Proof. First fix a pair of disk matching diagrams in the two half-cylinder vacua. Draw their common cut vertically upwards, with the sites in the stated order, and their matching arcs on the two sides of this line. The ket occupies the left side. Orient a left arc from \(+\) to \(-\), and a right arc from \(-\) to \(+\). Away from the marked ports the spin contractions concatenate directed arcs. At each marked port the two prescribed spins instead give the two endpoints of directed paths. Thus a nonzero assignment forces two disjoint simple directed arcs from \(p\) to \(p'\), constituting a single marked loop. Their directions are fixed, with no remaining factor of two.

Every other loop avoids the marked loop and cannot separate its two marked points. Its algebraic crossing number through the cut interval \(X\) is therefore zero. The twist in (4) does not change its orientation sum, which is the ordinary loop value \(k^2+k^{-2}=0\). It remains to compute the marked loop’s phase.

Let \(W\) be the sum of the tangent turns along its two directed arcs, and let \(F_X\) be their total number of westward minus eastward crossings of \(X\). The cup and cap coefficients contribute \(k^{-W/\pi}\), and the inserted twist contributes \(k^{-2F_X}\). We claim that \(W=-2\pi F_X\). One may smooth the semicircular matching arcs at their joins. For a simple regular arc \(\Gamma\) from \(p\) to \(p'\), consider the direction of the chord \(\Gamma(t)-\Gamma(s)\) on the triangle \(s\le t\). It extends continuously to its diagonal by the tangent direction. Since the triangle is simply connected, its argument has a continuous lift. Comparing its three boundary sides shows that the tangent turn equals the sum of the argument changes of \(\Gamma-p\) and \(p'-\Gamma\), with tangent limits at the vanishing endpoints.

For \(\Gamma-p\), put the argument cut downwards. There are no sites below \(p\), so the matching drawing has no crossings of this cut. The initial angles on the two arcs are \(0,\pi\), and the final angles are both \(\pi/2\); their summed change is zero. For \(p'-\Gamma\), use the upward cut and the range \((-3\pi/2,\pi/2)\). The sums of the initial and final one-sided angle limits are both \(-\pi\). Each algebraic crossing of the cut contributes \(-2\pi\), giving total change \(-2\pi F_X\). This proves the claim, so the marked loop has phase one.

The disk matching drawing is also valid when the resulting polygon winds around the cylinder: no vacuum strand runs to an infinite end. Finally the absolute convergence of the physical half-cylinder vacuum sums allows the diagram-by-diagram calculation to be summed. ◻

Proof. The root lists associated with a site and its star correspond under \(u\mapsto3\lambda-u\) with the same relative signs. The splitting relation transports an exchange of two underlying sites to the exchange of their two blocks of roots. After sorting, this agrees with the original sorting up to exchanges within a fixed sign class. The latter act on the projected spin by the nonzero scalar \(E\). The Yang–Baxter, unitarity, and disjoint-commutation identities show that different reduced sequences for the same permutation agree. This comparison may first be made with generic roots: none of the braids being compared exchanges the two roots of the same split, so specialization to a split is legitimate. The transports of the real and star lists cancel in their bilinear pairing by symmetry and unitarity. Consequently the vanishing or nonvanishing of \(D_\ell(D_r^*)^{\mathsf t}\) is unchanged when the same permutation is applied to the underlying sites of both lists.

If \(\ell_j>r_j\), put site \(j\) first. A leading plus root on one side and a leading minus root on the other are then untouched by sorting. At least one of these two sites is unsplit. It forces its input spin, whereas a leading root of a split can arise only from vacancy or from the input spin of that root. These conditions are incompatible. If \(\ell_j<r_j\), use the analogous argument with site \(j\) last. This proves the off-diagonal vanishing.

On the diagonal, replace each \(S_2S_2^{\mathsf t}\) by \(R(2\lambda)\). The remaining network sorts the minuses-first list into the pluses-first list. Each opposite-sign pair crosses once, and no equal-sign pair needs to cross. Follow the first plus through all the minuses. At any crossing it can only transfer charge to the new minus; because its final spin is again the maximal plus spin, it cannot make such a transfer. Each crossing therefore has the interchanged-spin coefficient \(F\). Removing this strand and repeating proves the product in (7). It is nonzero at generic arguments. Since there are exactly \(3^g\) covectors in a \(3^g\)-dimensional space, the two families are bases. Explicitly their identity resolution is \[\operatorname{Id} =\sum_{\ell}\frac{(D_\ell^*)^{\mathsf t}D_\ell}{d_\ell}.\] ◻

Proof. Within either equal-spin block, fusion at the ascending differences \(2\lambda,3\lambda\) makes the amplitude vanish. For nonadjacent sites move them together by braid covariance; equal-spin exchanges are scalars, so the same zeros hold. For any occupied slot the centered polynomial degree bound has exactly the degree supplied by these two factors for each of the other \(m-1\) sites in its block. Thus their product exhausts the degree in every slot, leaving a constant depending only on \(m,d_0\).

Determine the constant by deleting the adjacent pair across the two blocks at multiplicative arguments \(x,y=q^3x\). In the centered deletion quotient (3), each spectator factor is the negative of its removed same-block \(P_2P_3\) factor. There are \(2m-2\) spectators, so these signs multiply to one. The inserted cup contributes \(k^{d_0}\). The constant therefore satisfies \(C_m=(2c\,k^{d_0})C_{m-1}\), with \(C_0=1\), proving (8). Finally reciprocity of \(h_j\), together with its homogeneity, gives the asserted equality of the centered normalizations. ◻

Proof of Theorem 3: finite part. Resolve the contractions on \(X\) and \(Y\) with Lemma 4. The twist acts by a scalar on each charge subspace. Total spin conservation, including the marked spins, leaves exactly \[\sum_{i\in X}\ell_i=\sum_{i\in Y}\ell_i.\] Splitting and braid covariance of the normalized vacuum identify the resulting projected amplitudes with amplitudes on the sorted root lists. Write \(X_\pm,Y_\pm\) for the roots with the indicated relative signs. Move the ket tail \(Y_-\) to the front, and the bra tail \(Y_+\) to the front. The ket blocks are \([Y_-,p,X_+]\), \([X_-,p',Y_+]\), while the bra blocks are \([Y_+,p,X_-]\), \([X_+,p',Y_-]\). Both pairs have equal block sizes.

Cyclic covariance is an identity of matching diagrams. In the spin embedding, moving a tail to the front reverses the endpoint order exactly for cups joining the tail to its complement. If the tail endpoint has spin \(h\), its old coefficient divided by its new one is \(k^{-2h}\). Cups internal to the tail have total spin zero. Thus a moved tail of total spin \(H\) changes the amplitude, old order relative to new order, by \(k^{-2H}\). For the two tails above the product is \(k^{2\sum_Y\ell_i}\), canceling the original twist. The phases in the two constants of (8) also cancel. We next record all remaining factors, including the divisors from the change of normalization.

At an ordinary site of label \(\ell\), encode each root by its relative sign \(e\) and its shift \(h\), so that its log coordinate is \(a_i+ih\beta\): \[\mathcal R(1)=\{(1,0)\},\qquad \mathcal R(-1)=\{(-1,0)\},\qquad \mathcal R(0)=\{(1,-1),(-1,1)\}.\] Clearing \(\widetilde h_N^2\) on the original list, instead of the expanded list, divides by the squares of (2). Since \((2d)^2=2\) and \(P_4(\log(z/x))P_{-4}(\log(z/x))=(z+x)^2/(zx)\), each expanded site contributes a scalar divisor \(2\), and a spectator divisor \(P_4P_{-4}\). For two ordinary sites this divisor occurs once if exactly one is expanded; if both are expanded, it occurs at shifts \(-1,0,1\) of their log difference. The diagonal \(d_\ell\) also contains, within every expansion, \(F(2\lambda)=(2c)^2/2\). If \(z\) sites are expanded, the total number of occupied roots including the marks is \(N+z\). The two block constants contribute \((2c)^{N+z}\), and the scalar divisors contribute \((2c)^{2z}\). Their quotient is therefore \[(2c)^{N-z}=(2c)^2(2c)^{\sum_i\ell_i^2}.\]

For an ordered ordinary pair with groups \(\sigma,\omega\), multiply the following factor over \((e,h)\in\mathcal R(\ell)\), \((f,g)\in\mathcal R(r)\), evaluating it at shift \(h-g\) of the log difference: \[\begin{array}{c|l} \text{condition}&\text{factor}\\ \hline \sigma=\omega,\ f=e & P_2P_3P_{-2}P_{-3}\\ \sigma=\omega,\ f=-e & P_{-2e}P_{-3e}/(P_eP_0)\\ \sigma=1,\ \omega=-1,\ f=-e & P_{-2e}^{\,2}P_{-3e}^{\,2}\\ \text{otherwise}&1 . \end{array}\] A string of length \(a\) at a group-\(\sigma\) site \(a_i\) consists of particles at \(a_i+i\sigma j\beta\), \(1\le j\le a\); length zero means the empty string. Divide by the merge divisors just specified. The result is precisely the ordinary mutual string product with lengths \(1+\sigma\ell,1+\omega r\), multiplied by \(1\) within a group and \((-1)^{1-\ell r}\) between groups. Here is an exponent check that also fixes the signs. Encode \(\prod_j P_j(v)^{m_j}\) by the Laurent polynomial \(\sum_j m_j\xi^j\); shifting \(v\) by \(ih\beta\) multiplies this polynomial by \(\xi^h\). For string lengths \(a,b'\), put \[A=\sum_{h=1}^{a}\xi^{\sigma h},\qquad D=\sum_{h=1}^{b'}\xi^{-\omega h}.\] The string polynomial is \(B-F(A+D)+EAD\), where \[\begin{array}{c|ccc} &B&F&E\\ \hline \sigma=\omega& \xi^2+\xi^{-2}+\xi^3+\xi^{-3}&\xi+\xi^{-1}& 2+\xi^2+\xi^{-2}-\xi-\xi^{-1}\\ \sigma=1,\omega=-1& 2\xi^2+2\xi^3&2\xi&2+2\xi^2-2\xi . \end{array}\] Subtract this polynomial from the root-product polynomial and subtract \(\xi^4+\xi^{-4}\) at every divisor shift. The result is zero within either group. Across groups it is the following matrix, with rows and columns in the order \(+,0,-\): \[\begin{pmatrix} 0&b_4+2b_5&2b_5+2b_6\\ -b_4&b_4+b_3-b_5&b_4+2b_5\\ 0&-b_4&0 \end{pmatrix}, \qquad b_j=\xi^{j-8}-\xi^j.\] Since \(P_{j-8}=-P_j\), each \(b_j\) contributes a minus sign. The displayed matrix gives exactly \((-1)^{1-\ell r}\).

A root \((e,h)\) interacts with either marked spectator \(a\) by \(P_{-2e}P_{-3e}\), at shift \(h\). Dividing by the merge divisor when necessary gives, for \(\ell=1,0,-1\), respectively, \[P_{-2}P_{-3},\qquad P_{-3}P_3,\qquad P_2P_3\] at \(a_i-a\). Substituting the strings of lengths \(1+\sigma\ell\) in (5) gives these same three factors. There is no marked–marked pair factor: the two marks lie in different blocks in both amplitudes.

Set \(M=-\sum_X\ell_i=-\sum_Y\ell_i\). The particle totals are \(s-M,b+M\), and the cross signs multiply to \[(-1)^{sb-(\sum_X\ell_i)(\sum_Y\ell_i)} =(-1)^{sb-M^2}=(-1)^{sb+M}.\] To compute the string weights, move a group-\(\sigma\) main line toward its far line, in direction \(\sigma\). With measure \(dv/(2\pi)\), the old integral equals the new one plus \(i\sigma\) times the sum of enclosed residues. Directly from the \(P_j\) factors, \[ \operatorname*{Res}_{v=i\sigma\beta}F_s(v)=\frac1{2i\sigma d}, \qquad \operatorname*{Res}_{v=i\sigma\beta}E_s(v)=4i\sigma c^2, \qquad F_s(2i\sigma\beta)=-\frac1{2d}. \tag{9}\] Consequently the empty, one-root, and two-root residues are \[r_0=1,\qquad r_1=\frac1{2d},\qquad r_2=r_1(-4c^2)\left(-\frac1{2d}\right)=\frac{c^2}{d^2}.\] For two identical particles, the two choices of the first root cancel their factorial \(2!\). The numbers of strings of lengths zero and two are equal because the total elementary count is \(n\). Thus \((2d)^n\prod_i r_{1+\sigma_i\ell_i} =(2c)^{\sum_i\ell_i^2}\). This proves (6) for the finite string sum. The following contour argument completes the proof. ◻

Proof. Write \(A=25/8\), \(C'=7/2\). First move the double heights to \((A-1,A)\) in \(X\) and \((-C'+1,-C')\) in \(Y\). Then move the main contours to their far-singleton contours. The same-group site poles produce strings of lengths \(0,1,2\). There is room for both roots before the endpoint in each case, including for a site of height \(\beta\) in \(X\). An attempted third root is separated by \(2i\sigma\beta\) from the first; the zero of \(P_{-2\sigma}\) in their interaction cancels the pole from the second root. Thus no longer string occurs. The inequality \(A+C'<7\) excludes cross-group singleton poles. The interaction of a double with another particle has no poles: up to translation of the argument, the fused factors are \[E_s(v)E_s(v+i\beta)=P_0(v)P_1(v)P_{-2}(v)P_3(v), \qquad E_c(v)E_c(v+i\beta)=P_0(v)^2P_3(v)^2.\] Together with the stated heights, these identities make the double movements regular. Finally \(P_{\sigma_v}(v-a)^{-1}\) introduces no crossed pole on any of these paths. These observations also exclude poles during the subsequent deformation of the main \(X\) contour, including interactions of two moving main particles.

For completeness, the cancellation on the far contours is coefficientwise in \(p_+,p_-\). An unpicked main singleton cancels an original far singleton because their activities sum to zero. A main particle at \(v\) attaching to an original far singleton at \(w\) crosses the opposite pole, \(v-w=-i\sigma\beta\). Its oriented residue is \(i\sigma\operatorname*{Res}_{z=-i\sigma\beta}E_s(z)=4c^2\), by (9) and evenness of \(E_s\). It therefore produces a double canceling the original double: \[4c^2p_\sigma^2+4c^2(-p_\sigma)p_\sigma=0.\] The factorial measures give precisely the binomial coefficients for mixing these two origins of each final species. Thus only site-bound strings remain.

There are \(n\) ordinary sites and \(n\) elementary particles in each coefficient under consideration. The original confinement argument therefore gives simultaneous exponential integrability at infinity. Every factor from the marked insertion is bounded there, and fused residues have the same confinement. One may consequently perform the one-variable deformations and cancellations under all spectator integrals. Continuing from generic distinct offsets gives confluence: the physical transfer and its normalized vacuum are regular there, while the contour integrals remain absolutely convergent. Together with the finite calculation above, this completes the proof of Theorem 3. ◻

Proof. The choice at \(t=1\) has all the asserted positivity and mass properties. We construct the contour for \(t<1\), then prove the uniform estimates as \(t\uparrow1\). First \(\Re\rho(w)>0\) throughout \(|\Im w|<\beta\). To verify the sign away from the removable zero, write \(\gamma w=x+iy\), so \(|y|<\pi/3\). The numerator of the real part, up to a positive factor, is \[\cosh(5x)\cos y-\cosh x\cos(5y).\] Here \(\cos y>0\) and \(\cos y-\cos(5y)=2\sin(3y)\sin(2y)\ge0\); strict positivity follows except at the removable case \(x=y=0\), where \(\rho(0)>0\).

Set \(\Phi_t(u,y)=\Im[(1-t)\mathfrak R(u+iy) +t\mathfrak R(u+i(y-\beta))]\). Its derivative in \(y\) is strictly positive for \(0<y<\beta\). By conjugation symmetry, \(\Phi_t(u,\beta/2)\le0\) for \(t\ge1/2\). At \(y\uparrow\beta\), its second summand tends to zero and the first is strictly positive, possibly infinite at \(u=0\). The intermediate value theorem and strict monotonicity give the unique root. The analytic implicit-function theorem gives real analyticity, and symmetry in \(u\) gives evenness.

For positivity of \(\nu_X\), write \(A=(1-t)\rho(z)+t\rho(z-i\beta)=a+ib\), with \(a>0\). Differentiating \(\Phi_t(u,\beta\theta(u))=0\) gives \(\beta\theta'=-b/a\), and hence \[\nu_X=A(1+i\beta\theta')=\frac{|A|^2}{a}>0.\] Evenness follows from conjugation symmetry. Moving each translated density contour within its pole-free strip shows \(\int\nu_X=1\). Equivalently the primitive in (14) tends to \(1/2\) and \(-1/2\) at its two ends.

The exponential expansion of \(\rho\) at either end begins with a positive multiple of \(e^{-\gamma|u|}\). Substitution into the imaginary primitive equation, divided by this leading term, gives (16). Its derivative with respect to \(\theta_\infty\) is bounded below uniformly, since both relevant cosine arguments lie in \([-\pi/3,\pi/3]\). The analytic implicit-function theorem applied to the exponentially small remainder proves the uniform tail statements, including fixed-order derivatives. On compact \(t\)-subintervals below one, the same conclusions in the remaining bounded \(u\)-region follow by compactness and the strict derivative in \(y\).

It remains to examine the possible central degeneration as \(r=1-t\downarrow0\). At \(w=i\beta\), the density has a simple pole with negative imaginary residue, explicitly \(-i/(6\beta\gamma)\). On a sufficiently small fixed interval in \(u\), the defining equation therefore compares \[-(1-r)\Im\mathfrak R(u-i\Delta)\asymp\Delta\] with \[r\Im\mathfrak R(u+i\beta-i\Delta) \asymp r\left(1+\log^+\frac1{|u|+\Delta}\right).\] At \(u=0\) this gives \(d_0\asymp r\log(1/r)\). The same estimates, separating \(|u|\le d_0\) and \(|u|>d_0\), give \(|u|+\Delta(u)\asymp |u|+d_0\).

To obtain uniform derivative bounds, write the defining equation as \[F_r(u,\Delta) =r\Im\mathfrak R(u+i\beta-i\Delta) +(1-r)\Im\mathfrak R(u-i\Delta)=0.\] The regular function vanishes identically when \(\Delta=0\), and its Taylor expansion on this fixed central interval is \[\Im\mathfrak R(u-i\Delta)=-\Delta\rho(u)+O(\Delta^3).\] Thus its \(u\)-derivative is \(O(\Delta)\), whereas its \(\Delta\)-derivative is \(-\rho(u)+O(\Delta^2)\). Either first derivative of the pole summand, including its coefficient, is bounded by \[O\!\left(\frac{r}{|u|+d_0}\right) =O\!\left(\frac1{\log(1/r)}\right)=o(1).\] Since \(\rho\) is bounded below here and \(\Delta=o(1)\), it follows that \[|\partial_\Delta F_r|\ge c_3>0,\qquad |\partial_u F_r| \le C\left(\Delta+\frac{r}{|u|+d_0}\right)=o(1).\] In particular \(\Delta'=-\partial_u F_r/\partial_\Delta F_r=o(1)\). For higher derivatives, extend each imaginary part analytically using the two conjugate analytic terms. Around a real solution at \(u_0\), take complex discs in \(u,\Delta\) of radii \(c_2(|u_0|+d_0)\). The pole remains outside these discs when \(c_2\) is sufficiently small. The \(\Delta\)-derivative remains bounded away from zero, while varying \(u\) with \(\Delta=\Delta(u_0)\) changes \(F_r\) by \(o(1)(|u_0|+d_0)\), by the same estimates. The analytic implicit root therefore extends to a disc of comparable radius with oscillation \(o(1)(|u_0|+d_0)\). Cauchy’s estimates give \(z'-1=o(1)\) and the stated scaled higher-derivative bounds. The formula for \(\nu_X\) gives the same estimates for \(\nu_X-\rho\). In this central region \(\rho\) is bounded below; off it the uniform tail expansion supplies the relative estimate in (17). These statements prove, in particular, uniform Lipschitzness and uniform comparison \(\nu_X\asymp\rho\) throughout the parameter interval.

Finally the cumulative distributions are strictly increasing and have identical total mass. Analytic inversion on bounded intervals, and inversion of their normalized exponential tails, show that their matching map \(b_X\) is odd and obeys the claimed uniform displacement and derivative bounds. Differentiating \(\rho(b_X)b_X'=\nu_X\) on the same analytic discs gives the scaled higher-derivative estimates. ◻

Proof. We justify the contour continuations in the Fourier mean identities, since the \(X\) site measure now has two different heights. For a test on the \(X\) main contour, slide separately the two translated \(\rho\) contours to a horizontal line through the test point, keeping that line at distance strictly less than \(\beta\) from the relevant site height. For the diagonal logarithm, split the contour at the test point and deform its two halves with this common endpoint. Its logarithmic singularity is integrable, and the chosen continuous branch crosses no cut. For any other test species use a fixed admissible horizontal height between \(\beta/2\) and \(\beta\).

No additional singularity is met in these slides. The signed same-group separation of a far elementary particle from a main contour or its associated horizontal site line has absolute value strictly between \(2\beta\) and \(5\beta\). The cross separation, with the \(X\) coordinate first, is positive and less than \(6\beta\). All these bounds follow directly from the listed contours. They also hold for the slides used in a second pairing with a site’s translated density. A same-group \(V\) is always slid within distance \(\beta\) of its site. Coincident site–main heights in a cross \(V\) are regular.

On horizontal contours Fourier transforms use \(\widehat f(k)=\int e^{-iku}f(u)\,du\); put \(x=\beta k\). The self-main entry has transform \[\widehat H_{\rm self}(k) =\frac{2\pi c_*(k)\cosh(3x)}{k\sinh(\pi k)}.\] For a same-group far elementary particle at signed height separation \(j\beta\), replace the numerator \(c_*\cosh(3x)\) by \[c_*\cosh(x)e^{(4\operatorname{sign}j-j)x}.\] For a cross elementary pair at separation \(d\beta>0\), replace it by \(c_*e^{(3-d)x}\), transposing by \(k\mapsto-k\) when the row and column are reversed. Convolution with each translated \(\rho\) cancels \(c_*\); the translation multiplier adjusts the height from its main contour to its site. These are exactly the transforms of the corresponding \(V\) entries. This proves the one-body identity.

Pair once more with the translated density of a second site. For a horizontal same-group pair the symbol of \(\log B_s+\rho*H_{\rm self}*\rho\), divided by \(2\pi/(k\sinh(\pi k))\), is \[-\cosh(2x)-\cosh x+\frac{\cosh(3x)}{c_*(k)} =-2\cosh x+\cosh(4x)-\sinh(4x)\tanh(3x/2).\] The expression on the right is the symbol of \(\log H_*^2\) in the same normalization: factor \(H_*\) and use \(\widehat{\log|\tanh(\gamma u/2)|}(k) =-\pi\tanh(3\beta k/2)/k\). This is the horizontal centering calculation of [10]; the contour slides above extend it to the two site heights here. In the cross case one may first compute at positive main–main separation and undo that shift in the transforms. The cancellation needed there is \[\frac{\sinh(3x)}{c_*(k)}=\sinh(2x)+\sinh x.\] Endpoint identities at a height difference \(\beta\) follow by moving the heights slightly inward and taking the regular limit. The half self term for each site is fixed by the same-group identity at zero. With the affine terms fixed by the tails as below, this gives \[\exp\!\left(\tfrac12(\rho*H_{\rm self}*\rho)(0)\right) =\frac{H_*(0)}{\sqrt{B_s(0)}} =\frac{\gamma(1+d)}{4d\cos(\beta/2)}=C_*.\] Here \(H_*(0)=\gamma(1+d)\) and \(\sqrt{B_s(0)}=4\sin\beta\sin(3\beta/2)=4d\cos(\beta/2)\), with the positive square root. There is one such factor per ordinary site, proving (18) with its exact normalization.

For a single mark, slide the density in the same way. After removing the ordinary-site to marked-site translation multiplier, the combined log symbol, divided by \(-\pi/(k\sinh(\pi k))\), is \[e^{2\sigma x}+e^{\sigma x} +\frac{1-e^{3\sigma x}}{c_*(k)} = 2\left(\cosh(2x)+\cosh x-\frac{\cosh(3x)}{c_*(k)}\right).\] The equality follows by multiplying by \(2\cosh x-1\). This is exactly the symbol of \(\log H_*^2\), proving (19). These transform comparisons determine the second derivatives. To fix the affine ambiguity, subtract the linear real tails: the remaining real constants vanish by the exponential expansions and the centered real offsets; a translated \(\rho\) has purely imaginary first moment. On horizontal contours the imaginary parts are bounded and odd, so their affine ambiguity vanishes as well. Continuing the chosen logarithms fixes the stated signs.

Finally on a main contour the numerator angle of a marked particle factor is reduced to \(4\) plus the relevant shift, in units of \(\beta\), and the denominator angle is \(1\) or \(7\) plus that shift. These angles stay in the open interval \((0,8)\). The log-sinh tail expansion therefore gives zero real limit, exponential real decay with a derivative, and the imaginary limits \(\sigma(3\beta/2)\operatorname{sign}u\). The compact part remains uniformly regular because the main angles stay away from zeros and \(z'\) is uniformly bounded by Lemma 8. On far contours the denominator remains regular and the same tail expansion bounds the log modulus above by \(Ce^{-c_2|u|}\). A numerator zero only improves this upper bound. ◻

Proof. We will sweep the curved \(X\) charge harmonically to the two edges of the strip between heights \(\beta/2\) and \(\beta\). The boundary energy and the remaining positive Green energy together will control both the species norms and the weighted total primitive.

The horizontal boundary lines. Allow independent \(X\) main charges on the two boundary lines. This gives a translation-invariant system with seven species. At \(\varepsilon=0\), their elementary height lists, in units of \(\beta/8\), are \[(4),\quad(8),\quad(25),\quad(24,32),\quad (0),\quad(-28),\quad(-24,-32).\] The first four species belong to \(X\); the final three belong to \(Y\). Divide the real Fourier symbol by \(\pi/(k\sinh(\pi k))\), and divide its \(IJ\) entry by \(w_Iw_J\). Expand the resulting matrix in \(v=\cosh(\beta k/8)-1\). Its constant matrix has every entry equal to \(2\), and every coefficient of degree \(1\le l\le32\) is strictly positive definite. Here is a finite rational verification of this assertion.

The coefficient of \(v^l\) in \(\cosh(32\,\operatorname{arcosh}(1+v))\) is \[c_{32,l}=\frac{2^l}{(2l)!} \prod_{a=0}^{l-1}(32^2-a^2)>0.\] After division by this coefficient, define \[h_l(j)= \prod_{a=0}^{l-1}\frac{i_j^2-a^2}{32^2-a^2}, \qquad i_j=(j\bmod64)-32.\] For each pair of elementary heights, let \(j\) be their absolute difference. Average over the two species’ height lists the expression \[2h_l(j)+h_l(j+16)+h_l(j-16)-h_l(j-8)-h_l(j+8)\] when they are in the same group, and \[2\bigl(h_l(j)+h_l(j+16)-h_l(j+8)\bigr)\] when they are in opposite groups. Denote the resulting symmetric matrix by \(M_l\). These formulas follow by summing the log-factor angles modulo \(2\pi=8\beta\). The elementary cosh coefficient formula follows from \((2v+v^2)y''+(1+v)y'=i_j^2y\).

For \(l\le12\), symmetric elimination gives successive pivots strictly larger than the rational decimals in the following table: \[\begin{array}{c|rrrrrrr} l&1&2&3&4&5&6&7\\ \hline 1&1&.7&.8&.27&.49&.59&.09\\ 2,3&1&1&1&.32&.8&1&.30\\ 4,5,6&1.7&1.65&1.48&.35&1.53&1.65&.45\\ 7,\ldots,12&1.9&1.88&1.74&.35&1.89&1.88&.48 \end{array}\] The entries are obtained using exact rational substitution in \[m_{ij}^{(p)} =m_{ij}^{(p-1)} -\frac{m_{ip}^{(p-1)}m_{jp}^{(p-1)}}{m_{pp}^{(p-1)}}, \qquad i,j>p,\qquad M_l=(m_{ij}^{(0)}).\] Thus all these matrices are positive definite.

For \(13\le l\le32\), each \(h_l(j)\) is nonnegative and nonincreasing in \(l\): before a zero occurs, every additional factor lies in \([0,1]\), and after that zero the product stays zero. Split the displayed averages into their positive and negative parts, \(M_l=P_l-Q_l\). A lower bound for its diagonal is \(\operatorname{diag}(P_{32}-Q_{13})\), and an entrywise upper bound for the absolute off-diagonal entries is \[\max(P_{13}-Q_{32},\,Q_{13}-P_{32}).\] Weight column \(j\) relative to row \(i\) by \(v_j/v_i\), with \(v=(1,1,1,2,1,1,1)\). Subtracting the weighted off-diagonal row sums from the diagonal lower bounds gives exactly \[\begin{split} (&539085539017/288991072832,\ 153103458093/288991072832,\ 6754920483/37289170688,\\ &2413205499/288991072832,\ 8798842243/4515485513,\\ &496150561213/288991072832,\ 32952783649/1155964291328). \end{split}\] All seven numbers are positive. Weighted strict diagonal dominance of the symmetric \(M_l\) proves positive definiteness for every remaining degree.

At small frequencies the constant coefficient supplies the rank-one \(k^{-2}\) term for the weighted total density, and the strictly positive first coefficient supplies a uniformly positive bounded term on all species. At large frequencies the top coefficient supplies a positive multiple of \(1/|k|\), since \(\pi=4\beta\). The intervening frequencies are controlled by continuity and positivity. The real tail \(-w_Iw_J|u-v|\) contributes exactly \(\|D\|_2^2\) to \(B\). These observations prove the two-sided norm estimate on the seven boundary-line species at \(\varepsilon=0\).

Uniform movement of the far double. The estimate continues uniformly to sufficiently small \(\varepsilon>0\). At small frequencies use the strict positive quadratic coefficient, and on compact frequency intervals use continuity. To check uniformity at infinite frequency, divide the normalized symbol further by \(\cosh(4\beta k)\). For \(0\le\varepsilon<1/16\) it has the form \[\operatorname{diag}(2,2,2,\tfrac12,2,2,\tfrac12) +q_\varepsilon(k)(E_{24}+E_{42}) +O(e^{-c_3|k|}),\qquad q_\varepsilon(k)= \frac{\cosh((4-\varepsilon)\beta k)}{2\cosh(4\beta k)}.\] Here \(E_{ij}\) is the elementary matrix, and the error is uniform in \(\varepsilon\): after collecting the elementary factors of each double, all other contributing angles stay a fixed distance from zero modulo \(8\beta\). In particular the potentially small angles in the \(X\)-double/\(Y\)-double entry cancel with total coefficient \(2-4+2=0\), for every \(\varepsilon\). The exceptional pair couples the \(X\) main line of height \(\beta\) to the \(X\) double, and \(0\le q_\varepsilon\le1/2\). The only nontrivial \(2\) by \(2\) block is \(\left(\begin{smallmatrix}2&q\\q&1/2\end{smallmatrix}\right)\); its least eigenvalue is at least \((5-\sqrt{13})/4>0\). This proves uniform coercivity also in every joint limit \(\varepsilon\downarrow0\), \(|k|\to\infty\).

Harmonic sweeping to the boundary. We now pass from the two boundary lines to the graph. Let \(\mathcal S=\{z:\beta/2<\Im z<\beta\}\), and balayage the real charge on the graph harmonically to the two edges of \(\mathcal S\). Leave all other species unchanged, and denote the resulting seven-species charge by \(\widetilde\xi\). All interactions are harmonic in this strip, in either variable away from the source, and have at most linear growth. The only interior source in a graph self interaction is \[-2\log\left|2\sinh\frac{z-z'}2\right|,\] whose logarithmic singularity is \(-2\log|z-z'|\). Let \(G_{\mathcal S}\) be the Dirichlet Green kernel of \(-\Delta\), with singularity \(-(2\pi)^{-1}\log|z-z'|\), and put \[\mathcal G(\xi_X)= \iint G_{\mathcal S}(z(u),z(v))\,\xi_X(u)\xi_X(v)\,du\,dv.\] Harmonic interpolation first in one variable and then in the other therefore gives the exact identity \[ B(\xi)=B_{\partial\mathcal S}(\widetilde\xi) +2\pi\mathcal G(\xi_X). \tag{21}\] The coefficient \(2\pi\) includes the factor \(1/2\) in the definition of \(B\). The Green energy is nonnegative for signed real charges: by Dirichlet inversion and integration by parts it is the squared dual norm of the charge acting on traces of zero-boundary functions with gradient \(L^2\)-norm at most one. Harmonic measures in the strip decay exponentially; their Fourier kernels are ratios of the form \(\sinh(yk)/\sinh(\beta k/2)\). These facts justify the interpolations for rapidly decreasing densities and then by completion.

Uniform control as the graph approaches an edge. For clarity, we verify that (21) gives exactly the claimed norm, even when the graph approaches the upper edge. The Sobolev trace principle underlies this comparison [4]; we give the strip construction to retain uniform constants as the graph approaches an edge. The trace map \(H^1(\mathbb R^2)\to H^{1/2}(\mathbb R)\) on the graph and a right inverse both have bounds depending only on its uniform Lipschitz constant. Flatten the graph by a uniformly bilipschitz map. On a horizontal line, the trace bound follows by transverse Fourier Cauchy–Schwarz, and a right inverse extends a mode \(k\) by \(e^{-\sqrt{1+k^2}|y|}\). Reflection across the strip edges with a cutoff gives the same trace bound for strip functions. Harmonic extension from the two boundaries is bounded \(H^{1/2}\to H^1\); this follows either from the displayed Fourier kernels or the Dirichlet principle together with the fixed strip width.

Duality now bounds the two projected charges’ \(H^{-1/2}\) norms by the graph charge’s norm. It also bounds the Green energy above by that squared norm, since zero-boundary strip functions have their full \(H^1\) norm controlled by their gradient. Conversely, extend an arbitrary \(H^{1/2}\) test on the graph to the plane, restrict it to the strip, and decompose it into its harmonic boundary extension and a zero-boundary remainder. Pairing these two terms with the charge gives \[\|\xi_X\|_{-1/2}^2 \le C_3\left( \|\widetilde\xi_{X,\beta/2}\|_{-1/2}^2+ \|\widetilde\xi_{X,\beta}\|_{-1/2}^2+ \mathcal G(\xi_X)\right).\] Thus the Sobolev parts of the norms are equivalent in both directions, with the Green term included.

It remains to compare primitives of the weighted total densities. Let \(\widetilde D\) be the primitive after projection. For a smooth compactly supported function \(b\), pair the horizontal marginal of the difference between graph charge and projected charge with \(b\). This pairing equals that of the graph charge with \(b(u)\) minus the harmonic extension of the same boundary value \(b\) on both edges. The latter difference has zero boundary values and gradient norm at most \(C_4\|b'\|_2\): compare the harmonic extension to the constant-in-height function \(b(u)\) by the Dirichlet principle. Green-energy duality, followed by integration by parts in \(u\), therefore yields \[\|D-\widetilde D\|_2^2\le C_5\mathcal G(\xi_X).\] Together with (21), the boundary-line coercivity, and the Sobolev norm comparisons, this proves both inequalities in (20). All constants depend only on the fixed strip width and uniform Lipschitz bounds, so remain uniform as the graph reaches an edge, including the case \(t=1\). ◻

Proof. Positive count tests. We give the count tests explicitly. For each far label use translates of a nonnegative bump covering a unit cell to test the positive field \(x_I=\psi*\zeta_I\). For a main label use a grid beginning at its own wall and bumps supported outside that wall. The first bump rises from zero to one across distance \(h\), remains one through distance two, and then cuts off. Every particle in that first cell contributes a fixed positive amount after smoothing. The squared \(H^{1/2}\) norm of a bump is \(O(1+|\log h|)\), as follows from the difference-quotient seminorm, and bounded overlaps give the same bound for linear combinations with the sum of squared coefficients. The mean cell masses are summably bounded because \(s_g\bar\nu_g(u)=O(1)\) at its own wall. Duality and magnetic coercivity therefore give the first count estimate.

Beyond the larger wall, a count to the right of \(v\) is bounded by twice its positive smeared cumulative there. Its square integrates to \(O(B(x)+1)\) using the total primitive and the exponentially small mean tail; the left end is identical. Counts are integers, so this also bounds their distance sum \(\ell_e\). The remaining count mass lies in \(O(L)\) cells and is bounded by Cauchy–Schwarz. Finally the \(X\)-group total of \(x\) is \(-M\). Test it on a smooth plateau covering the larger wall interval, whose \(H^{1/2}\) norm is \(O(\sqrt L)\); its complement is bounded by the positive smeared tail masses. This proves the remaining count estimates. The covariance kernel of \((1+k^2)^{-1/2}\) is bounded by \(C(1+|\log|u||)e^{-c|u|}\); convolving twice with \(\psi\) also gives \(\|\psi*\zeta\|_{-1/2}^2\le C(1+|\log h|)\sum_j n_j^2\), which we use in the smoothing estimate below.

Horizontal singularity and curved residuals. We give the additional kernel estimates needed for curvature. Subtract \(S_0\delta_{IJ}\) from \(H_{IJ}\). This accounts for the full horizontal angle-zero terms, using double fusion; the residual curved self-log ratio on main has bounded size locally by divided differences and the graph estimates. Except for this ratio, all residual log singularities have horizontal offsets in imaginary height bounded away from zero by \(c_3\varepsilon\), after double fusion, for like near-diagonal real positions. This follows from \(1/2\le\theta\le1\), the singletons \(25/8,-7/2\), doubles \(3+\varepsilon,4+\varepsilon;-3,-4\), and the shifts in \(E_s,E_c\); for two curved main particles with different real positions use uniform Lipschitzness near their diagonal. All angle-nonzero sums thus have fixed-order local derivative bounds polynomial in \(1/\varepsilon\) (up to one derivative in each argument suffices, or fixed higher order where the curve has regular derivatives). In the far tails of the difference coordinate use the convergent log series. More precisely, the remaining leading terms after subtracting \(S_0\delta_{IJ}\) are \[-(w_Iw_J-\delta_{IJ})\log(2\cosh(u-v)) +i(c^0_{IJ}-\upsilon_I(u) w_J+w_I \upsilon_J(v))\tanh(u-v)\] with exponentially localized residuals. Here \(c^0_{IJ}\) are straight-contour sign coefficients, \(\upsilon_X(u)=\beta(\theta(u)-\theta_\infty)\) on main \(X\), and \(\upsilon_I=0\) otherwise. Indeed vary just the imaginary coordinate there: each reduced-angle log contributes its height translation to the linear tail, including the continued squared diagonal log via the same log ratio, and there is no zero crossed off a self diagonal. Derivatives up to one in each variable of the exponentially localized bounds also hold away from the curved same-main diagonal neighborhood. In that neighborhood the extra log ratio is smooth with uniform fixed-derivative bounds outside a fixed neighborhood of zero; near zero it is bounded with first derivative in \(u\) at most \(C/(|u|+d_0)\) when the graph approaches angle 1 as above. This follows by differentiating the divided difference: for \(|u-v|\le c_3(|u|+d_0)\) use the local derivative bounds on \(\mathfrak z'_X\), otherwise divide the bounded slopes by \(|u-v|\). Thus in this exceptional neighborhood its localized \(H^{1/2}\)-norm in \(u\) is at worst polynomial in \(L\) (recall \(1-t\ge 1/s\) if nonzero). Indeed in the difference quotient split translation lengths below and above \(c_3(|u|+d_0)\); the squared seminorm bound costs \(C(1+|\log d_0|)\). At \(t=1\) itself there is no such issue.

Smoothing the residual kernels. For (24) only the mixed error of \(\widetilde H_{ne}\) versus \(H_{ne}\psi\) has to be paired against \(\alpha-\mu\). The displayed smoothing transform gives the Fourier bound \(Ch^{c_2}(1+|k|)^{-1}\widehat\psi\) for the \(S_0\) term. For the decaying residuals just described, convolving in the explicit variable instead of evaluating at its point changes the kernel, as a test in the interior variable, by \(O(\varepsilon^{-C}h^{c_2}+o_L(1))\) in each localized \(H^{1/2}\)-norm, with exponential decay in the distance between unit cells. Indeed the first and mixed derivative bounds suffice by averaging shifts (the probability has small first moment and small exponential tails); when the explicit argument belongs to a curved same-main entry it starts beyond \(L_X\). To meet the exceptional neighborhood at zero it must thus shift by \(\gtrsim L_X\), gaining exponential smallness against the polynomial localized norm there. The smooth real leading tail gives decaying small errors too by even averaging and exponentially decaying second derivatives. For the imaginary leading terms the only change not exponentially localized is from varying \(w_I \upsilon_J(v)\) in the convolution. In (24) this only pairs against \(z\); summing with \(w_I\) and integrating by parts gains \(\tanh'\) against \(D_z\), so the small-error estimate uses just Lipschitz \(\upsilon_J\). Summing tests in spatial cells uses (23), Sobolev norm bounds from \(B\), and \(\|\psi*\zeta\|_{-1/2}^2\le C(1+|\log h|)\sum n_j^2\) from the previous argument. For clarity, local \(H^{1/2}\) bounds with exponential envelopes combine with loss at most constant in cell \(\ell^2\): insert a unit-scale partition with bounded overlap and use the difference-quotient norm. This proves (24).

For (25) the singular diagonal real summand \(S_0\) can only decrease on smoothing. Use the same localized residual estimates (only point values needed), now with any same-curved-label explicit points both outside their wall; the \(\log(2\cosh)\) tail again costs small decaying errors. On a same-variable smoothed diagonal the underlying shifts within a label stay regular uniformly after subtracting \(S_0\), giving the stated uniform self bound.

Summing the localized errors. To make the cell summation explicit, if \(a_j\) and \(b_j\) are the local negative Sobolev norms of the two fields, the error is bounded by \(C\epsilon_{\rm sm}\sum_{i,j}e^{-c|i-j|}a_i b_j\). The convolution operator with matrix \(e^{-c|i-j|}\) is bounded on \(\ell^2\). Cauchy–Schwarz, the magnetic energy norm equivalence, and the count bound then give \(C\epsilon_{\rm sm}(B(x)+B(z)+1)\). The omitted diagonal is treated separately, with one \(C(1+|\log h|)\) cost per explicit particle; it is never included in the off-diagonal estimate. ◻

Proof. The two anomaly logs subtracted from the action in (24) cost arbitrarily small positive fractions of \(B(x)\) plus constants depending on parameters for their real parts (a lower bound suffices). Indeed use the \(H^1\) bounds on main and decaying real upper bounds on far, with (23) and \(x=\Re\alpha-\mu+\psi*\zeta\). The imaginary correction is exactly \(+(\Im\mathfrak t_n,z)\).

Choose \(h\) after \(\varepsilon\) so that the smoothing error is absorbed by the positive real form. Equation (25) and the diagonal bound leave at most \(C_{h,\varepsilon}n_e\); the bounded anomaly terms are covered by \(C_{h,\varepsilon}L\). ◻

Proof. Indeed, on the straight comparison contours (main \(X\) at \(\theta_\infty\)) the term \(-(x,\Im H z)\) gives \[ -2p\int Dz_X-2\pi\int f_R^\psi z_X-(x,J_o z) \qquad(D_z=0,\ \int z_X=0). \tag{28}\] The main sign matrix has upper entry \(p\); each elementary far row, less the corresponding main row, has difference of its sign coefficients against the \(X,Y\) columns equal to \(4\beta=\pi\), directly by the far-row transform formulas. The residual \(J_o\) is the straight odd kernel after subtracting its sign tail. The log-factor Fourier formulas give its decaying multiplier bounds, uniformly here since against main columns we’re away from real log poles in the straight comparison at fixed parameters (constants polynomial in \(1/\varepsilon\)); near zero frequency these remainders are uniformly bounded also as \(\varepsilon\downarrow0\). For replacing a straight imaginary kernel by the curved kernel against tilts in macroscopic deep regions away from zero, the localized residual after the extra \(\upsilon\)-terms with \(\tanh\) is negligible: where the two arguments are close both are exponentially-asymptotic to straight with derivatives (divide out the self singularity first); elsewhere use the exponential difference-tail estimates with one derivative in each variable. Schur bounds then control the operator against \(H^{-1/2}\)-fields by an exponentially small norm, localizing the tilted variable by a slightly larger deep cutoff. In the extra \(\tanh\) terms \(w_I \upsilon_J(v)\) pairs against weighted-total \(x\) via \(D\), with \(\upsilon_J\) small on tilted support. The term \(-\upsilon_I(u)w_J\) vanishes when \(D_z=0\) (and for a deep-supported \(D_z\) gains exponential smallness against \(\tanh'\)). This justifies (28) up to negligible errors here. The far-count term is exactly \(2\pi t_0 U^2 I_\varphi\). All mixed terms besides the \(f_R^\psi\)-gain cost a fraction of \(B(x)\) plus \(O(t_0^2 U^2\delta L)+o(L)\) by these estimates and low-frequency concentration of the rescaled probes. The cost of \(B(z)\) is similarly bounded; the anomaly term is favorable up to negligible errors by its tails. This proves (27), taking \(h\) sufficiently small as indicated. In particular for bounded \(U\le U_0\) only \(O(U_0)\) far particles lie in \([-(1-2\delta)L_X,(1-2\delta)L_X]\).

For a particle in the interval between the probe supports, its contribution to the difference of the normalized averages is bounded below by a positive constant once \(L\) is large. The exponentially small smoothing tails do not alter this lower bound. Thus positivity of the counts gives the asserted \(O(U_0)\) bound on particles there. ◻

Proof. Indeed the real linear-growth tail \(-w_Iw_J|u-v|\) contributes just \(\int D^2\). In the remaining decaying kernels the extra term from differentiating masks in the main \(Y\) charge, of type \(w' D\), is negligible by Sobolev bounds (those decaying kernels are bounded forms on \(H^{-1/2}\); for main \(Y\) entries as needed here this follows already from log diagonal plus smooth localized terms with one derivative in each variable and the harmless \(|u-v|\) cusp). All other errors use the kernel times \(1-\sum w(u)w(v)\), vanishing quadratically at the diagonal, with uniformly integrable bounds \(O_\delta(L^{-1})\) also after up to one derivative in each variable. Indeed this factor kills diagonal log derivative singularities, and where both arguments are near the exceptionally varying central scales the masks are constant (off diagonal at large separation only first graph derivatives are needed). The identity \[1-\sum_w w(u)w(v)=\frac12\sum_w(w(u)-w(v))^2\] makes the diagonal cancellation explicit. A differentiated error kernel has integrable bounds in both variables; applying Schur’s test after half a derivative on each side bounds its pairing on \(H^{-1/2}\times H^{-1/2}\). The terms involving \(w'D\) are \(o(1)B(x)\), because \(\|w'\|_\infty=O_\delta(L^{-1})\) and \(\|D\|_2^2\le CB(x)\). Each localized form is positive by the magnetic norm equivalence. On \(w_c,w_s\) we may also replace it by the straight \(B_\infty\) with negligible errors using again the log ratios after diagonal division and the decaying residual estimates (real growth tail unchanged). Write the localized real fields on these masks as \(y^c,y^s\). ◻

Proof. In (28) the far part is favorable by monotonicity, and the anomaly gives \(6\beta A_0\int w_c^2+o(L)\). The odd remainder costs \(o(\sqrt L)\sqrt{B(x)}\) by Fourier bounds and low-frequency concentration. The cost \(B(z^c)\) is \((\kappa-p^2)A_0^2\int w_c^4+o(L)\) by the straight main symbol. At small frequency the real \(B_\infty\) symbol in variables \(D,x_{I\ne Y}\) has the \(D\) coefficient 1 and mixed \(D\) terms vanishing at zero; the complementary block stays strictly positive by the real norm bound. To spell out the square completion, let \(\mathcal B(k)\) be the real energy symbol in these variables, with the convention that its \(D,D\) entry tends to one. For \(f_L(u)=\operatorname{sign}(u)w_c(u)\), minimizing the quadratic form against the linear test at each frequency gives \[B_\infty(y^c)-2pA_0\int (w_cD)f_L \ge -\frac{p^2A_0^2}{2\pi} \int |\widehat f_L(k)|^2 [\mathcal B(k)^{-1}]_{DD}\,dk.\] The inverse entry tends to one at zero, since the mixed block vanishes there and the complementary block is strictly positive. Its growth away from zero is polynomial by the energy norm equivalence. The smooth scaled function \(f_L\) has Fourier mass concentrated at frequencies of order \(L^{-1}\), with arbitrarily rapid decay outside that scale. Splitting the integral at a small fixed frequency and then letting that cutoff tend to zero shows that the right side equals \(-p^2A_0^2\int w_c^2-o(L)\). Finally \(0\le w_c\le1\) and \(\kappa>p^2\), so the sum of the two negative quadratic costs is at least \(-\kappa A_0^2\int w_c^2-o(L)\). Since \(A_0=3\beta/\kappa\), \(6\beta A_0-\kappa A_0^2=9\beta^2/\kappa=1\), proving (29). ◻

Proof. Indeed the anomaly pairs against a derivative supported deep in each tail with vanishing integral there. The curved imaginary corrections are negligible as explained after (28); on straight kernels the mixed term is \(-(w_s x_e,K_1 V_0)\) up to negligible commutators (the differentiated kernels are smooth exponentially decreasing with derivatives). Also \(2 B(z^s)=(w_s V_0,H_p w_s V_0)\le(V_0,H_p V_0)+o(B(x)+1)\) by the same square partition argument on main \(Y\) alone. This recovers plus half the square of the imaginary coupling with \(H_p^{-1}\). In \(B_\infty(y^s)\) minimizing over the remaining real total primitive at each frequency gives the real Schur subtraction \((\Re H_{eY})H_{YY}^{-1}(\Re H_{Ye})\) from \(\Re H_{ee}\); here both are combined before integration, via the neutral form symbols, so no zero-frequency subtraction constants enter. The resulting bounded symbol (order \(-1\) at infinity) acts on \(w_s x_e\), replaceable by \(a\) as above. Combining gives (30). The cross term in \(B(z^c+z^s)\) is negligible by support separation and the zero common total, using exponentially decaying real residual kernels.

The sign in this completion is essential: if \(b=K_1^{\mathsf t}a\), then \(V_0=-H_p^{-1}b\) gives \[-\tfrac12(V_0,H_pV_0)-(b,V_0) =\tfrac12(b,H_p^{-1}b).\] Adding this positive term to the real Schur complement produces \(\operatorname{Re}(H_{ee}-H_{eY}H_{YY}^{-1}H_{Ye})\), rather than the Schur complement of \(\operatorname{Re}H\) alone. ◻

Proof. The reduced kernel. We first identify the reduced kernels, and then estimate their off-diagonal and diagonal parts separately. Use \(x=\beta k\). Define \[F_y(r)=2\operatorname{atanh}(\sin(\pi y/6)\operatorname{sech}(\pi r/(6\beta))), \qquad G_y=F_{y+1}-F_y+F_{y-1}.\] For \(|y|\le3\), \(F_y\) has symbol \(2\pi\sinh(yx)/(k\cosh(3x))\): differentiate in \(y\) and use the elementary sech transform by residues (imaginary shifts inside the first poles); extend to the endpoints by log integrability. Beyond that range use the foldings indicated by the sine.

For two elementary constituents let \(d\) be their absolute height difference in \(\beta\) units. Then the following give their contributions to \(\mathcal H\):

• across groups, \(G_{d-4}\);

• within a group, \((G_{2+d}+G_{2-d})/2\), plus only in group \(X\) a correction with symbol \[\frac{2\pi}{k}\frac{c_*(k)\sinh x\cosh(dx)}{4\cosh x\cosh(2x)\cosh(3x)} .\]

Indeed divide all \(H_{IY}\) symbols by \(2\pi c_*/(k\sinh(4x))\); the contributions are \(e^{(3-a)x}\) for \(X\) elementary at height \(a\), \(e^{(-4+j)x}\cosh x\) for \(Y\) far at height \(-j\), with main \(Y\)-entry \(\cosh(3x)\). For a cross pair, before folding at \(d+h=8,\ h=0,1,2\), its real \(H\)-numerator in the same normalization is \(\cosh((3-d)x)\). The subtraction uses \(\cosh((7-d)x)\cosh x/\cosh(3x)\), giving just the unfurled \(G_{d-4}\) expression by product-to-sum. Each crossing replaces a \(\cosh((4-d-h)x)/\sinh(4x)\) by its reflected expression, adding \(2\sinh((8-d-h)x)\) with the same coefficient; this is precisely the fold correction of \(F_{d+h-5}\). For a same-group pair before crossings at \(d=1,2\), the real symbol divided now by \(2\pi/k\) is \(c_* \cosh(3x)\cosh(dx)/\sinh(4x)-\sinh(dx)\); the last term corresponds to folding \(F_{3+d}/2\). Subtracting \(c_* \cosh^2(x)\cosh(dx)/(\sinh(4x)\cosh(3x))\) gives the stated \(Y\) expression, with crossings again exactly folds (additional \(+\sinh((d-1)x),-\sinh((d-2)x)\) respectively). For \(X\) remove the \(\cosh^2(x)\) in that subtraction, giving the correction. This covers the whole indicated ranges. Real tails cancel at infinity, excluding affine ambiguities; the prescription of \(\Re P\) with two derivatives given by the symbols is smooth analytic with exponentially decaying second derivatives as used above (heights here within fixed decay margins).

Positivity at zero far-contour parameter. All resulting fused entries are nonnegative at \(\varepsilon=0\), off point singularities. For the extra correction at \(|d|\le3\), use the three factors \[\frac{\sinh x}{k\cosh x},\qquad \frac{c_*}{\cosh(2x)},\qquad \frac{\cosh(dx)}{\cosh(3x)}.\] Each has a positive inverse transform as a measure: use averaging and real parts of imaginary-shifted sech (for \(c_*=2\cosh x-1\), the shifted real sech dominates \(\cos(\pi/4)\) times the unshifted). When one partner is a double, pair differences \(d,d+1\) signed; combining the two uses \(c_*\cosh(x/2)=\cosh(3x/2)\), also positive by the same test for \(|d+1/2|\le3\), which holds at zero parameter. Any perturbative deficit here is \(O(\varepsilon)e^{-c_2|r|}\) by the individual analytic symbols with decay margins.

For the \(G\)-terms note \(G_y\ge0\) for \(0\le y\le 21/8\): with \(t_1=\pi y/6,\ q_0=\operatorname{sech}(\pi r/(6\beta))\), the hyperbolic tangent addition comparison reduces to \(\sqrt3\ge1+q_0^2(\sin^2 t_1-1/4)\), true since \(\sin^2(7\pi/16)+3/4<\sqrt3\). This handles the singleton cross pairs, while double cross partners combine by \(G_v+G_{v+1}=F_{v-1}+F_{v+2}\ge0\) at \(-1/2\le v\le3\). For like-group singleton pairs only \(d=0\) or \(d\in[17/8,21/8]\) are needed. In the latter case \(G_{2+d}=G_{2-y}\ge G_y=-G_{2-d}\), \(y=d-2\): compare odd Taylor powers of atanh at \(t_1=\pi y/6\in[\pi/48,5\pi/48]\) and \(\pi/3-t_1\). The linear difference favors the latter; for third powers use \(\sin^3\alpha=(3\sin\alpha-\sin(3\alpha))/4\), whose triple-angle contribution is identical. For odd orders \(m\ge5\), the difference (apart from common coefficients) is \[\cos^m t_1-\sin^m(\pi/3-t_1)+2\sin^m(\pi/6-t_1)-\sin^m(\pi/6+t_1)+\sin^m t_1 \ \ge\ .945^m-.833^m-.754^m>0.\] The last strict inequality holds at \(m=5\), where the difference exceeds \(0.1088\). Dividing by \(.945^m\) shows that it persists for every larger \(m\), since both ratios \(.833/.945\) and \(.754/.945\) are less than one. For a same-group double combine at signed differences \(d,d+1\); the four \(G\)’s become \(F_{1+d}+F_{4+d}+F_{-d}+F_{3-d}\). The necessary lower differences (double heights counted upwards in \(X\), or in absolute value in \(Y\)) at zero parameter are \(d\in[2,5/2]\) or \(d\in[-1,0]\). In the first case \(F_{1+d}\ge F_d,\ F_{3-d}\ge F_{d-2}=-F_{4+d}\); in the second all terms are positive or zero.

Moving the far double. For small \(\varepsilon>0\) only combinations involving the moved \(X\)-double can change. Cross partners there continue positive: for \(1\le v\le3\) use the preceding identity, while for \(3\le v\le4\), \(\sin(\pi(v-1)/6)\ge-\sin(\pi(v+2)/6)\), so that \(F_{v-1}+F_{v+2}\ge0\). Thus the whole range \(1\le v\le4\) is covered; in the just indicated same-group comparisons, every loss costs only \(C\varepsilon e^{-c_2|r|}\) by differentiation wherever the comparison may fail (e.g. \(d\) near \(5/2\)), away from negative log divergences. Explicitly the case \(d\in[2+\varepsilon,5/2+\varepsilon]\) obeys the same inequalities until \(5/2\), and the other moving lower difference is \(d=-1/8+\varepsilon\). This proves the first bound in (31).

The diagonal self cost. Finally by tail normalization \[\Re P_{II}(0)=w_I^2\int_0^\infty \left[\frac{2 c_*(x/\beta) A_I(x)^2}{x\sinh(4x)\cosh(3x)}-\frac1{2x^2}\right]dx\] where \(A_I=1\) or \(\cosh x\) for \(X\) or \(Y\) respectively, times \(\cosh(x/2)\) for a double. We have \[4x(2\cosh x-1)\cosh^2 x\cosh^2(x/2)(1+4x^2) \ \le\ \sinh(4x)\cosh(3x),\qquad x\ge0.\] Indeed the coefficient condition by product-to-sum is \((2n+1)(b_n+4(2n)(2n-1)b_{n-1})\le7\cdot49^n+1\), \(b_n=\delta_{n0}+3+2\cdot4^n+9^n+16^n\), second summand absent at zero. For \(n=0,1,2,3\) the left sides are \(8,300,10500,347172\); thereafter both left summands grow by a factor at most \(16(2n+3)(2n+2)/(2n(2n-1))\) from \(n\) to \(n+1,\ n\ge3\). This factor is at most \(38.4\) for \(n\ge3\), whereas \((7\cdot49^{n+1}+1)/(7\cdot49^n+1)>48\); the four checked cases therefore initiate the induction. Thus the bracket is bounded above by \(-2/(1+4x^2)\), and \(\int_0^\infty 2/(1+4x^2)\,dx=\pi/2\), proving the second inequality in Equation (31). ◻

Proof. To use (29),(30) simultaneously while retaining control, fix small far parameter \(\varepsilon>0\), take \(h=\varepsilon^{K_2}\) for a sufficiently large fixed \(K_2\), set \(\eta_0=\sqrt{\varepsilon}\), and scale both tilts by \(v_0=1-\eta_0\). Keep \(v_0 B(x)\) for localization; the tilt cost is no worse than scaling the old costs by \(v_0\), leaving a real-energy reserve. The smoothing errors of (24) can use small portions of that reserve (and \(o_\varepsilon(1)L\)), since the unscaled \(B(z)\le C(B(x)+L)\). In \(\Delta_H\) split off the piece with entry weights \(v_0 w_s(u)w_s(v)\) on the explicit positions (same convention for missing diagonals). On the complement (25) costs \(o(\eta_0)B(x)+o(1)\), plus the diagonal omission bound \[-C(1+|\log h|)\sum_{v\in\zeta}(1-v_0 w_s(v)^2).\] For the piece split off only \(e\) labels intervene. All curvature changes there are negligible even after smoothing, since straight approximation holds exponentially well on deep support (with log ratios at the main diagonal), with buffers to any exceptional smearing regions where only pointwise residual bounds are needed. After combining with \(v_0(a,\mathcal H a)/2\) the result is \[\frac{v_0}2(w_s\zeta_e,\mathcal H w_s\zeta_e)_{\rm off} -\frac{v_0}2\sum_{v\in\zeta_e}w_s(v)^2\Re P_{I(v)I(v)}(0)\] up to negligible errors or \(o(\eta_0)(B(x)+1)\). Indeed one can replace \(a\) by \(\psi*(w_s\zeta_e)\) in that quadratic by slow variation, with error at most \(C_{\varepsilon,h,\delta} L^{-1}\sum n_j^2\) using the decaying kernel \(\mathcal H\) (at most local log singularities, as proved in Lemma 17). The remaining averaging errors use \(\Re(P-\psi P\psi)\), \(O(h^{c_2})\) with exponential spatial decay, since after two derivatives the symbol of \(\Re P\) is analytic and exponentially decreasing, uniformly here.

Thus (23) absorbs the first lower-bound error by the reserve. In this argument the off-diagonal deficit is at most \(C\varepsilon\sum_j n_j^2\). By Equation (23), its energy coefficient is \(O(\varepsilon(1+|\log h|))=o(\sqrt\varepsilon)\). The smoothing errors are smaller still when \(K_2\) is large. These estimates leave a fixed positive fraction of the \(\sqrt\varepsilon B(x)\) reserve. Applying Equation (23) to a further fraction of that reserve gives the tail and count terms in (32). Finally \(\int w_c^2=2L_X-O(\delta L)\), and scaling the gain by \(v_0\) loses only \(O(\sqrt\varepsilon L)\). ◻

Proof. Indeed in deep shoulders there are three singleton types and two double types (no main \(Y\)); \((\mu,R\delta_v)=o(1)\), curve Jacobian modulus \(1+o(1)\) where applicable, and the deep extra log penalty \(C(1+|\log h|)\eta_0=o(1)\) per particle. By (31) the limiting sum of grand activities per unit length there is bounded by \(3(d/c)e^{-\pi/4}/(2\pi)+2\cdot4c^2(d/c)^2 e^{-\pi}/(2\pi)=0.429757\ldots<0.44\). All boundary and transition buffers have only \(O(\delta L)\) volume; two infinite ends also use the \(\ell_e\) damping. Common interior far counts are bounded by (27); thus factorial integration with the indicated per-particle constants gives (33). The polynomially many allowed \(M\)’s cost only \(e^{o(L)}\). In using coarse bounds, summing with \(e^{C_{\varepsilon,h} n_e}\) and part of the retained damping costs at most \(e^{O_{\varepsilon,h}(L)}\); \(U_0\) can be chosen sufficiently large afterwards.

For example, for a species of activity \(a\) on a set of length \(V\), the factorial sum is \[\sum_{m\ge0}(aV)^m/m!=e^{aV}.\] This identity applied species by species explains both the shoulder contribution and the buffer-volume bound. On an infinite end, replace \(V\) by \(\int_0^\infty e^{-c_{\varepsilon,h}r}\,dr\), retaining the \(\ell_e\) penalty. In the common interior the bounded number of far particles contributes at most a fixed power of \(L\), hence \(e^{o(L)}\). ◻

Proof. Fix a dyadic diameter bin \(D/2<\operatorname{diam}P\le D\). We retain a fixed positive fraction of the ordered vertex pairs on every sufficiently long polygon in this bin, then compare their weights with marked cylinder correlations.

Short polygons and retained pairs. By Equation (34), polygons of length at most \(D^{4/3}\) contribute at most \(CD^{-2}D^{8/3}=O(D^{2/3})\) to the second-length mass in this bin. On a longer polygon, retain the ordered pairs whose separation exceeds \(D^{1/2}\) and whose displacement has perpendicular component greater than \(D^{1/8}\) in absolute value to each of the three triangular lattice directions. For each first vertex, the excluded disk contains \(O(D)\) lattice vertices, and the three excluded strips within distance \(D\) contain \(O(D^{1+1/8})\). Since \(|P|>D^{4/3}\), at least half of its ordered vertex pairs remain once \(D\) is large.

A row determined by the two ports. Choose a triangular angular basis whose positive \(60^\circ\) cone contains the directed displacement. At each endpoint triangle, at least one of the two occupied sides has an allowed direction, since only one of the three side directions is excluded. Move the endpoint to such an occupied port, using a fixed rule if there is a choice. These moves have bounded length and preserve the retained margins. Write \(e_1,e_2\) for the chosen unit basis sides and \(h_p,h_{p'}\) for the positive half-side vectors at the resulting ports. Then \[p'-p-h_p-h_{p'}=m_1e_1+m_2e_2,\qquad m_1,m_2\in\mathbb Z_{\ge0},\quad m_j\gg D^{1/8}.\] The left side joins vertices of the triangular tiling, which proves integrality. The angular margin gives the two positive lower bounds. Travel from \(p\) along \(h_p\), then along the \(m_1+m_2\) complete sides in a fixed order, and finally along \(h_{p'}\) to \(p'\). This gives a row with \(s=m_1+m_2\) ordinary bands and \(s\asymp|p-p'|\).

Choose among the finitely many rotations and reflections so that the larger coefficient is assigned to the upper site height. Let \(e_{\rm low}\) be the other basis direction. Continue from \(p'\) along its remaining half-side, then along \(b\) complete sides in direction \(e_{\rm low}\), and finish with a half-side parallel to \(h_p\). The period vector is \[T=m_1e_1+m_2e_2+b e_{\rm low}+2h_p+2h_{p'}.\] All increments lie in the same positive cone, so the repeated row is simple and \(|T|\ge T\cdot e_{\rm low}\ge b\). Choose \(b\) to be the smallest integer at least \(KD\) for which \(s+b+2\) is even, with \(K\) a fixed sufficiently large constant. Then \(b\asymp D\), \(s\le b\), and \(|T|>2D\). No two points of \(P\) can differ by a nonzero integer multiple of \(T\), so its cylinder projection is injective. This choice of row and period depends only on the port data, the chosen basis, and \(D\). Figure 2 shows the endpoint half-sides, a fixed order of the ordinary bands, and the complementary part of the period.

The retained pairs have \(s\gg D^{1/2}\), and the minority count is \(s(1-t)=\min(m_1,m_2)\gg D^{1/8}\). Consequently \[L_X\ge\frac{\log D}{2\gamma}-O(1),\qquad L_Y=\frac{\log D}{\gamma}+O(1).\] Every fixed \(0<c_1<1/2\) and \(0<c_4<\gamma/8\) therefore gives \(L_X\ge c_1L_Y\) and \(s(1-t)\ge e^{c_4L_Y}\) for all sufficiently large \(D\). These configurations meet the stated margins.

Counting the marked occurrences. Moving a vertex to an incident port has at most two inverse choices at each endpoint. The choices of basis, rotation, reflection, and port type also have bounded multiplicity. Normalize the first port modulo triangular-lattice translations. Once its lift is fixed, the projected polygon has a unique planar lift through that port, with its original vertex weight. Thus the retained pairs account for at least a fixed multiple of \(|P|^2\) marked occurrences, and any one marked cylinder polygon arises from at most a fixed number of these occurrences.

For a dyadic displacement scale \(S\), let \(\mathscr R_{D,S}\) be the row setups just constructed with \(S\le s<2S\), and write \(\mathcal C_{\mathfrak r}\) for the corresponding marked correlation. There are \(O(S^2)\) setups: after fixing the first port type, there are only \(O(S^2)\) possible second-port displacements, and the remaining choices are bounded. Positivity and the preceding multiplicity bounds give \[\sum_{\substack{[P]:\ D/2<\operatorname{diam}P\le D\\ |P|>D^{4/3}}} \rho_{\rm v}^{|P|}|P|^2 \le C\sum_{\substack{S\ \text{dyadic}\\D^{1/2}\ll S\ll D}} \ \sum_{\mathfrak r\in\mathscr R_{D,S}} \mathcal C_{\mathfrak r}.\] A bound of the stated form makes each displacement scale contribute at most \(D^{2/3+o(1)}\). The \(O(\log D)\) scales and the short-polygon estimate prove the claimed reduction. ◻

Proof. Trace bounds for the straight comparison. In the rare case let \(T^0\) denote the straight comparison using physical main coordinates \(W_X(v)+i\beta,\ W_Y(v)\), and exactly the same \(h_0\). In the regular case let \(T^0=T\). For \(T^0\) the mixed derivative of the true-minus-reference kernel has uniform \(C^j\) bounds of order \(C_j e^{-c|v-v'|}\), all fixed \(j\). Indeed on a regular curved main the same diagonal argument subtracts the horizontal log, the residual using the divided ratio for the curved \(\sinh\); it is uniformly regular including at caps by composition (the ratio of the two physical complex and horizontal differences before composing with \(W_X\) already extends smoothly with controlled log). Other near-diagonal entries are offset from their zero angles. At long differences the linear and sign asymptotes, including the terms from curved heights, are additive in individual coordinates in fixed order and vanish on mixed differentiation; the remaining log tails and the reference sine ratios obey the same exponential estimates. To obtain the trace bound, partition each interval into cells of lengths in \([1,2]\) and use their normalized cosine bases \(e_{a,n}\). Integrating twice by parts in each variable, with the first boundary term zero and the second controlled by a kernel derivative, gives \[|\langle e_{a,n},\mathscr K e_{b,m}\rangle| \le \frac{Ce^{-c|a-b|}}{(1+n^2)(1+m^2)}.\] Summing the trace norms of these rank-one entries gives \(O(L)\); Schur’s test gives an operator norm \(O(1)\). Boundedness of \(j\) transfers both statements to \(T^0\).

The common and shoulder symbols. Cyclic kernel integrals then give the word limits used below. More precisely, deep points \(|v|<L_X\), away from the center and edges by distances tending to infinity, see the common symbol \(T_\infty(k)\) there with \(p=(3-\lim\theta_\infty)\beta\) (angle 1 for the horizontal comparison in the rare case); see the main transform formulas. Indeed the reference densities in each tail are asymptotic with derivatives to a constant times \(e^{-\gamma |v|}\), and the curved height is exponentially asymptotic to \(\theta_\infty\). Deep points in \(L_X<|v|<L_Y\) see only its \(Y,Y\) entry. Their respective weights after division by \(2L\) are \(\lim L_X/L\) and the complement. Bounded-distance exceptional regions have zero limiting share, and relative-coordinate tails can be truncated uniformly by the kernel bounds. For completeness, its \(2\times2\) expression, for \(k>0\) and \(x=\beta k\), is \[T_\infty(k)=\frac{2\cosh x-1}{\sinh(4x)\coth(3x/2)} \begin{pmatrix}\cosh(3x)&e^{pk}\\ e^{-pk}&\cosh(3x)\end{pmatrix}; \qquad T_\infty(-k)=T_\infty(k)^{\mathsf t}.\] It is obtained by dividing the true primitive symbol \(k^2\widehat H\) by the limiting reference symbol \(2\pi k\coth(\pi k/(2\gamma))\). Its determinant is independent of \(p\), and, with \(y=e^{-x}\), \[\sqrt{\det T_\infty(k)} =\frac{(1-y^3)(1-y^6)}{(1+y)(1-y^8)}.\] The logarithmic integral is \[\int_0^\infty\log\sqrt{\det T_\infty(x/\beta)}\,dx =-\frac{\pi^2}{18}-\frac{\pi^2}{36} -\frac{\pi^2}{12}+\frac{\pi^2}{48} =-\frac{7\pi^2}{48}.\] Each term follows by integrating the convergent logarithmic series. Accounting for both signs of \(k\), the determinant square, and the common interval length \(2L_X\) yields the contribution \(7\gamma L_X/8\) to \(-\frac12\log|\det T|\). The scalar shoulder symbol for \(x=\beta|k|,\ y=e^{-x}\) is \[\frac{c_*(k)\cosh(3x)}{\sinh(4x)\coth(3x/2)} =\frac{(1-y^3)(1+y^6)}{(1+y)(1-y^8)}\] The integral of the logarithm of the shoulder symbol over \(x\in(0,\infty)\) is \[-\frac{\pi^2}{18}+\frac{\pi^2}{72} -\frac{\pi^2}{12}+\frac{\pi^2}{48} =-\frac{5\pi^2}{48},\] by expanding each logarithm in its absolutely integrable series. This gives (37) by the same truncated-log argument once trace convergence holds also for \(T\), since (35) was proved for \(T\).

Passing to the logarithmic determinant. Here is the truncated-log passage used in this calculation. For the singular values \(s_i(T)\), the small-value count gives \[\sum_{s_i<\eta}\log(\eta/s_i) =\int_0^\eta\frac{\#\{s_i<r\}}r\,dr\le CL\sqrt\eta.\] Clipping \(\log(T^*T)\) below \(\eta^2\) therefore loses only \(O(L\sqrt\eta)\). On the bounded spectral interval, approximate the clipped logarithm by polynomials vanishing at one, with error at most \(\epsilon|x-1|\). The trace error is at most \(\epsilon\|T^*T-I\|_1=O(\epsilon L)\). Cyclic kernel integration proves the polynomial trace limits: truncate relative separations, use the integrable logarithmic bound for \(G\) and the exponential bounds for \(\mathscr K\), and then discard bounded neighborhoods of the center and walls. The common interval occupies length \(2L_X\) and the shoulders length \(2(L_Y-L_X)\). The same clipping estimate holds for the displayed symbols. The common symbol has one singular value comparable to \(k^2\) near zero and its other singular value stays positive; the scalar shoulder symbol tends to \(3/8\) there. Both symbols approach identity exponentially at infinity. Take size to infinity, then \(\epsilon\downarrow0\) and \(\eta\downarrow0\). The two logarithmic integrals give exactly the coefficients in Equation (37).

The rare-center comparison. It remains to prove the comparison uniformly at the rare central scale. Ordinary derivative bounds in \(v\) degenerate at that scale, so we use a dyadic decomposition. In intermediate coordinates put \[\mathfrak z_X(b_X^{-1}(\xi))=\xi+i\beta+E(\xi).\] Then \(E,E'\) are \(o(1)\) uniformly, with fixed higher derivatives \(E^{(j)}=o(1)\,[\min(1,|\xi|+d_0)]^{1-j}\), by the graph and matching estimates. The kernel on primitives uses a mixed derivative of the difference \(K_d=H_{nn}-H_{nn}^0\) with the two sets of maps just described. Split \(X\), on \(|v|\le1\) (so \(W_X(v)=v\)), into sign-paired shells of radii \(r_j=2^{-j}\) down to \(r_J\asymp d_0\): \(r_{j+1}<|v|<r_j,\ j<J\), and the last central interval \(|v|<r_J\). In shell \(j\) encode \(Q_X\) by its oscillation from the single mean \(m_j\) over that shell, measured in \(L^2\) after rescaling the shell by \(r_j\), and keep the numbers \(m_0,\ m_j-m_{j-1}\). The means thus contribute to \(Q_X\) by these coefficients times the steps \(\mathbf1_{|v|<r_j}\). Outside this region and on \(Y\) keep \(Q\) itself in \(L^2\) on cells of lengths between 1 and 2. This joint encoding has squared norm bounded by \(C h_0(Q,Q)\). Indeed same- and adjacent-shell squared mean/oscillation differences are bounded in sum by the ordinary difference seminorm on \([-1,1]\), directly controlled there by the angular difference representation of \(h_0\); the outer mean and ordinary cells cost \(\|jQ\|^2\). For example, with \(A_j=\{r_{j+1}<|v|<r_j\}\), \[\frac1{r_j}\int_{A_j}|Q-m_j|^2 \le C\iint_{A_j\times A_j} \frac{|Q(u)-Q(v)|^2}{|u-v|^2}\,du\,dv,\] and the same calculation on \(A_j\times A_{j+1}\) bounds \(|m_j-m_{j+1}|^2\). These pair regions have bounded overlap. The last central interval satisfies the same scaled estimate. Thus the encoding map is bounded independently of the number of shells.

For oscillation or ordinary cells the matrix kernel on this encoding takes one derivative of \(K_d\) on the natural unit scale of each applicable variable (rescaled for shells). For a step, integrate that derivative over the step: take instead the difference of values at \(+r_j,-r_j\). Thus there is no cost from a large common primitive mean at tiny scales. Each cell pair, including derivatives through two further orders per continuous variable, has bounds \(C\epsilon_L\) times:

• \(\min(r_i/r_j,r_j/r_i)\) between shells (steps are included at their corresponding scale);

• \(r_j e^{-c|v'|}\) from shell \(j\) to an ordinary cell at \(v'\);

• \(e^{-c|v-v'|}\) between ordinary cells;

where \(\epsilon_L=o(1)\).

To verify them, on two central \(X\) arguments the diagonal log change is a multiple of the smooth log of \(1+(E(v)-E(v'))/(v-v')\) plus regular analytic changes. For comparable scales this has all required bounds by divided differences at nearby arguments in the same regular neighborhood of radius of that scale (including the innermost pair), or by separation otherwise. For a much smaller scale \(r_i\ll r_j\) differentiation on the smaller scale gains \(r_i/r_j\), since the denominator has size \(r_j\) and all positive-order normalized derivatives of \(E\) on the smaller shell have size \(C\epsilon_L r_i\); further derivatives on either scale preserve the bound. A step difference gains the same by varying its argument across \([-r_i,r_i]\), using only the uniform slope bound for \(E\) there. The larger step endpoints themselves stay on the larger scale. The argument across the interface to ordinary \(X\) cells is identical; a bounded-distance ordinary \(Y\) argument instead sees a nonsingular analytic kernel. In far-separated comparisons the pure one-variable log asymptotes drop out after the two derivatives/differences, including when one uses step endpoints (order with the other far argument is fixed); the log series remainders give exponential decay along with the small change bounds, with the scale gain from a central derivative or step. On ordinary \(X\) cells near a cap the divided differences for the change use \(E(\xi)-E(\xi')\) divided by \(\xi-\xi'\) before the cap map, so do not incur any endpoint derivative loss. These observations prove the stated estimates for the log branches continued by the graph deformation; additive constants do not enter.

By smooth cell expansion (e.g. the same cosine trace bound as before, on the two separate sign components where needed), these are also trace-norm bounds per pair. Their row and column sums are \(O(\epsilon_L)\), and their overall sum \(O(\epsilon_L(L+J))\). Factoring on both sides by the bounded encoding proves (38). Consequently the needed trace bounds and word limits transfer to \(T\) (trace-norm comparison of products), proving (37).

Since \(J=O(1+\log(1/d_0))=O(L)\), the shell-pair trace estimate is indeed \(o(L)\). To transfer a word of fixed length in \(T-I,T^*-I\), expand its difference from the corresponding word in \(T^0\) as a sum with one difference factor in each summand. The trace norm is \(o(L)\), because all other factors have bounded operator norms. The small-singular-value count in Equation (35) controls the removed portion of the logarithm uniformly, so no uniform pointwise inverse bound at the rare scale is required. ◻

Proof. Local scale and density. On rare \(X\), \[\frac{dv}{d\tau}\asymp\min(1,d_0+|v|).\] The added length near the center is \(O(1+\log(1/d_0))=O(L)\). Thus the intervals still have length comparable to \(L\). In a \(v\)-disk of radius \(c(|v|+d_0)\) near the center, the derivative of \(\tau\) has small relative variation when \(c\) is small. Analytic inversion, together with the graph and mean-matching estimates on that same scale, gives the claimed fixed \(\tau\)-disks. At the caps the stretch is smooth and respects the reflections; analytic continuation there is not needed.

The identity \(\lambda_g=\lambda_g^0\,dv/d\tau\) gives the logarithmic and relative derivative bounds. On \(|v|\le1\) in rare \(X\) it gives \[N_X\lambda_X\ge c s d_0\ge e^{c'L},\] using the minority margin. Away from this central region, throughout a core, \[N_g\lambda_g\ge c s_g e^{-\gamma(1-\delta_*)L_g} =c e^{\gamma\delta_*L_g}.\] These two estimates prove Equation (41). In particular one need not remove the center from the support of a primitive.

Compensated logarithms. Consider first two points on the same label. If their \(\tau\)-distance is at least a fixed \(c_0>0\), their horizontal physical separation is at least a constant times either local scale \(dv/d\tau\). This follows by integrating the local scale on a fixed subinterval; it remains true at a cap, where the physical derivative can vanish at the endpoint. Likewise the sine of half the angular difference bounds both local \(\lambda_g\) from below up to constants. The sine of the angular half-sum is at least as large. The logarithms and their derivatives therefore have the asserted bounds away from the local diagonal.

Near the diagonal, divide both physical and reference differences by \(\tau-\tau'\). Off the caps, the physical complex quotient has size comparable to \(dv/d\tau\), while the reference cosine quotient has size comparable to \(\lambda_g\sin t_g\); here \(\sin t_g\asymp\lambda_g^0\) away from the endpoints. Their relative analytic derivative bounds follow from divided differences on the fixed disks just constructed. At an endpoint, divide also by the reflected coordinate difference. The quadratic cap maps make both quotients smooth, nonzero on their natural scales, and even under reflection. Their logarithmic ratio is consequently smooth through both diagonals. Cross-label factors and the remaining same-label factors stay at regular imaginary offsets when their real arguments approach. Shrinking the core disks preserves those separations. This proves the bounds and core analyticity for \(H_{nn}-H_{\rm ref}\).

Linear sources and end corrections. For each label decompose the signed measure \(m'_gp_g-\mu_g\) into \((m'_gC_g-s_g)\bar\nu_g\,du\) plus \(m'_gC_g\) times the difference between the unnormalized capped push and \(\bar\nu_g\,du\). The first coefficient is polynomial. Its convolution is given by the mean identities, so it consists of ordinary \(V\)-logs evaluated on the contour. The only possible shrinking site-pole distance is at the rare \(X\) center. There it is bounded below by a constant times \(|u|+d_0\), exactly the local scale controlled by \(\tau\).

For the difference of measures, insert a fixed-width cutoff around the caps and their exterior tails. The measures agree off this set, and their densities in their respective parameters carry the factor \(e^{-\gamma L_g}\). In a same-label end convolution, subtract the ordinary local logarithm and its reflected term for the capped input. The curved divided-log remainder is regular. Derivatives of the singular logarithm transfer to the smooth cutoff density, using its even extension for the capped input; for the full-line input transfer derivatives in \(u\) before composition. Multiplication by \(m'_g\) thus leaves polynomial bounds. The end correction is analytic in core variables by spatial separation on the same label and by imaginary offsets on the other label.

The explicit-particle term \(\widetilde H_{ne}\zeta\) uses the regular offsets and horizontal subtraction from Equation (24). Its subtraction \(R_h\) is inactive on moving core variables. The anomaly logs have the uniform offsets established in the mean calculation. These observations prove the linear-test assertion. Its constant term has the stated lower bound by Equation (26).

Dual tests and polynomial approximation. Against a smooth density supported in the core, subtract the local \(\tau\)-diagonal logarithm of \(H_{\rm ref}\) and transfer its derivatives to the density. The remaining kernel has the bounds just proved. Thus \(H_{\rm ref}\) maps these densities to smooth even tests with polynomial derivatives. The basis fields were constructed with polynomial derivatives in \(v\), hence also in \(\tau\). The trial primitives have the same property by Equations (27), (29) and (30). Differentiating their density tests gives the odd-reflected primitive tests for \(h_0\). Adding the smooth compensated kernel gives those for the true \(H_{\rm form}\). This last step is why no polynomial derivative bound for the true form in the unstretched rare coordinate is needed.

We will also use physical sine approximation on the \(\tau\) intervals. Two estimates ensure that its errors remain polynomial. First, \(\mathcal X\hookrightarrow L^2(d\tau)\) has polynomial norm: the inverse kernel satisfies \[|G(\tau,\tau')|\le C\bigl(L+ |\log\min(1,|\tau-\tau'|)|\bigr)\] by the exponential lower bound on \(\lambda_g\), and Schur’s test applies on intervals of length \(O(L)\). Second, \(h_0\) is polynomially bounded on unit vectors in every polynomial Dirichlet sine bandwidth. For this use the angular difference kernel with its density factors. Its unreflected part is at most \(C(1+|\tau-\tau'|^{-2})\), by the relative density derivative bounds. The reflected part contributes an endpoint inverse-distance mass term, since \[\sin t_g\ge c\lambda_g\min(1,\widetilde A_g-|\tau|),\] and its density-weighted kernel is bounded outside pairs near the same endpoint. Sine functions vanish at the endpoints, so these local difference and boundary estimates give the asserted bandwidth bound. ◻

Proof. Coarse integration and count truncation. For any chosen tilt, the infimum \(m_z\) satisfies its deterministic trial lower bound: first test smooth real fields, then use continuity in \(\mathcal X\). For \(Z_0=0\), take absolute values and apply Proposition 24 with \(F=\operatorname{Re}T\). The smooth compensated data are defined at fixed sizes even before polynomial truncation. The spectral bounds and \(\|T-I\|_1=O(L)\) give an \(O(L)\) negative-log sum, and the reference masses \(J_{N_g}\) are bounded above. Thus the logarithm of the unshifted integral is at most \(-m_0+O(L)\).

Combining this with Equations (23) and (26) retains damping in \(\ell_e+(M^2+n_e^2)/L\), apart from \(C_{h,\varepsilon}(L+n_e)\). Sum the Gaussian damping in \(M\) and the position damping with the factorial measures as in Equation (33). We may thereby restrict \(n_e,|M|,\ell_e\) to sufficiently large fixed powers of \(L\) with error \(\exp(-\omega(L))\) at the extracted-product normalization. Fix these powers from now on. Then \(N_g=s_g+O(L^K)\asymp s_g\), and all kernels, dual tests, and lower bounds for constant terms have the polynomial controls proved above. The sharp source estimate (43) is available in \(\tau\).

The affine path and its real gap. Put \[\mathsf A=T_{SS},\qquad \mathsf C=T_{SR},\qquad \mathsf B=\mathsf A^{-1}\mathsf C, \qquad H_\vartheta=(1-\vartheta)I+\vartheta\mathsf A, \quad D_\vartheta=H_\vartheta^{-1/2}.\] For \(0\le\vartheta\le1\), the three successive output primitives are \[\begin{aligned} &Q+i\vartheta Z_0,\\ &iZ_0+Q_R+Q_S -\vartheta(\mathsf A^{-1}\Pi u_z+\mathsf BQ_R),\\ &iZ_0+Q_R+D_\vartheta Q_S -\mathsf A^{-1}\Pi u_z-\mathsf BQ_R. \end{aligned}\] The stages join at their endpoints, and the last endpoint is Equation (45). Accretivity places the spectrum of \(H_\vartheta\) in the right half-plane. The resolvent representation of its principal inverse square root gives polynomial bounds on \(D_\vartheta\) and its required parameter derivatives, using Equation (35). Flattening the parameter at the joins produces a smooth path.

We verify the real Hessian along this path. Write \(T[Y,Y]=h_0(Y,TY)\) with the complex bilinear convention. The first stage changes only the source. For real \(s\in S,r\in R\), the second-stage linear map satisfies the exact identity \[T[(s-\vartheta\mathsf Br,r),(s-\vartheta\mathsf Br,r)] =T[(s,(1-\vartheta)r),(s,(1-\vartheta)r)] +(2\vartheta-\vartheta^2)T_e[r,r].\] Both real forms on the right have the magnetic gap, and the two coefficients of \(\|r\|^2\) add to one. At the third stage the Hessian is block diagonal, with blocks \(T_e\) and \(\mathsf A H_\vartheta^{-1}\), because \(D_\vartheta\) commutes with \(\mathsf A\) and is symmetric. The Hermitian identity \[H_\vartheta^*\operatorname{Re}(\mathsf A H_\vartheta^{-1}) H_\vartheta =(1-\vartheta)\operatorname{Re}\mathsf A +\vartheta\mathsf A^*\mathsf A\] gives an inverse-polynomial gap throughout this stage as well. All nontrivial input uses of \(Q\) are through smooth dual tests; all output changes lie in \(S+\mathbb C Z_0\) and are core-supported. Their polynomial bounds follow from the core construction and the stretched regularity lemma. The path changes the Hessian by finite rank \(O(L)\), so its trace norms and lower bounds for minimizing costs are polynomial too.

At the endpoint, the real Hessian is \(I_S\oplus\operatorname{Re}T_e\). Its real infimum is at least \(m_z\). Indeed, for fixed real \(r\), the complex stationary point of \(C_z(s+r)\) in \(S\) is \(s_*=-\mathsf A^{-1}(\Pi u_z+\mathsf Cr)=s_1+is_2\). Completing that square gives \[\operatorname{Re}C_z(s_*+r) =\operatorname{Re}C_z(s_1+r) +\tfrac12h_0(s_2,\operatorname{Re}\mathsf A\,s_2) \ge m_z.\] The remaining endpoint input adds \(\|s\|^2/2\).

The particle chain and its pole distances. Write \(a_\vartheta(Q)\) for the output change along the flattened path. Choose a smooth cutoff \(\chi\), equal to one on \([0,1]\) and zero on \([4,\infty)\), and use the parameter \[\vartheta(Q)=\chi(\|Q\|_{-2,\widetilde A}^2/L^{2D}).\] Thus configurations in \(\mathcal G\) undergo the full path, while those outside \(\|Q\|_{-2,\widetilde A}\le2L^D\) remain fixed. For \(a(Q)=a_{\vartheta(Q)}(Q)\), move a particle at \(\tau\) by \[\Delta_g(\tau;Q)=-\frac{a_g(Q)(\tau)}{N_g\lambda_g(\tau)}.\] The homotopy uses \(a_{\upsilon\vartheta(Q)}(Q)\) for \(0\le\upsilon\le1\). The exponent \(D\) will be chosen after the polynomial truncations. For fixed \(D\), Equation (41) makes \(\Delta_g\) and its fixed-order spatial derivatives exponentially small, with \(Q\) and the cutoff parameter frozen for these derivatives.

On rare \(X\), a fixed small \(\tau\)-disk corresponds to physical increments on scale \(dv/d\tau\asymp\min(1,d_0+|u|)\) in the core. Hence the physical particle displacement is exponentially small relative to \(d_0+|u|\) near the center. The nearest site pole has at least that distance by the stretched regularity lemma. All other poles have fixed separation there. The compensated kernels and maps therefore remain analytic on the moving chains, including against inactive coordinates. Same-label coincidences are Vandermonde zeros; the divided difference of the compensated ratio extends analytically through them.

The exact contour argument of [10] now applies to these chains. Its maps need only be \(C^2\) in the real particle variables. This holds here because a particle’s first and second variations of its empirical primitive are \(-\delta_\tau\) and \(\delta_\tau'\), both continuous in the physical \(H^{-2}\) space. Holomorphy is required of the density times volume form, not of the smooth core fields defining the chain.

To see that the one nonholomorphic factor causes no difficulty, partition every coordinate into a slightly enlarged active core and its complement, and freeze the complementary coordinates. A like-main explicit particle lies outside its own wall, whereas \(R_h\) is supported at horizontal distance at most one. Thus \(A=\exp(-(X_{\rm int},R_h\zeta))\) is independent of every active coordinate. Factor it out before applying Stokes’ theorem to the remaining holomorphic top form. The displacements vanish near the core boundaries, so the side faces of the homotopy contribute zero. Only after this contour equality do we use \(0\le A\le1\).

Compensated cost and finite Jacobian. We indicate why the error estimates in that contour proof persist in \(\tau\). Freeze \(Q\) and \(a\), and multiply \(\Delta\) by a scalar complex parameter. Its analytic disk can have exponentially large radius: the motion stays inside the fixed \(\tau\)-disks, and the cosine-difference ratios stay near one by simultaneous divided differences and the small spatial derivatives of \(\Delta\). After removal of the unchanged explicit-pair constant, all measures have at most exponential total variation and all remaining kernels have polynomial bounds. Cauchy’s estimate on the growing disk, at a sufficiently high fixed Taylor order, therefore gives an \(o(1)\) remainder at scalar parameter one.

For real frozen displacements the \(k\)th mean-push coefficient is \[d_{g,k}=\frac{(-\partial_\tau)^k}{k!} (N_g\lambda_g\Delta_g^k).\] It has \(d_{g,1}=a_g'\), while the coefficients for \(k\ge2\) and their fixed derivatives are exponentially small, since \(N_g\lambda_g\Delta_g^k=-a_g\Delta_g^{k-1}\). Pairings of the centered push terms against smooth tests are also exponentially small on the polynomial event. Local-log subtraction controls \(H_{\rm ref}d_{g,k}\). The resulting finite Taylor identities are polynomial in the frozen field coefficients and hence hold for complex displacements as well. This is exactly the compensated substitution in the companion proof: after its one-coordinate density and diagonal cancellation, the cost is \[\mathscr C_0(Q+a(Q))-\tfrac12h_0(Q,Q)+o(1).\] As before, the displayed expression means its smooth compensated expansion; no empirical self energy is evaluated separately.

The remaining Jacobian differs from the identity by a matrix of rank \(O(L)\). On the full-shift event its reduced entries are empirical quadratures of smooth core tests divided by \(N_g\lambda_g\). Their centered errors are exponentially small by Equation (41) and the \(H^{-2}\) event bound. The matrix determinant lemma therefore gives its modulus as \[(1+o(1))|\det T_{SS}|^{-1/2},\] just as in [10].

The transition and restoration of masses. For each fixed path parameter the compensated real cost has the inverse-polynomial gap and polynomial bounds proved above. The negative-metric penalty \(L^{-E}\|Q\|_{-2,\widetilde A}^2\) has polynomial operator norm and trace norm by the stretched Sobolev estimates. Choose \(E\) large enough that subtracting it preserves half the gap. For this metric penalty use the sine-truncation extension of the coarse quadratic bound in [10]; its finite smooth truncations converge on empirical fields, and Fatou’s lemma passes the bound to the full penalty. It then suppresses \(\|Q\|_{-2,\widetilde A}>L^D\) by \[\exp(-L^{2D-E}+L^{E_1}),\] where \(E_1\) is fixed independently of \(D\).

The actual parameter depends on \(Q\). On the transition region, path derivatives of the cost are at most \(L^{C+2D}\), so a grid of mesh \(L^{-C-2D-1}\) compares every cost with a grid cost within one. Summing the preceding estimate over this polynomial grid controls that dependence. The cutoff adds one rank-one Jacobian term; the reduced matrix has size \(O(L)\) and entries at most \(e^{C_DL}\), so its determinant costs at most \(\exp(O_D(L^2))\). These estimates give a discarded contribution smaller than any prescribed \(\exp(-L^{E_2})\) by choosing \(D\) sufficiently large. Choose \(E_2\) after the fixed polynomial count and position cutoffs: their total factorial integration volume and number of sectors are at most \(\exp({\rm poly}(L))\), so summing the discarded contributions remains negligible. All these choices are fixed before \(L\to\infty\).

On the full-shift event apply the sharp quadratic bound using Equation (43) and the infimum comparison with \(m_z\). Equation (40) bounds the remaining negative-log sum by \(o_\xi(L)\) and replaces the core determinant by the full determinant. Finally restore \(J_{N_X}J_{N_Y}\), whose logarithm is \(-\gamma(L_X+L_Y)/4+O(1)\). Equation (37) now gives exactly the stated sector bound. Each \(N_g\) remained the same integer throughout the deformation. ◻

Proof. For \(U>U_0\), Equation (27), the count bound, and factorial summation make the contribution negligible once \(U_0\) is sufficiently large, after the smoothing and probe parameters have been fixed. For \(U\le U_0\), combine Proposition 25 with Equations (32) and (33), and divide by the cleared denominator of Theorem 7. The denominator estimate applies uniformly to the site mixture, and \(N\asymp b\).

The exponent can be checked before changing variables. The trial cost contributes \(-2L_X\), the fluctuation and reference factors contribute \(3\gamma L_X/8+\gamma(L_Y-L_X)/16\), and the explicit shoulders cost at most \(0.88(L_Y-L_X)\). The denominator contributes \(-(5\gamma/24)L_Y+o(L)\). Their sum is \[-\frac{16}{9}L_X+ \left(0.88+\frac{\gamma}{16}-\frac{5\gamma}{24}\right)(L_Y-L_X) +o(L).\] Since \(\gamma=4/3\), \(\gamma L_X=\log s\), and \(\gamma L_Y=\log b\), this is the claimed marked-pair exponent. To remove the error coefficients, first make the reserve and smoothing loss arbitrarily small, then make the buffer fraction sufficiently small, and finally improve the core projection. Each choice remains fixed while \(L\to\infty\). The regular and rare subsequence arguments cover every sequence of site mixtures and wall ratios, proving uniformity. ◻

Proof of Theorem 1. The coefficient \(617/1200\) is smaller than \(2/3\). For the configurations selected in Lemma 20, \(b\asymp D\) and \(s\le b\). The sharp theorem therefore bounds the retained port-pair weights by \(D^{o(1)}s^{-4/3}(D/s)^{2/3}\). At dyadic diameter \(D\), a dyadic displacement scale \(s\) has only \(O(s^2)\) possible placements after fixing the first port modulo translation. Its contribution is at most \[D^{o(1)}s^2s^{-4/3}(D/s)^{2/3}=D^{2/3+o(1)}.\] There are \(O(\log D)\) displacement scales. The short-polygon contribution is already bounded in Lemma 20 by the planar diameter tail. Summing the dyadic diameter bounds up to \(H\) proves Theorem 1. ◻

Proof. A closed loop disjoint from the marked arc cannot separate its endpoints, so its two orientations cancel as in the marked-pair calculation. Order the vertical matching from \(p\). The directed arc has westward initial and final tangents. With argument cut below \(p\), the angle change of \(\Gamma-p\) is \(-\pi/2\); with argument cut upward, that of \(p'-\Gamma\) is \(-3\pi/2-2\pi F_X\). The chord-angle identity therefore cancels the twist against the cup phases, leaving a constant phase. The summability and projection of the half-cylinder states are the same as for the marked-pair contraction. ◻

Proof. We first compute residues and then compare every exceptional spin factor. Keep real site offsets generic and distinct until taking confluence. Moving main contours to the far lines produces ordinary strings of length 0, 1 or 2; the cancellation excluding longer strings is unchanged. All singleton poles of \(Q_t\) lie outside the swept regions. For a double, multiplication of its elementary factors cancels the relevant particle-dependent denominator before moving contours. The additional real growth is bounded by \[\sum_{t=\pm1}\left\{ \frac12\sum_v|\Re v-\Re y_t| -\frac{n+1}{2}|\Re y_t|\right\}.\] There are \(n-1\) ordinary elementary particles. Up to a constant depending on the counts and fixed site offsets, their ordinary gas exponent is at most \[-n\sum_v|\Re v|+\sum_{\{v,w\}}|\Re v-\Re w| \le -2\sum_v|\Re v|,\] where the second sum is over unordered elementary pairs. Omitting an internal double pair does not change this bound, since its real difference is zero. By the same triangle inequality, the displayed probe contribution is at most \[\sum_v|\Re v|-\sum_{t=\pm1}|\Re y_t|.\] Adding the two bounds gives simultaneous exponential decay in all particles and both probes, justifying closure of the strips.

For a group-\(z\) site with string length \(a\), its site and string contribution to \(Q_t\), at argument site minus probe, is \[\begin{array}{c|ccc} a&0&1&2\\ \hline &1/P_J&P_{J+2z}/P_{J+z}&P_{J+3z} \end{array},\qquad J=z-t.\] The probe residues are therefore type \(\mathrm A\), at \(y_t=p_t\), or type \(\mathrm B_a\), at \(y_t=a_j+ita\beta\), where \(a_j\) is an ordinary site of group \(t\) with string length \(a=0,1\). There are no other poles in the indicated strips.

Expand in the full-group biorthogonal bases \(D_l,D_r^*\), including each initial mark, using Lemma 4. The summand contains \(D_lU,D_r^*U^*\) divided by their two norms; the remaining \(D_l^*,D_r\) meet the prescribed operator. An ordinary permutation in either group leaves the summand invariant: braid transport multiplies vectors and their norms by the same scalars, and the prescribed operator intertwines the ordinary braids. Place an ordinary site last. Its untouched last roots exclude \(r_j<l_j\). Projection gives \(\sum l=m\) in both groups and \(\sum(l-r)=1\) in each group. The plus mark has \(l_p=1\), while the minus mark has \(r_p=-1\), by their leading occupied roots. Thus the possibilities are precisely \[\begin{array}{c|cc|c} &t=+1&t=-1&\text{ordinary exceptions}\\ \hline \mathrm A&(l_p,r_p)=(1,0)&(l_p,r_p)=(0,-1)&\text{none}\\ \mathrm B_a&(l_p,r_p)=(1,-1)&(l_p,r_p)=(1,-1)& r_j=l_j+1\text{ at one site }j. \end{array}\] Other labels match. The string length at any ordinary site is \(\min(1+t l_j,1+t r_j)\), giving exactly the claimed particle totals.

Sort nonexceptional spectators first. On the plus group’s \(D_l^*\) the occupied mark traverses minus roots with saturated spins: each emerged minus forces nondecrease, and the initial spin is maximal. These crossings supply their \(F\)-factors. In type \(\mathrm A\) the vacancy on \(D_r\) splits to \(+,-\), since its leading root is plus; the second strand is again saturated. In type \(\mathrm B\) it starts vacant and ends minus while crossing plus roots, so exactly one crossing contributes the unit producing an emerged plus. This must be the leading plus root of site \(j\). If \(l_j=-1\), the actual spin on \(D_l^*\) is minus, forcing the real split to vacancy, minus: a minus could not emerge plus at the leading root. If \(l_j=0\), the actual spin must be vacant, and the starred split there is minus, plus. Their first relevant roots are untouched when the ordinary sites are sorted before inserting the mark. All remaining exceptional strands then cross spectators with saturated spins. Consequently there is exactly one output cup \(U\) in place of one \(F\)-factor. For the minus group exchange starred and real lists and negate relative spins; the physical cup phases agree. Spectator overlap gives the matching norm.

Here are explicit rules for the product comparison. Recall \[\mathcal R(\pm1)=\{(\pm1,0)\},\qquad \mathcal R(0)=\{(1,-1),(-1,1)\}.\] For ordered sites, evaluate factors at first coordinate minus second. Roots \((e,g),(f,h)\) shift that difference by \(g-h\). Put \(u=1\) for the \(l\)-half, \(u=-1\) for the \(r\)-half. Equal groups and equal signs contribute \(P_{2u}P_{3u}\); opposite groups and opposite signs contribute \(P_{-2e}P_{-3e}\). For equal groups and opposite signs an uncancelled norm contributes \[\frac{P_{-2e}P_{-3e}}{P_0P_e}.\] Include this once, from the \(l\)-list, for matching ordinary spectators. If the earlier site is exceptional, include it in each half only for \(e=-u\), as the saturated routes require. In each half divide by \(2\cosh((\cdot)/2)\) at shift zero if exactly one site label is zero, or at shifts \(-1,0,1\) if both are zero. Finally the special cup contributes \((-2itd)/P_{-ta}\) at mark minus changed site. These rules account for both occupied cyclic blocks and both centered merge quotients.

The ordinary one-site baseline is \((2c)^{(a-1)^2}\); use the same baseline at the changed site. The additional exceptional factors are \[C_0=\frac{\rho_{\rm v}(2c)^{3/2}}{2dF(2\lambda)}\quad(\mathrm A), \qquad C_0\frac{2c}{(2c)^{(a-1)^2}}\quad(\mathrm B_a).\] Each split supplies centered divisor \(2d\), one internal split norm is uncancelled, and the forced split coefficient is \(\rho_{\rm v}\), up to phase. Let \(z_{l,t},z_{r,t}\) be the split-site counts. Rotating the minus tail in both halves, including block phases and the twist, gives \[k^{1+(z_{l,+}-z_{r,+}+z_{r,-}-z_{l,-})/2}.\] The tail exponent is twice the number of relative plus roots in its \(r\)-list minus twice the number of relative minus roots in its \(l\)-list; the twist exponent is \(-2(m-1)\). The split and cup phases contribute \(k\) per group in \(\mathrm A,\mathrm B_0\), and 1 in \(\mathrm B_1\), leaving the constant \(k^2\).

For the remaining pair comparison, record phases in units of \(\pi/8\) modulo 16. Temporarily omit the cup numerator and own-probe residues. Remove the baseline ordinary cross-group sign \((-1)^{1-(a-1)(b'-1)}\), including changed sites; its total is sector-dependent only. Matching spectator pairs now have ratio 1. A full plus exceptional list of type \(\mathrm A\) or \(\mathrm B_0\), against a spectator of length \(j\), has phases \[\begin{array}{c|ccc} j&0&1&2\\ \hline \text{same group}&12&8&12\\ \text{opposite group}&4&8&4 \end{array}.\] Negate entries for \(\mathrm B_1\), and negate them for a minus exceptional list, combining both negations when applicable. The plus-list versus minus-list phases in order \(\mathrm A,\mathrm B_0,\mathrm B_1\), and the phases within one exceptional list, are \[\begin{pmatrix}0&4&0\\12&0&4\\0&12&0\end{pmatrix}, \qquad (0,4t,-4t).\] For example, place a plus exceptional list of type \(\mathrm A\) before a same-group spectator of string length zero, and write \(x\) for mark minus spectator. The mark has labels \(l=1,r=0\), whereas the spectator has \(l=r=-1\). The \(l\)-half contributes 1. The two mark roots in the \(r\)-half, including the single merge divisor, give \[\frac{P_{-3}(x)P_{-4}(x)}{P_{-1}(x)P_0(x)} \frac{P_{-1}(x)P_{-2}(x)}{2\cosh(x/2)} =i\frac{P_{-2}(x)P_{-3}(x)}{P_0(x)}.\] The corresponding ordinary-mark and probe factors on the gas side are \[\frac{P_2(-x)P_3(-x)}{P_0(-x)} =-\frac{P_{-2}(x)P_{-3}(x)}{P_0(x)}.\] Their ratio is \(-i\), the phase 12 in the table. Here we used \(P_{-4}(x)=2i\cosh(x/2)\) and \(P_j(-x)=-P_{-j}(x)\). The other entries use the same root and merge rules.

The following substitution verifies these tables and the constancy of the ratios. Encode powers of \(P_j\) by \(\xi^j\) in the signed spin-minus-gas polynomial of each pair. Insert the preceding root factors at shift \(g-h\), subtract \(\xi^4\) at each merge shift, and subtract \(\xi^{-ta}\) for the special denominator. Subtract ordinary string exponents for two ordinary sites. For an ordinary site of group \(z\), length \(a\), against a mark, subtract at ordinary minus mark \[\xi^{2z}+\xi^{3z} +\sum_{h=1}^a\xi^{zh}(\xi^{-4z}-\xi^z).\] For a probe shifted from its site by \(b=0\) in \(\mathrm A\), or \(b=ta\) in \(\mathrm B_a\), add for another ordinary site of group \(z\), length \(j\), at ordinary minus probe-site, \[\xi^{z-t-b}\left[1- \sum_{h=1}^j\xi^{zh}(\xi+\xi^{-1}-1)\right].\] For another mark add, at probe-site minus mark, \(\xi^b(1+\xi^{2t})\) in the same group and \(-\xi^{b-3t}\) in the other group. Reverse exponents for reversed arguments. Substituting the displayed \(\mathcal R\)’s gives zero modulo \(\xi^8-1\). To recover the constant send the ordered real difference to \(+\infty\): sum coefficients times exponent plus 8, omit the added 8 on reversed probe and ordinary-mark monomials, and replace it by \(-4\) on merge monomials. This gives the displayed tables after baseline removal.

The residues of the singular own-probe factors are \(-1/P_{2t}(0),1,P_t(0)\) for \(\mathrm A,\mathrm B_0,\mathrm B_1\); the remaining factors are evaluated at the pole. Write these residues as the vector \(R\). The exceptional scalar ratios relative to \(C_0\) are \(E=(1,1,2c)\), and the cup numerators are \(N=(1,-2itd,-2itd)\). In the spin-to-residue ratio their combined contribution is \(EN/R=(2itd,-2itd,2d)\). They have equal modulus \(2d\), and relative phases \((0,8,-4t)\), since \(\sin\lambda=c\). With the internal exceptional phases this is \((0,8+4t,-8t)\). Total ordinary length \(n-1\) makes the spectator contribution zero if exactly one \(\mathrm B_1\) occurs, and \(8(1+\#\mathrm B_0)\) otherwise. Adding the matrix above gives 8 in all nine cases. Thus the ratio is independent of pole sites and string lengths within a sector. Finally, replacing ordinary baselines by \(2d\) times their internal residues contributes only \((c/d)^{\sum(1-a)}=c/d\). All remaining magnitudes are fixed constants, proving the identity and uniform bound. ◻

Proof. Write \(L=L_Y\). Initially suppose that \(L_X\ge c_1L\) with fixed \(c_1>0\) and let \(L\to\infty\). We will identify all uses of this temporary restriction and remove them at the end.