, rational specialization, and nonvanishing of denominators generically there by (25). This reduces all projection terms to full occupancy.

Evaluating a fully occupied term.

Now work first on a generic fully occupied term (then specialize as above). Sort the variables on each block separately to put \(-\) before \(+\), retaining the two internal orders. Simultaneous covariance is by the ordinary exchanges, inverse-transposed in the inverse-site vacuum. In the sorted order the left eigenvector in each block is just its indicated spin bra: it is the lexicographic extreme of its charge with no inputs other than itself to that row. In reversed-run order (\(+\) before \(-\)) the right eigenvector similarly is the bare ket. Transport it to sorted order by the block-exchange braid; contracting with the inverse-site vacuum now reads the latter in reversed-run order, by transposed exchanges. The normalizing left-right pairing on each block is \(\prod_{a\ {\rm of\ color}\ -,\, b\ {\rm of\ color}\ +} W(z_b/z_a)\). Indeed move the minus-colored sites from right to left one at a time over the plus run. Each moved spin has to be \(p'\) on arrival (it stays fixed from then on); total charge forces the plus run still saturated then, leaving only the spin-preserving-along-labels swap at every crossing, of entry \(W\).

Thus use the two saturated amplitudes above: the lower one after starting its circle at the sorted \(R_+\) run (to read \(+\) first), the inverse upper one at \(R_-\) (to read \(-\) first), or at \(P\) if the indicated run is empty. Recovering the original amplitudes gives a factor \(\eta^{|R_+|-|R_-|}\) canceling \(\eta^{S_L}\). The constants now give \(d_0\) per site, and the denominators are \(\widetilde D(X)\widetilde D(X^{-1})\). For like colors on the same block the product contributes the \(S_l\) at \(l=\pm2,\pm3\); for unlike colors in order minus then plus the normalization gives \(1/W=S_{-2}S_{-3}/(S_0 S_1)\). For a pair first on \(R\) then \(L\), equal actual spins occur at opposite colors \(s,-s\); the two saturated orders place that plus-spin pair first as \(R,L\) in lower and \(L,R\) in inverse, or oppositely for minus spins. Thus the factors are \(S_{2s}^2 S_{3s}^2\); similarly with arguments reversed. Putting either mark first versus a color \(t\), it meets that color with like spin on exactly one side in order giving \(S_{-2t} S_{-3t}\) (two reversed-argument signs cancel). This proves the nonzero-color rules after \(S_l=-iP_l,\ P_{l+8}=-P_l\), with even-size inversion of \(\widetilde D\) as in the Pfaffian formula below.

Restoring the vacant colors.

Finally (25), normalized, gives \(\sqrt2\prod_y P_4(\tau_{xy})\) per split in each denominator. The internal new pair contributes \(1/W(Q^2)=2+\sqrt2\), so local color-zero weight after removing the constants is \(d_0^2(2+\sqrt2)/2=1\). All other factors just shift their lags as in the rules and pay \(2e_4\) from the denominator. For clarity, when expanding the first of a pair the new parity adds the two previous parities and \(\ell_0(-,t)+\ell_7(+,t)\) from wrapping the index range; for expanding the second it uses \(\ell_7(s,-)+\ell_0(s,+)\) instead. Applying this to the displayed base tables gives the within-block sign rows \((0,1,0),(0,0,1),(1,0,0)\), zero parities across blocks, and positive sign at a mark paired with a vacancy. This completes (24).

The exact formula and a comparison Pfaffian.

For even \(N\), the empty component of the \(p=1\) vacuum is \[D_1(X)=\prod_{i<j}\frac{p_K(x_i,x_j)}{x_j-x_i} \operatorname{Pf}\left[\frac{x_j^2-x_i^2}{p_K(x_i,x_j)}\right]_{i,j=1}^N, \qquad p_K(x,y)=x^2+\sqrt2xy+y^2.\] Here \(x_i=t_i\) in the vacuum notation. Skew alternation cancels the Vandermonde denominator, and each possible \(p_K\)-pole is canceled by the outside product, so this is a symmetric polynomial of the required degree. At \(x_j=q^3x_i=q^3u\), only the Pfaffian term contracting these two sites survives. With the pair adjacent, its prefactor is \((1+q^3)u\prod_{k\ne i,j}(x_k-q^{-3}u)(x_k-q^6u)\), exactly the empty-component recursion. Symmetry gives the reversed pair ratio as well. These values determine the polynomial by interpolation in any site, starting with \(D_1(x_1,x_2)=x_1+x_2\). Put \[\widetilde D(X)=D_1(X)\prod_i x_i^{-(N-1)/2}.\] Inverting all arguments preserves \(\widetilde D\), with consistent logarithms; the signs from the outside product and the Pfaffian cancel because \(N\) is even.

Take \(x_j=e^{i\theta_j}\), with all angles in a small fixed neighborhood of an interval of width \(\pi/4\). Fix that neighborhood throughout the argument. Its width \(\Theta\) is less than \(\pi b'\), where \(b'=3/4\), so after centering the angles the numbers \(p_j=e^{i\theta_j/b'}\) satisfy \(\Re p_j\ge a>0\) with a fixed \(a\). The estimates below are uniform over these arrays, without a minimum separation condition. Define the continuous function \[g(s)=\frac{(2\cos s+\sqrt2)\tan(s/(2b'))}{2\sin(s/2)}, \qquad g(0)=\frac{2+\sqrt2}{2b'}.\] We will prove \[ |\widetilde D(X)|^2\ge (2b')^{-N}\prod_{i<j}g(\theta_i-\theta_j)^2\, N^{5/24-o(1)}. \tag{26}\] The error depends on the fixed angular neighborhood, but not on the array or its separation. We first use distinct angles and pass to collisions at the end.

For \(w=\log(x_j/x_i)\), the Pfaffian kernel above has the representation \[\frac{\sinh w}{\cosh w+1/\sqrt2} =2\int_0^\infty \sin(wT) \frac{\cosh(\pi T/4)}{\sinh(\pi T)}\,dT.\] One can verify it for real \(w>0\) by integrating the full exponential in principal value in the upper half-plane, with upper edges at half-integer heights. The residues give \(i(1+2\sum_{n\ge1}(-1)^n\cos(n\pi/4)e^{-nw})\); analytic continuation gives the stated formula on the required strip. Replacing the density \(\cosh(\pi T/4)/\sinh(\pi T)\) by \(1/(2\sinh(\pi b'T))\) gives the comparison kernel \[K_0(w)=\frac1{2b'}\tanh\!\left(\frac{w}{2b'}\right).\] Its Pfaffian is \[\operatorname{Pf}[K_0(\log(x_j/x_i))] =(2b')^{-N/2}\prod_{i<j} \tanh\!\left(\frac{\log(x_j/x_i)}{2b'}\right).\] This is the Schur skew-Cauchy identity: induction compares the residues at \(x_i^{1/b'}=-x_j^{1/b'}\), and alternation removes the remaining constant. Substitution into the outside product defining \(\widetilde D\) gives a comparison magnitude whose square is exactly \((2b')^{-N}\prod_{i<j}g(\theta_i-\theta_j)^2\).

A positive ratio and its counting density.

Order the distinct angles once, and let \(\mathcal R_N\) be the ratio of the original kernel Pfaffian to the comparison kernel Pfaffian. Expanding the comparison Pfaffian by wedging its two-vector integral kernels gives a probability law of \(N/2\) variables \(T_k>0\), with density proportional to \[\det[e^{\pm\theta_jT_k}]_{j;(k,\pm)} \prod_k\frac{dT_k}{\sinh(\pi b'T_k)}.\] Choose the constant sign of the determinant to make this density positive. Its sign is constant: after ordering the exponent columns, total positivity follows because a nonzero linear combination of \(r\) distinct real exponentials has at most \(r-1\) distinct zeros (divide by one exponential, differentiate, and induct). Permuting entire column pairs has even sign. Alternation within each pair gives integrability at \(T_k=0\); the angular width \(\Theta<\pi b'\) gives integrability at infinity. The same fixed phase occurs in both Pfaffians and cancels in their ratio. Thus \[\mathcal R_N=\mathbb E\prod_{k=1}^{N/2}r(T_k)>0, \qquad r(T)=\frac{(1+e^{-u})(1-e^{-3u})}{1-e^{-4u}} =1+\frac1{2\cosh u}\ge1, \quad u=\pi T/2.\] Define the one-point counting density \(\rho_N\) by \(\mathbb E\sum_k f(T_k)=\int_0^\infty f(T)\rho_N(T)\,dT\). Jensen’s inequality now reads \[ \log\mathcal R_N\ge \int_0^\infty\rho_N(T)\log r(T)\,dT. \tag{27}\] We will show that for each fixed \(T>0\), \[\rho_N(T)=\frac{b'}\pi\log N+O_T(1),\] uniformly over distinct arrays. This concerns the density at bounded \(T\); the law itself always has exactly \(N/2\) variables.

Write \[K_{ij}=\frac{p_i-p_j}{p_i+p_j},\qquad w_j(y)=\frac{e^y-p_j}{e^y+p_j},\qquad B(y)=\prod_jw_j(y), \qquad k=b'T.\] The skew integrals of \(f_i^+f_j^--f_i^-f_j^+\), where \(f_j^\pm=e^{\pm\theta_jT}=p_j^{\mp ik}\), against \(dT/\sinh(\pi b'T)\) form the matrix \(K/(ib')\). Taking the half log-determinant derivative of this Pfaffian therefore gives the exact density formula \[\rho_N(T)=\frac{ib'}{\sinh(\pi k)} (p^{ik})^tK^{-1}p^{-ik}.\] The matrix is invertible for distinct angles in the fixed interval. Its inverse is controlled through the skew Schur complement in the same Pfaffian identity: \[w(y)^tK^{-1}w(y')= h(y-y')[B(y)B(y')-1],\qquad h(s)=\tanh(s/2).\] Also, contour shift followed by residues or the Euler integral gives \[\int_{\mathbb R}e^{iky}w_j'(y)\,dy=c(k)p_j^{ik}, \qquad c(k)=\frac{2\pi k}{\sinh(\pi k)}.\] These two identities reduce the density estimate to a uniform bound for a mixed derivative.

The logarithmic local density.

Put \(L_N=\log N\), \(I_N(y)={\bf1}_{\{|y|<L_N\}}\), and \(J_N=1-I_N\). Uniformly in the site array, \[|B(y)|\le e^{-cNe^{-|y|}},\qquad |B(y)-1|\le CNe^{-|y|}\quad (|y|>L_N),\qquad \|B'\|_1=O(1).\] The second estimate uses even \(N\) at the negative end. For the third, product differentiation gives \(|B'(y)|\le CNe^{-|y|}e^{-c(N-1)e^{-|y|}}\). Integrating the first bound on the central interval and the second on its complement yields \(\|B-J_N\|_1=O(1)\). All constants use only the fixed lower bound on \(\Re p_j\).

For \(F(y,y')=h(y-y')[B(y)B(y')-1]\), differentiation gives \[\begin{aligned} \partial_y\partial_{y'}F ={}&h''(y-y')[1-B(y)B(y')]\\ &+h'(y-y')[B(y)B'(y')-B'(y)B(y')]\\ &+h(y-y')B'(y)B'(y'). \end{aligned}\] The last two terms have bounded \(L^1(dy\,dy')\) norm, by \(|B|\le1\), \(\|B'\|_1=O(1)\), and integrability of \(h'\). In the first term replace both factors \(B\) by \(J_N\), at bounded \(L^1\) cost. The difference between the result and \(h''(y-y')I_N(y)\) is \(h''(y-y')J_N(y)I_N(y')\); it too has bounded \(L^1\) norm because \(h''\) has a finite first absolute moment. Consequently \[\partial_y\partial_{y'}F =h''(y-y')I_N(y)+\mathcal E_N(y,y'),\qquad \|\mathcal E_N\|_1=O(1).\] Integrate against \(e^{ik(y-y')}\). The interval selected by \(I_N\) has length \(2\log N\), and \(\int e^{iks}h''(s)\,ds=-ikc(k)\). The derivative of the left-hand Schur-complement expression therefore gives \[c(k)^2(p^{ik})^tK^{-1}p^{-ik} =-2ikc(k)\log N+O(1).\] Substitution in the density formula proves the asserted \(\rho_N(T)/\log N\to b'/\pi\), uniformly for each fixed \(T>0\).

Finally \(\rho_N\log r\ge0\), so Fatou applied to (27) gives, along any sequence of distinct site arrays, \[\liminf_{N\to\infty}\frac{\log\mathcal R_N}{\log N} \ge\frac{b'}\pi\int_0^\infty\log r(T)\,dT=\frac5{48}.\] For the integral expand the three logarithms in the displayed formula for \(r\): its value is \(\frac2\pi(\pi^2/12-\pi^2/18+\pi^2/24)=5\pi/36\). Since the argument applies to every sequence of arrays, the lower bound has a uniform \(o(1)\). Squaring the ratio and restoring the comparison magnitude proves (26) for distinct sites. For each fixed \(N\), continuity of \(\widetilde D\) and of the product of \(g\)’s then proves the same bound at colliding sites. No inverse matrix needs to be continued through a collision.

The charge space and its field phases.

The space has basis symbols \(\alpha_{ij}=C^j\alpha_{i0}\) (\(i=1,2\), denoting sectors \(R,L\), and \(0\le j<8\)), quotiented by the radical. The two diagonal blocks of the symmetric Gram are circulant with first row \((2,-1,1,0,0,0,1,-1)\), cross block \((0,0,0,-2,2,-2,0,0)\); \(C\) is the simultaneous forward shift. Write \(\mathcal L\) for the integral span as usual. Define \(h_1\) by pairings \((1,0,0,0,0,0,1,0)\) in its sector, \((0,0,0,-2,0,0,0,0)\) in the other, \(h_2\) symmetrically; \(v\) by \((0,0,0,-1,0,0,1,0)\) in both. Let \(J\) be the alternating form with within-sector first row \((0,1,1,0,0,0,-1,-1)\) in sector 1, its negative in sector 2, zero across. Put \(p_C=\sum C^j h_1/8\). These descend as indicated: the nullspace consists of span of sector-1-even plus sector-2-odd sums and the reverse sum (diagonalize by eighth roots); \(p_C^2=1/4,\ (p_C,\alpha_{ij})=(-1)^{i+1}/4\). The \(J\)-pairing and ordinary pairing coincide modulo 2 on \(\mathcal L\times(\mathcal L+\sum_i\mathbb Z h_i)\); and \(a^t J v+(a,v)+3(a,p_C)\in2\mathbb Z\) there for \(a\in\mathcal L\). All these checks just multiply the circulant data (in solving for pairings remove the two null vectors).

Use the same Fock states and \(W_b\) as before on \(\mathcal L^*\), but with momentum phase \[\exp(\pi i[b^t J P/2-3k_b(p_C,P)])\] (\(P\) momentum, \(k_b=0\) for lattice and color charges, \(k=1\) per mark \(v\), \(-2\) for compensator, extended additively and preserved by \(C\)). The oscillator trace is twisted initially just by \(\widehat C Q_L^H\) (no other sign), scaled as before; for stationary trace comparisons we also allow a character \(e^{2\pi i(\sigma,P)}\) with \(\sigma=0\) or \(2v\). Use \(X_b=[z^0]W_b\) for lattice roots of norm square 2. The same residue commutators hold with the new cocycle (coefficient \(e^{\pi i a^t J b/2}\) for root \(a\) acting when \((a,b)=-1\)), by the parity identities above.

The finite conjugation.

Within each sector \(i\) introduce labels \(j=1,2,3,4,E\) with \[\begin{gathered} r=(\alpha_{i0},0,-\alpha_{i1},\alpha_{i2},-\alpha_{i1}+\alpha_{i7}),\\ e_{jk}=r_j-r_k,\quad H_k=h_i+r_k-r_1,\quad E_{jk}=e^{-\pi i e_{jk}^t J H_k/2} X_{e_{jk}} . \end{gathered}\] Use this only for \(1234\) and the extra \(E3,E4\), which are roots. The \(1234\) differences have ordinary \(A_3\) root pairings, with \((e_{jk},H_l)=\delta_{jl}-\delta_{kl}\); across sectors these two \(A_3\)’s are orthogonal. Thus we again get commuting \(SL_4\) operators with matrix action on the four fields \(W_{H_l}\). Also \([E_{E3},E_{34}]=E_{E4}\), central for those three, by the residue formula. The exponentiation identities here work as before, using finite-dimensional modules along positive root subspaces (including the \(A_2\) form on the \(E3,34\) span). In particular no \(SL_5\) claim on the entire list including \(E\) is needed.

Set \(c=q^{-3}\). In each \(SL_4\) use lifted operators \(G_{0i},\widehat S_i,D_i\) for \[\begin{gathered} G_0=\begin{pmatrix}1&0&0&0\\c&1&0&0\\c^2&c&1&0\\0&0&0&1\end{pmatrix}, \qquad S=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\text{ on }12,\\ D_i=\exp(-\pi i(\alpha_{i0},P)/4)\quad(\operatorname{diag}(-1/c,-c,1,1)). \end{gathered}\] Put \(t=-c^2,\ k_1=-\mathrm{i}/c,\ k_2=\mathrm{i}/c,\ V_i=\exp(k_iE_{E3})\exp(t E_{34})\), \(g=G_{01}G_{02}V_1 V_2\). These two sector actions in \(g\) commute by nonnegative cross pairings (and orthogonality for \(A_3\) across sectors). Then \(g\) sends \(W_{h_i}\) by conjugation to the desired \(W_{H_1}+cW_{H_2}+c^2W_{H_3}\), leaving \(W_v,W_{-2v}\) fixed; indeed the \(V_i\) commute with these initial pure fields and \(G_0\) uses just the three-label subalgebra. Write \(A=S_{\alpha_{10}}S_{\alpha_{20}}C,\ \widehat A=\widehat S_1\widehat S_2\widehat C\).

We have \[g^{-1}\widehat Cg=\mathcal U D_1D_2\widehat A\] where the unipotent acts by a product of \(\exp(u X_b)\) (unit-modulus or fixed complex coefficients immaterial except as specified below), with rightmost-first root order across sectors grouped as: \(b=A e_{34}\) (call it \(e_{1F}\)), then \(e_{12}, e_{24}\), then \(e_{E3},e_{E4}\), each in both sectors. This fact is used with separate small contour perturbations within the groups. Further, if suppressing just \(1F,12,E3\) in sector 1, the resulting entire twist equals \[ \exp(-\mathrm{i}c E_{E4}^{(1)})\exp(c E_{24}^{(1)})\, g_2^{-1}D_1\widehat S_1\widehat C g_2,\qquad g_2=G_{02}V_2, \tag{28}\] where the initial two factors also commute with \(g_2\).

We verify the factorization, including the surviving coefficients in (28). Their values will matter in the partial trace. In either sector \(\widehat C\) sends the \(123\) algebra by conjugation to negative transpose with labels sent to \(324\), conjugated by \(\operatorname{diag}(-1,1,-1)\) on the renamed triple (indeed \(E_{12}\to E_{23}, E_{23}\to E_{42}\)); it sends \(E_{E3}\) to \(s_i E_{12}\), \(s_1=\mathrm{i},s_2=-\mathrm{i}\). These follow directly using the normalization: phase ratios for those three generators in units \(\pi i\) are \(0,0,(-1)^{i+1}/2\). Thus \(\widehat A\) sends \(E_{E3}\to-s_i E_{21}\). Also \(\widehat A\) sends \(E_{34}\) to a scalar root operator of charge \(e_{1F}\) (justified below). Now \(V^{-1}\prod_i U_{0i}(\widehat A V\widehat A^{-1})\), where \(V=V_1V_2\) and \(U_0=G_0^{-1}(\widehat C G_0\widehat C^{-1}) S^{-1}\) in each group, multiplies \(\widehat A\). Rearrange on the left first moving the \(E3\)’s outward: this gives \(\exp(-k_i E_{E3}-t k_i E_{E4})\) per sector on the far left, leaving \(\exp(-t E_{34})\). On the far right before \(\widehat A\) retain both \(1F\) root exponentials, leaving inner \(\exp(-E_{21}/c)\)’s. This ordering uses cross-commutation of the generating \(V\)-terms. In each group the middle computation now is \[(1-t E_{34})G_0^{-1}(\widehat C G_0\widehat C^{-1}) S^{-1} (1-E_{21}/c)=(1+cE_{24})(1-E_{12}/c)D_i\] in ordinary matrix notation. Move \(D_1D_2\) to the outgoing \(\widehat A\), just scaling the \(1F\)’s. This proves the order and coefficients. Keeping only \(E4,24\) in sector 1, move those surviving factors to the far left (nonnegative cross pairings); the calculation in sector 2 then reverses to (28). Indeed \(D_1\) commutes with the inner sector-2 algebra, \(g_2\) and the \(E3,E4\) row factors there; the \(D_1\)-conjugation of sector-2 \(1F\) is retained as prescribed. Likewise any indicated character commutes with the conjugations by integrality.

The Weyl phases.

For the Weyl operator \(\widehat S_i\) the action reflects oscillator modes and sends a pure momentum \(n\to S_\alpha n\) with phase \[\exp(\pi i[k\,\alpha^t J(h_i-n)+k^2-k]/2), \qquad \alpha=\alpha_{i0},\quad k=(\alpha,n).\] For \(k\ge0\) use \(E_{21}^k/k!\) on the highest state, as in the previous angular calculation; the Vandermonde contributes \((-1)^{\binom{k}{2}}\). For \(k<0\) use \((-E_{12})^{-k}/(-k)!\); the same formula results. Modes transform by their adjoint triples as before. This shows conjugation carries lattice root operators to scalar multiples of reflected ones: the possible extra momentum dependence in the phase relative to the new cocycle reduces to \(k[(\alpha,b)-\alpha^t J b]\in2\mathbb Z\). It also shows the total \(D_1D_2\widehat A\) uses oscillator action \(A\) and just the product of these two Weyl phases on \(Cm\) for starting momentum \(m\), and the displayed diagonals on \(A m\). Finally \(C h_i=A h_i,\ C v=A v\), and these images pair nonnegatively with all indicated unipotent roots, orthogonally with the Weyl roots. Thus the fields for a shifted color or mark group, conjugated by \(g^{-1}\widehat Cg\), are exactly the pure images under \(C\), without further scalar.

Fusing the site cluster.

Use \(\mathcal L^*\) for all momenta, cocycle as above \[\exp(\pi i[b^t J P/2-3 k_b(p_C,P)]),\quad p_C=\Pi_C h_1,\quad k_b\text{ the external }v\text{-label coefficient}.\] Thus \(J\) has first row \((0,1,1,0,0,0,-1,-1)\) in the first sector, opposite in second, zero across. Color fields are the combinations with coefficients \(1,q^{-3},q^{-6}\) on the angular side, of charges \(h_i,h_i-\alpha_{i0},h_i-\alpha_{i0}-\alpha_{i1}\), denoted \(b_{ic}\) with \(c=-1,0,1\). The two mark insertions have charge \(v\); add the compensator of charge \(-2v\) first, at \(s_*=u_*\) real positive. Work with \(0<u_*<L\). Use unshifted and shifted cluster positions \(s_j=2\pi i e_j+\epsilon\lambda_j\) (\(e_j=0,1\)), taking unshifted angles slightly positive and shifted slightly less than \(2\pi\), all distinct. The corresponding lattice site parameters are proportional to \(\exp(-s_j/8)\); in the limit their ratio between classes is \(q^{\pm1}\).

The two marked sites have a common shift \(e\in\{0,1\}\), as required in the two-mark bound. Write \(\mathcal V_j(s_j)\) for the color combination or mark field just specified, evaluated at \(z(s_j)=e^{-2\pi i s_j/L}\) and scaled field by field by \((2\pi/L)^{b^2/2}\). Let \(\mathcal V_*(u_*)\) be the similarly scaled compensator. The normalized angular trace is \[\mathcal T_\epsilon(L,u_*)= \frac{\mathop{\mathrm{Tr}}\!\left[\widehat C Q_L^H \left(\prod_j^{\leftarrow\mathrm{ang}}\mathcal V_j(s_j)\right) \mathcal V_*(u_*)\right]} {Z_C\Theta_C^{\mathrm{ang}}(L)},\qquad \Theta_C^{\mathrm{ang}}(L)= \sum_{t\in\mathbb Z}Q_L^{(4tp_C)^2/2}.\] The arrow orders the fields latest first in angle. We use the coefficientwise trace convention of Section 4, and then the continuation justified below.

Moving the shifted (latest) group cyclically over the twist brings them to positions \(s'_j=s_j-2\pi i e_j\), earliest, by \(Q_L^H W(Q_L^{-1}z)=W(z)Q_L^H\), with charges and fields now acted on by \(\widehat C\). There is no extra cocycle because \(J,p_C\) are preserved. Conjugate then as above. Thus the pure cluster charges are \(u_j=A^{e_j}h_i\) (or \(A^{e_j}v\) at the marks). With \(j\prec k\) denoting their fixed post-move chronological order, define the divided trace \[ \mathcal T(L,u_*)= \lim_{\epsilon\to0^+} \frac{\mathcal T_\epsilon(L,u_*)} {\prod_{j\prec k}(s'_j-s'_k)^{(u_j,u_k)}}. \tag{29}\] Fix the branches throughout. On the conjugated side the cluster fuses to its total charge. The division supplies exactly the vertex scaling as for (15); cluster cocycles contribute only a fixed unit factor.

As before all formal equalities are first read at fixed \(H\) and field powers (with the spread estimates for finiteness during cut changes), then evaluated near small \(Q_L\) by angular Gaussian and absolute oscillator bounds. The ordering for the post-conjugation root contours can put the later group on the far side of the external positions by commuting, as described in the ray estimate. Compensator-to-shifted angular separation in the original \(C\)-side tends to an entire turn, but \(0<u_*<L\) separates the points for continuation near confluence. Thus the \(C\)-side has a divided Laurent expansion with at worst finite pole, real-analytic coefficients at all positive \(L\) (keeping \(u_*/L\) fixed); negative coefficients vanish by the small-\(Q_L\) equality. The fused-vertex continuation follows by the analytic bound (39) on the radial series.

The angular limit.

For each fixed finite site array, we evaluate \(|\mathcal T(L,u_*)|\) by first letting \(L\to\infty\) at fixed \(u_*\), then \(u_*\to\infty\). The array size will grow only after these limits. Terms require \(\Pi_C d=0,\ d=\sum b_j\) now including compensator, i.e. equal color sums in both sectors since \((p_C,b_{ic})=(-1)^i c/4,\ (p_C,v)=0,\ p_C^2=1/4\). Conversely every such term has a momentum solution \(m=n+d,\ n=Cm,\ m=-K_C d+t p_C,\ K_C=(C-I)^{-1}\) off the invariant line, with a full arithmetic progression of \(t\) of spacing 4. Indeed the within-orbit differences of pairings \((\alpha_{is},K_Cd)\) are integers by balance and \(d\in\mathcal L^*\); also \((\alpha_{10}+\alpha_{20},K_C b_{ic})\) is integral (multiply the eight shifts). The two marks cancel the compensator in \(d\). All terms using \(J,k_b\) are independent of \(t\); the heat and angular-power sum divided by the invariant partition tends to 1 at bounded \(u\), uniformly locally. The products converge as in (16), now to pair products of \(P_l(\tau)=2\sin(\pi(l+\tau)/8)\) with \(\tau=i(s_j-s_k)/(2\pi)\) for chronological \(j<k\), raised to powers \((C^l b_j,b_k)\), and scaled self factors \(8^{-b_j^2/2}\prod_{l=1}^7 P_l(0)^{(C^l b_j,b_j)/2}\). Although the exponents can have nonzero mean, constants cancel globally by \((\Pi_C d)^2=0\) (subtract the same reference log-product for offsets \(8m+4,8m+4\) for each \(l\)). Branches are positive real for real \(0<\tau<1\). The limits commute with Laurent coefficient extraction by the same groupwise uniform bounds after factoring finitely many singular powers. The compensator pairs then tend to unit phases \(\exp(2\pi i(K_Cv,b))\) since its invariant projection is zero and \(\Im\tau\to+\infty\).

Pair powers and phases.

For explicit comparison with the colored sine products write \[H=(4,2,-2,-4,-2,-4,-2,2)/3,\qquad H'=(4,2,-2,-4,-2,2)/3 .\] Subtracting \(H_l\) from the \(C\)-pair exponents gives exactly the integral color exponents \(\ell_l\) above (multiply the circulant shifts); for two marks instead the difference is \(1/4\) constant. The signs for these integral factors in chronological angular order are \((-1)^{\sigma_{ij}}\) with \(\sigma=1\) on the directed within-sector cycle \(-\to0\to+\to-\), zero otherwise (for an external mark paired with an occupied color also a minus, canceling since there are two marks). This description works even if the color product was first given in another order: reversal negates \(P_0\) with parity exactly swapping the signs on the cycle and reflects the other indices without sign.

In units \(\pi i\), the \(J\)-phase is minus the ordered-pair sum of \(b_j^t J K_C C b_k\), including half of the self terms, by \(n=-K_C C d\) off the invariant line and skew symmetry. Add the correction \(-3\sum_{j<k} k_{b_k}(p_C,b_j)\). The net from these is precisely the required sign up to global phases. Here are checks: putting \(M=JK_CC\) the cross-sector color table is zero, and subtracting the central entry and \(D_i c c'\) in \(b_{ic}^t M b_{ic'}+\sigma_{cc'}\) leaves the tables \[\begin{pmatrix}0&0&-4\\0&0&0\\0&0&0\end{pmatrix},\qquad \begin{pmatrix}2&2&2\\0&0&2\\0&0&2\end{pmatrix},\qquad D_1=-9/4,\quad D_2=1/4.\] Thus the common-sum squares contribute \(-(\sum_{\text{sector }1} c)^2\) modulo 2 (sum here only over first-sector sites), which cancels the half-self parity in sector 2. For the external terms one may move \(v,v,-2v\) formally to the front of this phase sum since \(b_{ic}^t J v+3(p_C,b_{ic})=0\); their linear contributions cancel. Finally \((\alpha_{i0},K_Cv)=(\alpha_{i1},K_Cv)=-3/8\); the separated-compensator sine phases thus cancel color dependence of the local angular coefficients above. Relative to the scaled baseline self factor using \(H\), color self factors have ratio \(P_2(0)P_3(0)\) at \(\pm\), \(P_2(0)P_3(0)/P_1(0)\) at zero. This is \(2+\sqrt2\) times the local weights \((\sqrt{2-\sqrt2},1,\sqrt{2-\sqrt2})\).

The normalized comparison.

For a period-\(d'\) array \(D\), set \[M(D,r)= d'^{-D_{-r}} \prod_{\substack{0\le l<d'\\l\ne-r\bmod d'}}|1-e^{2\pi i(l+r)/d'}|^{D_l}.\] The baseline \(C\)-sine powers after cluster division give \(M(H,e_j-e_k)\) per pair and \(M(H,0)^{1/2}\) per site; indeed the zero sine divided by the merging coordinate difference tends in modulus to \(1/8\). Two marks with their extra \(1/4\) powers only change by a bounded positive factor (divide there with the appropriate pure exponents \((u_j,u_k)\)); similarly for mark self factors. Consequently \[ \lim_{u_*\to\infty}\lim_{L\to\infty}|\mathcal T(L,u_*)| \asymp |\mathscr N|(2+\sqrt2)^N M(H,0)^{N/2} \prod_{j<k} M(H,e_j-e_k). \tag{30}\] The comparison factors are bounded above and below independently of the site counts and shift proportions, including both common shifts of the two marks. Here color normalization uses the displayed local weights and the integral pair factors. Possible singular individual color terms are interpreted by confluence: the full rational lattice sum has a regular limit and common baseline factors are extracted before specialization.

The radial transform will use the heavy self scale \[ \mathcal S=M(H',0)^{N/2}\prod_{j<k}M(H',e_j-e_k)>0. \tag{31}\] Indeed \((A^k h_i,h_j)=(A^k h_i,v)=H'_k\) and \(A^2 h_i-Ah_i+h_i=0\), so the six-offset self calculation of (18) applies to every term involving \(h_i\); the remaining two-mark-only factors are bounded independently of the counts. Explicitly \[\begin{gathered} M(H',0)/M(H,0)=[(2+\sqrt2)/(2b')]^2,\\ M(H',1)/M(H,1)=[2\sqrt2/(\sqrt3\sqrt{2-\sqrt2})]^2,\qquad b'=3/4. \end{gathered}\] These are \(g(0)^2,g(\pi/4)^2\) of (26). Dividing (30) by \(\mathcal S\) therefore leaves, within bounded factors, \[|\mathscr N|(2b')^N\prod_{j<k}g(\theta_j-\theta_k)^{-2}, \qquad \theta_j-\theta_k\in\{0,\pm\pi/4\}.\] Since \(\mathscr H=\mathscr N/\widetilde D^2\), the Pfaffian lower bound gives the precise comparison used below: \[ |\mathscr H|\le C N^{-5/24+o(1)} \lim_{u_*\to\infty}\lim_{L\to\infty} \frac{|\mathcal T(L,u_*)|}{\mathcal S}. \tag{32}\] The constant and exponent error are uniform in the shift proportions. The next subsection supplies the radial expression for this same \(\mathcal T\). Section 9.8 bounds its absolute value uniformly through the two successive limits before letting the array sizes grow.

The phase identities needed for the transform.

The forms in (33) separate the phase of a single root from its residual pair phases. The following lemma gives the arithmetic properties needed to make those residual phases trivial, and to identify the momentum sum. We first use it to derive the radial rule; its full coordinate verification closes this subsection.

Poisson summation and local phases.

We derive the radial formula, using the earlier oscillator transformation (18)–(19) with \(R=A,\Pi,K\) as above for its mode functions. For each configuration of root charges and the indicated deterministic charges (or none), \(d=\sum b\), one requires \(n(d)=0\); then \(d\) is in the integral span of \(\mathscr E\). Thus all allowed pre-twist momenta \((I-A)m=d\) are exactly \[m=m_0(d)+t p,\qquad t\in\mathbb Z.\] The fields start from \(A m=m-d=K^*d+(t+s(d))p\), using \(K+K^*=-(I-\Pi)\). Before transforming, total momentum-dependent phase other than argument powers, in units \(\pi i\), is \[U(m,m)+F_1(m)+2(\sigma,A m),\] since total \(k_b\) vanishes. By Lemma 10, using \(t^2-t\in2\mathbb Z\), its value modulo 2 is \[g(d,d)+g_{\sigma,1}(d)+2t[1/4+(\sigma,p)] \pmod 2.\] The remaining cocycle phases are ordered pairs, value \(\mathcal D(b,a)-3 k_b(p_C,a)\) for angularly earlier \(a\) and later \(b\).

Apply the pair and self transforms exactly as for (19). They are valid on the same absolute-value-one eigenblocks (now also \(-1\)) with branch derivatives, by the derivation there. Their heat and order-independent linear argument corrections cancel off-invariant momentum powers/heat exactly since the post-twist state off-invariant has momentum \(K^*d\); the invariant Gaussian now uses \(t+s(d)\) with the extra \(t\)-phase above. Poisson summing gives initial dual label \(j_0\in1/4+(\sigma,p)+\mathbb Z\), with join scalar \(\sqrt{L/(2\pi p^2)}\) and phase \(-2 j_0 s(d)\) in the \(\pi i\) units. The real-position order terms give the same heat and angle rule: traverse rightward with \(j\) stepping by \(n(b)\) at each \(b\), apply \[ \exp\!\left(-\frac1{2p^2}\int j(u)^2du\right),\qquad \exp\!\left(i\phi_b n(b)\frac{j_{{\rm pre},b}+n(b)/2}{p^2}\right) \quad\text{per angle}. \tag{34}\] Here the base period contains \(0,u_*\) strictly inside when present. The image interactions are precisely (19) and self weights are \(\exp((H_0[A]b,b)/2)\) as before. All root coefficient/integration conventions remain those of the angular expansion (factorwise exponential of zero modes, scaled fields with \(du/(2\pi)\)).

Besides these there are the pair sign phases of the transform, \(\operatorname{sgn}(u_a-u_b)N(a,b)\) for the same angular order \(a,b\). Thus when passed to radial order \(a\) then \(b\) we can always use \[\mathcal D(b,a)-N(a,b)-3 k_b(p_C,a)\] up to an overall deterministic unit. Indeed \(N(x,y)+N(y,x)=(x,y)\); reversing from the other angular order changes parity (when at least one charge is a root) by zero, by the parity relation between \(J\) and the ordinary pairing, also valid for shifted external charges by \(C\)-invariance. Self sign constants already cancel as in the earlier transform. Now use \(g(d,d)=g^\#(d,d)\), and write \(-2j_0s(d)=-2\sum j_{{\rm pre},b}s(b)+2\sum_{a<b} n(a)s(b)\) in radial order. This gives local phases \[ g^\#(b,b)+g_{\sigma,1}(b)-2j_{{\rm pre},b}s(b) \tag{35}\] in the same units, and residual ordered pairs \(H^\#(a,b)-3k_b(p_C,a)\).

The two stationary sectors.

For stationary traces all those residual pairs vanish modulo 2. For the two deterministic vertices with \(\sigma=0\), root-root and \(b_H\)-root parts vanish again. For a root \(b\) outside the interval \(0<u<u_*\) the pair effects from marks add to zero modulo 2 by the \((C-I)v\) integrality; inside they are \[2[H^\#(w,b)-H^\#(b,v)+3(p_C,b)]\equiv 4(v,n_0 b)-2 r_e(\alpha)n(b) \pmod 2.\] Since \(n(B)=3/2=(2v,p)\), the slope sector inside is exactly that for the stationary \(\sigma=2v\) calculation, and the root factor phases there are (35) for that character, up to the diagonal-unit slope gauge with jump phase \(-2 r_e(\alpha)n(b)\). Thus they are stationary phases off and on the interval (with this fixed gauge), even through varied heavy profiles. Deterministic phases (including their mutual residual pair phase) only have modulus one.

The prefactor.

For the normalized prefactor put \(p_Q(r)=\mathfrak p(Q_L^r)\) as before. The characteristic polynomials give \[Z_C=p_Q(2)/p_Q(8)^2,\qquad Z_A=p_Q(2)^3p_Q(3)^2/[p_Q(1)^2p_Q(6)^4].\] Under the scalar transform of these products, \(Z_C\) times its invariant momentum sum becomes \(2\sqrt2\exp(-14\kappa/24-L/96)Z_C^r\Theta_C^r\), with \(\kappa=4\pi^2/L\), \(\Theta_C^r=\sum_{m\in\mathbb Z}\exp(-Lm^2/8)\); \(Z_A\) times the stated join scalar becomes \(2\sqrt2\exp(-14\kappa/24+L/48)Z_A^r\). Here radial oscillator products \(Z_R^r\) always use positive modes as in (18). Thus the full prefactor for (34)–(35) is \[ e^{L/32}\,\frac{Z_A^r(L)}{Z_C^r(L)\Theta_C^r(L)} . \tag{36}\] With deterministic vertices \(B,-2v\), these rules give the radial expression for \(\mathcal T(L,u_*)\), up to the fixed unit factor in the confluence convention. In the stationary cases, with no deterministic vertices, the radial series with this prefactor equals 1.

All identities use the formal-cut/absolute convergence and continuation bounds in the spread estimate: the angular quadratic bound suffices near small \(Q_L\) even with contour-to-mark commutations and then confluence as described above, and after momentum transformation the radial series has positive slope heat on the invariant line as required by (39), all residual phases unitary for real parameters. Thus the continuation argument in the final paragraph of Section 5.3, using (39) for the two-mark bound, applies to the fused vertex and the two stationary identities for every real \(L>0\).

In passing to long radial links one can now remove \(b_H\) from the central vertex, extracting the real self scale \(\mathcal S\) of (31), unit phases of bounded modulus even on varying slopes, interactions with mark/images tending to 1 in the successive limits at fixed site sizes (except for the counted self scale), and multipliers on root activities as in the spread estimate. Indeed \((A+A^{-1})h_i=h_i,\ n(b_H)=0\), and \((A^j h_i,v)=(A^j h_i,h_{i'})\) is exactly the displayed six-offset \(H'\). Thus all heavy self powers including cross terms with \(2w\) are the ones used there; \(H_0[A]\) contributions only among the two marked insertions themselves are bounded independent of site counts. Off the central inserting link the remaining root phases are literally the stationary ones on their slope sectors up to the unit gauge noted above. Independent of other phases, the entrance-to-exit slope across the central mark steps upward by \(3/2\).

Coordinate verification of the phase identities.

The matrix bound at the unperturbed rays.

The following strict matrix inequality is the main algebraic input. First take unperturbed angles, \(p=e^{-\pi|\xi|}\), and put \[B_0=2\begin{pmatrix} 0&0&0&0&0\\0&0&0&0&0\\0&0&0&1&1\\0&0&1&0&1\\0&0&1&1&2 \end{pmatrix},\quad B_1=\tfrac32(2,0,1,1,2)^t(2,0,1,1,2),\] and \(N_m\) the two-sector symmetric matrix with diagonal blocks zero and cross block \(B_m\). Then \[ M-(1-p)(N_0+pN_1)\ \ge\ (1-p) I/20. \tag{37}\] Here are details of one polynomial verification (rational entries throughout). Decompose by sector parity \(\delta=\pm1\). Multiply the difference after subtracting also \((1-p)I/20\) on the right, on that block, by \[D_+=(1-p^2)(p^4+p^2+1)^2(p^4-p^2+1),\qquad D_-=(1+p^2)(p^4-p^2+1)^3/(1-p)\] respectively. The resulting matrix \(P\) is polynomial of degree \(d=16,15\), respectively. All coefficient matrices \(T_j=[x^j](1+x)^d P(1/(1+x))/\binom dj\) are positive definite, except \(j=0,1\) in the plus block where the unnormalized coefficients are \(16,144\) times the Gram \((1,-1,1,1,2)^t(1,-1,1,1,2)\). For convenience the table gives lower bounds (rounded down to simple rationals) on the leading minors of \(T_j\) excluding just those two: \[\begin{array}{c|ccccc} &1&2&3&4&5\\\hline +&7/10&1&7/10&1/4&1/50\\ -&1&3&5&6&4 \end{array}\] To verify entries of the calculation, the following short coefficient formulas suffice. Use \(g_k=((b_{1s},A^k(b_{1t}+\delta b_{2t})))_{s,t=1}^5,\ h_{st}=|{\bf1}_{s=1}-{\bf1}_{t=1}|\), where for this formula order both groups with the first at angle 0 and others at \(\pi\). Form polynomial coefficients \(P_l\) (up to \(d\)) by multiplying \(D_\delta\) in Taylor coefficients at zero against \[\left(\sum_{k\le0}(g_k)_{st}p^{-2k+h_{st}} +\sum_{k>0}(g_k)_{st}p^{2k-h_{st}}\right)_{st} -(1-p)(\delta B_0+p\delta B_1+I/20).\] In this displayed recipe for unequal groups take \(s=1\) first and reflect across the diagonal for the transposed entry. Then the columns of the table are bounds on \[\frac{1}{\binom dj^r}\sum_{\sigma\in S_r}\operatorname{sgn}(\sigma) \prod_{a=1}^r\sum_{l=0}^{d-j}\binom{d-l}{j}(P_l)_{a,\sigma(a)}.\] Degrees (or cancellation of higher coefficients) follow by summing the two geometric matrix series using \(g_k\) (equivalently \((A^6-I)^3=0\) and sum the first six residues with their first two differences of step 6). This proves (37) by positive polynomial evaluation.

Distinct rays and a screened invariant mode.

Let angles now be perturbed by \(\Delta_i\). For sufficiently small fixed \(\omega>0,\kappa>0,|\Delta_i|\), the bound persists, in the form \[ \widehat H(\xi)+\frac{1-\omega}{\xi^2+\kappa^2} V\ \ge\ \frac{\pi(1-p)}{|\xi|}\left((N_0+pN_1)\odot(e^{-|\xi||\Delta_i-\Delta_j|})_{ij}+ I/100\right). \tag{38}\] Indeed at high frequency in units of \(\pi/|\xi|\), only the same-unperturbed-group direct entries survive, multiplied entrywise by the indicated exponential Gram; \(N_0\) also only uses same-group entries. The Gram Schur product retains the limiting strict gap since the Gram diagonal is 1. At low frequency \(\widehat H\) is continuous jointly in angles and frequency by the decaying radial series including the integrable translated-angle log singularities (or the omitted-zero resolvent formula). On \(\ker V\) the limiting gap of (37) is strict, while the added large positive matrix dominates on its complement. Compact remaining frequencies follow by continuity.

Coercivity for point configurations.

Consequently on the line, for nonnegative weighted finite root configurations, position ordered with weights \(t_x\), unit cells \(I\), there are constants \(c,C>0\) such that \[ \sum_{x<y}t_xt_y H_{xy}(u_y-u_x) +\frac{1-\omega}{2}\|Y\|_\kappa^2 \ \ge\ c\sum_I(\sum_{x\in I}t_x)^2-C\sum_x t_x^2,\qquad Y=\sum_x t_x\Pi b_x\delta_{u_x}, \tag{39}\] where the squared norm uses the Euclidean metric induced on the invariant line and covariance \((-\partial_u^2+\kappa^2)^{-1}\). The matrices \(N_m\) have nonnegative entries, and the scalar multipliers accompanying those entries have positive inverse kernels. Indeed, for \(0\le a<b\), \[\frac{e^{-a|\xi|}-e^{-b|\xi|}}{|\xi|} =\int_a^b e^{-t|\xi|}\,dt,\] whose inverse Fourier transform is an integral of positive Cauchy kernels. This applies to \(p^m(1-p)/|\xi|\), also after multiplication by the extra exponential in (38). Since the point weights are nonnegative, these terms can be discarded; the \(I/100\) term retains squared-cell control. No positive-semidefiniteness of the full matrices \(N_m\) is being asserted. To remove diagonal singularities first smooth each point by a Cauchy kernel of small width and apply the bound by Fourier quadratic forms. The smoothed Cauchy inverse kernels retain the local positive gap; discard smoothed \(H\)-diagonals at finite linear-in-\(\sum t_x^2\) cost. For fixed distinct angles (width \(<2\pi\)), each total kernel on the left before smoothing decays exponentially; it is Lipschitz piecewise at zero after subtracting, only on identical types, \(2\log(\sqrt{1+u^2}/|u|)\). Indeed differentiate the radial branch sum in \(|u|\) by geometric series as in (19). Cauchy smoothing this logarithm uses \(2\log|(u+i+i\eta)/(u+i\eta)|\) for width \(\eta\), bounded above by the original plus \(O(\eta/(1+u^2))\). On the remainder (also Lipschitz with decaying derivative at infinity) smoothing costs \(o(1)/(1+u^2)\) by splitting the convolution near and away from \(u\). These errors absorb into the gap, proving (39).

Retaining heat in each transfer link.

The quadratic bound now controls the root counts. We next explain why it can be combined with the link construction without spending all of the slope heat. Here are the details of adapting the link bounds to (39). Use the same intra-box and neighboring-box short fractions and Fock dampings as before, with \(f,g\) there (tails equal to a small \(\eta>0\)). On a link source-to-target use heat cost \(\int |m|^2 w^2/2\) on the two boxes, \(m\) the invariant-line slope momentum (jump \(\Pi b\)), \(w=f,g\) respectively. Add to the cost \[M'\eta^2\left(|m_L|^2+|m_R|^2-2|m_J|^2\right)\] where these are pre-step boundary slopes at left end, right end (computed from target), and joint. This telescopes, and the split heat sums exactly on cycles. \(M'\) can be fixed sufficiently large before choosing \(\eta\). For the roots weighted by \(w\), \(Y\) differs from the derivative of \(w m\) extended by zero by the regular \(w'm\), the weighted (bounded) mark atoms and the two boundary sources of strength \(\eta m_L,\eta m_R\). Thus using \(\omega>0\), their Yukawa cost in (39) is at most \[(1-\omega/2)\int |w m|^2/2 + C\int |m|^2/D^2 + M'\eta^2(|m_L|^2+|m_R|^2)+O(1)\] on bounded numbers/sizes of residual marks, \(D\) the large width. The negative cost at joint is absorbed into the spare heat and squared-cell gap since \(w\) there and over one adjoining unit is bounded below uniformly, and \(|m_J|^2\) is controlled by integrated square and counts squared there. Choose \(\eta\) sufficiently small then \(D\) large. The discrepancy between the \(w_xw_y\) short interactions and actual fractions is absorbed exactly as for the previous Hilbert–Schmidt links (exponential decay at large separation and identical-ray singularity favorable). Thus a squared-cell gap and a positive heat fraction remain, enough for Hilbert–Schmidt bounds, bounded-profile Lipschitz estimates and kernel continuity as before. Holomorphic convergence near every real period (for the continuation from small \(Q_L\), without first taking limits) also follows: repeat unweighted periods on the line using (39), on the circle the Yukawa form being bounded by a small-loss heat estimate via the periodic slope derivative, and scale positions including their period slightly complex as in the previous radial estimate. Oscillator interactions perturb by a small squared-cell cost, and angles to marks can be kept distinct.

Finiteness of the angular expansions.

The same matrix bound also justifies the coefficientwise angular conjugations. We prove this consequence separately, since its quadratic form uses the original indefinite charge pairing, whereas the transfer links above act on a positive Hilbert space.

Let \(m(\phi)\) be the piecewise constant angular momentum path, with nonnegative root counts \(k_i\) at the fixed rays \(\phi_i\), and with a fixed finite list of additional bounded jumps. Extend it by \(m(\phi+2\pi)=A^{-1}m(\phi)\). Put \[Z=m',\qquad \overline m=\frac1{2\pi}\int_0^{2\pi}\Pi m(\phi)\,d\phi.\] The total invariant charge of \(Z\) is zero by the gluing condition. Let \(G_\xi^A\) be the Green kernel of \((-\partial_\phi^2+\xi^2)^{-1}\) with this twisted boundary condition. Its pairing against root atoms is the exponential periodization \(M(\xi)/(2|\xi|)\) computed above. Integration by parts, with boundary terms cancelling by orthogonality of \(A\), gives \[\lim_{\xi\to0}\langle Z,G_\xi^A Z\rangle =\int_0^{2\pi}m(\phi)^2\,d\phi-2\pi\overline m^2.\] Here the brackets and squares use the original bilinear form. The subtracted constant mode lies on the positive invariant line.

For the root-only measure \(Z_{\rm root}\), the Fourier identity gives its regular Green form explicitly: \[\begin{split} \lim_{\xi\to0}\left( \langle Z_{\rm root},G_\xi^A Z_{\rm root}\rangle -\frac{|\sum_i k_i\Pi b_i|^2}{2\pi\xi^2}\right) &=\frac1{2\pi}\sum_{i,j}k_i k_j\widehat H_{ij}(0)\\ &\ge c\sum_i k_i^2-C\left|\sum_i k_i\Pi b_i\right|^2. \end{split}\] The inequality is (38) at zero frequency; its entrywise nonnegative terms can be discarded on the nonnegative count cone. Neutrality bounds \(\sum_i k_i\Pi b_i\) by the additional bounded jumps. The regular mixed terms with those jumps cost at most \(C(1+\sum_i k_i)\), while the invariant poles cancel in the total measure. Absorbing this linear error and restoring the invariant mean therefore yields \[\int_0^{2\pi}m(\phi)^2\,d\phi \ge c\sum_i k_i^2+2\pi\overline m^2-C.\] The gluing equation determines the off-invariant entrance momentum as a fixed linear function of the jumps. Its invariant part differs from \(\overline m\) by \(O(1+\sum_i k_i)\). After reducing the constants, the last display thus controls both the squared counts and the squared entrance momentum in any fixed Euclidean norm on the charge space.

The argument also works with coincident group angles for cut finiteness, provided all first types are grouped before all other types; the additional bounded insertions may have any order and angles. For fixed field powers and cut energy \(H\), all successive energies are bounded above. The angular average bound consequently gives finite support. For evaluation at small \(Q_L\), damping by that same average dominates the exponential-in-squared-count oscillator bounds exactly as in (15).

During coefficientwise conjugations cutting instead at \(C\), one only needs that the orthogonal-off-invariant \((1-C)^{-1}\) images of the jumps in \(g^{\pm1}\), on their nonnegative cone, control squared counts. Similarly cutting at \(B=S_{\alpha_{10}}C\) during the partial conjugation one uses \((1-B)^{-1}\) images of the neutral roots \(E4,24\) of sector 1 together with \(g_2^{\pm1}\). The comparisons below are lower bounds on the nonnegative count cone: replacing positive off-diagonal entries by zero decreases the quadratic form there, but need not do so on arbitrary vectors. For \(C\) the lists per sector are \(-\alpha_{i0},-\alpha_{i1},-\alpha_{i1}-\alpha_{i2},\alpha_{i7}\). The diagonal of the stated Gram times 16 is \(11,11,16,11\) twice, with within-sector negative entries only at \(12,24\) equal to \(-3\) (and transposes), cross-sector negatives at \(14,41\) equal to \(-5\). After the positive off-diagonal entries are removed, the strict diagonal-dominance margins are \((3,5,16,3)\) in each sector, so the true Gram is bounded below by \(3I/16\) on that cone. For \(B\), in the order indicated (using same list for sector 2), four times the Gram after dropping positive off-diagonals is \[\begin{pmatrix} 24&0&-4&0&0&0\\0&8&-4&0&0&0\\-4&-4&9&0&0&-5\\ 0&0&0&9&0&-5\\0&0&0&0&28&-6\\0&0&-5&-5&-6&9 \end{pmatrix}>0.\] To verify positivity, eliminate the first, second, fourth and fifth coordinates. The remaining Schur complement is \[\begin{pmatrix}19/3&-5\\-5&311/63\end{pmatrix}, \qquad \det=1184/189>0.\] These computations use just the circulant and reflections above. Thus squared-cut-momentum bounds hold with any bounded additional shifts; on the \(B\)-side for evaluated traces near small \(Q_L\) one can bunch the neutral-root contours closely.

The suppression profiles.

We finish by identifying which root activities the large fused charge suppresses. The two sector sizes enter only through these profiles; the coercivity constants above can be chosen uniformly. \((A+A^{-1})h_i=h_i\); for \(b_{js}\), \((x,y)=((h_i,b_{js}),(Ah_i,b_{js}))\) is zero across sectors and within equal respectively to \[(1,0),(1,0),(0,0),(0,1),(0,0).\] As in the single-mark case a charge \(a_i h_i+d_i Ah_i\) at angle and position zero gives real log-activity on its own axis (before images) \[(a_i x+d_i y) F_X+((a_i+d_i)y-d_i x) F_Y,\quad F_X=J(\psi)+J(\psi-\pi/3),\quad F_Y=J(\psi+2\pi/3)+J(\psi+\pi/3),\] where \(J(\theta)=\log|1-r e^{i\theta}|-\log|1+r e^{i\theta}|,\ r=e^{-|u|/6},\ \psi=\phi_b/6\); this follows by the six offsets in (19), in either position order. For \(d_i/a_i\) close in both sectors to a common \(\rho\in[0,1]\), choose the two group angles near \(-\pi\) and 0. For \(0<\rho<1\), precisely at \(-\pi\), \(F_X=F_Y<0\); precisely at 0, \(F_X<0,F_Y=0\); the inequalities for the required nonneutral types thus have uniform strict sign after division by \(r a_i\). Perturb away slightly, since a singularity at \(r=1\) only improves the cost. For \(\rho=0\) instead shift both angles first very slightly negative (\(F_Y/r\) has strictly positive \(\psi\)-derivative at 0). For \(\rho=1\) shift slightly positive (\((F_X-F_Y)/r\) has strictly negative derivative at \(-\pi/6\)). These statements follow directly from \(J=-\operatorname{atanh}(2r\cos\theta/(1+r^2))\). By neighborhoods and a finite cover one can thus choose distinct angles as needed above with a uniform strict bound \(-c a_i e^{-|u|/6}\), leaving angle 0 free; complex logs decay also in absolute value at the two infinities at that rate scale. In placing the compensator near 0, only the first root type needs to precede it; the later group can be put before or after the bounded-angle-near-zero cluster, using commutation through all external fields as necessary (their pairings there are nonnegative). This explains simultaneous availability of the damping profiles and the spread estimate.

Reducing the partially suppressed trace.

Suppression has exactly the same meaning (omit the corresponding exponential factors) before the radial transformation. Using (28) and removing \(g_2\) cyclically gives angular twist \[\exp(-\mathrm{i}c E_{E4}^{(1)})\exp(c E_{24}^{(1)}) \exp(2\pi i(\sigma,P))D_1\widehat S_1\widehat C.\] Indeed the two surviving factors shown on the left commute with the conjugation, and the indicated character can be moved through all the root actions used here. Put \(B'=S_\alpha C\), \(\alpha=\alpha_{10}\), to avoid using \(B\) for both twist and vertex. These manipulations can first be made power by power in the angular energy and evaluated for small positive \(Q_L\), by the cut finiteness and spread estimates (the \(B\) cut of that estimate is \(B'\)); equivalently expand the two root exponentials at the new cut, bunching their contours closely there. At a fixed cut energy each product has finite transitions on any input, identities of factors hold as operators by the angular calculation, and in cyclic cancellation of \(g_2\) the diagonal support is finite by the stated bound for the \(B'\) cut including \(g_2^{\pm1}\). On the radial side deleting factors still gives real-analytic convergence along \(0<L<\infty\).

Write \(Z_T=\prod_{k\ge1}\det(1-Q_L^k T)^{-1}\) with determinant on the pertinent charge space. In Euclidean four-space take \[\begin{gathered} D=\{x\in\mathbb Z^4:\ \sum x_i\in 2\mathbb Z\},\\ (f_1,f_2,f_3,f_4)=(e_1+e_2,e_1+e_3,e_1+e_4,e_1-e_3),\\ B_4=\tfrac12\begin{pmatrix}1&1&-1&1\\-1&1&1&1\\-1&1&-1&-1\\1&1&1&-1\end{pmatrix} \end{gathered}\] in orthonormal coordinates. The \(f\)’s form an integral basis of \(D\); \(B_4\) preserves \(D\), with characteristic polynomial \(\Phi_{12}\) (write \(\Phi_j\) for cyclotomic polynomials); \(K_4=(B_4-I)^{-1}\) preserves \(D\) and its dual since \(\det(B_4-I)=1\). Use the ordinary Fock states and fields on this lattice with cocycle \(\exp(\pi i(b,K_4P))\), always \(b\in D\), and \(X_b\) the zero coefficient for roots \(b^2=2\). Lift \(B_4\) by pure charge and oscillator transformation without a scalar. With \(U=\exp X_{f_1}\exp X_{f_2}\widehat B_4\), we claim \[ I_\sigma(L)= \frac{Z_{B'}}{Z_C\sum_{k\in\mathbb Z}Q_L^{(4kp_C)^2/2}}\, \frac{\operatorname{Tr}_{D}(UQ_L^H)}{Z_{B_4}} . \tag{41}\] Here the trace on \(D\) uses lattice momenta (we can still use the same operators on all dual momenta).

We verify both the contractions and the scalar phases in (41); matching the oscillator determinant alone would not identify the trace. The two charges \(u,z\), in the order of factors from left to right, are \(\alpha_{17}-\alpha_{11}-\alpha_{12},-\alpha_{12}\); in computations below they map to \(f_1,f_2\), intertwining the orbit pairings for \(B'\) with those for \(B_4\). Indeed for \(T=(u,z)\), \((B')^6 T=-T\), and the \(2\times2\) Gram blocks against \((B')^j T,\ j=0,\ldots,5\), in row-major tuples, are \[(2,1,1,2),\ (1,0,1,-1),\ (1,1,0,1),\ (0,-1,1,0),\ (-1,0,-1,-1),\ (-1,-1,0,1),\] exactly the Euclidean ones. Further \(B'\) has characteristic polynomial \(\Phi_4^2\Phi_8\Phi_6\Phi_{12}\) (use that of \(C\), \((x^8-1)^2/(x^2-1)\), times \(1+(\alpha,C(x-C)^{-1}\alpha)\)). Thus in each expanded term with counts \(r,s\) of \(u,z\), the pre-twist momentum \(m\) is uniquely \((1-B')^{-1}(r u+s z)\); it always lies in \(\mathcal L^*\). The Euclidean calculation likewise uses \(m'=(1-B_4)^{-1}(r f_1+s f_2)\) in \(D\). All resulting momentum powers (including heat) and oscillator pair/self contractions agree by the orbit pairings, even if the first orbit span has a radical. The free oscillator determinants are those indicated in (41).

For extra detail on the remaining scalars, let \(M=(1-B')^{-1}T\). Multiplication using the angular Gram and \(J\) gives its two columns (first-sector coefficients, second zero) \[(1,0,-1,-1,-1,-1,-1,0),\qquad (1,1,-1,-1,-1,-1,-1,-1)/2\] with integral pairings as claimed. With \(H_4=h_1+C^2\alpha-\alpha\), the other entries needed are \[\begin{gathered} (\alpha,CM)=(-1,-1),\quad J(\alpha,h_1)=-1,\quad J(\alpha,CM)=(3,2), \quad J(T,B'M)/2=\begin{pmatrix}9/2&2\\2&1\end{pmatrix},\\ (v,B'M)=0,\qquad J(u,z)=1,\qquad J(T,H_4)=(0,1)^t . \end{gathered}\] Thus the total coefficient phase in units \(\pi i\) is (put \(b=r u+s z,\ k=(\alpha,Cm)\)) \[\begin{aligned} &\tfrac12[kJ(\alpha,h_1-Cm)+k^2-k]-(\alpha,B'm)/4\\ &\qquad{}+J(b,B'm)/2+rs/2-J(b,H_4)/2-5r/4-3s/4\\ &\quad=13r^2/2+8rs+5s^2/2-(r+s)/2 . \end{aligned}\] On the Euclidean side \((f_i,K_4 f_j)_{i,j\le2}=\left(\begin{smallmatrix}-1&0\\-1&-1\end{smallmatrix}\right)\), and \((r f_1+s f_2,K_4 B_4 m')-\binom r2-\binom s2\) gives \(11r^2/2+6rs+3s^2/2+(r+s)/2\). Their difference is \((r+s)^2-(r+s)\), an even integer. This proves the reduction using the angular trace expansion at small \(Q_L\) (or formal zero modes). The closed Euclidean expression below is real-analytic for all real \(L>0\); thus (41) extends there by the radial convergence.

A finite Lie-algebra calculation.

We now calculate the Euclidean trace. Here \(D\) is the root lattice of type \(D_4\), but we include details of the automorphism argument. Its 24 roots are the vectors \(\pm e_i\pm e_j,\ i<j\), equivalently the list \[\mathcal R=(B_4^j f_l:0\le j<6,\ l=1,2)\ \text{ followed by its negatives}\] (\(l\) runs fastest). The Cartan operators \((h,P)\) together with \(X_b,\ b\in\mathcal R\), close into a Lie algebra of dimension at most 28: the residue calculation of the angular method applies verbatim with \(K_4\) since \(K_4+K_4^*=-I\) and the pairings are integral. In particular \([X_b,X_{-b}]=-(b,P)\); for \(b+d\) a root, \([X_b,X_d]=(-1)^{(b,K_4 d)}X_{b+d}\), with zero for other nonopposites. Commutation by each zero-mode generator has the same action as this adjoint calculation on the corresponding full fields, replacing \((h,P)\) by its Heisenberg current and \(X_b\) by \(W_b\); this follows from the same residue and oscillator brackets.

On momenta \(e_1+D\) at \(H=1/2\) there are just the eight states \(\pm e_i\); Cartans are diagonal and \(X_b\) takes \(m\to m+b\) with coefficient \((-1)^{(b,K_4m)}\) if possible. These matrices preserve infinitesimally the nondegenerate symmetric form pairing \(m,-m\) with coefficient 1 (use \((b,K_4 b)=-1\)). They are independent by their supports, so identify the Lie algebra exactly with the corresponding \(\mathfrak{so}(8,\mathbb C)\). Exponentials of any Lie elements are defined also on the whole Fock algebra, in finite energy spaces since now the form is positive.

The next finite calculation supplies the Cartan subalgebra and the two trace values from which we will determine the invariant character.

The invariant lattice and its character.

The fixed regular element allows us to read the infinite Fock trace from the action on a Cartan subalgebra. All conjugations and cyclic trace identities in this step are first performed on each finite-dimensional \(H\)-eigenspace of the positive \(D_4\) lattice. The zero modes preserve these spaces, so no boundedness of a conjugating operator on the completed Fock space is needed. We sum over energies only after identifying the unitary character below, when the resulting oscillator and theta series converge absolutely.

An orthogonal conjugation of determinant one puts \(x\) in the Cartan as an element \(y\) with nonzero purely imaginary pairings against all roots: choose paired eigenvectors on opposite eigenspaces, a paired isotropic basis on the kernel, adjusting orientation by interchanging a pair if necessary. This conjugation is a product of exponentials in the Lie representation and hence can be performed also on the Fock operators using zero modes.

To recall the elementary argument, an orthogonal matrix for a nondegenerate complex symmetric form is a product of reflections in nonisotropic vectors, by matching an orthonormal basis successively (for two unit vectors, their sum or their difference is nonisotropic, allowing at most two reflections). Thus an even product is joined to identity by deforming each reflecting line to a common one, possible along complex paths avoiding zero norm. Near identity the logarithm is skew by orthogonality, giving the claimed product. Bracket compatibility ensures that lifting any such Lie exponentials gives the required action on the currents at once.

After conjugation the new twist fixes \(y\), hence preserves the Cartan (its centralizer) and permutes root spaces, inducing an orthogonal root permutation \(W\) also preserving the positive system cut out by the imaginary pairings. This same action holds on the current fields. Thus the twist sends an empty oscillator state \(m\) to \(t_m|Wm\rangle\) by energy and Cartan weights, with \(t_m\ne0,\ t_0=1\) (the zero modes and lattice lift used here preserve the vacuum), and acts compatibly by \(W\) on the oscillators. It permutes the classes of dual momenta modulo \(D\) exactly as \(B_4\) since root rotations preserve those classes. In particular the permutation has order divisible by 3: \(B_4 e_1,B_4^2 e_1,e_1\) are in distinct classes and \(B_4^3 e_1\) is in \(e_1+D\). On the other hand \(W\) acts by a permutation of the three outer simple roots of the positive system, fixing the central one. Indeed signed permutation of coordinates puts the defining regular vector for positives in the usual chamber \(a_1>a_2>a_3>|a_4|\); the positive roots there have as indecomposables \(e_1-e_2,e_2-e_3,e_3-e_4,e_3+e_4\), with the second central. Thus \(W\) cycles the three outer ones.

Write \(\beta\) for that central simple root, \(\gamma\) for the sum of the three outer ones. The invariant sublattice is \(\mathbb Z\beta+\mathbb Z\gamma\), with \(\gamma^2=6,\ \beta^2=2,\ (\gamma,\beta)=-3\); its roots are \(\pm\beta,\pm(\gamma+\beta),\pm(\gamma+2\beta)\). For each invariant root \(b\), if its adjoint eigenvalue is \(c_b\), full field covariance implies \(t_{m+b}=c_b t_m\) on invariant \(m\), simply by the pure-output coefficient of \(W_b\). Hence \(t\) there is a character, generated using \(b=\beta,\gamma+\beta\), and \(c_b=t_b\). Put \(\lambda_i=t_b\) for the three positive invariant roots in displayed order; \(\lambda_3=\lambda_1\lambda_2\). Only fixed roots contribute to the Lie traces of the first two powers (Cartan trace 1 both times), so \(p_i=\lambda_i+\lambda_i^{-1}\) have \(\sum p_i=0,\ \sum p_i^2=4\) by the checked traces. Since \(\sum p_i^2-p_1p_2p_3=4\), their values are \(0,\pm\sqrt2\). In particular the character is unitary and the absolute argument residues of \(\lambda_1,\lambda_2,\lambda_3\) modulo 8 in units \(2\pi/8\) are \(1,2,3\) in some order.

We have proved \[\operatorname{Tr}_D(UQ_L^H) =Z_W\sum_{m\in\mathbb Z\gamma+\mathbb Z\beta} t_m Q_L^{m^2/2}.\]

The exponential rate.

Put \(\Lambda=\mathbb Z\beta+\mathbb Z(\gamma+\beta)\). In this basis its Gram matrix is \(\left(\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}\right)\), so its covolume is \(\sqrt3\). Since the character is unitary, choose a real vector \(\vartheta\) with \(t_m=e^{2\pi i(\vartheta,m)}\) on \(\Lambda\). Ordinary Gaussian Poisson summation, using \(Q_L=e^{-4\pi^2/L}\), gives \[\sum_{m\in\Lambda}t_m Q_L^{m^2/2} =\frac{L}{2\pi\sqrt3} \sum_{q\in\Lambda^*}e^{-L|q-\vartheta|^2/2}.\] The right side has positive terms, so its exponential rate is determined by the nearest point of this dual coset, without cancellation. Its minimum squared norm is the minimum of \((B^2+BC+C^2)/96\), where \(B,C\) range over representatives of the angle residues of \(\lambda_1,\lambda_2\); this is \(7/96\). For nearest representatives the absolute residues of the sum enforce equal sign for \(1,2\), opposite sign when one is 3; other lifts have a coordinate at least 5 in modulus, and \(B^2+BC+C^2\ge 3\max(B^2,C^2)/4>7\) for those lifts. Thus the momentum growth rate in \(\log(\cdot)/L\) is \(-7/192\); the Poisson prefactor is only polynomial.

Finally the rates for \(Z_{B'},Z_C,Z_W,Z_{B_4}\) are \(-1/192,-1/96,1/18,1/144\), respectively. These follow from the Euler-product transformation in the radial calculation by factoring the characteristic polynomials above (\((x-1)(x^3-1)\) for \(W\)); a cyclotomic factor \(\Phi_j(x)=\prod_{d|j}(x^d-1)^{m_d}\) contributes \(\sum_{d|j}m_d/(24d)\). The invariant \(C\)-sum has zero exponential rate. Substitution into (41) proves (40), as required.

Summing the marked displacements.

Return to the plane translation classes used in (23). Write \(\ell(\gamma)=|\gamma|\) and \(w(\gamma)=x_*^{\ell(\gamma)}\).

An exact expression for the cylinder arch mass

We use the \(p=1\) vacuum constructed in Section 2.3, at \(q=Q^2=e^{i\pi/4}\) and twist \(\eta=-i=q^{-2}\). Only even circumferences are needed here. The polynomial tensor \(V_N\) is the raw vacuum; its empty component is \(D=D_1\), and the physical stationary state is \(V_N/D\).

For an arc at ordered endpoints \(i<j\), the link-map coefficient is \(q^{s_i}\) for either winding class. Thus the link diagrams may be read as noncrossing diagrams in a disk. Write the component of the raw \(V_N\) supported just at \(i,j\) as \(q^{s_i}T_{ij}\), where \(s_j=-s_i\). Define the normalized scalar \[F_N:=D^{-1}\sum_{i<j}z_i z_jT_{ij},\qquad t_i=z_i^2, \qquad \mathbf t=(t_i)_i.\] We shall prove the rational identity \[ F_N=h_*\left[\sum_i t_i+\mathbf t^t\mathcal K^{-1}\mathbf1\right], \qquad \mathcal K_{ij}=\frac{t_i^2-t_j^2}{t_i^2+\sqrt2t_it_j+t_j^2}. \tag{47}\] The main step is to prove that \(F_N\) is symmetric in the site variables. A null identity for the vacuum gives this symmetry; the even Pfaffian formula of Section 9.2 then determines the scalar by interpolation.

The size of the homogeneous correction

The exact formula reduces the cylinder loss to one scalar. Write \(m_N\) for the homogeneous limit of \(\mathbf t^t\mathcal K^{-1}\mathbf1\). For \(N=2l\), \[ -m_N\asymp N^{3/4}. \tag{48}\] In log-coordinates the kernel is \(\sinh w/(\cosh w+1/\sqrt2)\), \(w=\log(t_i/t_j)\); its derivative is the cosine transform of the positive even weight \(\rho(k)=k\cosh(\pi k/4)/\sinh(\pi k)\) on \(\mathbb R\) by the integral used above for (26). Take derivative bases in each variable at confluence (equivalently change rows and columns and both contracted vectors by factorial-scaled divided differences before taking the limit). Ordering evens then odds from orders \(0,\ldots,N-1\), the kernel matrix becomes \(\left(\begin{smallmatrix}0&-H\\ H&0\end{smallmatrix}\right)\), \(H_{ab}=(-1)^{a+b}\int k^{2(a+b)}\rho(k)dk,\ 0\le a,b<l\), invertible. The two vectors become all ones and the order-zero unit vector. Hence \(-m_N=\mathcal P_{l-1}(-1,0)\) for the reproducing kernel of polynomials of degree \(\le l-1\) in \(s=k^2\) under \(\rho(k)dk\).

Here are details of this kernel’s size. Put \(A_0=3/4\), and compare even probability weights \(\sigma_{1/2},\sigma_1\) proportional to \(1/\cosh(\pi A_0 k)\), \(k/\sinh(\pi A_0 k)\) respectively. The ratio \(\rho/\sigma_1\) is uniformly bounded above and below and decreasing with \(s\), proportional to \(1+\sinh(2u)/\sinh(4u)\), \(u=\pi |k|/4\). The ratio \(\rho/\sigma_{1/2}\) is increasing, proportional to \(u/(\tanh u+\tanh(3u))\). The diagonal kernel \(\mathcal P_{l-1}(0,0)\) is comparable to its \(\sigma_1\) counterpart by the extremal evaluation norm. And \(\mathcal P_{l-1}(s,0)/\mathcal P_{l-1}(0,0)=p(s)/p(0)\) with \(p\) monic orthogonal of degree \(l-1\) for \(s\rho\), by reproducing. At \(s=-1\) this last ratio is bounded below by that for \(s\sigma_1\) and above by that for \(s\sigma_{1/2}\).

We use the classical Markov monotonicity principle for zeros under an increasing weight ratio [13], and give the derivative argument in the present normalization. The zeros of a monic orthogonal polynomial on \(s>0\) are positive simple by the sign-change test. They all shift increasingly when tilting by an increasing positive ratio \(r(s)\), so the product \(\prod_j(1+1/s_j)=p(-1)/p(0)\) decreases. To see this differentiate along weights \(d\mu_\tau=r(s)^\tau d\mu_0\) (moments and coefficients are smooth here), at a zero \(s_j\) putting \(h_j=p/(s-s_j)\). For any polynomial \(f\) of degree below \(\deg p\), \(\int f h_j\) is \(f(s_j)/h_j(s_j)\) times \(\int h_j^2\) by orthogonality. Differentiating \(\int p h_j=0\) thus gives \[s'_j \int h_j^2\,d\mu_\tau =\int (s-s_j)h_j(s)^2(\log r(s)-\log r(s_j))\,d\mu_\tau .\] Decreasing ratios similarly reverse the shift.

The comparison polynomials are the Meixner–Pollaczek polynomials \(P_d^{(\lambda)}(3k/4;\pi/2)\) in their standard generating-function normalization [15]. We give the orthogonality and norm computation for the probability measures used here.

For \(\sigma_\lambda,\lambda=1/2,1\), orthogonal polynomials \(g_d(k)\) are the coefficients of \(u^d\) in \[(1+u^2)^{-\lambda}\exp(2A_0 k\arctan u),\] with squared norms \((2\lambda)_d/d!\) (rising factorial). In fact the moment generating function of \(A_0 k\) at \(y\) is \(\cos(y/2)^{-2\lambda}\): for the hyperbolic secant integrate \(e^{yz}/\cosh(\pi z)\) shifting by \(i\) across \(i/2\); its self-convolution gives density proportional to \(z/\sinh(\pi z)\) by the tanh addition identity. Integrating the product of two generating functions thus gives \((1-uv)^{-2\lambda}\) near zero. These polynomials have the indicated degrees with fixed parity. For \(\lambda=1\), \(g_{2a}(0)=(-1)^a\); thus the diagonal kernel at zero for even degrees through \(2l-2\) is of order \(\log(2+l)\). For \(s\sigma_\lambda\) the desired orthogonal polynomial is proportional to \(g_{2l-1}(k)/k\). At \(k=i\), \(g_d(i)\) has modulus of order \((1+d)^{A_0+\lambda-1}\) at large degrees, from the coefficients of \((1+iu)^{A_0-\lambda}(1-iu)^{-A_0-\lambda}\). Indeed after removing \(i^d\) and dividing by \((A_0+\lambda)_d/d!\), the convolution tends to \(2^{A_0-\lambda}\) since the binomial series of the first factor at 1 converges, and the second sequence is increasing with fixed-shift ratios tending to 1 (summation by parts). At zero, \[|g'_{2l-1}(0)|=2A_0\sum_{a=0}^{l-1} \frac{(\lambda)_a}{a!}\frac1{2(l-1-a)+1} \asymp l^{\lambda-1}\log(2+l)\] by splitting the sum at halfway (the rising-factorial estimates follow by logarithms). Thus both comparisons give normalized ratio of order \(l^{A_0}/\log(2+l)\). This proves (48).

Recovering the half-plane mass

At homogeneous sites \(F_N/N\) is the summed arch weight from a specified port on the half-infinite cylinder to ports at strictly positive lifted displacement, by translation, the link fixed-vector identification and counting both lifts (negative endpoints have equal total by reversal/translation). Reflection can put the half-plane above its line. These are exactly the retained arches from that source in the plane half-plane with endpoint to the right, when projecting to a simple arch between distinct ports on the cylinder. Necessarily the span between ends in spacings is \(<N\) by interlacing. The plane total on that side is \(\le h_*\) by the single boundary estimate from the transfer argument. Since \(F_N/N=h_*(1+m_N/N)\), the plane total must be \(h_*\), and the deleted mass for even \(N\) is \[ d_N\asymp N^{-1/4}. \tag{49}\] Set \(X_n=\sum_{j>n}h_j\). An arch whose endpoint displacement is at least the cylinder period cannot be retained. Consequently \(X_n\le Cn^{-1/4}\), first by comparison with an even period comparable to \(n\), and then for every \(n\ge1\). At this stage this is only an upper bound: a path can be deleted by meeting a translate even when its endpoint displacement is small. The next argument controls this additional mechanism.

Collision mass between separated boundary intervals

Since weights are positive, the deficit \(P_{uv;xy}\) is exactly the product mass of pairs that are not mutually vertex-disjoint. This remains true when their endpoints interlace: in that case disjoint pairs do not exist. We shall prove the noninterlaced-interval collision bound \[ L(n):=\sum_{d>c>n}P_{0 n;cd}\le(\sqrt2-1)X_n^2 . \tag{51}\] Separate partner locations into \(M=(0,n)\) and \(R=(n,\infty)\), and put \[\Pi(E,F)=\sum_{A\in E,\ B\in F}P_{0A;nB},\qquad \begin{array}{ll} p=\Pi(R,R),&v=\Pi(M,R),\\ g=\Pi(R,M),&w=\Pi(M,M). \end{array}\] Also write \(H_M=h_*-X_n-h_n\) and \(t_0=\sqrt2-1\). We sum the three-root relations over two families of triples.

First take \((0,n,c)\) with \(c\in R\). Summing \(D_0^+=D_n^-\) gives \[p+v+h_*h_n=g+h_nX_n+X_n^2.\] The partners on opposite exterior sides in the right-hand sum interlace, so their deficits are their full product masses. The sums of \(D_n^+\) and \(D_c^+\) are \(p\) and \(L(n)\), respectively; the other three-root relation therefore gives \[L(n)=t_0p-v-h_*h_n.\]

Next take \((0,c,n)\) with \(c\in M\). Reflection makes the sums of \(D_c^-\) and \(D_c^+\) equal; call their common value \(J\). The sums of \(D_0^+\) and \(D_0^-\) are \(g+w+h_nH_M\) and \(v\), respectively, again using reflection. Hence \[J=g+w+h_nH_M,\qquad v=t_0J.\] Substitution into the first family of equalities yields \[p-J=X_n^2-w-t_0J-(2H_M+h_n)h_n.\] Every subtracted term is nonnegative. Thus \(L(n)=t_0(p-J)-h_*h_n\le t_0X_n^2\), proving (51).

A matching arch tail and pointwise decay

Consider (49) at a large even period \(m\). Let \(d_m^{\rm small}\) be the deleted mass with endpoint \(1\le n\le\epsilon m\). Each such arch \(\gamma\) collides with a horizontal translate \(\gamma+pm\) for some \(p\ge1\); fix one \(p=p(\gamma)\) deterministically. Otherwise its cylinder projection would remain simple. Joining at an intersection gives a simple crosscut in their union from \(n\) to \(pm\), above the base and no higher than the maximum height of \(\gamma\).

Take a second independent arch of the same deleted small-endpoint class, with endpoint \(k\), and shift its source to \(\delta\). If \(n<\delta<\delta+k<pm\) and its height is at least that of \(\gamma\), it must meet this crosscut: the region cut off against the base has interior strictly below the crosscut’s maximum height. Thus it shares a vertex with \(\gamma\) or \(\gamma+pm\). There are at least \(cm\) allowed integer shifts when \(\epsilon<1/4\) and \(m\) is large. By exchange of the two independent arches, the required height ordering holds on at least half the product mass.

Charge intersections with \(\gamma\) to the corresponding terms of \(L(n)\). For intersections with \(\gamma+pm\), shift the second arch back by \(pm\); both of its endpoints then lie left of zero. Reflection about \(n/2\), with reversal of the first arch, puts these pairs into the reflected version of the same collision sum. For each fixed first arch, \(p(\gamma)\) is fixed and \(\delta\mapsto\delta-pm\) is injective. The charged pair retains that first arch, so varying \(p\) between first arches causes no additional multiplicity. Summing and using \(X_n\le Cn^{-1/4}\) gives \[m(d_m^{\rm small})^2\le C\sum_{n\le\epsilon m} L(n) \le C(\epsilon m)^{1/2}.\] Thus \(d_m^{\rm small}\le C\epsilon^{1/4}m^{-1/4}\). Choose a fixed \(\epsilon>0\) so small that this is less than half the lower bound for \(d_m\) in (49). All remaining deleted paths have endpoint greater than \(\epsilon m\), whence \[X_{\lfloor\epsilon m\rfloor}\ge d_m-d_m^{\rm small} \ge c m^{-1/4}.\] Comparison with even \(m\) gives \(X_n\asymp n^{-1/4}\) for every \(n\). To extract a pointwise estimate, monotonicity first gives \[\lfloor n/2\rfloor h_n \le\sum_{j=\lceil n/2\rceil}^{n}h_j \le X_{\lceil n/2\rceil-1}\le Cn^{-1/4}\] for large \(n\). Conversely, choose a fixed \(A>1\) so large that \(X_n-X_{\lfloor An\rfloor}\ge c n^{-1/4}\). Then \[c n^{-1/4}\le\sum_{j=n+1}^{\lfloor An\rfloor}h_j \le An\,h_{n+1}.\] Absorbing bounded values of \(n\) proves \(h_n\asymp n^{-5/4}\). This controls where an arch ends. We next control where it travels.

Horizontal and vertical excursions

We first bound horizontal excursion. Let \(U_r\) denote summed arch mass from a given source in its flat half-plane reaching absolute tangential displacement at least \(r\). Endpoints of such order already cost \(O(r^{-1/4})\). Use spacing units, and consider just endpoints \(0<n\le r/4\), excursions to the right reaching \(r\). In each such \(\gamma\) choose a far point of coordinate \(x\ge r\) at a vertex, so reflection across this vertical axis preserves the honeycomb and its boundary ports. Indeed in port-spacing units the vertex coordinates are \(i+j/2\) and \(i+j/2+1/2\); twice the chosen coordinate is integral, and reflection preserves both sublattices. Take a second independent arch in that class reaching at least the first arch’s maximum height, start it at integer \(\delta\) with \(|\delta-x|\le r/8\), and also consider the mirror image of this shifted arch across the chosen vertical axis. Both pairs of endpoints are strictly to the right of \(n\). At least one variant intersects \(\gamma\): up to first reaching the specified height the shifted arch forms a bottom-to-top barrier in the corresponding closed horizontal strip. The chosen point lies on a barrier or to the right of at least one of the two mirrored barriers, while \(\gamma\) starts left of both. Thus by (51) again, \(r\) times the square of this class mass costs at most \(C\sum_{n\le r/4} L(n)\) (shifts and their mirrors counted separately for each first arch, reversing orientation as needed). The leftward case is identical with barriers left of both ends, and negative endpoints follow by symmetry. This proves \(U_r\le C r^{-1/4}\).

We next bound the normal-height excursion mass. Let \(Y_r\) be the total weight of arches from a fixed port in its supporting half-plane that reach normal height at least \(r\). Endpoints are free on the boundary line, and weights still count visited vertices. We work in Euclidean units; fixed lattice rounding is absorbed into constants.

Take an upward-open wedge \(\mathcal W\) of angle \(2\pi/3\). In counterclockwise boundary order its successive side tangents have angles \(-\pi/6\) and \(\pi/6\). Choose ports \(a\) on the left side and \(c\) on the right side, each at distance in \([.25r,.30r]\) from the apex. For either source, the total wedge mass to the other side is \(O(r^{-1/4})\). Indeed, a normal tube of width a sufficiently small constant times \(r\) from that source stays in the wedge indefinitely. In the supporting half-plane, an own-side arch lost on restricting to the wedge must leave this tube. Its mass is at most the already bounded horizontal excursion mass. Comparing the positive real single-root identities in the wedge and half-plane therefore bounds the new opposite-side exits by \(Cr^{-1/4}\).

We use this cross-mass bound in the three-root identity. Write \(I_1,I_2,I_3,I_4\) for the successive open boundary intervals obtained by splitting the left side at \(a\) and the right side at \(c\). Thus the boundary order runs down the left side through \(I_1,a,I_2\), then up the right side through \(I_3,c,I_4\). Put \[M_{ij}=\sum_{v\in I_i,\ w\in I_j}P_{av;cw},\] where the deficits are taken in \(\mathcal W\), and write \(H(a,c)\) for the wedge arch mass between the two roots. All entries of \(M\) are nonnegative. With \(f(k)=e^{i\pi k/16}\) applied entrywise, put \[A=f(5,-7,-5,-5),\qquad C=f(5,5,7,-5),\qquad R=f(4,8,8,4).\] For a third root \(z\in I_u\), Equation (50) gives \(S_c=R_uS_a\), with partner phases \(C\) and \(A\). Summing over \(z\) gives \[\sum_j C_jM_{uj}-R_u\sum_i A_iM_{iu}=O(H(a,c)).\] The error consists of omitted coincident partners at \(a\) or \(c\); their product masses are bounded by the uniformly finite row totals.

The entries with \(i=3,4\) and \(j=1,2\) have total \(O(r^{-1/2})\), because both arches then cross to the opposite side and their product is bounded by the two cross-mass totals. For the other entries, take the dot product of the real parts of the four equations with \((10,7,-1,-2)\) and of the imaginary parts with \((0,6,0,0)\). The resulting coefficient matrix is, to within \(1/10\) entrywise, \[\begin{pmatrix} 7.51&8.04&1.12&5.17\\ .56&.84&8.23&.56\\ -10.36&-9.71&.64&1.41\\ -10.92&-10.26&.44&.85 \end{pmatrix}.\] The exact lower bound used here follows directly from the phase formula. If \(a_i,c_j,r_j\) are the displayed integer phase indices and \(\alpha,\nu\) are the real and imaginary multipliers, the coefficient of \(M_{ij}\) is \[\alpha_i\cos\frac{\pi c_j}{16} +\nu_i\sin\frac{\pi c_j}{16} -\alpha_j\cos\frac{\pi(r_j+a_i)}{16} -\nu_j\sin\frac{\pi(r_j+a_i)}{16}.\] The half-angle identities show that every coefficient outside the lower-left \(2\times2\) block is at least \(2/5\). Positivity of the entries, together with the bound on that exceptional block, yields \[\sum_{i,j}M_{ij}\le C\bigl(r^{-1/2}+H(a,c)\bigr).\] For fixed \(a\), the allowed interval for \(c\) contains order \(r\) ports. Its total \(H(a,c)\) mass is at most \(Cr^{-1/4}\) by the cross-mass bound. We may therefore choose an allowed \(c\) with \(H(a,c)\le C r^{-5/4}\).

It remains to put height excursions into this deficit estimate. Fix a sufficiently small constant \(\epsilon>0\). Let \(\mathcal E_a\) be the own-half-plane arches from \(a\) that reach normal height \(r\) and whose points all have tangential displacement less than \(\epsilon r\) from \(a\). Define \(\mathcal E_c\) in the same way on the other side, and let \(e_a,e_c\) be their raw masses. By symmetry and the definition of \(U_{\epsilon r}\), writing \(x_+=\max\{x,0\}\), \[e_a,e_c\ge (Y_r-U_{\epsilon r})_+.\] The two narrow normal tubes lie in \(\mathcal W\), so these arches are wedge arches ending on their own sides. The inward normal rays from \(a\) and \(c\) cross at normal heights between \(.4r\) and \(.6r\) on each ray. Their tubes of tangential half-width \(2\epsilon r\) have an overlap parallelogram. A path in \(\mathcal E_a\), being confined to the narrower tube and reaching height \(r\), contains a subpath across one pair of opposite sides of this parallelogram. A path in \(\mathcal E_c\) crosses the other pair. To see each crossing, follow the signed tangential coordinate relative to the other normal ray. At the source and at the first point of normal height \(r\), this coordinate lies on opposite sides of the other wider tube, because the ray crossing occurs between heights \(.4r\) and \(.6r\) and the path stays in its narrower tube. Continuity gives the required subpath. The two side pairs alternate, so the two subpaths intersect. In the planar honeycomb embedding such an intersection includes a shared vertex.

Every pair from \(\mathcal E_a\times\mathcal E_c\) is therefore counted by the deficit for its two endpoints. Those endpoints belong to unique intervals among the \(I_i\), and hence \[(Y_r-U_{\epsilon r})_+^2 \le e_ae_c \le \sum_{i,j}M_{ij} \le C\bigl(r^{-1/2}+r^{-5/4}\bigr).\] Since \(U_{\epsilon r}\le C_\epsilon r^{-1/4}\) for the fixed \(\epsilon\), this proves \(Y_r\le C r^{-1/4}\). Finally, if an arch from a fixed source has diameter greater than \(R\), some point of it is at distance greater than \(R/2\) from the source, up to a bounded port offset. Its tangential displacement or normal height is then at least a fixed multiple of \(R\). The bounds for \(U\) and \(Y\) give the raw all-endpoint diameter tail \(O(R^{-1/4})\), with bounded \(R\) absorbed into the constant.

Confinement at a prescribed endpoint

The radial tail controls the total mass removed by a large truncation. A finite-domain monotonicity identity will distribute the mass left in that truncation to each prescribed endpoint. This is the final step in Theorem 15. In any finite lattice-cut convex truncation with flat side containing \(b<a_1<a_2\) in counterclockwise order there, \(a_1,a_2\) adjacent, cut the ordered list just after \(a_2\), writing \(\theta_A\) for their common angle. In (50), rewrite \(S_{a_i}\) as the single-identity product minus the disjoint terms. Free root \(a_2\) can have a disjoint partner only \(x<b\); free \(a_1\) only \(b<x<a_1\) by noninterlacing, where \(\theta_x=\theta_A\). Thus taking the imaginary part after multiplying \(S_{a_2}=q S_{a_1}\) by \(e^{i\phi-i\beta\theta_A}\) gives \[\sin\phi\,[H(b,a_1)-H(b,a_2)] =\sum_{x<b}\sin(\beta(\theta_x-\theta_A+2\pi))Z(ba_1;xa_2)\ge0\] since \(q e^{2i\phi}=-1\) and by convexity. Now truncate around a given flat source \(b\) at sufficiently large constant times \(n\), containing that size half-neighborhood up to bounded rounding. The just-proved radial tail and \(h_j\asymp j^{-5/4}\) leave mass \(\gtrsim n^{-1/4}\) to same-side endpoints \(b+j,\ j\in[n,2n]\). Monotonicity along the side in this truncation yields (46), increasing the diameter constant for bounded \(n\) as needed.

Total mass and diameter tightness

Slab total bridge weight \(B_h\) from a bottom port across \(h\) levels is \(\le C h^{-1/4}\), by the single identity comparing to the full half-plane (the loss of same-side arches). Its portion of diameter \(>M h\) is \(O((M h)^{-1/4})\) by subtracting the identity in a slab truncated by the hexagon (both bottom and top losses are positive). For lower bounds, in a wedge from a port at distance comparable to \(k\) from the apex on one side, mass of walks to the other side is \(\asymp k^{-1/4}\), still with lower bound for diameter \(\le C k\). Indeed all own-side endpoints beyond the apex are lost from the half-plane, whereas a fixed-radius fraction of a \(k\)-neighborhood at the port stays in the wedge on its half-plane side; compare the single identities as before. Truncation loses at most \(C'(C k)^{-1/4}\). Summing such cross weights over ports at distances in \([k,2k]\) on one side gives \(\gtrsim k^{3/4}\); by reversal and the upper bound, ports on the other side within distance \(\delta k\) of the apex cost at most \(C(\delta k)^{3/4}\). Take \(\delta>0\) small. Cutting the wedge by a parallel to the first side \(h\) levels inward, for \(k\) a sufficiently large fixed multiple of \(h\), deletes those targets further than \(\delta k\). Thus by the single identities the new top exits, of weight at most \(B_h\) per source, compensate the lost weight with coefficients bounded positively. This gives \(B_h\gtrsim h^{-1/4}\).

A bound uniform in the upper endpoint

For the pointwise upper bound fix opposite roots \(a,c\) (bottom, top). In boundary order the four open classes labeled successively \(1,2,3,4\) are the bottom split at \(a\) (order left to right) and the top split at \(c\) (right to left), normal angles \(-\pi/2,\pi/2\). Write \(P_{av;cw}\) for the deficit, product of the two arch weights minus their joint disjoint weight, and \(M_{ij}=\sum_{v\in i,w\in j}P_{av;cw}\). Put \(F(k)=e^{i\pi k/16}\) entrywise, \(A=F(3,-9,-3,-3)\), \(C=F(3,3,9,-3)\), \(r=F(4,8,8,4)\). The three-root identity at \(a,c,z\) has integrated complex deficits \(S_j\) for each choice of free root, i.e. the sum over its partner with factors \(\exp(it\theta_{\rm partner}\pm i\phi)\) (plus before the root); for increasing roots these \(S\)’s are in ratio \(1:q:q^2,\ q=e^{i\pi/4}\). Thus for \(z\in u\), \(S_c=r_u S_a\). Summing, \[\sum_j C_j M_{uj}-r_u\sum_i A_i M_{iu}=O(H(a,c)),\] where \(H(a,c)\) is the point-to-point bridge weight; the errors are omitted coincident partners \(a\) or \(c\) weighted by products, bounded using the uniformly finite row totals. These equations bound all entries by \(O(B_h^2+H(a,c))\). Indeed exceptions \(i=3,4,\ j=1,2\) already cost \(B_h^2\); take the dot product with \((2,2,-2,-7)\) on the real parts and \((0,2,-3,0)\) on imaginary parts. The coefficient of \(M_{ij}\) is given by the same explicit trigonometric formula used in the wedge, with the present phase indices and multipliers. Every coefficient outside the lower-left \(2\times2\) block exceeds \(1/2\). Half-angle substitution proves this finite list of inequalities; thus the nonnegative entries have total \(O(B_h^2+H(a,c))\). For a third root \(z\) on the bottom, positivity after removal of its root phase makes \(|S_z|\) comparable to the sum of its deficits. Place \(a\) at horizontal coordinate zero and choose \(z\) in a bottom interval of length comparable to \(\delta h\) to the left of \(a\), and its partner \(w\) in such an interval to the right. Here \(\delta>0\) is a sufficiently small fixed constant. The endpoints of \(zw\) and \(ac\) interlace, so their deficit is the full product. The confined arch bound gives mass \(c h^{-5/4}\) for each pair \((z,w)\) inside the slab: its diameter is at most \(C\delta h<h\). There are order \(h^2\) such pairs. Consequently \[\sum_z |S_z|\ge c_1 h^{3/4}H(a,c).\] For each triple the three-root identity gives \(|S_z|=|S_c|\). The previous entry bound, with the omitted endpoint terms restored, therefore gives \[c_1h^{3/4}H(a,c)\le C_1\bigl(B_h^2+H(a,c)\bigr).\] For large \(h\) absorb the last term into the left side and use \(B_h^2\le Ch^{-1/2}\). This proves \(H(a,c)\le Ch^{-5/4}\) uniformly in \(c\); the uniformly bounded row sums cover the finitely many smaller heights.

Corridors in a fixed normal direction

We first confine bridges around the upward normal. It is important here to remove the multiplicity caused by choosing intermediate cuts; merely multiplying layer masses would count the same bridge many times.

Choose a large fixed integer \(m\), and let each of the \(m-1\) intermediate seams vary in a short lattice interval around its height \(jh/m\). These intervals are disjoint, have lengths comparable to \(h/m\), and ensure that every successive gap is comparable to \(h/m\). The diameter estimate and the lower bound for \(B_t\) show that, for a sufficiently large fixed \(C_2\), bridges across a gap \(t\) with diameter at most \(C_2t\) have mass at least \(c t^{-1/4}\). Reflection lets us retain at least half this mass with endpoint displacement directed toward the vertical axis. At every seam the endpoint is then within \(C_3h/m\) of that axis, and each intervening piece stays within \(C_4h/m\). Choosing \(m\) sufficiently large places the concatenation in the desired corridor. The interiors of its pieces lie in disjoint open layers, so the concatenation is self-avoiding.

Let \(K(\gamma)\) be the number of allowed choices of the \(m-1\) marked seams in an output bridge. Summing over the seams and then over the layer bridges yields \[\sum_\gamma w(\gamma)K(\gamma) \ge c_m h^{m-1-m/4}.\] To bound the multiplicity we use only the total bridge upper bound. Write \(N(\gamma)\) for the number of internal cuts crossed exactly once. For every fixed integer \(s\ge0\) (with \(N^0=1\)), \[ \sum_{\gamma\in\mathcal B_h}w(\gamma)N(\gamma)^s \le C_s h^{-1/4+3s/4}. \tag{53}\] Indeed, after ordering \(j\le s\) distinct marked cuts, splitting the path at them produces \(j+1\) bridges of positive heights summing to \(h\). Summing their intermediate endpoints is bounded by the product of their total bridge masses. One height is at least \(h/(j+1)\), hence contributes at most \(C_jh^{-1/4}\). Summing each of the other heights up to \(h\) contributes at most \(\sum_{t\le h}B_t\le Ch^{3/4}\). The bounded number of orderings and coincidences gives (53). Every marked seam of the construction is crossed once, so \(K\le N^{m-1}\). Cauchy–Schwarz now gives the mass of its unmarked outputs as at least \[\frac{\bigl(c_mh^{m-1-m/4}\bigr)^2} {C_mh^{-1/4+3(2m-2)/4}}=c'_m h^{-1/4}.\] This proves the corridor lower bound for the normal direction.

The same argument concatenates any fixed finite list of directions for which corridor lower bounds have already been obtained. For a convex combination of two slopes, use sufficiently fine but fixed alternating layers in the corresponding proportions. The polygonal staircase stays within the desired tolerance of the intermediate slope; choose each constituent tolerance smaller than the remaining tolerance. Applying this separately to the finitely many linear pieces gives piecewise linear guides, with small relative perturbations of the seams allowed. It remains to obtain sufficiently many slopes.

Changing the cut normal

Let \(n,n'\) be upward unit cut normals at angle \(\pi/3\), and suppose narrow corridors have weight \(\gtrsim h^{-1/4}\) in slabs normal to \(n'\) for a forward direction with projected height on \(n\) equal to \(\alpha>0\) per unit height on \(n'\). We transfer that forward direction to a slab on \(n\) of physical height \(H\). Take two internal parallel \(n'\)-cuts. Choose a starting port \(x_1\) of the first at \(n\)-height in \([\delta H,2\delta H]\); separate the second cut by a physical \(n'\)-height \(D\) with \(\alpha D/H\in[1-5\delta,1-4\delta]\). Between them use the directed corridor bridge from \(x_1\) with very small tolerance compared to \(\delta\); its endpoint \(x_2\) is now within \([c\delta H,C\delta H]\) in height distance from the top and the corridor stays inside the outer slab. Join \(x_1\) to the bottom in the wedge before the first cut and above the bottom, and \(x_2\) to the top after the second cut and below the top. Junction masses are \(\gtrsim H^{-1/4}\) each with diameters \(\le C_3\delta H\), by the wedge estimates. Pieces cannot intersect by the tilted cuts. Taking \(\delta\) sufficiently small depending on the desired outer tolerance and slopes confines all this to the required guide corridor relative to the bottom endpoint.

Count the three-piece constructions modulo translations along the bottom cut. In this quotient, the first crossing port \(x_1\) has only finitely many lattice residues at each permitted height, and there are order \(H\) such heights. Independently, the permitted interval for \(D\) contains order \(H\) spacings. For each such port and gap, the product of the three piece masses is at least a constant times \(H^{-3/4}\). Translate each resulting bottom endpoint to the prescribed root, retaining the two tilted cuts as marks. The output with those marks recovers the crossing ports and all three pieces, so this translation is bijective on the quotient. If \(K(\gamma)\) denotes the number of resulting marked representations of the rooted output \(\gamma\), then \[\sum_\gamma w(\gamma)K(\gamma)\ge cH^{5/4}.\]

For the second moment, take two representations of one output and order their \(k\le4\) distinct tilted cuts. The first is one of the two first cuts and the last is one of the two second cuts; their crossing ports therefore remain at distances comparable to \(\delta H\) from the bottom and top, respectively. The rooted path starts before every marked cut and finishes after every marked cut. Since each is crossed exactly once, the crossings occur in the cut order, and the intervening pieces stay in their open tilted slabs.

Again count modulo bottom translations. There are \(O(H)\) choices of the first crossing-port orbit. Reverse the first exterior piece: its bottom endpoint is then free, and the wedge row bound sums that endpoint with mass at most \(CH^{-1/4}\). The last wedge has the same uniform bound on its retained top-distance band. The original junction diameters are \(O(\delta H)\) and each original middle gap is \(O(H)\), so every spacing between consecutive cuts in the combined list is at most \(CH\). For each fixed spacing, intermediate crossing ports are already summed by the total bridge mass \(B_t\). Summing positive spacings therefore costs at most \[\sum_{1\le t\le CH}B_t\le C'H^{3/4}\] per internal layer. Keep the last-port distance restriction until applying its uniform wedge bound, and drop the remaining restrictions for an upper bound. Thus the contribution from \(k\) distinct cuts is at most \[CH\,(H^{-1/4})^2(H^{3/4})^{k-1} =CH^{-1/4+3k/4}.\] There are only finitely many ways to assign these cuts to the two ordered mark pairs, including coincident cuts. Consequently \[\sum_\gamma w(\gamma)K(\gamma)^2\le CH^{11/4}.\] Cauchy–Schwarz removes the marks and leaves total mass at least \[\frac{(cH^{5/4})^2}{CH^{11/4}}=c'H^{-1/4}.\] All seam choices use their lattice spacings, which agree in physical units by symmetry. Bounded rounding inside fixed positive relative intervals changes only the constants.

From the straight corridor and this step we have guides at \(\pm60^\circ\) to any normal, hence also between them by mixtures. Transferring again from the appropriate tilted normal now covers any slope strictly forward on the original one. Concatenation as above gives the piecewise linear form. For uniformity, fix the tolerance \(\varepsilon\) and cover the compact family by finitely many uniform balls of radius \(\varepsilon/3\) centered at its graphs. Carry out the construction for each center with tolerance \(\varepsilon/3\), and take the minimum of the finitely many lower-bound constants and the maximum of their lower height thresholds. Each resulting corridor lies in the \(\varepsilon\)-corridor of any graph in its ball. This proves Theorem 16.