Proof. The local identity is \(R_{aj}(\lambda/2+i(x-b_j))=Z_aL_{aj}(x-b_j)Z_j\). Both local matrices commute with \(Z_aZ_j\). Moving the auxiliary \(Z\)’s through the product, and using that \(L\) is even, gives (10). This also proves that the gauge is independent of the roots, spectral arguments, and inhomogeneities.

If \(N_{\alpha\beta}(x)\) is one auxiliary block of a physical local matrix, direct conjugation of its entries gives \(N_{\alpha\beta}(x)^*=N_{\bar\alpha\bar\beta}(\bar x)\), where a bar on a spin index exchanges its two values. In an entry of a monodromy, different local factors act on different quantum sites. They therefore commute when the adjoint reverses their order. Consequently \[(M_{{\rm ph},\alpha\beta}(x))^* =M_{{\rm ph},\bar\alpha\bar\beta}(\bar x).\] Taking the marked trace proves the second identity in (10); no reversal of the spatial inhomogeneities is involved.

The standard six-vertex Yang–Baxter identity gives \(\mathcal R_{ab}(p-q)M_a(p)M_b(q) =M_b(q)M_a(p)\mathcal R_{ab}(p-q)\). It is a local identity of the three displayed six-vertex matrices: spin conservation reduces its nonconstant entries to the addition formula for \(\sinh\). Moreover, \[M_{{\rm ph},a}(p)M_{{\rm ph},b}(q) =Z_b\mathcal P_L M_a(p)M_b(q)\mathcal P_L Z_b,\] and the analogous reversed product has \(Z_a\) in place of \(Z_b\). Substitution yields (11). Taking auxiliary traces proves the assertion about marks. For a common diagonal matrix \(D\), \(D\otimes D\) commutes with the intertwiner, so the two twisted rows commute. Finally \(R(0)\) is the permutation matrix, and tracing a chain of such permutations gives the cyclic translation. The other endpoint follows by the adjoint identity. ◻

Proof. Logarithmic differentiation gives \(q_Y'/q_Y=2\pi i\xi_Y'\). Comparing the limits as \(u\to+\infty\) fixes the integration constant: \[ q_Y(u)=(-1)^n e^{2\pi i\xi_Y(u)}\qquad(u\in\mathbb R). \tag{23}\] Indeed \(S(+\infty)=(1-\eta)/2\) and \(\Theta(+\infty)=(1-2\eta)/2\), while \(q_Y(+\infty)=e^{-i\lambda L+2i\lambda n}\) and \(L\) is even. Equation (19) now gives the Bethe equations. The two diagonal action formulas proved in Lemma 4 below give the coefficient replacing a root \(p\) by \(x\) as \[g(x-p)\left( e^{it}a(p)\prod_{y\ne p}f(y-p) -e^{-it}l(p)\prod_{y\ne p}f(p-y)\right).\] The Bethe equation, with the self factor \(-1\) removed, makes the parenthesis zero. The unchanged coefficient is (15). The physical eigenvalue follows from (10), since \(\mathcal Q_L=(-1)^n\) on the sector with \(n\) minus spins.

Differentiating the left side of (19) with respect to every root gives (20); the \(K(0)\) term cancels the derivative of the self interaction. The analytic implicit-function theorem gives the local branch, and differentiation with respect to \(t\) gives (21). When \(K\le0\), \[G_{jj}=\sum_a s(y_j-b_a)+\sum_{k\ne j}|K(y_j-y_k)| >\sum_{k\ne j}|G_{jk}|.\] The matrix is real symmetric, so strict diagonal dominance proves positive definiteness. The formula for \(G\mathop{\mathrm{diag}}(\ell)\) follows entry by entry. ◻

Proof. The standard RTT relation gives commuting creation entries and \[\begin{align*} \mathsf A(p)\mathsf B(u) &=f(u-p)\mathsf B(u)\mathsf A(p) +g(p-u)\mathsf B(p)\mathsf A(u),\\ \mathsf L(p)\mathsf B(u) &=f(p-u)\mathsf B(u)\mathsf L(p) +g(u-p)\mathsf B(p)\mathsf L(u),\\ [\mathsf C(p),\mathsf B(u)] &=g(p-u)\bigl(\mathsf A(u)\mathsf L(p) -\mathsf A(p)\mathsf L(u)\bigr). \end{align*}\] Move the diagonal entry successively through the creation entries and apply its vacuum value. Combining terms that replace the same root uses \[f(u-h)-f(u-p)=g(p-u)\frac{[p-h]}{[u-h]}.\] This gives (27); reversing differences gives (28). These formulas include the unchanged term \(p=v'\) because \(\beta(0)=1\).

For completeness, the two-removal induction can be checked before any on-shell specialization. Induct on \(|U|\). Strip one creation at \(u\) from the product, use the commutator, and apply the two diagonal formulas. For fixed upper and lower removals \(h,k\), divide the coefficients by \[a(h)l(k)f(k-h)\prod_{s\ {\mathrm{remaining}}}f(s-h)f(k-s).\] The required addition of the three resulting contributions is \[\begin{align*} \beta(v'-h)\beta(k-v') &=\frac{\beta(v'-h)\beta(k-v')}{f(u-h)f(k-u)}\\ &\quad+g(v'-u)\left( \frac{\beta(u-h)\beta(k-v')}{f(k-u)} -\frac{\beta(v'-h)\beta(k-u)}{f(k-v')}\right). \end{align*}\] Substituting (13) and multiplying by the displayed denominators reduces this identity to the addition formula for \(\sinh\). The initial case is \(\mathsf C(v')|0\rangle=0\); for one creation the two ordered removals give the displayed commutator on the vacuum. The induction proves (29), including cases where \(p\) or \(q\) is the newly added argument. Finally, there are only \(m\) newly added arguments. If the final cardinality equals the initial one, the number of absent original roots equals the number of surviving added arguments and is therefore at most \(m\). ◻

Proof. We apply the trigonometric scalar-product theorem to the fundamental six-vertex monodromy (8), with highest vector \(|0\rangle\), vacuum functions \(a,l\), and diagonal twist \(\kappa=e^{-2it}\). To match the conventions of (Kitanine et al. 2005), take its anisotropy to be \(i\lambda\) and its inhomogeneities to be \(b_j+i\lambda/2\), with a relabeling of sites if necessary. Its monodromy is \(s_\lambda^L M(x)\), so the scalar product and the vacuum-times-derivative determinant both acquire the same factor \(s_\lambda^{2Ln}\). After cancelling this factor, the prefactor is \(\prod_{y\in Y}l(y)\) for the eigenvalue \(\mathsf A+\kappa\mathsf L\); this is (Kitanine et al. 2005, Proposition 3.1, equations (3.8)–(3.9)). Replacing that eigenvalue by \(\tau_t=e^{it}\tau_\kappa\) multiplies its \(n\) derivative columns by \(e^{it}\) and therefore gives the compensating factor \(e^{-int}\) in (34). The untwisted derivative-over-Cauchy version, with its equal-spin-normalized monodromy, is (Kitanine et al. 1999, Theorem 3.1, equations (3.4)–(3.5)). These are finite scalar-product identities; they make no assertion about completeness or about a thermodynamic limit.

Differentiation of (15) gives \[\partial_{y_j}\tau_t(u;Y) =e^{it}a(u)F_Y(u)(-2\pi i)m_u^t(y_j).\] The factor \(e^{int}\) cancels that in (34). When \(u\to y_k\), the product \((1+e^{-2it}q_Y(u))\coth(u-y_k)\) tends to \(-2\pi i\xi_Y'(y_k)\), because \(q_Y'(y_k)=-2\pi i e^{2it}\xi_Y'(y_k)\). The two remaining coth terms give \(K(y_k-y_j)\) after division by \(-2\pi i\). This proves (33), including its diagonal.

At a root \(p\) the residue of the extracted vacuum factor is \[ \lim_{u\to p}[u-p]a(u)F_Y(u)=-s_\lambda A_Y(p). \tag{38}\] Cancel these poles with the corresponding Cauchy poles. If every column tends to its root, the remaining determinant is \(\det G_Y\), giving (35). For partial coincidence first cancel the unchanged columns. The determinant of the matrix obtained from \(G_Y\) by replacing columns \(H\) by \(m_{v'}^t\), divided by \(\det G_Y\), is \(\det((G_Y^{-1}m_{v'}^t)_p)_{H,V}\): multiply the matrix on the left by \(G_Y^{-1}\) and expand along its unchanged unit columns. Expanding the Cauchy determinant along its unchanged poles leaves exactly \(\det(1/[v'-p])_{H,V}\) with the same slot order. Each deleted root loses the factor in (38). This proves (36), including its sign and constant. Finally \(G_Y^{-1}=\mathop{\mathrm{diag}}(\ell)(I+K\ell)^{-1}\) proves (37).

All factors in the norm outside \(\det G_Y\) are nonzero: real \(y,b\) give \([y-b\pm i\lambda/2]\ne0\), and distinct real roots give \([y-y']\ne0\) and \([y-y'\pm i\lambda]\ne0\). The scalar-product theorem is first used at generic variables. At any regular on-shell specialization, the nonsingular counting Jacobian permits a local analytic perturbation of the roots along with the parameters. Meromorphic continuation and the pole cancellations above then give the asserted formulas there as well. In particular, no exclusion at a root of unity is necessary. ◻

Proof. Polarize the adjoint identity in Lemma 2 over the four auxiliary marks. It gives \(\mathsf B_{\rm ph}(x)^*=\mathsf C_{\rm ph}(\bar x)\). Using the gauge identity and commuting \(\mathcal P_L\) with \(\mathcal Q_L\) gives \(\mathsf B(x)^*=\mathcal Q_L\mathsf C(\bar x)\mathcal Q_L\). Since \(\mathsf C\) changes the number of minus spins by one, it anticommutes with \(\mathcal Q_L\), proving (39). Creation entries commute, so for real roots \[(\mathcal P_L|Y\rangle)^* =(-1)^n\langle0|\prod_{y\in Y}\mathsf C(y)\mathcal P_L.\] Lemma 5 makes this norm nonzero, and positivity is then Hilbert-space positivity. Applying the same identity to both cross scalar products proves (41). Absorbing the twist into the monodromy merely replaces \(\mathsf B\) by \(e^{it}\mathsf B\) and \(\mathsf C\) by \(e^{-it}\mathsf C\), preserving the adjoint relation and the normalized ratio. ◻

Proof. For this proof extend the vacuum base to added arguments by \[\mathcal A_Y(p)= \begin{cases} A_Y(p),&p\in Y,\\ a(p)F_Y(p),&p\notin Y. \end{cases}\] The coefficient of an upper removal is \(\mathcal A_Y(p)\beta(v'-p)U_Y(T,p)\). The corresponding lower coefficient is \(\mathcal A_Y(p)\chi_Y(p)\beta(p-v')L_Y(T,p)\): on a root the equality of the two vacuum bases is precisely (16) at \(t=0\), including its self factor \(-1\); off a root their ratio is \(q_Y(p)\). For two removals the extra factor is \(f(q-p)\), exactly as in (29). Thus the coefficient of a path before taking its scalar product is the product of its corrections \(\gamma_j\) times one \(\mathcal A_Y(p)\) for every removed element.

Every original root is either retained or removed once. Similarly, each new spectral argument is either retained in \(V\) or removed once, possibly in the very action that introduced it. Consequently the product of vacuum bases along the path is \[\left(\prod_{p\in H}A_Y(p)\right) \left(\prod_{v'\in\{v'_1,\ldots,v'_m\}\setminus V} a(v')F_Y(v')\right).\] Multiply by (36). The deleted-root bases cancel, and the surviving arguments supply the missing vacuum bases. Dividing by \(\prod_j\tau_0(v'_j;Y)\) leaves exactly \(\prod_j(1+q_Y(v'_j))^{-1}\) and \((-s_\lambda)^{-k}\). Equation (37) supplies the determinant and weights.

The Hilbert dual contributes the same \((-1)^n\) to the scalar product and norm, so it cancels by Lemma 6. Each physical entry, after division by \((-1)^n\), contributes \((-1)^{n_{j-1}-n}\) from \(\mathcal Q_L\) on its input sector; adjacent \(\mathcal P_L\) factors cancel. This proves the explicit parity sign. Different final spin sectors are orthogonal, proving the zero assertion.

Finally consider a pair of deleted original roots. If removed at different steps, the earlier root occurs in the denominator of (43) at the later removal. If removed together, the two denominators contribute \(f(q-p)^{-2}\) and the extra factor \(f(q-p)\) leaves one reciprocal. There are no other factors involving two original roots. The factor \(\beta\) at each original removal is explicit in (45). This proves the last claims. ◻

Proof. We first construct the exactly half-filled lists and control their sparse ends. We then compare the finite-interval density with the full-line density and use its quantiles to construct the dense lists. Both constructions will give the same local quadrature estimate.

Exactly half-filled roots. Suppose \(K<0\) and write \(k=-K>0\). The Jacobian of the counting equations is \[G_{ii}=\sum_j s(y_i-b_j)+\sum_{j\ne i}k(y_i-y_j), \qquad G_{ij}=-k(y_i-y_j)\quad(i\ne j).\] Its quadratic form is the sum of a positive diagonal form and \(\sum_{i<j}k(y_i-y_j)|u_i-u_j|^2\). It is strictly positive, so the equations are strictly convex gradient equations. Start at \(\lambda=\pi/2\), where \(K=0\) and each counting level has one real preimage, and continue in a compact parameter interval. There can be no coincident roots because their prescribed counting levels are distinct.

There can be no escaping roots either. If \(k_0\) roots tend to \(+\infty\), sum their equations. The interaction primitives within that set cancel by oddness. Dividing by \(k_0\), the limiting excess of the left side over the prescribed levels is \[(1-\eta)(L/2-n+k_0)>0.\] The same argument applies at \(-\infty\). All site parameters here are finite in each continuation problem. Nonsingularity and this compactness continue the solution to the required parameter.

At half filling the total mass of \(\rho_L=\xi'\) is \(n\). The two outer counting half-levels occur at infinity. Consequently the signed measure and its primitive \[e_L=\sum_j\delta_{y_j}-\rho_L(u)\,\mathrm du,\qquad F_L(u)=e_L((-\infty,u])\] satisfy \(|F_L|\le1/2\) and \(F_L(\pm\infty)=0\). The density equation and convolution inversion give \[ \rho_L-p_L=-R_0*e_L=-R_0'*F_L. \tag{53}\] Integrating once also bounds the difference of the tail integrals by \(\|R_0\|_1/2\). This proves the first assertion. Moreover \(|\rho_L'|\le C\rho_L\): both \(s\) and \(k\) are positive kernels with bounded logarithmic derivatives.

The sparse terminal cells. The preceding discrepancy estimate identifies the dense bulk, but a uniform inverse must also control the last few roots. We therefore estimate their densities and gaps directly. Number the roots from the right as \(x_1>x_2>\cdots\) and write \(T(u)=\int_u^\infty\rho_L(v)\,\mathrm dv\). Then \(T(x_j)=j-1/2\). For either \(q=s\) or \(q=k\), put \(\bar q(u)=\int_u^\infty q(v)\,\mathrm dv\). The displayed hyperbolic formulas for the kernels imply \[\begin{align*} q(u)&\le C\bar q(u) &&(u\in\mathbb R),\\ c\bar q(u)&\le q(u),\qquad \bar q(u+t)\le C e^{-2t}\bar q(u) &&(u,t\ge0). \tag{54}\end{align*}\] At a root \(x_j\) lying beyond every bare center, its own kernel and the \(j-1\) farther-out kernels contribute at most \(h/2+h(j-1)=hT(x_j)\) to this tail. The remaining centers are inward. Their tail is at least \((1-h)T(x_j)\), so \[ c(j-\tfrac12)\le\rho_L(x_j)\le C(j-\tfrac12). \tag{55}\] This bounds each fixed-index reciprocal density away from both zero and infinity.

Terminal gaps cannot become long. If \(x_{j+1}\) also lies beyond the bare centers and \(\Delta=x_j-x_{j+1}\), compare the inward tail at \(x_j\) with its tail at \(x_{j+1}\) using (54). It gives \[(1-h)(j-\tfrac12) \le C e^{-2\Delta}(j+\tfrac12), \qquad \Delta\le\frac12\log\frac{3C}{1-h}.\] Conversely \(1=\int_{x_{j+1}}^{x_j}\rho_L\) and the upper tail comparison give \(\Delta\ge c/(j+1)\). Thus each fixed-index gap has positive finite subsequential limits.

The explicit tail of \(p_L\), together with its bounded integrated error, locates any sufficiently large fixed edge index at \(e_L^{\rm edge}+O(1)\), where \[ e_L^{\rm edge}=\eta\log L \quad\hbox{or}\quad e_{d,B}^{\rm edge}=B/2+\eta\log(1/d) \tag{56}\] in the homogeneous and array cases, respectively. For example, choose an index larger than the integrated error bound; the \(p_L\) tail is comparable to \(\exp(-(u-e_L^{\rm edge})/\eta)\) near the displayed edge. That root lies beyond the bare centers in the limiting procedures. The gap estimate then places the last root no more than a bounded distance beyond that edge. The left end is identical. For each fixed density threshold \(M\), the portions with density below \(M\) start within a distance \(C_M\) of the nominal edges and contain only a bounded number of root indices. Both the number and the distance may depend on \(M\).

These statements have their asserted array uniformity. In a purported failure sequence we may take \(d\downarrow0\) and \(B\) arbitrarily large depending on \(d\). Every fixed edge index is then eventually beyond all bare centers, and (55) applies to every such index. One should not instead take a fixed-\(d\) edge limit and assert that its reciprocal weights tend to zero inward: the subsequent \(d\downarrow0\) limit is essential for that assertion.

Finite-interval densities. We turn to the dense construction. Its reference configuration will consist of quantiles of \(Lr_Q\), so we first need positivity and a relative comparison with \(r\), uniformly as \(Q\) grows. The inverse of \(I+K_Q\) exists on bounded functions because its perturbation has norm at most \(h<1\). Comparison with the full-line equation gives \[ r_Q-r=(I+K_Q)^{-1}K_{\rm out}r, \tag{57}\] where \(K_{\rm out}\) integrates over \(\mathbb R\setminus[-Q,Q]\). When \(K<0\), write the right side as the negative of the exit kernel \((I-k_Q)^{-1}k_{\rm out}\) applied to \(r\). That positive kernel has mass at most \(h\): the constant \(h\) is a supersolution of its mass equation, since \(h-k_Qh-k_{\rm out}1=(1-h)k_Q1\ge0\). For exterior \(v\) and interior \(u\), we have \(r(v)\le r(Q)\le r(u)\). Applying the exit-kernel mass bound to this inequality gives \((1-h)r\le r_Q\le r\) on the interval. The positive Neumann series also gives \(1\le H_Q\le(1-h)^{-1}\) in this case.

For \(0<\lambda<\pi/2\), a direct bound on the alternating Neumann series for \(I+K_Q\) would lose positivity when \(h\) is large. Instead the full-line dressed kernel is positive. To see this explicitly, put \[\vartheta=\frac{\pi\lambda}{\pi-\lambda},\qquad J_\lambda(u)= \frac{\sin\vartheta} {(\pi-\lambda) \bigl(\cosh(2\pi u/(\pi-\lambda))-\cos\vartheta\bigr)}.\] Here \(0<\vartheta<\pi\), so \(J_\lambda\ge0\). Rescaling the bare-kernel transform in (24) gives \[\widehat J_\lambda(p) =\frac{\sinh((\pi-2\lambda)p/2)} {\sinh((\pi-\lambda)p/2)}.\] Together with \(\widehat r(p)=(2\cosh(\lambda p/2))^{-1}\), this identifies \(R_0=J_\lambda*r\). In particular \[ R_0\ge0,\qquad \int_\mathbb RR_0=\frac{h}{1+h}=:\tau<\frac12. \tag{58}\] This argument uses integrable kernels, so Fourier uniqueness identifies the actual convolution kernel, not just its multiplier.

Let \(I_Q=[-Q,Q]\) and \(O_Q=\mathbb R\setminus I_Q\). The positive operator \(u\mapsto R_0*(\mathbf 1_{O_Q}u)\) has sup norm at most \(\tau\). Its Neumann series defines bounded nonnegative functions on the entire line by \[ \begin{aligned} \overline r_Q&=r+R_0*(\mathbf 1_{O_Q}\overline r_Q),\\ \overline H_Q&=\frac1{1+h} +R_0*(\mathbf 1_{O_Q}\overline H_Q). \end{aligned} \tag{59}\] Multiplying by \(I+K*\) and using \((I+K*)R_0*=K*\) gives \[\overline r_Q+K*(\mathbf 1_{I_Q}\overline r_Q)=s, \qquad \overline H_Q+K*(\mathbf 1_{I_Q}\overline H_Q)=1.\] Their restrictions are therefore \(r_Q,H_Q\), by uniqueness for \(I+K_Q\). Since \(r\) decreases on \([0,\infty)\), the first equation in (59), restricted to \(O_Q\), gives \[\|\overline r_Q\|_{L^\infty(O_Q)} \le r(Q)+\tau\|\overline r_Q\|_{L^\infty(O_Q)}, \qquad \|\overline r_Q\|_{L^\infty(O_Q)}\le\frac{r(Q)}{1-\tau}.\] Applying the same equation inside \(I_Q\) and the second equation on the full line now yields \[ \begin{aligned} r(u)&\le r_Q(u)\le r(u)+\frac{\tau r(Q)}{1-\tau} \le\frac{r(u)}{1-\tau},\\ \frac1{1+h}&\le H_Q(u)\le1\qquad(|u|\le Q). \end{aligned} \tag{60}\] At \(\lambda=\pi/2\), \(K=R_0=0\), \(r_Q=r=s\), and \(H_Q=1\). We have thus proved (50) for every fixed \(0<\lambda<\pi\).

These arguments also give the relative derivative bounds needed for quadrature. In the positive-kernel case, differentiate the convolution in (59); each fixed derivative of \(R_0\) is integrable, and the exterior source is bounded by \(r(Q)/(1-\tau)\). Thus every fixed derivative of \(r_Q-r\) is \(O(r(Q))\) on \(I_Q\). For \(K<0\), put \(d_Q=r_Q-r\) and differentiate \(d_Q=K_{\rm out}r-K_Qd_Q\) in its output variable; the same bound follows from \(\|d_Q\|_\infty\le h r(Q)\) and the integrability of all fixed derivatives of \(K\). Since \(|r^{(j)}|\le C_jr\) and \(r(u)\ge r(Q)\) on the interval, in both cases \(|r_Q^{(j)}(u)|\le C_jr_Q(u)\). All these constants can be chosen uniformly when \(\lambda\) ranges over a compact subinterval of \((0,\pi)\). At an endpoint we extend \(r_Q\) by \(s-K_Qr_Q\) outside the interval. For \(K\ge0\) this is the positive function \(\overline r_Q\) just constructed; for \(K<0\) it is the positive sum \(s+k_Qr_Q\). The same derivative bounds hold on an extension of bounded counting length at either endpoint, since its physical width is \(O((Lr(Q))^{-1})\) in the dense limits used below.

Differentiation with respect to the interval endpoint, followed by symmetry of the kernel, gives \[ \frac{\,\mathrm d}{\,\mathrm dQ}\int_{-Q}^Q r_Q(u)\,\mathrm du =2r_Q(Q)H_Q(Q)>0. \tag{61}\] Indeed \((I+K_Q)\partial_Qr_Q =-r_Q(Q)(K(\,\cdot-Q)+K(\,\cdot+Q))\); pair this with \(H_Q\). The just proved bound \(|r_Q-r|\le C r(Q)\) shows that the integrated difference on \([-Q,Q]\) is at most \(2CQr(Q)\) and tends to zero. Thus the mass tends to \(\int_\mathbb Rr=1/2\). By (50), the right side of (61) is bounded above and below by positive constants times \(r(Q)\). Integrating (61) from \(Q\) to infinity proves the asserted mass-deficit estimate.

From dense quantiles to exact roots. The interval density now has a strictly increasing mass and uniform relative derivative bounds. Choose \(Q\) to have the desired mass \(n/L\), and let \(y_j^0\) be the mid-quantiles of \(Lr_Q\). The continuum counting function has derivative \(Lr_Q\) on the interval and, by oddness, gives the required levels at these quantiles. Midpoint quadrature against a function of bounded variation bounds the discrete residual of each root equation by a constant. The same holds for every fixed derivative of the counting function: apply that quadrature to translates of the interaction kernel or its derivatives.

Put \(m_L=Lr(Q)\), and use displacement coordinates \(\zeta_j=Lr_Q(y_j^0)(y_j-y_j^0)\). At the quantiles the Jacobian in these coordinates is \[I+\left(\frac{K(y_i^0-y_j^0)}{Lr_Q(y_j^0)}\right)_{ij} +O(m_L^{-1})\] in the sup row norm. The middle matrix has absolute row norm at most \(h+o(1)\) by quadrature and the relative derivative bounds just proved. Its inverse is uniformly bounded. On any fixed-radius ball in the \(\zeta\) coordinates, the physical displacements are \(O(m_L^{-1})\). Neighbor sums of kernel derivatives, weighted by \(1/(Lr_Q)\), are bounded. Comparing the bare-minus-root density to \(Lr_Q\) before displacing the roots also bounds the change of the divided diagonal entries by \(O(m_L^{-1})\). Thus the Jacobian varies by \(o(1)\) on that ball.

For clarity, Newton’s map here may be taken with the inverse Jacobian frozen at the quantiles. Choose the ball radius larger than twice that inverse bound times the residual bound. For large \(L\) its derivative has norm less than \(1/2\), so it maps the ball into itself and is a contraction. The fixed point gives the exact roots and (51). Positivity of the resulting counting derivative forces their ordering. The same estimates hold uniformly on every fixed compact continuation interval joining \(\lambda>0\) to \(\pi/2\): throughout that interval \(h\) is bounded strictly below one, and the density and derivative comparisons above are uniform. For the near-half-filled sequence, the mass-deficit comparison also gives \(Lr(Q)\asymp L^\epsilon\) uniformly there. The Jacobian is nonsingular; equivalently its symmetrization has positive eigenvalues, since the absolute row bound for the perturbation of the identity is below one.

Quadrature and the Perron vector. Both root constructions are now available, with positive counting density and a nonsingular Gaudin matrix. Their common local quadrature estimate follows from the counting cells. In a cell of density at least \(M\), its length \(\Delta_j\) is \(O(M^{-1})\). The relative derivative bound gives \(|\ell_j-\Delta_j|\le C\Delta_j^2\). Integrate a Lipschitz test cell by cell and include the two partial boundary cells to obtain (52). Uniform local mass follows in the dense region from the same argument and at the ends from (55) and the gap bounds. In regular filling every cell is dense, including the bounded counting extension of the interval.

The Gaudin determinant and Lemma 6 make each Bethe vector nonzero. In the homogeneous chain the logarithmic row derivative at the swap has, up to an additive scalar, local link matrix equal to the swap times \(-\eta R'(0)\). Its hopping coefficient on an opposite-spin pair is \(-\eta/\sin\lambda<0\). Within each nontrivial particle sector the hopping graph is connected, so the sector minimum is simple and has a positive vector. At \(\lambda=\pi/2\) this is the free nearest-neighbor exclusion Hamiltonian. In ordered particle coordinates its eigenvectors are antisymmetrized plane waves with circle sign \((-1)^{n-1}\), and the minimum fills the consecutive momenta about zero. These are precisely the consecutive Bethe roots: their physical shift phase is \(e^{ip}=-F_{\{y\}}(i\lambda/2)\), and differentiation of the eigenvalue gives \(-2\pi\eta s(y)=-\cos p\) at the free point. The sign gauge thus centers the occupied momenta at zero.

The real-root branch remains this simple minimum by continuation. The physical transfer matrix is primitive in the sector: its diagonal is positive, and successive allowed auxiliary swaps move any particle to a neighboring hole, connecting the sector. It commutes with the link Hamiltonian. The positive simple minimum is consequently its Perron vector. Taking the long cylinder limit gives its normalized vacuum expectations. Finally nonsingularity of \(G\) gives the finite analytic twist branch by the implicit function theorem. This last statement does not require a twist neighborhood uniform in \(L\). ◻

Proof. For regular filling, local quadrature up to the endpoints bounds the absolute row norm of \(K\ell\) by \(h+o(1)<1\). This proves (62) there. For \(K<0\) at half filling, finite strict diagonal dominance does not give a uniform inverse: its bare-source deficit may shrink at the last roots. We first isolate those roots from the contracting bulk, then use their reciprocal densities to obtain a decaying solution of a one-sided averaging equation.

Truncate the exponentially small kernel tail at a fixed distance and then choose a high density threshold. Local mass and (52) show that every row sufficiently far inside the dense portion has absolute row sum at most a fixed \(q<1\). The edge-index bounds confine all other rows to a fixed number of indices at either end.

If the inverse bound failed, there would be vectors \(W_L\) with \(\|W_L\|_\infty=1\) and \((I-k\ell)W_L\to0\) in sup norm. A near-unit coordinate must lie among those fixed edge indices. Retain one end, translate its terminal root to zero, and extract all fixed-index gaps and weights. At the right end write the limiting points as \(x_1=0>x_2>\cdots\); the left end is treated by reflection. The bounds of Lemma 8 make the limiting weights positive and finite; at every fixed index they satisfy \(c/j\le\ell_j\le C/j\). The limiting points have no finite accumulation, since otherwise this lower bound would contradict the uniform local mass bound. In the array case this extraction is along \(d\downarrow0\) with \(B\) sufficiently large, exactly as in (49).

Put \(v_j=\ell_jW_j\) in the resulting one-sided limit. It is nonzero, and \(v_j\to0\) as \(j\to\infty\). Uniform local mass and exponential tails justify passing the equation to this limit. It reads \[(\rho_j-k(0))v_j=\sum_{i\ne j}k(x_j-x_i)v_i.\] The left coefficient is positive: finite row dominance bounds it below by \(k(x_j-x_i)>0\) for any other retained index \(i\), whose limiting gap is finite. After division by this coefficient, the equation is an averaging equation with strictly positive off-diagonal weights whose sum is at most one. The latter follows from finite Gaudin row dominance and Fatou’s lemma; a strictly positive limiting deficit is not required. The absolute maximum of \(v\) is attained, and some other index has smaller modulus. Its positive weight strictly decreases this maximum, a contradiction.

We spell out the mixed case. The measure \[\widetilde\nu=\,\mathrm du+D^0+D^+ =\chi_-\,\,\mathrm du+(1-\chi_-)\nu_L\] is nonnegative. Exponential decay and bounded local variation justify Fubini, and \(R_0+K*R_0=K\) gives the exact identity \[ (I+K*)(I+R_0*(D^0+D^+))u =u+K*(\widetilde\nu u). \tag{64}\] The line convolution \(I+K*\) is boundedly invertible because \(h<1\).

For \(K<0\), suppose the undressed operator had normalized approximate null functions \(u_L\). They satisfy \(u_L=k*(\widetilde\nu u_L)+o(1)\) uniformly. The quantity \(\sup_x\int k(x-y)\widetilde\nu(\,\mathrm dy)\) is uniformly bounded. It follows that the supremum of \(|u_L|\) on \(\mathop{\mathrm{supp}}\widetilde\nu\) is bounded below. Divide by that supremum. The support supremum is now one and the full supremum remains bounded. On the support, every row except a fixed number of right-edge atoms has sum at most \(q<1\): the replaced side is Lebesgue, the blend is dense, and the remaining noncontracting part is the original discrete right end.

Thus a near-unit support value survives at one of those atoms. The modified left side is arbitrarily far away from each retained right-edge index, so its contribution disappears in the extracted equation. The preceding harmonic-maximum argument applies. There is no additional approximate mode between the atoms: \(k*(\widetilde\nu u_L)\) has a uniformly bounded derivative, so the approximate equation makes \(u_L\) equicontinuous up to a vanishing sup error. In the regular-filling case the entire mixed measure instead has absolute kernel row norm \(h+o(1)\), by quadrature, and contracts directly. Equation (64) proves the claimed lower bound.

Finally exponential conjugation changes the discrete kernel row norm by at most \[\sup_i\sum_j |K(y_i-y_j)|\ell_j |e^{a(y_i-y_j)}-1|\le C|a|.\] This follows by summing over unit intervals and using local mass and exponential kernel decay. A sufficiently small \(a\) therefore preserves the inverse bound by the inverse identity. The same perturbation estimate preserves the mixed lower bounds. The weighted middle contraction is proved explicitly in Lemma 11. ◻

Proof. On \(\operatorname{Im}x=\lambda/2\), local inversion says that each site matrix divided by its equal-spin entry is unitary. Taking the auxiliary trace costs at most \(2\|D\|\), and gives the boundary estimate with \(|a(x)|\). Crossing gives the reflected lower boundary estimate. For \(0<\operatorname{Im}x<\lambda/2\) define \[ U(x)=\log\left|\frac{[x+i\lambda/2]}{[i\lambda]}\right| +\int_\mathbb Rr(u) \log\left|\frac{[u-x+i\lambda]}{[u-x]}\right|\,\mathrm du. \tag{68}\] The integrand has no singularities in that open half-strip and decays exponentially at both real ends. Its boundary value on \(\operatorname{Im}x=\lambda/2\) is zero, because the two sinh moduli coincide. Reflect (68) conjugately to the lower half-strip. The two values agree on the real line. Differentiating the logarithmic sinh kernels at that line, with the denominator term interpreted by its Poisson-kernel limit, gives the upper normal derivative \[\pi(s-K*r-r)=0.\] Thus the two harmonic functions glue harmonically. Their growth at the real ends is at most linear.

For unit vectors \(u,v\), apply the strip maximum principle to \(\log|\langle u,T^D(x)v\rangle|-\sum_jU(x-b_j)\). The boundary bound is \(\log(2\|D\|)\) and the growth is at most linear. One may justify the unbounded-strip principle by subtracting \(\varepsilon\cosh(a\operatorname{Re}x)\cos(a\operatorname{Im}x)\), with \(0<a<\pi/\lambda\), applying the principle in finite rectangles, and then sending \(\varepsilon\) to zero. Taking the supremum over \(u,v\) proves (65). The strip is simply connected, so a harmonic conjugate gives the nonvanishing \(\mathcal N_L\).

It remains to compare it with the actual eigenvalue. It suffices first to work in a fixed small open neighborhood inside the upper swap point \(i\lambda/2\). There \(q_Y(x)\to0\). Homogeneously, for each fixed \(0<\lambda<\pi\) choose a sufficiently small disc about \(i\lambda/2\). The root ratio \([x-y+i\lambda]/[x-y-i\lambda]\) is bounded by a constant \(M_\lambda\) for every real \(y\) on that disc: the imaginary part of its denominator stays away from \(\pi\mathbb Z\), and the ratio has finite limits at both real ends. The one-site ratio \([x-i\lambda/2]/[x+i\lambda/2]\) tends uniformly to zero as the disc shrinks. Shrink it until its supremum times \(\max(1,\sqrt{M_\lambda})\) is less than some \(q_*<1\). Then \(n\le L/2\) gives \(|q_Y(x)|\le q_*^L\) on the disc, and hence on its intersection with the strict strip. The disc and \(q_*\) may depend on the fixed \(\lambda\). For the array, \(\lambda>2\pi/3\) and the neighborhood can be chosen so that each site and root ratio has modulus at most one. Indeed the difference of the squared sinh moduli is governed, respectively, by \(\sin(2\operatorname{Im}x)\sin\lambda>0\) and \(\sin(2\operatorname{Im}x)\sin(2\lambda)<0\). There are order \(1/d\) sites at bounded distance from the moving bulk center, with site ratios uniformly below one. Hence \(q_Y(x)=O(e^{-c/d})\) after \(B\to\infty\), uniformly in that bulk.

Put \[H_x(u)=\log\left|\frac{[u-x+i\lambda]}{[u-x]}\right|, \qquad \widetilde H_x=(I+K*)^{-1}H_x.\] These are smooth tests in \(u\) on the chosen neighborhood. The exponentially weighted Neumann series gives uniform exponential decay for \(\widetilde H_x\) and all its fixed derivatives, also after real recentering. The density equation gives the exact identity \[ \sum_{y\in Y}H_x(y)-\int_\mathbb Rp_L(u)H_x(u)\,\mathrm du =\int_\mathbb R\widetilde H_x(u)e_L(\,\mathrm du). \tag{69}\] At half filling, integrate by parts against the bounded counting sawtooth \(F_L\). In counting coordinates its mean over every complete cell is zero. On a dense fixed window, the relative derivative bound for \(\rho_L\) makes the physical cell averages converge to that mean. Thus \(F_L\) tends weakly to zero there. The exponential tail of the test and the uniform bound \(|F_L|\le1/2\) discard the complement of an increasing window. This proves that (69) tends to zero, including the array limit after bulk recentering.

For the homogeneous near-half-filled sequence, use the true outer counting half-levels as cell endpoints. They are a bounded counting distance from \(\pm Q\) by (51), and the positive comparison extends to them. The cellwise primitive is zero at the endpoints and is bounded by \(1/2\) inside. The same midpoint argument therefore works within these cells. Outside them, the root measure vanishes and \[|\xi'(u)|\le C Lr(Q)=O(L^\epsilon).\] To see this uniform exterior bound, replace the root sum of any translate of \(K\) by its \(Lr_Q\) integral. Midpoint quadrature and bounded counting displacement cost \(O(1)\) by variation and reciprocal-density summation. The remaining exterior expression \(L(s-K_Qr_Q)\) is \(O(Lr(Q))\), by the full-line equation and (57). Its pairing with the dressed test outside the cells is consequently \(O(L^\epsilon e^{-aQ})\) for a fixed \(a>0\). Since \(Q=\eta(1-\epsilon)\log L+O(1)\), it tends to zero when \[ 0<\epsilon<\frac{a\eta}{1+a\eta}. \tag{70}\] Decreasing \(a\) if necessary gives one such choice on the compact continuation interval used to construct the roots.

The first-term eigenvalue formula, \(q_Y\to0\), and (69) now prove (66) on this open neighborhood. On the whole strip, \(\tau_L/\mathcal N_L\) is bounded by two. Every subsequential holomorphic limit therefore has modulus one on the neighborhood and is a unimodular constant by the maximum modulus principle. Normal-family compactness gives local uniform modulus convergence throughout the strip. The same argument works for any purported failing moving-bulk array sequence. Dividing (65) by \(|\tau_L|\) proves the last assertions. ◻

Proof. Separate the two sparse ends. Choose the fixed-profile partition in Lemma 9 so that \[\mathop{\mathrm{supp}}D^-\subset(-\infty,a_L],\qquad \mathop{\mathrm{supp}}D^+\subset[b_L,\infty),\qquad a_L<0<b_L.\] The word center is deep between these boundaries. The middle support is compact and dense. At a fixed density threshold, Lemma 8 gives \[ \max Y\le b_L+C_0,\qquad \min Y\ge a_L-C_0, \tag{79}\] with a threshold-dependent constant uniform in the limits. For regular filling take the transitions just inside \([-Q,Q]\). For the array both the last root and the transition contain the same \(\eta\log(1/d)\) shift in (56). All three signed pieces have uniformly bounded variation mass on intervals of length one.

Use the pointwise Banach space \[X_\beta=\left\{u\in C^1(\mathbb R): \|u\|_\beta:=\sup_x e^{\beta x} (|u(x)|+|u'(x)|)<\infty\right\}.\] The quadrature error on each unit-scale test is as small as desired on the middle support and its taper. To bound \(R_{0,\beta}*(D^0u)\), insert a smooth unit-scale partition in the integration variable and apply (52). For the output and its first derivative, the test derivatives involve \(R_{0,\beta},R_{0,\beta}',R_{0,\beta}''\) and \(u,u'\). The envelopes (77) sum after weighting to give \[\|R_{0,\beta}*(D^0u)\|_\beta \le C_\beta\varepsilon_{\rm quad}\|u\|_\beta.\] Choose the density threshold so this norm is below \(1/2\) for the fixed finite list of shifts, including zero.

We first remove the right-end discrepancy from the column equation, keeping its true left-end input. Let \[W^p(c)=(I+R_0*D^0)^{-1} (W^0(c)-R_0*(D^-W(c))),\] using the shift-zero contraction on \(X_0\). Our task is to characterize the combinations \(c\) for which this bounded column also belongs to \(X_\beta\). The omitted right end will then be restored with the mixed-operator lower bound.

Solve after subtracting the slow residues. For \(c\in E\) solve in \(X_\beta\) \[ (I+R_{0,\beta}*D^0)U_\beta(c) =W^0_\beta(c)-R_{0,\beta}*(D^-W(c)). \tag{80}\] The true-column bound and local variation imply \[\|R_{0,\beta}*(D^-W(c))\|_\beta \le C_\beta e^{\beta a_L}\|c\|_2.\] For example multiply the convolution integrand by \(e^{\beta(x-y)}e^{\beta y}\), use \(y\le a_L\), and sum the exponentially decaying weighted kernel envelopes over unit intervals. The same argument applies to the first derivative. Consequently \[ \|U_\beta(c)\|_\beta\le C_\beta\|c\|_2. \tag{81}\] This estimate holds for every coefficient vector before imposing any residue condition.

Impose only the residue conditions. Define the linear map \[ \Lambda_\beta(c) =P_\beta*(D^0U_\beta(c)+D^-W(c))-p_\beta(c). \tag{82}\] Its values lie in \(\mathcal E_\beta\). Indeed convolution of an exponential polynomial with a source preserves that span: expand each \((x-y)^q\). The middle source is compactly supported, and the left moments converge because \(d>0\) and the left variation mass is locally bounded. Rank-nullity therefore gives \(\operatorname{codim}\ker\Lambda_\beta\le r_\beta\). No inverse or condition-number bound for \(\Lambda_\beta\) is used.

Restoring the residues in (80) gives \[(I+R_0*D^0)U_\beta(c) =W^0(c)-R_0*(D^-W(c))+\Lambda_\beta(c).\] For \(c\in\ker\Lambda_\beta\), this identity first proves \(U_\beta(c)\in X_0\): in the rearranged equation the middle source \(D^0U_\beta\) is a finite compactly supported measure, so its convolution and first derivative are bounded at both ends. The other two terms are already bounded. This membership assertion for a fixed chain does not yet require a uniform estimate for that compact source. Only now apply the shift-zero inverse to identify \(U_\beta(c)=W^p(c)\) and obtain its uniform unweighted bound.

Conversely, if \(W^p(c)\in X_\beta\), rewrite its equation with the shifted kernels. The defect is an element of \(\mathcal E_\beta\) which also belongs to \(X_\beta\). It is zero: every nonzero exponential polynomial in (76) violates the \(\beta\)-weighted bound at \(+\infty\). Weighted uniqueness then gives \(W^p(c)=U_\beta(c)\) and \(\Lambda_\beta(c)=0\). Hence \[ F_{\beta,L}:=\ker\Lambda_\beta =\{c:W^p(c)\in X_\beta\}. \tag{83}\] The spaces are nested, because \(W^p(c)\) is already unweighted bounded at \(-\infty\) and a larger positive weight at \(+\infty\) implies every smaller one. Estimates (81) hold uniformly for every unit vector in these subspaces.

Restore the right end at the actual nodes. For such a vector, subtract the true and middle equations: \[(I+R_0*(D^0+D^+))(W(c)-W^p(c)) =-R_0*(D^+W^p(c)).\] The right side has sup norm at most \(C_\beta e^{-\beta b_L}\). The infinite Lebesgue portion of \(D^+\) causes no extra factor: sum its uniformly bounded local variation against \(e^{-\beta(b_L+j)}\) over \(j\ge0\). The mixed lower bound therefore gives the same estimate for \(W-W^p\). At every actual positive root, (79) converts it to \(C_\beta e^{-\beta u}\), which proves (78). Reflecting proves the negative-end assertion. Finally restriction of (82) to a coordinate subspace still has rank at most \(r_\beta\), and a unit vector there is a unit vector in \(E\). All the asserted restriction properties follow. ◻

Proof. Choose pole-free numbers \(\beta_j\in(2j-2,2j-1)\) for \(1\le j\le k\). By (75), the corresponding flag has codimension at most \(j-1\). For a positive-row minor with \(r\) columns, order its rows as \(p_1\ge\cdots\ge p_r\ge0\). Choose an orthonormal basis of its coefficient space backwards: the \(j\)th basis vector belongs to \(F_{\beta_j,L}\). This is possible because these spaces are nested and have dimension at least \(r-j+1\). A unitary basis change does not alter the determinant modulus. Applying (78) to its columns and using the rearrangement inequality in the determinant expansion gives \[ |\det W|\le C\exp\left(-\sum_{j=1}^r\beta_jp_j\right) =C\exp\left(-\int_0^\infty \sum_{j=1}^{k_+(t)}\beta_j\,\mathrm dt\right), \tag{85}\] where \(k_+(t)=\#\{i:p_i>t\}\). For a determinant with both signs, first expand by its positive and negative rows. Every Laplace minor has a column coordinate subspace, on which the same codimension and unit-vector estimates hold. Choose its flag basis separately. Thus no simultaneous adaptation of the two tail flags is required. The finite Laplace sum gives the product of the positive bound and its negative counterpart.

Set \(\varepsilon=\min_{j\le k}(\beta_j-(2j-2))>0\). For every \(r\le k\), \(\sum_{j=1}^r\beta_j\ge r(r-1)+\varepsilon r\). Combining (85) at both ends with (73) proves (84). The case \(k=0\) is the empty determinant and needs no tail estimate. Uniform local quadrature mass now bounds the sums of the majorant by convergent geometric series on unit boxes. Discarding boxes with some coordinate beyond \(R\) proves the last assertion, with a harmless smaller exponent. ◻

Proof. First keep all arguments in the generic neighborhood used above. The finite row expansion has finitely many action patterns, each with at most \(m\) deleted original roots. Words with a net change of particle number have coefficient zero. On each bounded rapidity window, the line extensions of the dressed columns and their derivatives are uniformly bounded. Local quadrature and their convolution equation therefore give, along any subsequence, a smooth local limit satisfying \[W+K*W=m^{(0)}_{v'}, \qquad m^{(0)}_{v'}(u)= \frac{\coth(v'-u)-\coth(v'-u-i\lambda)}{-2\pi i}.\] The tails in this equation are negligible by exponential kernel decay and bounded local mass. Since \(h<1\), the bounded solution is unique. The elementary identity \(m^{(0)}_{v'}(u)=s(u-v'+i\lambda/2)\) shows that dressing replaces \(s\) by \(r\). Equivalently its Fourier transform is \(e^{-ip(v'-i\lambda/2)}/(2\cosh(\lambda p/2))\). This gives \[ W_{v'}(u)\longrightarrow \frac{1}{2\lambda \cosh((u-v'+i\lambda/2)/\eta)}. \tag{87}\] For the array, the same conclusion holds along every bulk-recentered sequence in its prescribed double-limit sense.

Apply (46) on a bounded rapidity window. Its weights are exactly the product quadrature measure. Local quadrature, (87), and \(q_Y(v')\to0\) turn its sum into the corresponding line integral. The apparent near-collision singularities have already been canceled by the root-pair factors; deleting diagonal tuples has no effect in the limit, since the maximal atom on such a window tends to zero. Lemma 12 then permits the window to grow to the whole line. This proves convergence of every path and hence of the word. It also proves independence of the choice between homogeneous and array limits. All surviving ingredients depend only on rapidity differences. Changing all variables of integration by a common real amount therefore proves real-translation invariance in the \(x\) coordinate.

Lemma 10 bounds all the finite words locally uniformly on the full strip. Normal-family compactness and the identity theorem extend the convergence from the generic open set to \(\mathcal S_\lambda^m\), with uniqueness of every subsequential limit. The derivative in the common translation direction vanishes on that open set and hence everywhere. Integrating it along translations staying in the strip gives the stated complex-translation invariance. For any finite linear combination of word vectors, apply the finite operator inequality (67) to its Gram norm. Passing to the limit gives the bound \(2\|D\|\) for the induced row action and also shows that it respects zero Gram vectors.

For the remaining equal-time assertion, first fix \(Q<\infty\) and send \(L\to\infty\) with filling \(\int_{-Q}^Qr_Q\). The same finite path calculation is valid, now with dense nodes on \([-Q,Q]\) and the interval dressing operator. Then send \(Q\to\infty\). The inverse bounds, local convergence, and joint tail bound are uniform in this second limit, so the same full-line integrals result.

This calculation can be made on small contours surrounding \(i\lambda/2\), using nested distinct radii to keep all arguments generic. For the regular-filling assertion we only use \(\lambda>2\pi/3\); choose the contour radii so small that all the fixed shifted poles stay outside them. On these contours the homogeneous \(q_Y\) still tends to zero exponentially, and the eigenvalue first term is analytic and nonzero. Thus each complete finite normalized word is holomorphic on the product of the enclosed discs for sufficiently large \(L\). The same determinant bounds apply on their boundary contours, including the portions just outside the strict strip. Convergence on these contours is uniform: first take the fixed-\(Q\) chain limit on a bounded rapidity window, then use the uniform joint tail bound when taking \(Q\to\infty\) and removing that window. The iterated Cauchy formula therefore gives the coincident endpoint values.

At the swap, a marked row implements an arbitrary matrix on one site together with the cyclic shift. Multiplying marked swap rows conjugates the successive site matrices by that shift; their product reads any finite set of sites, with identity marks at unused sites. The residual shift fixes the homogeneous Perron vector, so its vacuum expectation is exactly the desired local-matrix expectation. The half-filled homogeneous calculation has the same endpoint contours and integrals. This proves equality of the equal-time local states in the stated order of limits, without asserting an unrestricted row-strip limit at a fixed nonzero magnetization. ◻

Proof. We first use generic arguments near the upper boundary in the \(x\) coordinate. Their real parts may be separated by arbitrary translations. The integral formula of Proposition 13 applies there by analytic continuation of its difference integrals; keep the imaginary parts in a fixed small generic box and avoid the finitely many coincidence poles during continuation.

For a path with deleted roots \(H\) and surviving arguments \(V\), write \(N_H(t)\) and \(N_V(t)\) for the numbers of their real parts to the left of \(t\). These two lists have equal length. The quotient of the continuum determinant (87) and the original hyperbolic Cauchy determinant, including the root-pair cancellations, is bounded by \[ C\exp\left(-(\eta^{-1}-1) \int_\mathbb R(N_H(t)-N_V(t))^2\,\mathrm dt\right). \tag{88}\] Here the constant is uniform as the fixed internal lists separate. To verify the exponent, use \[\cosh((p-v'+i\lambda/2)/\eta) =i\sinh((p-v')/\eta)\] and apply the sinh Cauchy identity at scales \(1/\eta\) and \(1\). The exponent in their ratio is \(-(\eta^{-1}-1)\) times \[\sum_{p\in H,v'\in V}|p-\operatorname{Re}v'| -\sum_{p<p'}|p-p'| -\sum_{v'<v''}|\operatorname{Re}v'-\operatorname{Re}v''|.\] This expression is exactly the integral in (88). The bounded imaginary offsets and fixed within-group generic factors change only the constant. The reciprocal \(f\) factors remove the near-root-collision denominators as in Lemma 12.

The integral in (88) controls ordered matching distance. Indeed \(N_H-N_V\) is integer-valued, so its square is at least its absolute value, and \[\int_\mathbb R|N_H-N_V|\,\mathrm dt =\sum_j|p_j-\operatorname{Re}v'_j|\] for the two ordered lists. Since \(\eta<1\), the estimate localizes all integration variables integrably around the surviving arguments, uniformly in their separations.

For diagonal words every action has one removal. Each removal also has its factor \(\beta\) decaying exponentially in the distance between the adding argument and the removed argument or root. If a path couples two different groups, either this factor or (88) makes its integral vanish as the gap grows. More explicitly, first restrict every matched root to a fixed radius of its matching argument using the uniform exponential bound. On this restricted region any removal between distinct groups has distance tending to infinity. Discard these paths and then increase the matching radius. The surviving paths have their additions and removals inside individual groups; in each group the numbers of deleted original roots and surviving added arguments are equal.

On a bounded matching region the two determinants are block diagonal in the limit, so their quotient tends to the product of the group quotients. Cross-group correction products tend to one. In fact \(f(u)\to e^{\pm i\lambda}\) as \(u\to\pm\infty\), and at every removal the numbers of extra surviving arguments and missing original roots in another group agree. Their limiting phases cancel. The constants \((-[i\lambda])^{-1}\) occur once per deleted column and multiply in the same way. Termwise limits, the integrable matching bound, and the finite sum over paths therefore give exactly the product of the separate group answers.

If different groups were interlaced in the original word, they may first be gathered while retaining each internal order. The normalized intertwiner in RTT tends exponentially to an invertible diagonal matrix at large real separation. Its conjugation of a pair of diagonal marks therefore differs from the unchanged pair by an exponentially small matrix. The bounded limiting rows make the corresponding exchange error tend to zero.

Finally the row norm bound makes these translated kernels a normal family as functions of the internal strict arguments. Every subsequential holomorphic limit equals the product on the generic open box just treated. The identity theorem determines it throughout the product strip, proving the proposition and its local uniformity. ◻

Proof. The uniform normalized row bounds of Lemma 10, applied before taking each Gram limit, show that left multiplication by a row preserves null vectors and has the asserted bound. Thus it extends to the completion. Adjoint and RTT identities pass to all word pairings and then to bounded operators. Matrix coefficients on words are holomorphic by Proposition 13; local boundedness extends this to all matrix coefficients and hence to operator holomorphy.

Translate all arguments of a word by \(ib\). In its Gram product the adjoint arguments are translated by the same amount, because \(-\overline{z+ib}=-\overline z+ib\). Common-translation invariance of the word kernels therefore makes this map an isometry. Translation by \(-ib\) is its inverse. Continuity of the kernels gives strong continuity on words, and the isometry bound extends it to \(\mathcal H\). This proves (89). Notice that positive dilation of \(w\) corresponds to one of the two signs of this unitary parameter.

If a vector is orthogonal to every word whose real arguments lie in a fixed open interval, successive applications of the identity theorem to the word arguments show that it is orthogonal to all strict words. This proves density. Commutation of the untwisted rows and their adjoint rule put them in one abelian von Neumann algebra. Generate that algebra using a countable dense set of arguments. A multiplication representation, together with the uniformly bounded Taylor series on a countable cover of the strip, gives scalar analytic representatives on one common set of full measure. This also proves (90). The spectral measure class is invariant under imaginary translation by (89). ◻

Proof. Write \(u=\eta(v-z)\), so \(u=0\) at the upper endpoint, and put \(b(u)=\sin u/\sin\lambda\) and \(r(u)=1-b(u)^2\). Define the reversed unmarked row by \[\widetilde T^I(-u) =\mathop{\mathrm{tr}}_{a'}\bigl(R_{a',L-1}(-u)\cdots R_{a',0}(-u)\bigr).\] Multiply \(T^D(u)\) by this row. Transposition in \(a'\) reverses the second auxiliary product; factors at different quantum sites then commute past one another. The product is therefore the two-auxiliary partial trace of \[(D\otimes I)G_0(u)\cdots G_{L-1}(u),\qquad G_j(u)=R_{aj}(u)R_{a'j}(-u)^{t_{a'}}.\] The auxiliary line spanned by \(\Phi=2^{-1/2}(|++\rangle+|--\rangle)\) is invariant. Inversion of \(R\) makes the action on this line equal to \(r(u)I\). At \(u=0\) the swap identity makes the range of \(G_j(0)\) lie in \(\mathbb C\Phi\otimes\mathbb C^2\). Thus, in the same auxiliary decomposition for every site, \[ r(u)^{-1}G_j(u)= \begin{pmatrix}I&B_j(u)\\0&C_j(u)\end{pmatrix},\qquad \|B_j(u)\|\le C_\lambda,\quad \|C_j(u)\|\le C_\lambda|u|. \tag{93}\] These bounds follow directly from the entries of \(R\), since \(\sin\lambda\ne0\). For example, on a sufficiently small disc of radius proportional to \(\sin\lambda\) one has \(|r|\ge3/4\) and \(\|C_j\|\le1/4\). The upper-right block of the product in (93) is bounded by a geometric series, while its lower-right block has norm at most \(4^{-L}\). Taking the unmarked auxiliary trace therefore gives \[r(u)^{-L}T^I(u)\widetilde T^I(-u)=I+F_L(u), \qquad \|F_L(u)\|\le3\cdot4^{-L}.\] The off-diagonal triangular block has zero auxiliary trace. The marked partial trace is uniformly bounded by \(C_\lambda\|D\|\). The displayed identity is invertible, so both finite square factors are invertible and \[T^D(u)T^I(u)^{-1} =\bigl(r(u)^{-L}T^D(u)\widetilde T^I(-u)\bigr)(I+F_L(u))^{-1}\] has the required uniform bound. Reversing the local product gives the same argument with an invariant covector and proves the other quotient. Crossing transfers it to the lower endpoint. None of these steps uses the sign of \(\cos\lambda\).

The common finite vacuum remains an eigenvector of \(T^I\) in these discs by analytic continuation. Its eigenvalue is nonzero there because the row is invertible. Thus \[A_L^D(z)\Omega_L =\bigl(T^D(z)T^I(z)^{-1}\bigr)\Omega_L\] is uniformly bounded and holomorphic on a common endpoint disc. Cauchy’s estimates control every fixed Taylor coefficient and the norm of each Taylor remainder on a smaller disc, uniformly in \(L\).

We first use these bounds to construct continued vacuum vectors inside \(\mathcal H\). Take a subsequential Gram enlargement containing the vectors obtained by applying finite words of strict rows, endpoint rows and shifts to the vacuum and to these Taylor coefficients. Use countable dense strict arguments and a basis of marks. The coefficient bounds define a holomorphic vector series on the common disc in this enlargement. On its intersection with the strict strip, uniform Taylor remainder bounds identify that series with the already constructed strict vacuum vector. Its orthogonal projection onto \(\mathcal H^\perp\) is consequently zero on this open intersection, and hence on the entire disc by the identity theorem. The continued vacuum vectors therefore belong to \(\mathcal H\). Their pairings with strict words also show uniqueness, independent of the enlargement. At the center, the finite swap and crossing identities give the endpoint vectors in (91); the Cauchy bounds approximate them in norm by nearby strict vacuum vectors.

Choose a closed interval \(I\subset(-v,v)\) with nonempty interior such that every RTT intertwiner between either endpoint and an angle in \(I\) is invertible. Such an interval exists: the excluded angles form a finite set. More explicitly, the possible zeros of the intertwiner determinant occur at \(s\in\pi\mathbb Z\pm\lambda\), where \(s=\eta(z-\alpha)\). When \(\lambda>\pi/2\) the possible interior exceptions at the endpoints are \(\alpha=\pm(v-(\pi-\lambda)/\eta)\); avoiding these points suffices. There is then a common endpoint disc on which all these inverses are bounded. Commute an endpoint-near row through a fixed word with angles in \(I\). RTT expresses the result as a finite sum of bounded strict rows acting on continued marked vacuum vectors. The number and bounds of the terms may depend on the word, but the radius of analyticity does not. The same finite-chain estimates give norm convergence to the endpoint action. Density of the \(I\)-words and the uniform interior row bound extend this to strong convergence on all of \(\mathcal H\). In particular both \(S\) and \(S^{-1}\) preserve \(\mathcal H\); their restrictions are unitary inverses. Successive strong approximations give the same Gram limits for products containing several endpoint rows. Define \(D_0=S^{-1}A^D(v)\) and \(D_j=S^{-j}D_0S^j\). The finite site-matrix identities pass to these products: at each site this is a unital star representation of the \(2\times2\) matrix algebra, and matrices at distinct sites commute. Products of these matrices are the local spin operators used below.

To pass from this vector continuation to scalar continuation with one radius, choose countably many dense \(I\)-words \(h_j\) and a finite measure equivalent to the scalar spectral measure. On a fixed endpoint disc write the continued vector \(P(z)h_j\) as \(\sum_{k\ge0}a_{jk}(z-v)^k\). For \(r'<R\) smaller than its common radius, Cauchy’s estimate gives \[\|a_{jk}\|_{L^2}\le M_{j,R}R^{-k},\qquad \sum_{k\ge0}(r')^k\|a_{jk}\|_{L^2}<\infty.\] The series of fiber norms is therefore summable almost everywhere. Intersecting over the countable words and radii produces pointwise holomorphic vector series on the same disc. On the interior part each series equals \(f(z)h_j\). Its continuation is consequently proportional to \(h_j\). At almost every spectral point at least one \(h_j\) is nonzero, and the resulting scalar ratios agree by the identity theorem. This constructs the scalar continuation on a common disc.

Equation (91) gives unitary values at both endpoints. Covariance and strong convergence give unitary boundary values at \(\pm v+ib\). Applying this first on a countable dense set of \(b\)’s and then using continuity shows that \(|f|=1\) on each continued boundary diameter. Schwarz reflection gives \(f(z)\overline{f(2v-\overline z)}=1\) there, and the analogous formula at \(-v\). Thus the continued functions are nonzero. Rational imaginary translations cover the boundary lines by discs of one radius, with a common null set. The bounded strip maximum principle now yields \(|f|\le1\). Under \(w=e^{iz}\) the excluded boundary bands give (92). Finally each scalar function has isolated zeros. Fubini on an imaginary-translation orbit gives a null zero set at almost every point of that orbit, and invariance of the spectral measure class transfers the assertion to every specified point. ◻

Proof. At fixed even transverse length, primitivity of the physical transfer in the chosen sector identifies the long-torus limit with its normalized Perron vector. A finite collection of arrows is read by physical row steps on a seam and by spins on finitely many equal-time sections. Repeated seam marks are combined, and the spins are expressed by (91). Its joint probabilities are therefore finite linear combinations of strict and endpoint word coefficients. Their limits exist by Proposition 13 and Lemma 16. When \(c<1\) the root construction used the exactly balanced sector, so these are precisely the two limits, in their prescribed order, defining \(\mathbb P_c\).

The two axes when \(c<1\). Here the exactly balanced roots already identify the cylinder limit. To compare its two axial quantizations, fix a rectangle. Orient its input path upward along the left side and then eastward along the top side, and express all spins relative to those directed steps. Let \(K\) be the tensor of its unmarked physical tiles, viewed as a matrix from the input sides to the other two sides. It is a product of physical \(R(\lambda/2)\) matrices and permutations of tensor factors. The local eigenvalues before the harmless factor \(c\) are \(1/c,1/c,1/c+1,1/c-1\). Thus \(K\) is invertible for \(c<1\).

For a finite surrounding cylinder, cut on all four sides and contract the exterior with its normalized vacuum data. This gives a linear functional \(L\) of the inside matrix, with \(L(K)=1\). The normalization includes the exterior transfer eigenvalues appropriate to that cylinder. Putting elementary matrices \(D_1,\ldots,D_q\) on the input path tests the functional on \(K(D_1\otimes\cdots\otimes D_q)\): the row index of each mark is inside and its column index outside. Since tensor products of elementary matrices span the input matrix algebra and \(K\) is fixed and invertible, the limiting marked path values determine every entry of the limiting exterior functional. They also prove that those entries have limits.

In the eastward representation the successive path angles are \(v,0\); in the northward representation, with space pointing west, they are \(0,-v\). To compare them, first replace the first pair by \((v-\varepsilon,\varepsilon)\) and translate both angles by \(-v\). The resulting pair is \((-\varepsilon,-v+\varepsilon)\), still strict. Common-translation invariance identifies the marked word values. Strong boundary convergence then permits \(\varepsilon\downarrow0\). The spin and matrix-index conventions agree: at a reversed collapsed step the mark is \(XD^tX\), exactly as in (91). Crossing only relabels those spins; the interior physical tensor remains \(K\). Thus the two normalized limiting functionals agree on \(KD\) for every elementary tensor \(D\), and hence on every inside matrix. All local arrow probabilities agree. This argument compares normalized exterior functionals; it requires no equality between eigenvalues of cylinders of different circumferences.

Identification when \(c\ge1\). In this range we must identify the state also when the root construction uses sectors tending to half filling. Let \(W_c\) be the physical vertex weight and, for a stationary probability measure \(\rho\) on allowed arrow configurations, put \[ \mathcal F_c(\rho)=-h(\rho)-\mathbb E_\rho\log W_c, \tag{95}\] where \(h\) is specific entropy per vertex. The equivalent gradient formulation fixes the height parity and forgets an additive constant; these changes have zero entropy cost per unit area. Spatial translation acts on gradients: after translating an anchored height representative, one subtracts its new reference value. Thus fixing the reference parity does not remove any physical translation from the gradient action. The domain and reference-height convention are those of (Sheffield 2005, sec. 2.3 and Appendix B); the entropy–energy decomposition is also explicit in (Lammers and Tassy 2024, secs. 2.3.2–2.3.4). Since the alphabet is finite and \(0<c<\infty\), the entropy and local energy in (95) are finite. Hard constraints are included by restricting to allowed configurations.

Write \(U(0)\) for the homogeneous harmonic normalization in Lemma 10, and set \(p_*=U(0)+\log c\). The same boundary-unitarity and strip argument used there applies to a row with specified auxiliary input and output spins: it is a matrix-unit auxiliary mark. A rectangle of width \(a\) and height \(b\), with any boundary data, therefore has partition sum at most \[\exp\{abp_*+C(a+b)\}.\] Indeed each open row has operator norm at most \(2e^{ap_*}\), and summing the boundary spins costs only an exponential in its perimeter. The finite entropy variational inequality on the rectangle then gives \(h(\rho)+\mathbb E_\rho\log W_c\le p_*\) for every stationary allowed state.

For the near-balanced tori, choose their longitudinal lengths sufficiently large to realize the Perron asymptotic. Their constrained log partition functions per vertex tend to \(p_*\) by Lemma 10. Tile each torus by fixed-size squares. Entropy subadditivity, with a negligible uncovered fraction, bounds its entropy per vertex above by the entropy of a fixed-square marginal divided by the square area, plus a vanishing boundary error. Pass to the local limit at this fixed square size, then let the square size grow. The exact finite identity “entropy plus mean log weight equals log partition function” proves the reverse inequality \(h(\rho)+\mathbb E_\rho\log W_c\ge p_*\). Hence the limiting state attains the global minimum of (95). Its expected spatial tilt is zero by the sector sizes. The other expected tilt is zero as well: at the physical angle the current mark is skew-adjoint, whereas its finite physical expectation is real. This observation specifies the tilt; it is not being used as a uniqueness argument.

The equilibrium characterization following equation (6) in (Duminil-Copin, Kozlowski, Lammers, et al. 2026), for \(1\le c\le2\), identifies the finite-specific-free-energy minimizer with the balanced plane law. We spell out the variational-domain reduction needed to apply this characterization, since a zero expected tilt alone would not suffice. For a fixed spatial translation lattice \(\mathcal L\), let \(\mathcal P_{\mathcal L}\) denote invariant laws of allowed gradients and write \[\sigma_{\mathcal L}(a) =\inf\{\mathcal F_c(\rho):\rho\in\mathcal P_{\mathcal L},\ S(\rho)=a\}.\] Here \(S(\rho)\) is the expected physical slope. Both the preceding perimeter bound and the tiling proof apply on this domain, using the average energy over a fundamental cell if necessary. We keep the same \(\mathcal L\) when forming the infimum and the ergodic decomposition. The surface tension is unchanged on passing from full translations to a finite-index sublattice: average any sublattice-invariant law over the finitely many translation cosets. This preserves its slope and its entropy–energy density and makes it fully invariant; the reverse inequality between the infima follows from inclusion of the domains. Thus one may use the full physical gradient translation action. If a periodic height-potential representation uses a parity-preserving sublattice, the argument is performed throughout on that sublattice and is then projected back to arrows.

Specific free energy is affine under this ergodic decomposition (Sheffield 2005, chap. 3). Thus every component of a globally minimizing law is itself a global minimizer: each component has free energy at least the global infimum, and their average equals it. For \(c\ge1\), strict convexity of the six-vertex surface tension on the interior of its slope domain is (Lammers and Tassy 2024, Theorem 13.16 of the 2020 preprint). Its stated condition is \(c^2\ge\max(1,1)\), and the accompanying simply-attractive-potential description supplies the height-model setting. Arrow reversal makes \(\sigma_{\mathcal L}\) even. The origin is an interior slope: height functions formed as sums of two one-dimensional unit-increment paths give allowed stationary gradient laws at every slope in \([-1,1]^2\), all of finite specific free energy. Hence strict convexity implies \(\sigma_{\mathcal L}(a)>\sigma_{\mathcal L}(0)\) for every nonzero interior \(a\). A boundary slope cannot be another minimizer: otherwise convexity along the segment from \(0\) to that slope would contradict this strict inequality at an interior point. Every minimizing ergodic component therefore has slope zero and is a fixed-slope minimizer. The variational principle makes it a minimal gradient phase.

At this last step we use the zero-slope uniqueness supplied by the combined equilibrium characterization in (Duminil-Copin, Kozlowski, Lammers, et al. 2026), with its Sheffield and delocalization inputs and its range \(1\le c\le2\). This is not an assertion of uniqueness for every Gibbs state, nor an extension to \(c>2\). The domain reduction above explains why the possibly nonergodic auxiliary state is covered, even when a formulation of the underlying height result is stated for minimal phases. The gradient-to-arrow correspondence is one-to-one after forgetting the additive height constant, so their common law is \(\mathbb P_c\). The same argument applies to either axial quantization, and therefore identifies the two orientations as well.

Reflection across a coordinate cut. In either homogeneous axial quantization, transposing the finite row product at a coordinate cut reflects its tiles and changes the spin mark by the stated adjoint rule. The finite Gram square is therefore the expectation of the observable and its conjugate reflected copy. Pass to the local and Gram limits to obtain the assertion. Neutral height differences can be read along paths on the prescribed side of the cut, so the same identity applies to their finite products. ◻

Proof. As an auxiliary argument is dilated far from all arguments in a fixed word, the scalar-normalized RTT intertwiner tends to an invertible diagonal matrix, with error \(O(e^{-\eta|s|})\). Its limiting diagonal matrix commutes with the diagonal joint auxiliary marks. Move \(P(e^s)\) through the word to the vacuum, where it equals the identity. The exchange errors are finite sums of bounded marked-row products, proving (100). Moving through several separated groups gives the stated sum, since the number of factors is fixed. These are operator-norm bounds on each finite exchange error, not just bounds on its vacuum expectation.

On a countable total set in \(\mathcal D\), the spectral integral of \(|1-f(e^s)|^2\) over \(s\) is consequently finite. Apply the full-space factorization (94). If \(a>0\) or \(b>0\), then \(|f(e^s)|\) tends to zero at one end, contrary to the integrability just proved. For a zero \(\zeta\) with \(r=|\zeta|\), the identity \[\left|\frac{t-\zeta}{t+\overline\zeta}\right|^2 =1-\frac{4t\operatorname{Re}\zeta}{|t+\overline\zeta|^2},\qquad t>0,\] and (92) imply \(1-|f(e^s)|\ge\varepsilon_0\) whenever \(|s-\log r|\le\delta_0\), with fixed positive \(\varepsilon_0,\delta_0\). Infinitely many zeros would escape to one end and supply infinitely many disjoint such intervals, again contradicting integrability. The resulting finite product has limits at both ends, and integrability forces both limits to be one. Reality was already established. Nonreal zeros occur in conjugate pairs, and the endpoint quotient is \((-1)^{\deg f}\), so the degree is even. This proves (96) and the dip assertion.

Differentiate commutation of equal diagonal-twist transfers at twist zero and apply it to \(\Omega\). Since \(P(w)\Omega=\Omega\), it gives \[ (1-P(w'))J(w)\Omega=(1-P(w))J(w')\Omega. \tag{101}\] There is no component of \(J(w)\Omega\) on \(f=1\). Indeed, let \(Q\) be the projection to that spectral set, which contains \(\Omega\). The endpoint identities and their dilates glue \(QJ(w)Q\) to the left half-plane by \(QJ(-w)Q=-QJ(w)Q\), because \(P(w)=I\) on \(Q\). It is bounded on the punctured plane, so the singularities at zero and infinity are removable. Liouville’s theorem and oddness make it zero. On every nonconstant fiber \(|f(1)|<1\), so one may set \(\mathsf h=J(1)\Omega/(1-f(1))\). Equation (101) gives (97), first at a countable set of arguments and then everywhere, including endpoints by strong convergence. Covariance, with \(U_b\Omega=\Omega\), makes this quotient transform without a scalar multiplier. Its squared fiber norm therefore gives a dilation-invariant measure.

Every nonconstant finite product has a unique smallest zero radius. Rescale it to one to obtain a measurable cross-section. Translation invariance in its logarithmic dilation coordinate gives (98), initially as a sigma-finite disintegration. Each orbit has a dip of fixed width at its first zero, where \(|1-f(1)|\ge\varepsilon_0\). Since \[\int |1-f(1)|^2\,\mathrm d\mu=\|J(1)\Omega\|^2<\infty,\] this disintegration is locally finite and \(\kappa<\infty\).

Applying (100) to \(J(1)\Omega\) gives \[ I(l):=\int |1-f(1)|^2|1-f(e^l)|^2\,\mathrm d\mu \le Ce^{-2\eta l}\qquad(l\ge1). \tag{102}\] On an orbit of width \(\mathcal L\), integrate first over the dip window of its smallest zero and then over \(l\) placing the second argument in the dip window of its largest zero. Both intervals have a fixed width and both factors a fixed lower bound. Thus \[c\,\nu\{\mathcal L>t\} \le \int_{t-2\delta_0}^{\infty} I(l)\,\mathrm dl,\] after changing the constant for bounded \(t\). This proves (99). ◻

Proof. The zero budget and analytic annulus. Evaluate (94) at \(1\). On the band \(H_0\le u\) it gives \[a+b+\frac12\sum_j \log\left(1+\frac{4\operatorname{Re}\zeta_j}{|1-\zeta_j|^2}\right)\le u.\] The cone condition implies the zero budget \[ a+b+\sum_j\min(|\zeta_j|,|\zeta_j|^{-1})\le Cu. \tag{109}\] Indeed \(\operatorname{Re}\zeta_j\ge c_0|\zeta_j|\) and \(|1-\zeta_j|\le1+|\zeta_j|\). Each zero is therefore either of radius at most \(Cu\) or of reciprocal radius at most \(Cu\). Comparing each factor at \(w\) and at \(1\) shows, on \(Au<|w|<a/u\), \(\operatorname{Re}w\ge0\), that \[\left|\log\frac{f(w)}{f(1)}\right| \le Cu(|w|+|w|^{-1}).\] Choose \(A\) large and \(a\) small once and for all. The function and its inverse are then bounded on this half-annulus. More precisely, \[ P(w)Q_u^\sigma=\sigma Q_u^\sigma +O\bigl(u(|w|+|w|^{-1})\bigr), \tag{110}\] and the same estimate holds for the inverse restricted to the band.

The endpoint formulas give \[J(i)=SZ_0,\qquad J(-i)=-Z_0S^{-1},\qquad J(-i)=-P(i)^{-1}J(i)P(i)^{-1}.\] The other endpoint gives the analogous identity on the other ray. Dilate these uncompressed identities using (89), and only then compress. This is important because \(Q_u\) need not commute with the dilation group. The formula \[ J_u(-w)=-P(w)^{-1}J_u(w)P(w)^{-1},\qquad J_u(w)=Q_uJ(w)Q_u, \tag{111}\] defines the left half of the annulus; both inverses here and below are only on \(Q_u\mathcal H\). Boundedness and strong boundary convergence make the two definitions agree on both imaginary rays. Scalar Morera applied to every matrix coefficient gives holomorphic gluing. Thus \(\|J_u(w)\|\le C\) on the full annulus. Cauchy’s formula on circles near its inner and outer boundaries proves (104) and its coefficient bounds. In particular, uniformly on \(|w|=1\), \[ \tfrac12(J_u(w)-J_u(-w)) =wJ_{u,1}+w^{-1}J_{u,-1}+O(u^3)=O(u). \tag{112}\] The nonconstant even modes are \(O(u^2)\), but this observation alone does not control the constant coefficient.

Sign blocks and the leading modes. For \(X_{\sigma\tau}(w)=Q_u^\sigma J(w)Q_u^\tau\), (111) and (110) give \[X_{\sigma\tau}(-w)=-\sigma\tau X_{\sigma\tau}(w) +O(u)\|X_{\sigma\tau}(w)\|.\] On an equal-sign block its even part is therefore bounded by \(Cu\|X_{\sigma\sigma}(w)\|\). Its odd part is \(O(u)\) by (112). Absorbing gives \(\|X_{\sigma\sigma}\|\le Cu\) and then an \(O(u^2)\) bound for its whole even part, including the constant coefficient. This proves the first estimate in (105).

For opposite signs use an additional RTT comparison at \(r=\sqrt u\). The normalized intertwiner for arguments \(r,w\) is \(R_\infty+O(r^\eta)\) with bounded inverse, uniformly for \(w\) on the closed right unit semicircle; \(R_\infty\) is diagonal and invertible. It commutes with the joint mark \(I\otimes Z\). Conjugating this mark and expanding the \(O(r^\eta)\) error in tensor matrix units, each producing a bounded marked-row product, proves \[\|[P(r),J(w)]\|\le Cr^\eta.\] Strong approximation includes the endpoints. On \(Q_u^\sigma\), \(P(r)=\sigma I+O(\sqrt u)\) by (110). Compression of the commutator yields \[(2-C\sqrt u)\|X_{\sigma,-\sigma}(w)\| \le Cu^{\eta/2},\] which proves the second estimate in (105). Inserting \(P(w)=\sigma I+O(u)\) in (103) now proves (106)–(107): equal signs add, whereas opposite signs cancel. In the latter case the odd part of \(X\) is itself \(O(u^{1+\theta})\) by the gluing identity. Thus the leading expression can equivalently be written \[\tfrac12\{\mathop{\mathrm{sgn}}(P(1))Q_u, wJ_{u,1}+w^{-1}J_{u,-1}\} +O(u^{1+\theta})\] on the whole band. Higher modes need not vanish at a fixed cutoff; the displayed error is what removes them after rescaling.

Time damping and shift phase. For unequal bands embed both in their larger band. Here is the precise meaning of the damped leading-mode remainder. For large \(m\), put \(t_q=2^q/m\) and let \(q_m\) be the largest integer with \(t_{q_m}<u_0\). Let \(B_0=\mathbf 1_{[0,t_0]}(H_0)\), \(B_q=\mathbf 1_{(t_{q-1},t_q]}(H_0)\) for \(1\le q\le q_m\), and let \(B_*=I-\sum_{q=0}^{q_m}B_q\). Write \(\Sigma=\mathop{\mathrm{sgn}}P(1)\) and define \[L_u(w)=\tfrac12\{\Sigma Q_u, wJ_{u,1}+w^{-1}J_{u,-1}\},\qquad u_{qr}=\max(t_q,t_r).\] On the pair of low bands \(B_q,B_r\) the preceding estimates give \[\|B_q(N(w)-L_{u_{qr}}(w))B_r\| \le C u_{qr}^{1+\theta}.\] Thus the leading operator after damping is the finite block sum \[L_m^{\rm damp}(w) =\sum_{q,r=0}^{q_m}P(1)^m B_q L_{u_{qr}}(w) B_rP(1)^m.\] Each block uses its own larger cutoff; no cutoff-independent Laurent coefficient is asserted. On \(B_q\) with \(q\ge1\), time damping contributes at most \(e^{-mt_q/2}\). For every \(\beta>0\) the corresponding double sum is bounded by \[C_\beta m^{-\beta} \sum_{q,r\ge0}\max(2^q,2^r)^\beta e^{-c(2^q+2^r)} \le C'_\beta m^{-\beta},\] where the lowest bands are absorbed into the constant. Taking \(\beta=1+\theta\) gives \[\|P(1)^mN(w)P(1)^m-L_m^{\rm damp}(w)\| \le C m^{-1-\theta}.\] Indeed any term involving \(B_*\) is exponentially small: its spectrum lies above \(t_{q_m}\ge u_0/2\), and \(N\) is globally bounded. The same block sum with \(\beta=1\) bounds the damped \(N\) itself, and with \(\beta=\theta\) bounds a sign-opposite \(J\). These arguments also work when the damping rows range in a fixed compact strict-angle interval. Schwarz–Pick compares their modulus loss \(1-|f(w)|\) uniformly with \(1-|f(1)|\). On low bands this is comparable to \(H_0\); on the remaining spectrum it is bounded below by a positive constant.

On a sufficiently small band, (110) at \(w=i\) gives \(|\arg f(i)^2|\le C H_0(f)\), by applying the estimate with cutoffs decreasing to the fiber value. On its complement the principal argument is at most \(\pi\), which is also bounded by a fixed multiple of \(H_0\). Spectral calculus proves (108). ◻

Proof. Differentiate the inner factors along the imaginary boundary. The singular factors in (94) contribute \(a+b\), and a zero \(\zeta\) contributes \[ \frac{2\operatorname{Re}\zeta}{|i-\zeta|^2}. \tag{117}\] All contributions are nonnegative. Scalar continuation makes the derivative finite almost everywhere and justifies the sum, or one may first differentiate finite partial products and use positivity. Since \(\partial_z=iw\partial_w\), this is exactly the spectral derivative in (113). The cone condition bounds (117) below by a constant times \(\min(|\zeta|,|\zeta|^{-1})\). Thus small \(E\) implies the small/large zero budget in (109). On that budget the negative logarithm of each factor at \(1\) is bounded above by the same quantity. This proves (115).

At finite length, differentiate one factor in the swap row and divide by its shift. The inserted link is \(-\eta\operatorname{swap}R'(0)\), proving (114). Dividing by the vacuum eigenvalue subtracts its vacuum energy. For a fixed local matrix \(O\), the finite action on \(O\Omega_L\) is therefore \[\left[\sum_jh_{j,j+1},O\right]\Omega_L,\] where only links meeting the support of \(O\) occur. To justify passing this derivative to the limit, write the local matrix as a linear combination of products of one-site matrices, inserting identity marks at gaps. Put \(B_L(D)=A_L^D(v)=S_LD_0\). The finite identity \[B_L(D_m)\cdots B_L(D_1) =S_L^m(D_m)_{m-1}\cdots(D_1)_0\] expresses \(O\Omega_L\), after a fixed translation if needed, as a finite sum of upper-endpoint words with a single power of \(S_L\) outside the word. First commute the unmarked row \(P_L(z)\) through that outer shift. Then move it through the upper-endpoint rows by RTT. The intertwiner at coincident upper-endpoint arguments is the invertible permutation matrix, so all these exchanges are analytic on a fixed small disc about \(v\). Changed auxiliary marks still belong to the same finite-dimensional matrix algebra. Each resulting term is a bounded endpoint word applied to a continued marked vacuum vector, which has the uniform Taylor bounds from the finite division estimate. This proves wordwise vector Taylor bounds for \(P_L(z)O\Omega_L\) on that disc. The same Gram and uniform-remainder argument as in Lemma 16 identifies the continued vector in \(\mathcal H\). Cauchy’s formula therefore passes its derivative, in norm and in all Gram pairings, to the spectral derivative. Consequently \[ E(O\Omega)=\left[\sum_jh_{j,j+1},O\right]\Omega. \tag{118}\] Iterating is legitimate because the commutator remains local. If \(O\) is supported on an interval of length \(\ell\), each new commutator enlarges the support by at most two. Its norm is bounded by \(2\|h\|\) times the number of interacting links. Hence \[\|E^kO\Omega\|\le C_O C_O^k k!.\] Local vectors are analytic for the symmetric operator specified by (118). Analytic-vector uniqueness makes its closure self-adjoint on their closed span. Since \(E\) extends this operator, the span reduces \(E\) and the two self-adjoint operators agree there. This also identifies real-time evolution. Finally \(J(i)=SZ_0\) and \(P(i)=S\) in (103) give (116). ◻

Proof. Since \(Z_0\Omega=S^{-1}J(i)\Omega\), the spectral measure of the local density vector is \[ \|Z_0\Omega\|^2(\,\mathrm df)=|1-f(i)|^2\,\mathrm d\mu(f). \tag{121}\] Set \(c(j)=\langle\Omega,[Z_0,[E,Z_j]]\Omega\rangle\) and take its spatial Fourier transform. By \(E\Omega=0\), the transform is twice the symmetrized energy-weighted spectral measure of \(Z_0\Omega\). On the other hand (118) and conservation of \(Z_0+Z_1\) leave only the three coefficients at separations \(0,1,-1\), namely \(2D_0,-D_0,-D_0\). Their transform is \(|1-e^{ip}|^2D_0\). Thus \[2\operatorname{sym}\bigl((p)_*(E|1-f(i)|^2\mu)\bigr) =|1-e^{ip}|^2D_0\frac{\,\mathrm dp}{2\pi}.\] Testing on compact subsets of the circle without zero permits division by the squared factor and proves (119).

On a nonconstant finite-product orbit, direct differentiation gives \(-\,\mathrm dp/\,\mathrm ds=E(F(e^s\cdot))>0\). The boundary phase change on the positive imaginary ray is one half of the full boundary change: this follows either by pairing conjugate zeros in (96) or by integrating (117) with its dilated argument. It is \(2\pi n_F\). At each nonzero phase there are consequently \(n_F\) crossings, counted with multiplicity, and none has zero derivative. The change of variables from \(s\) to phase gives \[(p)_*(E\,\,\mathrm ds\,\nu)=\left(\int n_F\,\mathrm d\nu(F)\right)\,\mathrm dp\] away from zero, first with truncated orbit measures and then by monotone convergence. Equation (119) implies finiteness; in these conventions it also gives \(4\pi\int n_F\,\mathrm d\nu=D_0\). The endpoints of an orbit and its finitely many zero-phase crossings have zero \(\,\mathrm ds\) measure, so no missing phase atom is involved in this computation. ◻

Proof. First consider a directed path of physical or endpoint steps, and write their spectral multipliers as \(t_1,\ldots,t_m\). Reflection changes the sign of a current. Equations (97) and (91) therefore express the covariance of two distinct increments, in their path order, as the negative real part of the integral of \[(1-t_a)t_{a+1}\cdots t_{b-1}(1-t_b),\qquad a<b,\] with the conjugate term supplying the real covariance. A single increment has variance one. At an endpoint this equals \(2\operatorname{Re}\int(1-f(i))\,\mathrm d\mu\), because \(|f(i)|=1\) and (121) integrates to \(\|Z_0\Omega\|^2=1\). The same identity holds at the physical step \(w=1\): on each orbit move the ray from \(i\mathbb R_+\) to \(\mathbb R_+\) by Cauchy’s theorem for \((1-F(w))\,\mathrm dw/w\). Its small and large circular arcs vanish, since \(F(0)=F(\infty)=1\). The integrated identity is justified by the bounds below.

The elementary telescoping identity \[1-\prod_{a=1}^m t_a =\sum_{a=1}^m(1-t_a) -\sum_{a<b}(1-t_a)t_{a+1}\cdots t_{b-1}(1-t_b)\] therefore gives the first equality of (122), taking \(k\) horizontal and \(|j|\) vertical steps. It is equally valid by truncating the orbit measure first and then passing to the absolutely convergent integrals below.

Here are bounds that both justify this passage and give the asymptotic. On an orbit choose one zero radius as origin in logarithmic coordinates and let \(\mathcal L\) be the range of the log radii. Beyond either extreme by distance \(d\), the factor formulas and the endpoint normalizations give \[ |1-F(e^s)|+|1-F(\pm i e^s)|\le Cn_F e^{-d}, \tag{123}\] or the trivial constant bound when it is smaller. The factors on the imaginary ray have modulus one; those on the positive ray have modulus at most one. Hence the integrand in (122) is absolutely integrable on each orbit. Outside the zero range enlarged by \(\log k+\log(1+n_F)\), the inequality \(|1-a^kb^{|j|}|\le k|1-a|+|j||1-b|\), with \(|j|\le k\), and (123) give a bounded integral. The remaining extra interval lengths cost at most \(C(\mathcal L+\log(1+n_F)+1)\).

For the main interval, use any single zero. Its cone bound gives \[|F(e^s)|^k\le \exp\{-c k e^{-|s-s_0|}\} \quad\text{for } |s-s_0|\le\log k,\] where \(s_0\) is its logarithmic radius (with the sign dictated by the chosen orbit coordinate). The integral of the right side over this interval is bounded uniformly in \(k\). Multiplication by \(F(\pm i e^s)^{|j|}\) does not change that bound. Consequently the integral of the real part of \(1-F(e^s)^kF(\pm i e^s)^{|j|}\) is \[2\log k+O\bigl(1+\mathcal L+\log(1+n_F)\bigr).\] The remainder is integrable against \(\nu\) by (99), (120), and \(\kappa<\infty\). Multiplication by the factor \(2\) in the variogram proves (122). The same majorants justify the orbitwise contour move and the telescoping passage above. Axial symmetry, already proved for the physical law, supplies the final uniform displacement bound. ◻

Proof. The root-dependent part of the endpoint energy is \(\sum_j e(y_j)\), where \[e(u)=-2\pi\eta s(u).\] Differentiating the counting equations at zero twist gives \[G y'=\frac{\mathbf 1}{\pi},\qquad (G y'')_j=-L s'(y_j)(y'_j)^2 +\sum_kK'(y_j-y_k)(y'_j-y'_k)^2.\] All derivatives here are taken on the regular real-root branch at fixed finite \(L\). Lemma 9 and the fixed-\(Q\) quadratures imply, at the nodes, \[L y_j'\longrightarrow W(y_j),\qquad W=\frac{H_Q}{\pi r_Q}.\] Set \(\varepsilon_Q=(I+K_Q)^{-1}e'\). The symmetry of \(G\) gives \(L G^{-1}e'\to\varepsilon_Q/r_Q\). Substituting the equation for \(y''\) into the second energy derivative and expanding its square therefore gives \[\begin{align*} \lim_{L\to\infty}L e_L''(0) &=\int_{-Q}^{Q} \left\{(r_Q\varepsilon_Q'-\varepsilon_Qr_Q')W^2 +2\varepsilon_Q W(r_QW)'\right\}\,\,\mathrm du \\ &=2\bigl(\varepsilon_Qr_QW^2\bigr)(Q). \tag{127}\end{align*}\] Here is the cancellation in the first equality. The direct squared displacement term is \(\int r_Q e''W^2\). The term \(-s'+K_Q'r_Q\) in the implicit derivative is \(-r_Q'\), so it contributes \(-\int\varepsilon_Qr_Q'W^2\). Transposing the other \(K_Q'\) term replaces \(e''\) by \(\varepsilon_Q'\). The remaining cross term uses \((I+K_Q)(r_QW)=1/\pi\). The integrand is now the derivative of \(\varepsilon_Qr_QW^2\). The density and \(W\) are even and \(\varepsilon_Q\) is odd, proving the boundary expression.

We next evaluate its boundary factors, rather than differentiating a bulk asymptotic. Write \[r(u)=\frac1{2\lambda\cosh(u/\eta)}.\] If \(g\) solves the full-line equation with the relevant source, its restriction is corrected to the interval solution by \[g\ \longmapsto\ g+(I+K_Q)^{-1}K_{\mathrm{out}}g,\] where \(K_{\mathrm{out}}g\) integrates over \(\mathbb R\setminus[-Q,Q]\). Apply this map to \(r\) and to \(-2\pi\eta r'\). On the outer tail near \(Q\) their ratio tends uniformly to \(2\pi\). The contribution of the opposite tail is negligible relative to \(r(Q)\): insert a sufficiently small exponential weight in the convergent Neumann series for \(K_Q\), and use the separation of the two ends. The same comparison gives \(r_Q(Q)\asymp r(Q)\), hence \[\frac{\varepsilon_Q(Q)}{r_Q(Q)}\longrightarrow2\pi.\]

After translating the endpoint \(Q\) to zero, the Neumann series for \(H_Q\) converges term by term to its half-line series and is dominated by the geometric series with ratio \(\|K\|_1<1\). Thus \(H_Q(Q)\) has a limit. Symmetry of the resolvent, or differentiation of its series, gives the exact identity \[ \frac{\,\mathrm d}{\,\mathrm dQ}\int_{-Q}^{Q}H_Q(u)\,\,\mathrm du=2H_Q(Q)^2. \tag{128}\] The bulk series also gives \[\frac1{2Q}\int_{-Q}^{Q}H_Q(u)\,\,\mathrm du \longrightarrow\frac1{1+\widehat K(0)}.\] Since the right side of (128) has a limit, averaging that identity identifies \[H_Q(Q)^2\longrightarrow\frac1{1+\widehat K(0)} =\frac{\pi}{2(\pi-\lambda)}.\] Finally (127) equals \(2\varepsilon_Q(Q)H_Q(Q)^2/[\pi^2r_Q(Q)]\). The two boundary limits yield (126). ◻

Proof. Replace the inverse-energy quadratic form in (125) by the smaller form with \(g_u(E_{\mathrm{exc},L})\), keeping \(u\) fixed. Translation invariance rewrites its factor \(2/L\) as the sum over link separations of twice a symmetrized single-source pairing.

An explicit real-time representation is \[g_u(E)=\int_0^\infty \operatorname{erfc}\!\left(\frac{t}{2\sqrt u}\right) \sin(tE)\,\,\mathrm dt,\qquad \operatorname{erfc}(x)=\frac2{\sqrt\pi}\int_x^\infty e^{-s^2}\,\,\mathrm ds.\] Integration by parts gives this identity. Its sine kernel is bounded at zero, integrable, and has a Gaussian tail at large real times. Oddness expresses the symmetrized pairing through expectations of real-time commutators of the local currents. The endpoint-Hamiltonian identification in Lemma 20 makes this the same real-time evolution as that generated by the finite-range link Hamiltonian. For link separation \(r\), split the time integral at a fixed small multiple of \(r\). In the shorter-time part the finite-range Lieb–Robinson estimate gives exponential decay in \(r\); in the longer-time part the filter tail is summable without using locality (Lieb and Robinson 1972). The estimates are uniform in the circle distance. For bounded times the local evolutions are norm approximated on a bounded neighborhood, so local convergence of the ground states gives convergence of each such commutator.

Let \(\beta_b\) be the spectral measure of the plane vector \(b_0\Omega\) for the commuting angular rows; the endpoint energy \(E\) and shift phase \(p\) are functions on this spectrum. The preceding summability shows that \[2\operatorname{sym}\bigl((p)_*(g_u(E)\beta_b)\bigr) =\mathcal S_u(p)\,\frac{\,\mathrm dp}{2\pi}\] has a continuous density. The finite regularized term converges to \(\mathcal S_u(0)\), first as \(L\to\infty\) and then as \(Q\to\infty\). Throughout this passage \(u\) is fixed. Consequently (125) and Lemma 24 imply \[ \frac2{\pi-\lambda}\le D_0-\mathcal S_u(0). \tag{130}\] The current and local spin preserve the magnetization sector, so the nonnegativity used in this comparison is sector nonnegativity. The real-time commutator estimate itself needs no ground-state hypothesis.

At nonzero momentum, the continuity equation \[i[E,Z_j]=b_j-b_{j-1}\] identifies the current spectral density. On the vacuum, \(\|Z_0\Omega\|^2(\,\mathrm df)=|1-f(i)|^2\mu(\,\mathrm df)\). The continuity equation implies, away from \(p=0\), that the energy-weighted current measure satisfies \(g_u(E)\beta_b=E(1-e^{-uE^2})\mu\). Indeed the spin measure carries \(|1-f(i)|^2\), which cancels the squared shift difference in the continuity equation. Subtract this identity from the sum rule in Lemma 21. On the punctured phase circle this gives \[ \bigl(D_0-\mathcal S_u(p)\bigr)\frac{\,\mathrm dp}{2\pi} =2\operatorname{sym}\bigl((p)_*(Ee^{-uE^2}\mu)\bigr). \tag{131}\] The equality extends to the limit \(p\to0\) because the left side is continuous. Indeed, the summable commutator bounds show that the fixed-\(u\) susceptibility measure has an absolutely convergent spatial Fourier series, hence no atom at phase zero. The regulator annihilates zero energy, and the \(f=1\) current component was removed in the construction of \(\mu\). Thus the off-zero division has omitted no additional contribution to this continuous density.

On a dilation orbit write \(p(s)=\arg f(ie^s)\). Its absolute derivative is \(E\), and its winding number is \(n_f\) when the Blaschke degree is \(2n_f\). For small nonzero \(p\) of either sign, changing variables from \(s\) to \(p\) in (131) leaves \(n_f\) crossings, weighted by \(e^{-uE^2}\), and the density convention contributes \(4\pi\). As \(p\to0\), precisely one crossing escapes to the corresponding orbit end, where \(E\to0\); the other crossings have positive limiting energy. The sum is bounded by \(n_f\), and \(\int n_f\,\,\mathrm d\nu<\infty\) by the sum rule. Dominated convergence, first at \(p\to0\) and then at \(u\to\infty\), therefore gives \[\lim_{u\to\infty}\bigl(D_0-\mathcal S_u(0)\bigr)=4\pi\kappa.\] Combining this with (130) proves (129). ◻

Proof. Choose finitely many real samples in a window, and let \(Q(f)=\sum_j(1-f_{s_j})^2\). Any bounded continuous detector \(F\) supported away from the all-ones point can be written \[F=\sum_j \frac{F}{Q}(1-f_{s_j})^2,\] with each quotient extended continuously by zero near that point. Thus (136) gives convergence for such detectors without assuming convergence of the full unweighted mass. Similarly, products of two detectors are bounded by finite sums of two-dip factors. The constants depend on the fixed window and mesh of samples, not on the array center in the permitted bulk.

The cone condition and the dip estimate for inner functions provide a fixed dip width and depth around each zero scale, independent of the degree. Choose a fixed fine sampling mesh and a continuous detector \(0\le F_T\le1\) in a window of length \(T\), equal to one when a prescribed interior dip is detected and zero near all ones. Each dilation orbit contributes an interval of centers of length at least \(T-C\). Hence \[\int F_T\,\,\mathrm d\mu\ge(T-C)\kappa.\] Take the detector to be a continuous cutoff of \(\max_j|1-f_{s_j}|\) at a fixed positive dip threshold \(a\). Then \(F_T\le C_a\sum_j(1-f_{s_j})^2\), with \(C_a\) independent of \(T\). Equation (137) bounds the product for two windows separated by a gap \(g\) by \(CT^2e^{-2\eta g}\). For \(N\) regularly spaced windows, their pair integrals sum to at most \[\frac{C N T^2e^{-2\eta g}}{1-e^{-2\eta(T+g)}}.\]

Pack windows of length \(T\) and gaps \(g\) into a bulk interval of logarithmic length \((1-\gamma)B/\eta\). Their number \(N_B\) satisfies \[\frac{N_B}{B}\longrightarrow \frac{1-\gamma}{\eta(T+g)}.\] For numbers \(a_j\in[0,1]\), \(\sum_j a_j\le1+\sum_{j<k}a_ja_k\). Integrating this inequality against \(\zeta_L\), using bulk-uniform detector convergence and the finite two-dip estimate, gives \[\frac{1-\gamma}{\eta(T+g)}(T-C)\kappa \le \lim_{d\downarrow0}\limsup_{B\to\infty}\frac{I_L}{B} +\frac{CT^2 e^{-2\eta g}}{T+g}.\] For fixed \(T,g,\gamma\), all detectors use only a fixed finite number of samples, so this application does not ask for convergence of a growing word. The uniform estimates allow the inequalities to be summed over their translated windows. The single-window convergence error, after division by \(B\), is \((N_B/B)\varepsilon_{d,B,T}\), where \(\lim_{d\downarrow0}\limsup_{B\to\infty}\varepsilon_{d,B,T}=0\). Shortening the bulk interval at its ends by \(O(T+g)\) keeps every sample in the permitted interior and does not change the limiting count.

Use Lemma 26, choose \(g=A\log T\) with \(A\) sufficiently large, let \(T\to\infty\), and then let \(\gamma\downarrow0\). This proves (138). ◻

Proof of Proposition 23. Since \(\eta=\lambda/\pi\), the upper bound (138) equals the lower bound (129). Thus (124) holds. The covariance conversion follows directly from (1). ◻

Proof of Lemma 26. In the product part of (133), the quadratic coefficient is a sum of \((\log[x+i\lambda])''\) times the two first displacements. Its integral on the real line is \(2\), from the two limits of \(\coth(x+i\lambda)\). The constant accelerations cancel between cross and within-set terms. With root density \(1/d\) and displacement \(dz_*\), the coefficient per unit length is \(2z_*^2\), with a vanishing two-limit error.

For the determinants, the bulk sequence of \(dM_{AC}\), at row-column difference \(y=d(k+z_*)\), is \(d\) times \[ \frac{(1-e^{2\pi i z_*})\coth y-\coth(y-i\lambda) +e^{2\pi i z_*}\coth(y+i\lambda)} {-2\pi i}. \tag{148}\] For \(M_{CA}\) replace \(z_*\) by \(-z_*\); for the two Gaudin denominators use \(z_*=0\). The coincident pole is continued by its removable limit. The preceding trace comparison reduces the computation to the quadratic coefficient of these bulk symbols.

Use Fourier exponent \(-ipd(k+z_*)\). The simple pole contributes the shifted-sampling identity \[\sum_{k\in\mathbb Z}\frac{e^{-ipd(k+z_*)}}{k+z_*} =\pi\bigl(\cot(\pi z_*)-i\mathop{\mathrm{sgn}}p\bigr), \qquad 0<|p|<\pi/d,\] with symmetric summation. Subtract \(1/y\) from the singular cotangent before treating the remaining part. Its derivative is integrable and analytic in every strip \(|\operatorname{Im}y|<a<\min(\lambda,\pi-\lambda)\). Contour shift therefore makes every nonzero alias exponentially small in \(1/d\), including its fixed Taylor coefficients. This remains uniform at the band ends since a nonzero alias has absolute frequency at least \(\pi/d\); the value at \(p=0\) is irrelevant to the integral.

Write \(a_0=e^{-\lambda p}\) and \(b_0=e^{-(\pi-\lambda)p}\). The continuous transform together with the sampled pole gives \[\frac{(1-e^{2\pi i z_*})(1+a_0b_0)/2-b_0 +e^{2\pi i z_*}a_0} {1-a_0b_0} +\frac{1+e^{2\pi i z_*}}2 =\frac{(1-b_0)(1+e^{2\pi i z_*}a_0)} {1-a_0b_0}.\] Dividing by the value at zero twist, the relative symbol is \[\frac{1+e^{2\pi i z_*}a_0}{1+a_0}.\] The offset Fourier phases cancel between the two orientations. Their combined quadratic logarithmic coefficient is \[-4\pi^2 z_*^2\frac{a_0}{(1+a_0)^2}.\] Since \[\int_{\mathbb R}\frac{e^{-\lambda p}}{(1+e^{-\lambda p})^2}\, \frac{\,\mathrm dp}{2\pi}=\frac1{2\pi\lambda},\] the determinant contribution per length is \(-2\pi z_*^2/\lambda\). For each finite \(L\), let \([t^2]\) denote the Taylor coefficient at \(t=0\) of the local logarithm fixed above. Adding the product contribution proves the explicit coefficient limit \[\lim_{d\downarrow0}\limsup_{\substack{B\to\infty\\ B=Ld/2}} \left|\frac1B[t^2]\log|\langle\Omega_{L,0},\Omega_{L,t}\rangle|^2 +\frac1{2\lambda(\pi-\lambda)}\right|=0.\] The Taylor coefficient is taken before either array limit; no common analytic neighborhood or uniform finite-\(t\) expansion is asserted. By (134) that coefficient is \(-I_L\), which proves the lemma. ◻

Proof. We give the construction at zero. The construction at infinity uses the coordinate \(v=1/w\) and the same estimates. The reciprocal-radius budget in (150), together with the cone condition, gives, for a sufficiently small constant \(c_*>0\), uniformly on the band \(E_+(R)\) \[ P(w)=I+O(R|w|),\qquad \|P(w)^{-1}\|\le C, \quad \operatorname{Re}w\ge0,\quad |w|<c_*/R. \tag{154}\] For example, normalize each Blaschke factor to one at zero. Its difference from one is at most \(C|w|\operatorname{Re}(\zeta^{-1})\) in this disc, and the sum of these bounds is at most \(CR|w|\). Taking \(c_*\) small bounds the product and its inverse. All inverses in this proof are restricted to the stated band.

On \(E_+(R)\mathcal D\) write \(P_R(w)\) for the restriction of \(P(w)\). Compress \(J\) by \(E_+(R)\) and continue it to the other half-disc by the reversal identity \[ J_R(-w)=-P_R(w)^{-1}J_R(w)P_R(w)^{-1}, \qquad J_R=E_+(R)JE_+(R). \tag{155}\] For this boundary gluing, first dilate the uncompressed endpoint reversal identity and then compress by \(E_+(R)\), which commutes with every \(P(w)\). Thus the argument of Lemma 19 applies without moving a cutoff through a boost. The resulting function is bounded and holomorphic on the punctured disc. Its center is removable: the negative Laurent coefficients vanish by the Cauchy bounds. At the center (155) says \(J_R(0)=-J_R(0)\), so that constant coefficient is zero. Define \(E_+(R)j_+E_+(R)\) to be the linear Taylor coefficient. The Cauchy estimate on a circle of radius proportional to \(R^{-1}\) proves the bound \(CR\).

Compression from a larger band to a smaller band commutes with taking this coefficient. This proves consistency and defines the form on the union of bands. For positive real \(w\), \(J(w)^*=-J(w)\), so the coefficient is skew-adjoint as a form. The same argument gives \(j_-\).

Dilation multiplies \(a_+\) by the dilation factor and \(a_-\) by its reciprocal. On each nonconstant orbit either substitution changes Haar measure \(ds\) to \(dt/t\). Equation (149) therefore gives (153). In particular, \(\int_{a_\pm\le R}a_\pm^2\,\mathrm d\mu=\kappa R^2/2\). Dividing the vacuum identity by \(w\), or by \(v\) at infinity, and taking the band Taylor coefficient now gives the vacuum assertion in (151). Its inner product with \(\Omega\) is zero: \(\Omega\) lies in the zero-energy band, and its compressed current mean is bounded by \(CR\) for every \(R>0\).

For the damping assertion, divide the spectrum of \(a_\pm\) into \([0,s^{-1}]\) and \((2^{q-1}s^{-1},2^qs^{-1}]\), \(q\ge1\). The form bound controls a block between levels \(q,r\) by \(Cs^{-1}\max(2^q,2^r)\). Damping on both sides supplies \(e^{-2^{q-1}}e^{-2^{r-1}}\) when the respective index is positive. The sum of these block norms is at most \(C/s\), which proves both existence of the bounded extension and (152). The same argument shows that tails of these damped block sums tend to zero in operator norm. ◻

Proof. If \(\|T_\delta\|\le M\), then (156) implies \(\|Tu\|\le M\|u\|\) on the core. Approximate \(v_\delta\) by \(u_\delta\) as in the definition. The norm of \(T_\delta(v_\delta-u_\delta)\) is at most \(M\) times that approximation error, while \(T_\delta u_\delta\) has the asserted limit. Taking the approximation error to zero proves the first assertion and then composition. For the last assertion, use \[\|v_\delta-u_\delta\|^2 =\|v_\delta\|^2+\|u_\delta\|^2 -2\operatorname{Re}\langle v_\delta,u_\delta\rangle \longrightarrow\|v-u\|^2\] and approximate \(v\) by a core vector \(u\). ◻

Proof. Spectral functions and exclusion of exterior zeros. The strong row actions preceding Lemma 29 give convergence of every polynomial in finitely many samples of the two spectral functions and their complex conjugates. The functions are bounded by one on the right half-plane; normal-family compactness therefore gives the joint compact-open convergence. The limiting functions are the original finite Blaschke spectral functions on the two factors, and are not identically zero. For a vector \(v\), its state spectral measure is \(\sigma_v(B)=\|\mathbf 1_Bv\|^2\) on measurable sets of spectral functions. These measures have total masses \(\|u_\delta\|^2\to\|u\|^2\); normalization to probability is harmless when \(u\ne0\).

The rescaled quantities in (157) require control of every zero, including zeros between the two scales. For a fixed core representative, the exterior RTT estimate from Lemma 18 gives, outside fixed word offsets, \[ \|(I-P(e^s))u_{r_1,r_2}\|^2 \le C_u\sum_{i=1}^2e^{-2\eta|s-\log r_i|}. \tag{160}\] Every zero \(\zeta\) forces a fixed-width interval about \(\log|\zeta|\) on which \(|1-f(e^s)|\) is bounded below by a positive constant. This follows from the cone condition and the modulus loss of its single Blaschke factor; all other factors have modulus at most one. Integrating (160) outside slightly smaller windows thus yields \[ \begin{split} &\sigma_{u_{r_1,r_2}} \left\{\text{some zero lies outside } \bigcup_{i=1}^2\{\zeta:|\log(|\zeta|/r_i)|\le D\}\right\}\\ &\hspace{40mm}\le C_u e^{-c_*D}. \end{split} \tag{161}\] An additive fixed change in \(D\) absorbs the dip width and the word offsets. To justify the event bound explicitly, if any zero lies outside the enlarged windows, one entire dip interval lies in the integration region. The integral of \(|1-f(e^s)|^2\) there is then at least a fixed positive number. Markov’s inequality after integration over the state spectral measure gives (161).

Complete zero clusters and the physical-scale expansion. Here is the order of truncations. Given a desired exceptional mass, first choose \(D\) large enough in (161) and so that the finite zero sets of \(F_1,F_2\) lie in their annuli with correspondingly high probability. Choose their boundary circles to carry no limiting zero almost surely. Only countably many radii are excluded: first restrict to a bounded zero count, then apply this assertion to its finite intensity measure, and finally take the union over count bounds. On these fixed annuli, the cone keeps the zeros in a compact subset of the open right half-plane. The argument principle and compact-open convergence give joint convergence of the complete zero multisets, including multiplicities. Next choose a finite degree bound \(M\) so that larger limiting counts have arbitrarily small mass. Take the scale limit only after \(D\) and \(M\) have been fixed. Zero counts and locations now lie in a compact, bounded-count set, so every product expansion below is uniform. Finally let the allowed exceptional mass tend to zero, increasing \(D\) and \(M\) if necessary.

Each limiting product has even degree: conjugate zeros pair, real zeros contribute a factor \(-1\) between the two ends, and both end values are one. On the preceding high-mass event the complete limiting zero counts occur in the chosen annuli. Multiplicity convergence therefore makes both finite-scale clusters even with probability tending to one. This proves the needed parity; it is not an assumption on a truncated cluster.

For clarity, write the large zeros as \(\delta^{-1}\alpha_j\) and the small zeros as \(\delta\beta_j\). On the retained event each normalized zero has modulus between \(e^{-D}\) and \(e^D\), and there are at most \(M\) zeros. Normalize each factor to one at zero. The two even clusters then give, for \(w=1,i\), \[ f(w)=1-\delta\left( w\sum_j2\operatorname{Re}\alpha_j^{-1} +w^{-1}\sum_j2\operatorname{Re}\beta_j\right)+O_{D,M}(\delta^2). \tag{162}\] The two sums converge jointly to \(a_+(F_1)\) and \(a_-(F_2)\). Taking logarithmic modulus at \(1\) and phase at \(i\) gives (157). The middle sign is positive on this event. Values of \(H_0/\delta\) can be arbitrarily large on the discarded event; convergence in law and bounded multiplier convergence require no first-moment estimate for that quantity.

Strong multiplier convergence and propagators. We next strengthen bounded spectral convergence to convergence of vectors. Choose a countable total word core, including its rational linear combinations, and exclude the union of the atoms of \(A\) in all their state measures. That union is countable; any other \(R\) is admissible in (158). At such \(R\), continuous cutoffs approximate the indicator in each state \(L^2\) norm. More generally, let \(b_\delta\) be a uniformly bounded spectral multiplier with the stated joint limit \(b\). The algebra of polynomials in a countable dense set of interior samples and their conjugates generates the spectral functions and is dense in the state \(L^2\) space. Given a state \(u\), choose such a polynomial \(p\) with \(\|(b-p)u\|\) small. Joint convergence gives \[ \|(b_\delta-p_\delta)u_\delta\|^2 \longrightarrow\|(b-p)u\|^2. \tag{163}\] For indicators, take the continuous approximations first. Polynomial actions already converge strongly. Equation (163) proves strong convergence of the multiplier actions; polarization gives the corresponding mixed pairings. Lemma 29 extends these actions to all strongly converging families and permits their compositions. No assertion that a current preserves the exceptional-zero event is needed.

Finally, \(|f(1)|\le1\) and \(|f(i)|=1\). The bounded multipliers \(f(1)^{m_\delta}f(i)^{l_\delta}\) converge jointly by (157) to \(e^{-xA+iy(-a_+\otimes I+I\otimes a_-)}\), which is (159). The same vector argument applies. ◻

Proof. Write \(Q_\delta=Q_\delta(R)\) and \(Q=Q(R)\). Glue \(J_\delta(w)=Q_\delta J(w)Q_\delta\) to the full annulus \[CR\delta<|w|<c_* /(R\delta)\] using Lemma 19. Let \(C_{k,\delta}\) be its Laurent coefficients. Fix \(\rho>0\) sufficiently small compared with \(R^{-1}\). At the outer circle, the substitution \(v=\delta w\) gives \[ \delta^{-1}C_{1,\delta} =\frac1{2\pi i}\int_{|v|=\rho} J_\delta(v/\delta)\frac{dv}{v^2}. \tag{165}\] At the inner circle, put \(w=\delta v\) to obtain \[ \delta^{-1}C_{-1,\delta} =\frac1{2\pi i}\int_{|v|=\rho^{-1}} J_\delta(\delta v)\,dv. \tag{166}\] Both contours are counterclockwise. The rescaled contours are fixed, and both integrands are uniformly bounded in operator norm at fixed \(R,\rho\).

On the strict right semicircle the marked rows converge on their respective tensor factors. The cutoffs converge strongly by (158). On the reflected semicircle the integrand is the product specified by reversal, for example \[-\bigl(Q_\delta P(v/\delta)^{-1}Q_\delta\bigr) J_\delta(v/\delta) \bigl(Q_\delta P(v/\delta)^{-1}Q_\delta\bigr).\] The inverse factors are uniformly bounded spectral multipliers on \(Q_\delta\mathcal D\) only. Their joint limits follow from Proposition 30, since on \(Q\mathcal D^{\otimes2}\) the first energy is at most \(R\) and (154) applies. The second contour uses the analogous estimate at infinity. Lemma 29 therefore proves convergence of these noncommuting products at every strict contour point.

Here is the norm justification for passing through the contour integrals. For a representative \(u_\delta\), let \(v_\delta(t)\) be the integrand applied to it, including the scalar contour weight, and let \(v(t)\) be its limit. The vectors are bounded uniformly in \(t,\delta\). Core test pairings of \(\int v_\delta(t)dt\) converge by a single dominated integral. Their squared norms converge because \[ \left\|\int v_\delta(t)dt\right\|^2 =\iint\langle v_\delta(t),v_\delta(s)\rangle\,dt\,ds \longrightarrow \iint\langle v(t),v(s)\rangle\,dt\,ds. \tag{167}\] Pairings of strongly convergent representatives converge, so dominated convergence applies to this double integral. The two imaginary-axis points have contour measure zero. The norm criterion in Lemma 29 now gives strong convergence of each coefficient action. By the definitions of the chiral forms, the limits of (165) and (166) are respectively \[Q(j_+\otimes I)Q, \qquad Q(I\otimes j_-)Q.\]

It remains to pass from these coefficients to \(N\). Put \(\Sigma=\mathop{\mathrm{sgn}}P(1)\) and \[T_\delta(w)=w\delta^{-1}C_{1,\delta} +w^{-1}\delta^{-1}C_{-1,\delta}.\] The same-sign and opposite-sign estimates of Lemma 19 give, for some \(\theta>0\), \[ \delta^{-1}Q_\delta N(w)Q_\delta =\tfrac12\bigl(\Sigma T_\delta(w)+T_\delta(w)\Sigma\bigr) +O_R(\delta^\theta) \tag{168}\] in operator norm, uniformly on the unit semicircle. Indeed the anticommutator is \(+T_\delta\) on the positive-sign block, \(-T_\delta\) on the negative-sign block, and zero between opposite signs. On same-sign blocks the even Laurent part and all odd modes of order larger than one vanish after rescaling, with the stated remainder. The opposite-sign block of the rescaled \(N\) has the same vanishing bound. Proposition 30 gives \(\Sigma\to I\) strongly on representatives. The coefficient operators are uniformly bounded and have the strong limits just proved. Composition in (168) proves (164). The low-band bounds include the endpoints \(w=\pm i\).

We finish with a summable bound for removing cutoffs. For all \(U>0\) and all sufficiently small \(\delta\), \[ \|\mathbf 1_{H_0/\delta\le U}\,\delta^{-1}N(w) \mathbf 1_{H_0/\delta\le U}\|\le CU. \tag{169}\] If \(U\delta\) is small this is Lemma 19; otherwise the global bound on \(N\) gives it by \(\delta^{-1}\le C U\). Decompose \(H_0/\delta\) into bands \([0,1]\) and \((2^{q-1},2^q]\), \(q\ge1\). A current block between levels \(q,r\) costs at most \(C\max(2^q,2^r)\). Each internal propagator with \(\delta m\ge b>0\) supplies at level \(q\ge1\) the factor \(e^{-b2^{q-1}}\), since \(\|P(1)^m\mathbf 1_{H_0/\delta>2^{q-1}}\|\le e^{-b2^{q-1}}\); unit shifts do not affect this norm. The outer states are the vacuum, of energy zero. For a product of \(k\) currents, a bound for the sum over its \(k-1\) internal indices is a constant times \[ \sum_{q_1,\ldots,q_{k-1}\ge0} \prod_{j=1}^{k-1}(1+2^{q_j})^2e^{-b'2^{q_j}}, \qquad b'>0. \tag{170}\] This converges, and its tails with any \(2^{q_j}>R\) tend to zero uniformly in the mesh. The limiting forms have the identical bound using \(A\) and (151). Fixed-cutoff convergence, followed by this tail estimate, proves the product assertion. The constants are uniform when the finitely many limiting time gaps have a fixed positive lower bound. ◻

Proof. On a compact subset of an eastward ordering chamber, give each current a fixed positive amount of damping on both sides. The remaining semigroups have parameters with real parts bounded below. These semigroups are holomorphic in operator norm there: for instance \(\|a^m e^{-sa}\|\le m!s^{-m}\) after reducing \(s\) by a fixed factor. The resulting bounded product is jointly holomorphic in all its parameters. This proves joint holomorphy on \(U_x\), rather than merely separate holomorphy.

By Proposition 17, northward quantization describes the same physical law. In its coordinates \(z'=-iz\), the centered differences transform as \(D_{x',\delta}=D_{y,\delta}\) and \(D_{y',\delta}=-D_{x,\delta}\), hence \(D_\delta^+=-iD_\delta^{\prime+}\). The possible face-origin shift is one lattice spacing and is covered by moving-face convergence. The northward formula is thus holomorphic on \(U_y\). On \(U_x\cap U_y\) both formulas are limits of the same physical moments, so they agree and glue. Permuting the insertions does not change a physical product, which proves symmetry.

With the other points as in the statement, failure of both axial orderings occurs only at the grid points (172). Choose a small neighborhood of the fixed configuration in which all its real gaps and imaginary gaps remain at least \(b>0\). Near \(p_{jh}\), the eastward representation has only one possibly small gap, \(g_x=|\operatorname{Re}(z_1-p_{jh})|\), between \(z_1\) and \(z_j\). The northward representation has only one possibly small gap, \(g_y=|\operatorname{Im}(z_1-p_{jh})|\), between \(z_1\) and \(z_h\). Use the ordering with \(g=\max(g_x,g_y)\ge|z_1-p_{jh}|/\sqrt2\). Only the two currents bordering that gap need shrinking damping. Give each damping \(g/4\) on both sides: the inner dampings consume \(g/2\), and the outer dampings come from neighboring fixed gaps or from a vacuum endpoint. All other currents receive fixed damping. Equation (152) gives a cost \(C/g\) for each of the two currents and a bounded cost for the rest. This proves the uniform quadratic bound. A point sharing the other coordinate is separated in the chosen ordering.

Finally, when \(|z_1|\) is large, one coordinate makes \(z_1\) extreme in the corresponding ordering, with a gap \(g\) of order \(|z_1|\) to all other points. Give the isolated current damping of order \(g\) on both sides, using free damping on its vacuum side. Keep fixed damping around all remaining currents. The isolated factor costs \(C/g\); imaginary semigroup factors have norm one. Thus \(F_k\to0\) at infinity. ◻

Proof. Split the middle damping of (173) into two halves. Each of the resulting damped current factors has norm bounded by a constant, by (152) at scale \(\epsilon/2\). This gives uniform boundedness.

Apply the representative construction with scales \(r_1=\epsilon\) and \(r_2=1\). Its Gram and row limits use only separation of the two scales, so they are unchanged. The all-zero event estimate (161), with the same order of fixed windows, bounded degree, and then scale limit, gives \[ \epsilon a(f)\longrightarrow a(F_1) \tag{176}\] jointly with the two spectral functions. To see the scaling directly, a zero \(\epsilon\alpha\) contributes \(2\operatorname{Re}\alpha^{-1}\) to \(\epsilon a\), whereas a zero \(\beta\) at the unshifted scale contributes \(2\epsilon\operatorname{Re}\beta^{-1}\). On the bounded-count annuli the latter sum tends uniformly to zero. The exceptional-zero event has arbitrarily small mass and is discarded before making this expansion.

Choose a non-atomic band endpoint and put \(B_\epsilon=\mathbf 1_{\epsilon a\le R}\) and \(B=\mathbf 1_{a\otimes I\le R}\). The spectral multiplier argument proves \(B_\epsilon\to B\) strongly. The definition of \(j\) by its Taylor coefficient gives, for a sufficiently small fixed \(\rho<c_*/R\), \[ \epsilon B_\epsilon jB_\epsilon =\frac1{2\pi i}\int_{|v|=\rho} B_\epsilon J(\epsilon v)B_\epsilon\,\frac{dv}{v^2} \longrightarrow B(j\otimes I)B. \tag{177}\] Indeed, the unscaled band is \(a\le R/\epsilon\), whose Taylor disc has radius proportional to \(\epsilon/R\). On the right semicircle the row acts on the first factor; on the left semicircle the bounded compressed inverse and reversal apply. The single- and double-contour argument (167) proves convergence in representative norm. Thus (176) and (177) give the cutoff limit of \(K_\epsilon\).

These cutoffs can be removed in operator norm uniformly in \(\epsilon\). Decompose \(\epsilon a\) into \([0,1]\) and the dyadic bands above one. A block of \(\epsilon j\) between levels \(q,r\) costs at most \(C\max(2^q,2^r)\). The three damping factors of \(K_\epsilon\) give an absolutely summable majorant \[C\sum_{q_0,q_1,q_2\ge0} \max(2^{q_0},2^{q_1})\max(2^{q_1},2^{q_2}) e^{-b(2^{q_0}+2^{q_1}+2^{q_2})}.\] Absorbing the lowest-band constants into \(C\) gives some fixed \(b>0\). The part with any \(2^{q_i}>R\) tends to zero as \(R\to\infty\). The same estimate holds for the limiting tensor product. Therefore, on separated-scale representatives, \[K_\epsilon\longrightarrow \bigl(e^{-a}je^{-a}je^{-a}\bigr)\otimes I.\]

A fixed unboosted word vector \(u\) is exactly the representative of \(\Omega\otimes u\): its small-scale word is the identity. Hence for fixed word vectors \(u,v\) the last limit contracts the first factor to \[\langle\Omega,je^{-a}j\Omega\rangle =-\int a(f)^2e^{-a(f)}\,\mathrm d\mu(f) =-\kappa\int_0^\infty t e^{-t}\,dt=-\kappa.\] The sign uses the skew-adjointness of \(j\); the integrals are justified by (153). This proves (174) on a dense set of test vectors. Uniform boundedness extends it to every pair of vectors in \(\mathcal D\).

For (175), take \(z_1=z_j+\epsilon\) in the eastward ordering. The two colliding currents are consecutive. After multiplication by \(\epsilon^2\), their part of the product becomes \(K_\epsilon\), provided its two outer dampings are subtracted from the adjacent exterior gaps. Those gaps remain positive for small \(\epsilon\); at an endpoint, insert damping freely on the vacuum. The two exterior vectors converge strongly, because all their damping gaps remain positive. They may therefore be replaced by their limits in the weak convergence of the uniformly bounded \(K_\epsilon\). The two insertion minus signs cancel. Omitting the colliding currents recombines the adjoining semigroups, giving exactly the right side of (175). No dependence on the collision position remains. ◻

Proof. The preceding lemmas already give the limits as products of pure chiral functions and their conjugates. It remains to determine \(F_k\). Fix generic spectators \(z_2,\ldots,z_k\) as in Lemma 32. Its quadratic bound makes every grid puncture of \(F_k\) a pole of order at most two: the Cauchy bound for a Laurent coefficient of order \(-m\), \(m>2\), is \(Cr^{m-2}\) on a circle of radius \(r\), and tends to zero.

First consider an artificial puncture \(p=p_{jh}=\operatorname{Re}z_j+i\operatorname{Im}z_h\) with \(j\ne h\). Take a fixed small circle \(\Gamma\) about its position for the chosen spectators. In a sufficiently small complex neighborhood of the spectators, the circle encloses just this moving grid point and remains in \(U_x\cup U_y\). The contour moments \[m_\ell=\frac1{2\pi i}\int_\Gamma \zeta^\ell F_k(\zeta,z_2,\ldots,z_k)\,d\zeta\] are jointly holomorphic in the spectator variables, by the uniform bounds and joint holomorphy on the circle. If the principal part is \(A/(\zeta-p)^2+B/(\zeta-p)\), residue calculation gives \[ m_\ell=Bp^\ell+A\ell p^{\ell-1},\qquad m_0m_2-m_1^2=-A^2,\qquad m_0m_3-m_1m_2=-2pA^2. \tag{181}\] If \(A\) is nonzero at any parameter value, the nonzero denominator persists locally and \[p=\frac{m_0m_3-m_1m_2} {2(m_0m_2-m_1^2)}\] is holomorphic there. This is impossible, since \[p=\frac{z_j+\bar z_j}{2}+\frac{z_h-\bar z_h}{2}, \qquad \partial_{\bar z_j}p=\tfrac12.\] Thus \(A\) vanishes. If \(B\) is nonzero, \(p=m_1/m_0\) gives the same contradiction. Every artificial puncture is removable. This removal is joint in the parameters: inside \(\Gamma\), the Cauchy integral of \(F_k(\eta,\mathbf z)/(\eta-\zeta)\) supplies a jointly holomorphic extension and agrees with each one-variable removable extension.

At a true collision, put \(\xi=(z_1+z_j)/2\) and \(d=z_1-z_j\), retaining generic exterior points. Locally the union \(U_x\cup U_y\) contains every \(d\ne0\). The function \(H(\xi,d,\mathbf z)=d^2F_k\) is uniformly bounded there and therefore extends jointly holomorphically to \(d=0\). One can see this directly by its Cauchy coefficients in \(d\): boundedness removes the negative coefficients, uniformly on smaller spectator neighborhoods. Symmetry under exchanging the two colliding insertions makes \(H\) even in \(d\). Lemma 33 identifies its constant term as \[H(\xi,0,\mathbf z) =-\kappa F_{k-2}(\mathbf z),\] independent of \(\xi\). Subtracting this term leaves an even holomorphic function with zero constant term, hence a function divisible by \(d^2\). It follows that \[F_k+\frac{\kappa F_{k-2}(\mathbf z)}{(z_1-z_j)^2}\] is holomorphic at this diagonal. In particular there is no simple residue. Independence of the collision center is needed here: a varying coefficient of \(d^{-2}\) could otherwise create a simple term when \(z_j\) is held fixed.

Subtract all these true double-pole terms as a function of \(z_1\). The artificial points are removable, the true points have no remaining principal parts, and the resulting function is entire. Lemma 32 gives decay at infinity, and the subtracted rational terms also decay there. Liouville’s theorem yields \[ F_k(z_1,\ldots,z_k) =-\kappa\sum_{j=2}^k \frac{F_{k-2}(z_2,\ldots,\widehat z_j,\ldots,z_k)} {(z_1-z_j)^2}. \tag{182}\] Together with \(F_0=1\) and \(F_1=0\), this is Wick’s recursion with contraction \(-\kappa/(z_i-z_j)^2\). Continuity extends it from the generic spectators to every strict axial chamber. Factorization of the two chiral sectors and reality then give (178).

Finally, \(\partial_z\partial_\xi K_\kappa(z,\xi)=-\kappa/(z-\xi)^2\) and \(\partial_z\partial_{\bar\xi}K_\kappa(z,\xi)=0\) away from collisions. The conjugate identity gives the other pure contraction. This proves the covariance assertion and, by linearity, all the real derivative formulas. The uniform moving-face convergence established above gives the asserted local uniformity. Estimates across a time-ordering wall and for undifferentiated fields will be supplied in Section 7; none is needed for this proposition. ◻

Proof. Work throughout this proof in \(\mathcal H_{\mathrm{spin}}\), the closed span of \(O\Omega\) with \(O\) a finite local spin matrix. By Lemma 20, this subspace contains \(\Omega\) and reduces \(E\). Every finite local spin matrix and its adjoint preserve the local generating vectors and hence their closed span. In particular \(D=D_g=\sum_lg_lZ_l\) is bounded and self-adjoint on \(\mathcal H_{\mathrm{spin}}\). The shift preserves this subspace in both directions, since it translates local matrices and fixes \(\Omega\). In this proof \(E\) denotes its self-adjoint restriction to \(\mathcal H_{\mathrm{spin}}\), \(B_M=\mathbf 1_{\{E\le M/R\}}\) is its spectral projection there, and all Hilbert-space norms are taken on this subspace.

First we prove a two-sided compression bound. For fixed \(M\), Lemma 20 places the range of \(B_M\) inside the ambient spectral band \(H_0\le C M/R\), when \(R\) is sufficiently large. The full-space estimates of Lemma 19 therefore apply to these compressions without increasing their norms. The identity \[N(i)=\tfrac12S^2(Z_0+Z_1)\] and translation by the unitary shift therefore give \(\|B_M(Z_l+Z_{l+1})B_M\|\le C_M/R\). Pair the consecutive sites in the sum for \(D\). The coefficients of the paired sums have total absolute value at most \(AR\); the remaining single-site terms have total absolute coefficient at most \(\sum_l|g_{l+1}-g_l|\). Consequently \[ \|B_MDB_M\|\le C_{A,M}. \tag{186}\] For bounded \(R\), this follows instead from \(\|D\|\le\sum_l|g_l|\).

The local-link description of \(E\) in Lemma 20 gives \[ \|[D,[E,D]]\|\le C\sum_l|g_{l+1}-g_l|^2\le C_A/R. \tag{187}\] Indeed the diagonal part of a link commutes with \(D\), and its spin exchange preserves \(Z_l+Z_{l+1}\); only the coefficient difference \(g_l-g_{l+1}\) enters each commutator. These identities initially hold on the local analytic core in \(\mathcal H_{\mathrm{spin}}\), which is a core for the restricted \(E\). On that core, \([E,D]\) is a finite sum of bounded local operators. To extend it, approximate a vector in \(\mathop{\mathrm{Dom}}(E)\subset\mathcal H_{\mathrm{spin}}\) in the graph norm by local core vectors and use \(EDu=DEu+[E,D]u\) there. Boundedness of \(D\) and of this commutator, followed by closedness of \(E\), shows that \(D\) preserves the restricted domain and that the identity extends to it. The same holds for every fixed local power of \(D\), justifying the following energy-projection and vacuum calculations on this subspace.

Set \(m_* =\|DB_1\|\), which is finite since \(D\) is bounded. For a unit vector \(u\in B_1\mathcal H_{\mathrm{spin}}\), expanding the double commutator yields \[\langle Du,EDu\rangle =\tfrac12\langle u,[D,[E,D]]u\rangle +\operatorname{Re}\langle Du,DEu\rangle \le \frac{C_A+m_*^2}{R}.\] Here \(Eu\in B_1\mathcal H_{\mathrm{spin}}\) and \(\|Eu\|\le R^{-1}\). The component of \(Du\) above energy \(2/R\) thus has squared norm at most \((C_A+m_*^2)/2\). Its component below \(2/R\) is bounded by (186), with \(M=2\). Taking a supremum gives \[m_*^2\le C_A+\tfrac12m_*^2, \qquad\text{and therefore}\qquad \|DB_1\|=\|B_1D\|\le C_A.\]

We next bound powers of \(D\) on the vacuum. Since \(E\Omega=0\), \[2\langle D^k\Omega,ED^k\Omega\rangle =\langle\Omega,[D^k,[E,D^k]]\Omega\rangle.\] In the site-diagonal representation, let \(D^{\mathrm{sw},l}\) be the value of \(D\) after exchanging the spins at \(l,l+1\). An exchange term in this double commutator has, up to a bounded link coefficient and sign, the multiplier \(\bigl(D^k-(D^{\mathrm{sw},l})^k\bigr)^2\). This nonnegative multiplier commutes with the exchange. Furthermore, \[|D-D^{\mathrm{sw},l}|\le2|g_l-g_{l+1}|, \qquad \bigl|D^k-(D^{\mathrm{sw},l})^k\bigr|^2 \le C_{A,k}|g_l-g_{l+1}|^2(1+|D|^{2k-2}).\] For the second inequality use the factorization of \(a^k-b^k\) and \(|a-b|\le2A\). Taking the absolute value of an exchange expectation is bounded by the expectation of its nonnegative multiplier, since the exchange is a norm-one involution commuting with that multiplier. Summing the links proves \[ \langle D^k\Omega,ED^k\Omega\rangle \le \frac{C_{A,k}}R \bigl(1+\|D^{k-1}\Omega\|^2\bigr). \tag{188}\] The low-energy component satisfies \(\|B_1D^k\Omega\|\le C_A\|D^{k-1}\Omega\|\); on its complement, positivity of \(E-R^{-1}\) bounds the squared norm by \(R\langle D^k\Omega,ED^k\Omega\rangle\). Induction in \(k\) now bounds every \(\|D^k\Omega\|\). These are the even ordinary moments of \(D\); interpolation gives all finite orders. Axial symmetry gives the other orientation. ◻

Proof. Suppose first \(r/\delta\ge1\), and put \(R=r/\delta\). Integrate \(f\) on mesh cells, giving masses \(a_{jl}\) of size \(O(R^{-2})\), with \(O(R)\) possible indices in each direction. Let \(m_j=\sum_l a_{jl}\), and choose a mass-one discrete profile \((p_l)\), of width \(O(R)\), bounded by \(C/R\). The masses \(a_{jl}-m_jp_l\) are neutral in each column. Their discrete primitives are bounded by \(C/R\), with consecutive differences \(O(R^{-2})\). Summation by parts and Lemma 35, applied after multiplication by \(R\), bound the \(L^q\) norm of each column contribution by \(C_q/R\). Their sum is bounded by Minkowski’s inequality. For the remaining masses \(m_jp_l\), note that \(\sum_jm_j=0\), \(|m_j|\le C/R\); sum by parts in the other axis, then average against \(p_l\). This gives the same bound. If \(r<\delta\), the support meets only a bounded number of neighboring faces. Neutrality, bounded total variation, and unit height jumps give a deterministic bound. Finally apply Hölder’s inequality to products, using the required higher moments. No disjointness is used. ◻

Proof. First assume that \(\int\phi(x,y)\,dy=0\) for every \(x\). For fixed continuum time \(x\), put \(F(x,y)=\int_{-\infty}^y\phi(x,t)\,dt\). Discrete summation by parts writes the line pairing as \[\int h_\delta(x,y)\phi(x,y)\,dy =-\sum_{l\in\mathbb Z}F(x,(l+1)\delta) \bigl(h(\lfloor x/\delta\rfloor,l+1) -h(\lfloor x/\delta\rfloor,l)\bigr).\] The primitive is compactly supported, bounded, and changes by \(O(\delta)\) at consecutive sites. Lemma 35 therefore bounds this pairing in every \(L^q\), uniformly in \(x\) and \(\delta\). The same remains true for any finite product of line pairings.

Replace the forward height differences in this expression by the centered differences of Proposition 34, shifting the coefficients accordingly. The coefficient error has size \(O(\delta)\) and consecutive differences \(O(\delta^2)\), on \(O(\delta^{-1})\) sites. Dividing this error by \(\delta\) puts it under Lemma 35; the resulting \(L^q\) error is \(O(\delta)\), uniformly in the line coordinate.

Expand a joint moment as an integral over its time coordinates. For distinct fixed times, Proposition 34 applies to the transverse Riemann sums, locally uniformly on their compact supports and including the rounding of faces. Their limit is Wick’s formula for transverse derivatives of \(K_\kappa\). Integration by parts against the compact primitives gives the undifferentiated test densities and the kernel (183). The line moment bounds dominate the integrand uniformly. Equal-time walls have Lebesgue measure zero, so dominated convergence completes the time integrations. The resulting products of logarithmic kernels are integrable: in each Wick pairing the two-point variables are disjoint from those of every other pair, and the logarithm is locally integrable in two dimensions. Gaussian moments determine the law. Axial symmetry gives the same result for horizontally line-neutral tests.

For an arbitrary smooth neutral \(\phi\), let \[q(x)=\int\phi(x,y)\,dy,\qquad p_R(y)=R^{-1}p(y/R),\qquad \phi_R(x,y)=\phi(x,y)-q(x)p_R(y),\] where \(p\) is a fixed smooth compact probability density. Then \(\phi_R\) is vertically line-neutral, while \(q p_R\) is horizontally line-neutral. For fixed \(R\), the already proved result gives \[\lim_{\delta\downarrow0}\mathbb E|\langle h_\delta,q p_R\rangle|^2 =\iint K_\kappa(z,z')q(x)p_R(y)q(x')p_R(y')\,dz\,dz'.\] This tends to zero as \(R\to\infty\). To see this, set \(y=Rt\), \(y'=Rt'\). The additive \(-2\kappa\log R\) contributes zero because \(\int q=0\). The remaining logarithm is \(\log\sqrt{((x-x')/R)^2+(t-t')^2}\), which converges almost everywhere to \(\log|t-t'|\), independently of \(x,x'\). On the fixed supports it is bounded in absolute value by \(C+\log_+(1/|t-t'|)\). Dominated convergence and neutrality of \(q\) prove the assertion. The identical approximation holds for \(\mathcal H_\kappa\). Thus first taking the mesh limit at fixed \(R\), and only then taking \(R\to\infty\), proves joint convergence in law.

For any fixed compact support, Corollary 36 bounds all higher moments of each fixed smooth pairing. Uniform integrability therefore upgrades convergence in law to all joint moments. Finally, small changes in a smooth test on a fixed support give small \(L^q\) changes by the same corollary, after dividing by the sup norm of the change. A finite covering of a compact family proves the asserted uniformity. ◻

Proof. If empty time slabs separate the selected factor from all the others, the intervening rows damp its short height paths. General endpoint geometry need not have such a slab in either axial ordering. We will create one by translating the groups lying to the left and right of the selected factor’s horizontal cluster. Those groups remain separated from the cluster by horizontal gaps; bounded analytic continuation across these gaps then returns them, one at a time, to their original positions.

The estimate with empty time slabs. First suppose that the selected support has an empty axial slab on each side, of width at least \(c_2d_0\), separating it from all other factors. Insert time cuts in these two slabs. The vectors defined by the exterior factors have bounded Hilbert norms. Here is the connection with ordinary moments. Read all heights on one side of a cut relative to a common face, along a common seam and spatial paths contained on that side. Spatial steps use equal-time diagonal spins; each time-seam step uses a time spin, with repeated occurrences allowed when several paths share the step. In a finite homogeneous system, reflection across the cut transposes the entire product and complex-conjugates its scalar coefficients. The crossing identity reverses the time-step spins and preserves the spatial-step spins, exactly as for the reflected height differences. The squared norm is therefore the ordinary expectation of the exterior product times its conjugate reflected product. This finite identity passes to the plane by Proposition 17. Hölder’s inequality and the ordinary moments of the individual factors bound that expectation. The factors on a side may overlap; Hölder’s inequality requires no separation among them.

Inside the slab, use neutrality to express the selected factor as an average of differences from a fixed reference face in its support. Each difference has a path of \(O(r/\delta)\) steps, with a bounded number of straight segments. Choose the direction of the difference so that time runs forward; spatial steps may have either sign. There remain \(m\ge c_3d_0/\delta\) unmarked time rows on both sides. Pair consecutive steps along each straight segment. The paired current identity and Lemma 19 give the sandwiched norm bounds \[ \begin{aligned} \|P(1)^m(\text{two consecutive step sum})P(1)^m\| &\le C m^{-1},\\ \|P(1)^m(\text{one step})P(1)^m\| &\le C m^{-\theta_0} \end{aligned} \tag{191}\] for some \(0<\theta_0\le1\). To verify the passage from band bounds to these inequalities, split both sides into the bands \(H_0\le m^{-1}\) and \(2^{j-1}/m<H_0\le2^j/m\). A block between two such bands has current bound a power of the larger cutoff, while the time rows contribute \(\exp(-c2^j-c2^{j'})\). The resulting double geometric-exponential sum is \(O(m^{-1})\) for a paired current. For a single current use the equal-sign \(O(u)\) and opposite-sign \(O(u^\theta)\) estimates, with \(\theta_0=\min(1,\theta)\). Bands above a fixed cutoff are exponentially damped and the uncompressed row norms are bounded. Unit shifts commute with the damping and do not enlarge these norms.

There are \(O(r/\delta)\) pairs and only a bounded number of unpaired steps. The bounded total variation of the selected factor and the exterior norm bounds give \[\left|\mathbb E\prod_iA_i\right| \le C\left(\frac r{d_0} +\left(\frac\delta{d_0}\right)^{\theta_0}\right) \le C(r/d_0)^{\tau_0},\qquad \tau_0=\min(1,\theta_0).\] This includes a terminal residual: its comparison paths have bounded length and the same damping margins. Rounding the cuts changes their widths by only a bounded number of sites.

Creating a slab by two translations. Cluster the centers by their first-coordinate projections: consecutive projections belong to the same cluster if their gap is at most \(a d_0/k\), for a sufficiently small fixed \(a\). Every cluster has width at most \(a d_0\). Choose \(a_k\) much smaller than \(a/k\), so that the supports preserve a gap of size \(c_kd_0\) at each cluster boundary. In the cluster containing the selected center, every other center has second-coordinate distance at least \(c_kd_0\) from it. Indeed its first-coordinate distance is at most \(a d_0\), whereas its Euclidean distance has the lower bound in the statement.

Translate the clusters strictly to the left as one group vertically, and do the same independently for the clusters strictly to the right. For each group there is an interval of translation amounts, of length at least \(c_kd_0\) and at distance at most \(C_kd_0\) from zero, such that every translation in the interval leaves a vertical gap of size \(c_kd_0\) around the selected support. To construct one, exclude the intervals of translations putting any of its at most \(k\) centers within a fixed small multiple of \(d_0\) of the selected height. Their total length is bounded by \(C_kd_0\); choose a larger bounded translation range and take a complementary interval of the asserted size. The choices for the two groups are independent. Restrict to translations by even numbers of lattice sites. Lattice stationarity preserves each individual \(L^{2k}\) bound under these translations, so Hölder’s inequality still controls the exterior products. Throughout the resulting rectangle of choices the selected factor has a free slab in the northward time ordering, so the preceding estimate applies.

Returning the two groups. The first-coordinate gaps have remained open throughout these vertical translations. They supply the damping needed to transfer the estimate back to zero translation. Write \[K=\operatorname{Arg}(S^2),\qquad -\pi\le K\le\pi.\] This is bounded spectral functional calculus. The low-band estimate \(S=\pm1+O(H_0)\) gives \(|K|\le CH_0\) for small \(H_0\); on the complement the same inequality follows by increasing \(C\), since \(|K|\le\pi\). Thus \[ |K|\le CH_0, \qquad \|P(1)^N e^{izK}\|\le1 \quad\text{if}\quad |\operatorname{Im}z|\le N/C. \tag{192}\] For integer \(m\), \(e^{imK}=S^{2m}\). Across a first-coordinate gap containing \(N\ge c_kd_0/\delta\) unmarked rows, translating an entire exterior group by \(2m\) sites therefore inserts \(P(1)^NS^{\pm2m}\) between fixed exterior vectors. This identity follows by conjugating every site and seam insertion on that side by the site shift, which fixes the vacuum. Reflection and the ordinary moment estimates bound both exterior vectors uniformly. Equation (192) extends the pairing to a bounded holomorphic function of \(z/(d_0/\delta)\) on a strip of fixed positive width. If one of the exterior groups is absent there is no restoration step for that group.

For completeness, the elementary analytic estimate used here is the following. A holomorphic function bounded by one on a fixed-width strip, whose absolute value is at most \(a\le1\) on a fixed-length real interval within a fixed bounded distance of zero, has absolute value at zero at most \(C a^\vartheta\), where \(\vartheta>0\) depends only on those geometric constants. Take a half-disc based on a shorter part of the interval. The two-constants theorem bounds the function by \(a^{\omega}\) on an interior disc, since the harmonic measure of the base there is bounded below by \(\omega>0\). Cover the remaining path to zero by finitely many overlapping discs inside the strip. The three-circles inequality transfers a positive power bound at each step. The number and radii of these discs depend only on the fixed strip, interval, and distance bounds.

Our free-slab estimate initially holds only at the allowed integer values of \(m\). Cauchy’s estimate on a slightly smaller strip bounds the derivative in the normalized real coordinate. The gap between successive allowed values is \(O(\delta/d_0)\), so throughout a shorter real interval the bound becomes \[C(r/d_0)^{\tau_0}+C\delta/d_0 \le C'(r/d_0)^{\tau_0},\] using \(r\ge c_0\delta\) and \(\tau_0\le1\). First restore the left group while keeping the right translation at any one of its allowed values. The analytic estimate gives the same bound with a possibly smaller positive exponent, uniformly over those right translations. Then restore the right group. The two reductions of the exponent still leave a fixed \(\tau>0\). This proves (190) in the original geometry. ◻

Proof. The first moment is zero by arrow-reversal symmetry. For \(k\ge2\), fix such a compact set of tuples and let \(d_0>0\) be a lower bound on all cross-pair endpoint distances. Endpoints within a single pair may coincide. Choose \(r_0=a_kd_0\) as in Lemma 38, and put \(r_j=2^{-j}r_0\). For each endpoint, telescope its bump averages down to the last scale \(r_J\asymp\delta\), and include the residual from that average to the specified face value. This is a finite exact identity, including for points on grid lines under the fixed floor convention.

Expand the product of increments. A factor is either a coarse increment at scale \(r_0\), or one of the two endpoint blocks at a dyadic scale, including the terminal residual. All individual blocks have uniformly bounded moments. The coarse increments also do: join their endpoints by a bounded-length chain of overlapping scale-\(r_0\) bumps and telescope, applying Corollary 36 to each difference. The number of links is bounded by the fixed compact size and separation data.

In a summand containing an endpoint block, select one of minimum scale \(r\). Decompose every other coarse increment by a bump chain avoiding a disc of radius \(d_0/3\) about the selected endpoint. Such a chain exists with bounded length because both endpoints of that other pair are at least \(d_0\) away; a polygonal detour around the disc has length bounded in terms of the same fixed data. After subdividing into steps of size \(O(r_0)\), every factor other than the selected one has center at distance at least \(c d_0\) from it. There are still at most \(k\) factors in each resulting product; the number of resulting products is bounded independently of the mesh and dyadic levels. Apply Lemma 38 to each of them.

At fixed deepest dyadic index \(j\), there are at most \(C_k(1+j)^{k-1}\) choices of the other levels and endpoint labels. Consequently the error from keeping only scales coarser than \(r_{J_0}\) is bounded by \[ C\sum_{j\ge J_0}(1+j)^{k-1}2^{-\tau j}, \tag{193}\] uniformly once the mesh is fine enough to reach that level. This tends to zero as \(J_0\to\infty\). Notice that the terminal residual obeys the same bound; dropping it before applying the estimate would not justify point evaluation.

For fixed \(J_0\), Proposition 37 gives the joint moments of the smoothed increments, uniformly in their centers. The corresponding Gaussian Wick expressions converge uniformly as the smoothing radius tends to zero: each paired kernel involves endpoints from two different increments, so its arguments stay a positive distance apart. Combine this with (193). If the two endpoints within a pair coincide, that increment and its smoothed versions are identically zero; no additional exclusion is needed. ◻

Proof. The total variation of \(\mu\otimes\mu\) is \(|\mu|\otimes|\mu|\). Ordinary integrability is integrability with respect to this variation. On a fixed compact support, the negative part of \(-\log|x-y|\) is bounded, which proves (194). An atom would give positive product mass to the logarithmic diagonal singularity, so there are no atoms.

For the mixed assertion take dyadic squares of side \(2^{-j}\) in a box containing both supports. Up to an additive bounded kernel, \(\log_+(1/|x-y|)\) is bounded above by a constant times \(\sum_{j\ge0}\mathbf 1_{\{|x-y|\le C2^{-j}\}}\). Write \(a=|\mu|\), \(b=|\nu|\). At a fixed level the mixed mass of these nearby pairs is at most \(\sum_Q\sum_{Q'\sim Q}a(Q)b(Q')\), where each square has a bounded number of neighbors. Cauchy–Schwarz bounds this by \[C\left(\sum_Qa(Q)^2\right)^{1/2} \left(\sum_Qb(Q)^2\right)^{1/2}.\] The sums of \(\sum_Qa(Q)^2\) and \(\sum_Qb(Q)^2\) over all levels are finite by their self-integrability: pairs in one square are within \(\sqrt2\,2^{-j}\), and summing their indicators is bounded by \(C(1+\log_+(1/|x-y|))\). A second Cauchy–Schwarz inequality over \(j\) proves the claim. ◻

Proof. Let \(V(a)=\mathbb E(h(a)-h(0))^2\) for lattice-face displacement \(a\in\mathbb Z^2\), and put \(V(0)=0\). Proposition 22 and axial symmetry imply \[V(a)=4\kappa\log(1\vee|a|)+O(1),\] with a uniform bounded error over all lattice displacements. For distinct continuum points \(x,y\) in a fixed compact set, write \[V_\delta(x,y)= V\bigl(\lfloor x/\delta\rfloor-\lfloor y/\delta\rfloor\bigr).\] For \(0<\delta\le1\) this gives \[ \left|V_\delta(x,y)-4\kappa\log\delta^{-1}\right| \le C\bigl(1+|\log|x-y||\bigr). \tag{195}\] Indeed when \(|x-y|\) exceeds a fixed multiple of \(\delta\), the lattice separation is comparable to \(|x-y|/\delta\). In the other case its size is bounded, and \(\log\delta^{-1}\le C+|\log|x-y||\). This latter case includes distinct points rounded to the same face.

For any four points, stationarity of increments and zero mean give \[\begin{align*} &\mathbb E[(h_\delta(x')-h_\delta(x)) (h_\delta(y')-h_\delta(y))]\tag{196}\\ &\qquad=\tfrac12\bigl( V_\delta(x',y)+V_\delta(x,y') -V_\delta(x',y')-V_\delta(x,y)\bigr). \end{align*}\] The four copies of \(4\kappa\log\delta^{-1}\) cancel exactly. Equation (195) therefore supplies a majorant by a constant and four continuum logarithmic singularities.

For a neutral \(\mu\ne0\), its Jordan parts have the same mass \(M>0\). The measure \(M^{-1}\mu^+(dx')\mu^-(dx)\) represents its pairing as an integral of \(h_\delta(x')-h_\delta(x)\). Use this representation for two pairings. By Lemma 40, the four-logarithm majorant is integrable against the resulting product measure. Geometric coincidences of endpoints belonging to different pairings have measure zero because the Jordan parts are nonatomic. Distinct endpoints rounded to a common lattice face remain covered by the same majorant. Proposition 39 for two increments and dominated convergence thus give, for every such \(\mu,\nu\), \[ \lim_{\delta\downarrow0} \mathbb E[\langle h_\delta,\mu\rangle \langle h_\delta,\nu\rangle] =\iint K_\kappa(x,y)\,\mu(dx)\nu(dy). \tag{197}\] The zero measure causes no exception.

It remains to identify laws beyond second moments. Let \(\mu_\epsilon=\mu*\varrho_\epsilon\), where the mollifier is smooth, nonnegative and compactly supported. These measures are smooth and neutral. Mollification converges in logarithmic energy: \[ \iint K_\kappa(x,y) (\mu-\mu_\epsilon)(dx)(\mu-\mu_\epsilon)(dy) \longrightarrow0. \tag{198}\] To check this for the present ordinary integral, the logarithm convolved in one or both variables converges pointwise off the diagonal. On the fixed enlarged compact support its absolute value is bounded by \(C(1+|\log|x-y||)\), uniformly in small \(\epsilon\). For \(|x-y|\ge C\epsilon\) this follows by comparison of distances; for \(|x-y|<C\epsilon\), scaling the bounded compact mollifier gives \(C+|\log\epsilon|\le C'(1+|\log|x-y||)\). Lemma 40 permits dominated convergence in each of the four energy terms, proving (198).

Apply (197) to \(\mu-\mu_\epsilon\). It gives \[\lim_{\epsilon\downarrow0}\limsup_{\delta\downarrow0} \|\langle h_\delta,\mu-\mu_\epsilon\rangle\|_{L^2}=0.\] At fixed \(\epsilon\), Proposition 37 identifies the Gaussian limit. The same energy estimate approximates the Gaussian pairing. Applying this argument to a finite family, or to each of its linear combinations, proves the asserted joint law and cross covariance. ◻

Proof. Let \(m=\int\chi\psi_{b,\ell}\), so \(|m|\le C\). Subtract \(m\) times a smooth mass-one bump of scale \(\ell\) at \(b\). The difference obeys Corollary 36. Next telescope that bump through the scales \(\ell,2\ell,\ldots,1\), and finally compare the scale-one bump with \(\rho\). Each difference is neutral, has bounded total variation, and is supported and bounded at its stated scale. There are \(O(1+|\log\ell|)\) differences. The same argument works below the mesh scale because the final clause of Corollary 36 has no lower bound on \(\ell/\delta\). ◻

Proof. Choose a compactly supported \(\chi\) equal to one on a box containing \(\overline U\) and \(\mathop{\mathrm{supp}}\rho\). Use compactly supported wavelets of sufficient finite regularity, including their coarse-scale scaling functions (Daubechies 1988, sec. 4.C). We use the full-space Besov coefficient characterization in (Hairer and Labbé 2017, Proposition 2.3). Normalize the scale-\(2^{-j}\) test functions as densities of size \(O(2^{2j})\), and denote their averages of \(X_\delta^\chi\) by \(a_{j,k}\). There are \(O(2^{2j})\) relevant indices \(k\). For \(\alpha<\beta<0\), choose a moment order \(b\) with \(2+b\beta<0\). Lemma 42 gives \[ \mathbb E\sum_{j\ge0}\sum_k|2^{j\beta}a_{j,k}|^b \le C_b\sum_{j\ge0}2^{(2+b\beta)j}(1+j)^b<\infty. \tag{200}\] The coarse block has the same bound. This bounds the stronger \(B^\beta_{\infty,\infty}\) norm in probability. On a fixed compact support its bounded sets have uniformly vanishing tails in \(B^\alpha_{\infty,\infty}\), since \[\sup_{j>J,k}2^{j\alpha}|a_{j,k}| \le 2^{(\alpha-\beta)J} \sup_{j,k}2^{j\beta}|a_{j,k}|.\] For finitely many levels only finitely many coefficients remain. This proves compactness by finite-dimensional truncation, and also puts the fields and limits in the separable smooth closure.

The same gain proves the other asserted norm compactness. Indeed a compactly supported distribution bounded in \(B^\beta_{\infty,\infty}\) has scale blocks satisfying \(\|\Delta_j X\|_{L^p}\le C2^{-j\beta}\) for every finite \(p\). Thus \[\sum_{j>J}2^{j\alpha}\|\Delta_jX\|_{L^p} \le C\sum_{j>J}2^{j(\alpha-\beta)}\longrightarrow0.\] This controls every \(\ell^q\) Besov block norm, including \(q=\infty\). It also controls the negative Bessel-potential norm: on a scale block, \((1-\Delta)^{\alpha/2}\) has an \(L^1\)-bounded convolution kernel after extracting \(2^{j\alpha}\), so summing these bounds works for all \(p\ge1\). The same summable block bound controls the usual full-space negative Slobodeckij realization. Passing to its restriction uses the defining quotient norm. Restricting a global extension to \(U\) is continuous by definition; no extension operator from \(U\) is being invoked. Proposition 37 identifies every subsequential limit, and hence proves the full mesh convergence.

For clarity, retain also the source’s printed outward convention. Let \(Q_0=(-1,1)^2\), and let \(\mathcal T_1(Q_0)\) consist of compactly supported neutral densities whose zero extension is one-Lipschitz. For smooth functions its seminorm is \[ \|f\|_{\mathrm{out},\alpha,U} =\sup_{\substack{0<\varepsilon\le1,\ \varphi\in\mathcal T_1(Q_0)\\ \mathop{\mathrm{supp}}\varphi\subset U/\varepsilon}} \varepsilon^{-\alpha} \left|\int f(x/\varepsilon)\varphi(x)\,dx\right|. \tag{201}\] The absolute value is immaterial for the symmetric test family. Constants vanish in this seminorm. Its completion is understood modulo its kernel, as in the source.

Write \(\varepsilon=2^{-j}a\), \(1/2\le a\le1\). The exact mesh identity, including the floor convention, is \[h_\delta(x/\varepsilon)=h_{2^{-j}\delta}(x/a).\] After the change of variables \(x=ay\), the test becomes \(a^2\varphi(ay)\), supported in \(2Q_0\), neutral, with a uniformly bounded Lipschitz norm. The condition involving \(U\) only reduces this family. For any fixed \(-1<\gamma<0\), local \(\mathcal C^\gamma\) control bounds all these pairings. One can see this directly from a wavelet expansion: cancellation of fine wavelets against a Lipschitz test gains \(2^{-j}\), and the resulting series converges because \(1+\gamma>0\). The preceding moment argument, with a still stronger negative exponent and arbitrarily high moment order, therefore gives random variables \(Z_{j,\delta}\) such that \[\sup_\delta\|Z_{j,\delta}\|_{L^b}\le C_b, \qquad \sup_{2^{-j-1}\le\varepsilon\le2^{-j}}(\text{quantity in } \eqref{eq:prob-outward-norm}) \le C2^{j\alpha}Z_{j,\delta}.\] A local pin may be inserted in the rescaled field since every test is neutral. In particular the supremum over \(j\ge J\) tends to zero in \(L^b\), uniformly in \(\delta\), by the summability of \(\sum_{j\ge J}2^{j\alpha}\). On finitely many bands, the tested original coordinates lie in one bounded box. Their test norms are bounded by constants depending on the number of bands, so the stronger local compactness already proved applies.

These estimates also give the required smooth completion, rather than only a bound on a larger space of functionals. Take a smooth compact cutoff \(\chi_0\) equal to one on a sufficiently large fixed box and a compact smooth mollifier \(\theta_\eta\), and use \[f_{\delta,J,\eta}(x) =\chi_0(2^{-J}x)(X_\delta*\theta_\eta)(x).\] Convolution preserves the local negative Hölder bound, uniformly for \(0<\eta\le1\), since its rescaled radius on band \(j\) is at most \(2^{-j}\). For the outward bands \(j\ge J\), the cutoff becomes a bump of scale at most \(2^{J-j}\) in the rescaled coordinates. The local \(\mathcal C^\gamma\) test bound gives its contribution the additional factor \(2^{(J-j)(2+\gamma)}\le1\); comparison of the fixed pin with the rescaled pin costs at most \(C(1+j)\) by the same dyadic telescoping as in Lemma 42. Thus the cutoff approximations have tails bounded by a constant times \(\sum_{j\ge J}(1+j)2^{j\alpha}\), uniformly in probability and in the mollification parameter. On finitely many bands send \(\eta\downarrow0\); then send \(J\to\infty\). These smooth approximations and the vanishing tail prove membership and tightness in the completion of (201). Its limits agree with the smooth-test limit on each finite band. This proves the literal outward assertion too. ◻

Proof. We first refine the comparison of line weights. Suppose nonnegative weights \(a_l\), supported on \(O(R)\) consecutive sites, satisfy \[\sum_l a_l=m>0,\qquad a_l\le M/R,\qquad R\ge1.\] Let \(b_l\) be a mass-one profile bounded by \(C/R\), and assume that the union of the supports of \(a\) and \(b\) lies in one interval of length \(CR\). We claim \[ \left\|\sum_l(a_l-mb_l)h_l\right\|_{L^b} \le C_bm\bigl(1+\log_+(M/m)\bigr). \tag{203}\] Increasing a fixed constant in \(M\) if needed, assume \(M\ge m\). For dyadic integer \(v\ge1\), let \(A_va\) replace the weights in each consecutive block of \(v\) sites by their block average. The difference \(d=A_va-A_{2v}a\) is neutral in every \(2v\)-block. For its normalized primitive \(g_l=m^{-1}\sum_{i\le l}d_i\), this gives \[\sum_l|g_l|\le Cv,\qquad \sum_l|g_{l+1}-g_l|\le C,\qquad \sum_l|g_{l+1}-g_l|^2\le C/v.\] The first bound sums a block length times its variation mass. For the last, use \(\|d\|_\infty\le Cm/v\) and \(\|d\|_1\le2m\). Lemma 35 and summation by parts give cost at most \(C_bm\) for each flattening step. This use of the lemma explains why its statement retained the three summability conditions, without a support-diameter hypothesis.

If \(\min(R,Rm/M)>1\), choose a dyadic \(v_0\) comparable from below to that number; otherwise put \(v_0=1\). When \(v_0>1\), one can start directly with \(a-A_{v_0}a\). Its primitive has the same first two bounds, and \[m^{-2}\|a-A_{v_0}a\|_2^2 \le C m^{-2}\|a\|_2^2 \le CM/(Rm)\le C/v_0.\] When \(v_0=1\), this initial difference vanishes. Continue flattening until \(v\asymp R\), where comparison with \(mb\) has bounded cost by the original width-\(R\) case of the lemma. The number of steps is \(O(1+\log_+(M/m))\), proving (203).

Now rescale the square \(Q\) to a unit square. For any probability density \(f\le M\) on it, set \(m(x)=\int f(x,y)\,dy\). Apply (203) to its mesh masses on each vertical line, comparing them with \(m(x)\) times one bounded profile. Minkowski’s inequality bounds the total cost by \[C_b\int m(x)\bigl(1+\log(M/m(x))\bigr)\,dx \le C_b(1+\log M).\] The last inequality is Jensen’s inequality for the concave function \(u\mapsto u\log(M/u)\), with \(\int m=1\) on an interval of length one. Compare the remaining horizontal marginal with a bounded profile by the same line estimate. We have therefore compared the original density with a fixed scale-\(\ell\) product smoothing, at cost \(C_b(1+\log M)\). Telescoping that smoothing to scale one and then to the pin costs \(C_b(1+|\log\ell|)\), as in Lemma 42. If \(\ell<\delta\), all comparisons inside a comparable square use only a bounded number of faces and have a deterministic bound; the remaining pin comparison is unchanged. Taking \(f=\ell^2|A|^{-1}\mathbf 1_A\) in rescaled coordinates gives \(M=t^{-1}\) and proves (202). ◻

Proof. Take a fixed square containing \(U\), subdivided dyadically at side lengths \(\ell_j\asymp2^{-j}\). For a square \(Q\) at level \(j\), put \[U_Q=Q\cap U,\qquad t_Q=|U_Q|/|Q|,\qquad X_{\delta,Q}=|U_Q|^{-1}\int_{U_Q}X_\delta.\] Ignore cells of zero area. Let \[Z_{\delta,j}=\sum_{Q\text{ at level }j} X_{\delta,Q}\mathbf 1_{U_Q},\qquad F_{\delta,j}(\phi)=\int_U Z_{\delta,j}\phi,\] and let \(P_j\phi\) denote cell averaging of \(\phi\) on the same partition. Consistency of averaging implies \[ (F_{\delta,j+1}-F_{\delta,j})(\phi) =\int_U Z_{\delta,j+1}(\phi-P_j\phi). \tag{204}\]

Suppose first \(p>1\), so \(q<\infty\), and choose \(0<\zeta<sq\). Call a parent \(P\) of side \(\ell=\ell_j\) thick when \(t_P\ge\ell^\zeta\). Jensen’s inequality gives \[\begin{align*} \int_{U_P}|\phi-\phi_P|^q &\le |U_P|^{-1} \iint_{U_P\times U_P}|\phi(x)-\phi(y)|^q\,dx\,dy\\ &\le C\ell^{sq}t_P^{-1} \iint_{U_P\times U_P} \frac{|\phi(x)-\phi(y)|^q}{|x-y|^{2+sq}}\,dx\,dy. \end{align*}\] Summing over thick parents yields \[ \|\phi-P_j\phi\|_{L^q(\mathrm{thick})} \le C\ell^{s-\zeta/q}[\phi]_{W^{s,q}(U)}. \tag{205}\] On all cells, averaging is an \(L^q\) contraction, so the same difference has \(L^q\) norm at most \(2\|\phi\|_{L^q}\).

Lemma 44, with \(b=p\), implies \[\mathbb E\|Z_{\delta,j+1}\|_{L^p(U)}^p\le C_p(1+j)^p.\] For this inequality sum \(|Q|t_Q(1+j+\log(1/t_Q))^p\) over children, and use \(t(1+\log(1/t))^p\le C_p\) for \(0<t\le1\). A child of a thin parent has \(t_Q\le4\ell^\zeta\). The sharper elementary inequality \[t(A+\log(1/t))^p\le C_p A^p t^{1/2},\qquad A\ge1,\] therefore gives \[ \mathbb E\|Z_{\delta,j+1}\mathbf 1_{\mathrm{thin}}\|_{L^p(U)}^p \le C_p(1+j)^p\ell^{\zeta/2}. \tag{206}\] The sum of the areas \(|Q|\) is bounded by the area of the surrounding square, independently of the geometry of \(U\). Equations (204)–(206) and Hölder’s inequality now give \[ \mathbb E\|F_{\delta,j+1}-F_{\delta,j}\|_{(W^{s,q}(U))^*} \le C(1+j) \left(\ell_j^{s-\zeta/q}+\ell_j^{\zeta/(2p)}\right). \tag{207}\] Both exponents are positive.

At \(p=1\), \(q=\infty\), the intrinsic Hölder seminorm gives directly \[\|\phi-P_j\phi\|_\infty \le C\ell_j^s[\phi]_{C^s(U)}.\] There is no need to distinguish thin parents in this case. The first moment version of the cell estimate gives \[ \mathbb E\|F_{\delta,j+1}-F_{\delta,j}\|_{(W^{s,\infty}(U))^*} \le C(1+j)\ell_j^s. \tag{208}\] Thus at every finite \(p\) the right-hand side, denoted by \(a_j\), is summable over \(j\), uniformly in \(\delta\).

Let \(\mathcal E_0\) be the closed linear span in \(\mathcal E=(W^{s,q}(U))^*\) of the countable collection of cell indicator pairing functionals. It is a separable Banach space. For each fixed mesh, \(X_\delta\) is a bounded function on the bounded set \(U\), and its conditional cell averages converge to it in \(L^p(U)\). Since \(L^p(U)\) maps continuously into \(\mathcal E\), they also converge in \(\mathcal E_0\). Summing the preceding estimates proves \[ \sup_\delta\mathbb E\|X_\delta-F_{\delta,J}\|_{\mathcal E} \le\sum_{j\ge J}a_j\longrightarrow0. \tag{209}\]

Every coefficient \(X_{\delta,Q}\) is the pairing with the neutral measure \[\mu_Q=|U_Q|^{-1}\mathbf 1_{U_Q}\,dx-\rho\,dx.\] This bounded compact density has ordinary finite logarithmic energy, regardless of the shape or area of \(U_Q\). Proposition 41 therefore gives joint convergence of the coefficients at each finite level. Couple their Gaussian limits using the countable family of Gaussian pairings with \(\mu_Q\), and denote the corresponding finite-level elements of \(\mathcal E_0\) by \(F_J\). Finite-dimensional convergence and the lower semicontinuity of the norm transfer the bound \(\mathbb E\|F_{j+1}-F_j\|_{\mathcal E}\le a_j\). The sum of these norms is almost surely finite and integrable, so \(F_J\) converges almost surely and in \(L^1(\mathcal E_0)\) to an \(\mathcal E_0\)-valued random variable \(F\). Finite-level convergence and (209) then prove \(X_\delta\Rightarrow F\) in \(\mathcal E_0\), including norm tightness.

We finally identify the limit on the whole intrinsic test space, not only on interior smooth tests. For \(\phi\in W^{s,q}(U)\), the measure \[\mu_\phi=\phi\mathbf 1_U\,dx-\left(\int_U\phi\right)\rho\,dx\] has ordinary finite logarithmic energy. Indeed \(q>1\) or \(q=\infty\), and on a fixed compact set Hölder’s inequality for the logarithmic kernel gives \[\iint |\log|x-y||\,|f(x)f(y)|\,dx\,dy \le C_q\|f\|_{L^1}\|f\|_{L^q}.\] For finite \(q\), \(P_J\phi\to\phi\) in \(L^q(U)\); for \(q=\infty\) the intrinsic Hölder bound gives uniform essential convergence. The associated neutral measures consequently converge in logarithmic energy. Hence \(F_J(\phi)\) converges to the canonical pinned Gaussian pairing with \(\mu_\phi\). This identifies \(F\) on every intrinsic test and rules out an unidentified boundary functional in the larger dual. The proof is complete. ◻

Proof. We obtain the required moment bounds directly from (Duminil-Copin, Kozlowski, Lammers, et al. 2026, Theorem 4.5 and equations (28)–(30)), which applies at \(c=2\). For \(2n\) increments whose continuum endpoints lie in a fixed bounded box, that theorem implies, for \(0<\delta\le1\), \[ \left|\mathbb E_2\prod_{i=1}^{2n} \bigl(h_\delta(x_i)-h_\delta(y_i)\bigr)\right| \le C_n\sum_\pi\prod_{\{i,j\}\in\pi} \left(1+\sum_{u\in\{x_i,y_i\}} \sum_{v\in\{x_j,y_j\}} \log_+\frac1{|u-v|}\right). \tag{210}\] Here and below inequalities with a logarithmic singularity are used off the corresponding coincidence diagonals. To check the mesh cutoff, let \(d_{ij}\) be the minimum cross-endpoint distance after rounding to lattice faces, and \(D_{ij}\) the continuum distance. Rounding gives \(D_{ij}\le\delta(d_{ij}+C)\), while each lattice increment length is at most \(C/\delta\). The logarithmic branch of the cited bound therefore satisfies \[1+\log_+\frac{C/\delta}{1\vee d_{ij}} \le C'\left(1+\log_+\frac1{D_{ij}}\right).\] Its other branch is at most one. The logarithm of the minimum of four distances is bounded by the sum of their positive logarithms, giving (210). If a rounded increment is zero, the entire moment vanishes; otherwise every scale ratio is well-defined. Thus the estimate also handles distinct continuum endpoints that round to the same face.

Let \(f\) be a probability density supported in a square of side \(0<\ell\le1\) and bounded by \(C/(\ell^2t)\), where \(0<t\le1\). For every point \(v\), the disc-mass bound gives \[\begin{align*} \int\log_+\frac1{|u-v|}\,f(u)\,du &=\int_0^\infty\!\left(\int_{|u-v|<e^{-s}}f(u)\,du\right)ds\\ &\le\int_0^\infty\min\left(1, \frac{C e^{-2s}}{\ell^2t}\right)ds \le C'\bigl(1+|\log\ell|+|\log t|\bigr). \end{align*}\] Write \(\langle X_\delta,f\rangle\) as the integral of \(h_\delta(x)-h_\delta(y)\) against \(f(x)\rho(y)\,dx\,dy\). After expanding its \(2n\)-th moment, each pair in (210) involves endpoint variables disjoint from those in every other pair. The pair integrals therefore multiply. The fixed smooth pin has a uniform logarithmic integral, and the last display bounds the other ones. Consequently \[\mathbb E_2|\langle X_\delta,f\rangle|^{2n} \le C_n\bigl(1+|\log\ell|+|\log t|\bigr)^n.\] This is stronger than the bound needed in (202). Taking \(f=|A|^{-1}\mathbf 1_A\), for an arbitrary positive-area subset \(A\) of the square, proves that estimate at the endpoint. Signed densities bounded by \(C\ell^{-2}\) are dominated by a constant times the uniform probability density on their supporting square. The same moment expansion therefore proves (199), including its smooth cutoff. Interpolation gives all finite moment orders. Coarser meshes have uniform deterministic bounds on the fixed box.

We also need scale-uniform neutral blocks for the outward completion. If both endpoints of every increment lie in a square of side \(r\), the preceding rounding argument replaces \(1+\log_+(1/D_{ij})\) by \(1+\log_+(r/D_{ij})\). Integration against densities bounded by \(C r^{-2}\) gives a constant independent of \(r\) and \(\delta\) when \(\delta\le r\). When \(\delta>r\), unit jumps give the same bound deterministically. For a neutral signed density \(g\) on that square, choose a probability density \(p\le C r^{-2}\) there and use the exact representation \[\langle h_\delta,g\rangle =\iint\bigl(h_\delta(x)-h_\delta(y)\bigr)g(x)p(y)\,dx\,dy.\] Domination by \(|g(x)|p(y)\le C r^{-4}\) gives the required moment bound without dividing by a possibly small Jordan mass. This proves the neutral-block estimate (189) at \(c=2\).

We may now repeat the wavelet proof of Proposition 43: for \(\alpha<\beta<0\) choose \(b\) with \(2+b\beta<0\). The endpoint pinned-average bound gives (200); finite-level truncations and the gain \(2^{j(\alpha-\beta)}\) yield compactness in every asserted restriction norm. Smooth neutral pairings identify the limits by the hypothesis. The global localization and continuous restriction to \(U\) require no boundary extension theorem. The neutral-block bound and the exact mesh-rescaling identity give the outward tails and smooth approximations by the same proof.

Finally the endpoint cell-average estimate gives every thick- and thin-cell bound in the proof of Proposition 45, including the case \(p=1\), \(q=\infty\). Its finite-level coefficients are pairings with ordinary finite-energy measures, so their joint convergence is part of the hypothesis. The summable dual norm tail proves convergence in the same separable closed subspace; energy approximation identifies the limit on the entire intrinsic test space. This proves the proposition. ◻

Proof. Bulk RSW in a sequence of buffered annuli gives a polynomial bound for an actual primal or dual arm between their extreme rims: each annulus has a uniformly positive conditional probability of a blocking circuit of the opposite color.

We also need a count consequence of RSW. In an annulus, an actual component consists of vertices connected by open paths inside the annulus; boundary wiring alone does not identify its vertices. The number of such components crossing a fixed-aspect annulus has a geometric tail. To see this, explore complete actual components from the inner rim and reveal all their closed exiting edges. Remove the explored components, including noncrossing ones, until a crossing component has been found. The domain Markov property leaves on the surviving graph only the boundary identifications inherited from the original rims: the closed exiting edges prevent an explored component from wiring surviving bulk vertices. Extend the surviving configuration by closed edges on the deleted graph. Its conditional law is stochastically dominated by FK on the original annulus with every vertex on both original rims wired together. Even under this maximal boundary condition, bulk RSW gives a dual circuit in a buffered middle annulus with probability bounded below. Such a circuit blocks a further actual primal crossing, irrespective of the identifications at the original rims. Repeating this exploration bounds the conditional chance of each additional crossing by a fixed number below one. Apply the same argument with the colors exchanged.

A loop has a primal cluster on one side and a dual cluster on the other. The cluster on its bounded side is its interior-side cluster; an open path in this cluster running alongside an arc of the contour is an interior bank. Each loop is the unique exterior contour of its interior-side cluster, so distinct loops give distinct colored clusters. Cover \(B(0,M)\) by \(O((1+M/a)^2)\) balls of radius \(a/100\). A loop of diameter at least \(a\) meeting one of these balls has an interior bank crossing an annulus about the ball’s center whose two radii are fixed multiples of \(a\). Choosing one such annular component for each loop gives distinct actual components for distinct global clusters. The preceding geometric tail and Minkowski’s inequality prove (215).

For (216), a bank of an escaping loop gives an actual arm from scale \(M\) to scale \(R\). For (217), a bank of the indicated loop gives an arm from scale \(r+C\delta\) about \(z\) to a fixed fraction of \(a\). The polynomial arm bound proves both assertions. ◻

Proof. This is the construction preceding the metric conclusion, with its parameters retained. In the notation of (Duminil-Copin, Kozlowski, Krachun, et al. 2026, Definition 3.8 and Section 3.6, pp. 49–50), choose \(\eta=N^{-b}\) and \(t=N^{-b^2}\), for \(b>0\) sufficiently small. On an event of probability \(1-O(N^{-c})\), the selected nail clusters survive the whole extrema-resampling and track-exchange chain. The proof in the cited section retains a bijection of the non-trivial loops throughout this chain. During an extrema-resampling step, each such loop either avoids the resampled boxes or the configuration remains unchanged. During a track exchange, the local changes preserve the loops without breaking them and preserve their homotopy classes relative to the surviving nails. Thus the oriented reduced words are unchanged at every step, for both colors of contour.

Initially, the nail grid spacing is \(t^{1/2}N\), nail diameters are \(O(tN)\), and the nail window has size comparable to \(t^{1/4}N\). The terminal nail positions differ by these errors from their images under \(M_{\pi/2,\alpha}\). The independent FK argument in (Duminil-Copin, Kozlowski, Krachun, et al. 2026, sec. 4.2, proof of Theorem 1.7) proves \(M_{\pi/2,\alpha}=\mathrm{id}\). We may therefore use the already constructed coupling with corresponding nails at the same limiting positions.

For clarity, these scales can be arranged for every small physical mesh \(\delta\). Set \(a=b^2\) and choose \(N=2\lfloor\tfrac12\delta^{-1/(1-3a/8)}\rfloor\), so \(N\) is even as required by the track construction. Then the physical grid spacing, nail diameter, and nail-window size are respectively of orders \[N^{-a/8},\qquad N^{-5a/8},\qquad N^{a/8}.\] The homogeneous observation window has scale \(\delta N\), which also diverges. Center the nail window by translating each endpoint sample by a period vector of its own homogeneous lattice nearest the same centering vector. The two chosen translations differ by \(O(\delta)\); this only adds a vanishing error to corresponding nail positions. The endpoint restrictions have exactly the homogeneous laws, by (Duminil-Copin, Kozlowski, Krachun, et al. 2026, Proposition 2.11). After centering, both the nail window and the homogeneous observation windows exhaust the plane. The probability \(O(N^{-c})\) tends to zero. This proves the stated extraction; no conclusion about curve-set convergence is used to recover the bijection. ◻

Proof of Theorem 48. The rectangular model on \(\mathbb L(\alpha)\) is invariant under reflection \(S_{\alpha/2}\) about \(e^{i\alpha/2}\mathbb R\) (Duminil-Copin, Kozlowski, Krachun, et al. 2026, Remark 1.8). Its fair-sign height correlations inherit that symmetry. If the chosen endpoint rounding is not reflection equivariant, the reflected representatives differ by \(O(\delta)\), which is harmless by (219). Apply (222) at the original and reflected tuples. The square model is consequently asymptotically invariant under \(S_{\alpha/2}\), uniformly on every compact subset of \(\mathcal D_k\). Composing with its exact horizontal-reflection symmetry yields invariance under rotation through \(\alpha\). Square symmetries cover the endpoints and the other half of the circle. This proves the assertion for each fixed rotation.

The estimate (219) upgrades this to uniformity in the rotation angle. Indeed, the orthogonal images of \(K\) form a compact subset of \(\mathcal D_k\), and a finite angular net approximates every rotation with endpoint displacement at most \(C_K\) times its angular mesh. First use convergence at the finitely many net angles, and then let the angular mesh decrease. Reflections follow by composition with an exact square symmetry. This argument requires no estimate uniform in a degenerating isoradial angle.

Finally, any translation differs from a period translation of the mesh-\(\delta\) square lattice by a vector of length \(O(\delta)\). Exact period invariance and the location-uniform estimate (219) handle this residual displacement, uniformly over all translation vectors. Every plane isometry is a translation followed by an orthogonal map. The fixed rotation and dilation identifying \(\mathbb L(\pi/2)\) with \(\mathbb Z^2\) merely change the mesh convention and compact set. These observations prove (212) for the original square lattice. ◻

Proof. Lemma 49 identifies the fair unit-jump height with the prescribed balanced plane law. Theorem 48 supplies (Duminil-Copin, Kozlowski, Lammers, et al. 2026, Theorem 4.1) at \(c=2\); odd correlations vanish by arrow reversal. We can therefore follow the isotropic argument of that paper with its symmetry premise now proved here.

Specifically, (Duminil-Copin, Kozlowski, Lammers, et al. 2026, Theorems 6.1, 7.1 and 8.1) give this alternative: either the field converges to a fixed GFF multiple in all three senses of its Definitions 2.5–2.7, or two mesh sequences give GFF limits with distinct nonnegative multipliers. These senses include separated correlations, finite-energy tests and the outward completion. The free-energy identification in its Theorems 4.3–4.4 gives every possible squared multiplier as \[\sigma^2=-\frac1{f''(0)} =\frac2{\arccos(-1)}=\frac2\pi.\] Thus the alternative with two distinct multipliers is excluded, and convergence holds along the full mesh limit.

Proposition 47 now applies to the finite-energy convergence. Using the lattice-cutoff estimate of (Duminil-Copin, Kozlowski, Lammers, et al. 2026, Theorem 4.5), it proves convergence and tightness in the ordinary restriction and intrinsic negative topologies on every bounded open set, with the common chosen pin. Its proof retains endpoints rounded to the same face and arbitrarily small positive-area cells at the boundary.

Finally, the fixed similarity identifying the FK medial lattice with \(\mathbb Z^2\) changes the logarithmic Green kernel only by an additive constant. Neutrality cancels that constant, so the coefficient and test conclusions agree with Theorem 1. ◻