Proof. For real \(f\), make the change of variables \(s_x^{\mathrm{old}}=\exp(-tf_xg)s_x^{\mathrm{new}}\) at every integrated vertex. The fixed spins are unchanged because \(f|_P=0\). Commutation of rotations about the same axis gives \(W(t(df)g)=1\). Its second derivative at zero is \[0=\bigl\langle H_2((df)g,(df)g)+X((df)g)^2\bigr\rangle.\] The coefficient bound in (5) proves (8) for real \(f\). For \(f=f_1+if_2\), with \(f_1,f_2\) real, the squared modulus of \(X((df)g)\) is the sum of the squares for \(f_1\) and \(f_2\). Adding their bounds proves the complex case with the same constant.

Differentiating (4) twice gives the equality in (9). The local term is bounded in absolute value by \(\beta\|h\|_2^2\), and the other term is nonnegative before its minus sign, proving the inequality. Finally, the same change of variables gives \(W((h+df)c)=W(hc)\) for real \(h,f\). Differentiation in independent scalar multiples of \(h\) and \(df\), followed by complex linear extension, gives \(R(h,df)=R(df,h)=0\). This proves the last assertion. ◻

Proof. We first derive an exact identity relating the response \(R\) to a fourth derivative of \(W\), and sum it over the family. Gradient invariance will control the resulting cross terms; a nonnegative square will supply the remaining fourth-derivative lower bound.

The noncommuting gauge identity. Let \(F_x\) be a real matrix potential in the span of \(a,b\), zero on \(P\). The substitution \(s_x^{\mathrm{old}}=\exp(-F_x)s_x^{\mathrm{new}}\) gives \[ W(dF)=W(B),\qquad B_e=\log\bigl(\exp(F_x)\exp(F_y-F_x)\exp(-F_y)\bigr) \tag{13}\] for \(F\) sufficiently small. The product inside the logarithm is orthogonal and near the identity; hence its local logarithm is skew symmetric. The logarithm also commutes with conjugation. Multiplying the exponential series to degree two, and then taking the logarithm, gives \[ B_e=\tfrac12[F_x,F_y]+O(F^3),\qquad DB_e(0)=0. \tag{14}\] For directions \(fa\) and \(gb\), the mixed second derivative of \(B_e\) is therefore \(S(f,g)_e c\); it is zero for two directions using the same generator.

We also need a fourth-order cancellation. Take four independent parameters whose matrix directions have generator pattern \((a,b,a,b)\), with arbitrary spatial potentials. The corresponding mixed fourth derivative of \(B_e\) is zero. To see this, conjugate by the half-turn about the first axis. This fixes \(a\) and negates \(b\), so the coefficient involving all four parameters is fixed by conjugation: its two \(b\)-directions contribute two minus signs. Conjugation by the half-turn about the second axis likewise fixes this coefficient. On \(\mathfrak{so}(3)\), the fixed spaces of these two conjugations are respectively \(\mathbb R a\) and \(\mathbb R b\), with zero intersection. Thus \[ D_{1234}B_e=0 \quad\text{for the generator pattern }(a,b,a,b). \tag{15}\] Both (14) and (15) extend complex multilinearly. No complex change of spin variables is involved in this extension.

For fixed \(i,j\), take the four edge directions \[ A_1=u_i a,\qquad A_2=\bar u_i b,\qquad A_3=\bar u_j a,\qquad A_4=u_j b. \tag{16}\] The fourth derivative of the composition in (13) has no term containing \(DB\), by (14). Its term \(DW[D_{1234}B]\) also vanishes, by (15). Only pairings of two second derivatives remain. The pairing \(13|24\) is zero because both pairs have the same generator. For the other two pairings, put \(S_i=S(f_i,\bar f_i)\) and \(S_{ij}=S(f_i,f_j)\). Their arguments are \[12|34:\quad(S_i c,\overline{S_j}c),\qquad 14|23:\quad(S_{ij}c,-\overline{S_{ij}}c).\] Here \(S(\bar f_j,f_j)=\overline{S_j}\) and \(S(\bar f_j,\bar f_i)=-\overline{S_{ij}}\). Each labeled pairing occurs once in the composition rule, so \[ D^4W(A_1,A_2,A_3,A_4) =-R(S_i,\overline{S_j})+R(S_{ij},\overline{S_{ij}}). \tag{17}\] This also holds when some spatial functions coincide: the derivative identity is multilinear in the four labeled directions.

Summing over \(i,j\) and rearranging gives the identity to be estimated: \[ R(Q,\bar Q) =\sum_{i,j}R(S_{ij},\overline{S_{ij}}) -\sum_{i,j}D^4W(u_i a,\bar u_i b,\bar u_j a,u_j b). \tag{18}\] We bound the first sum from above using gradient invariance, and the second from below by retaining a nonnegative square.

The cross terms. For functions \(f,g\) vanishing on \(P\), the product \(fg\) also vanishes there, and the edge identities \[ \begin{split} S(f,g)+\tfrac12d(fg)&=\tfrac12(f_x+f_y)\,dg,\\ S(f,g)-\tfrac12d(fg)&=-\tfrac12(g_x+g_y)\,df \end{split} \tag{19}\] hold on \(e=(x,y)\). Applying gradient invariance in both arguments of \(R\), followed by (9), gives \[ R(S(f,g),\overline{S(f,g)}) \le\beta\min\{\|f\|_\infty^2\|dg\|_2^2, \|g\|_\infty^2\|df\|_2^2\}. \tag{20}\]

The fourth-derivative lower bound. We claim that \[ \sum_{i,j}D^4W(u_i a,\bar u_i b,\bar u_j a,u_j b) \ge-K_{\mathrm{rot}}(\beta)\|T\|_{\mathrm{HS}}^2. \tag{21}\] To enumerate its terms, set \(X_r=X(A_r)\), and write \(H_{r_1\cdots r_k}=H_k(A_{r_1},\ldots,A_{r_k})\). Direct differentiation of \(\exp H\) gives all fifteen set partitions: \[ \begin{aligned} D^4W(A_1,A_2,A_3,A_4)=\bigl\langle &X_1X_2X_3X_4\\ &+H_{12}X_3X_4+H_{13}X_2X_4+H_{14}X_2X_3\\ &+H_{23}X_1X_4+H_{24}X_1X_3+H_{34}X_1X_2\\ &+H_{12}H_{34}+H_{13}H_{24}+H_{14}H_{23}\\ &+H_{123}X_4+H_{124}X_3+H_{134}X_2+H_{234}X_1\\ &+H_{1234}\bigr\rangle. \end{aligned} \tag{22}\] The four terms comprising \((X_1X_2+H_{12})(X_3X_4+H_{34})\), after summation over \(i,j\), have expectation \[ \left\langle\left|\sum_i \bigl[X(u_i a)X(\bar u_i b)+H_2(u_i a,\bar u_i b)\bigr] \right|^2\right\rangle\ge0. \tag{23}\] We bound the other eleven terms in absolute value.

For an edge \(e\), let \(t_e(l)=T_{el}\) and \(\rho_e=(\sum_l|T_{el}|^2)^{1/2}\), and abbreviate \(X_g(v)=X(vg)\). Crucially, \[ t_e=d\left(\sum_i u_i(e)\overline{f_i}\right). \tag{24}\] The potential on the right, and its conjugate, vanish on \(P\). Lemma 3 therefore gives \[ \langle|X_g(t_e)|^2\rangle\le\beta\rho_e^2, \qquad \langle|X_g(\bar t_e)|^2\rangle\le\beta\rho_e^2. \tag{25}\]

The following table records every remaining contraction. Expand each \(H_k\) using (6), sum over \(i,j\), and multiply the last two columns. One then sums over \(e\), except in the two \(H_2H_2\) rows, where one sums over \(e,l\).

For example, the first row follows from \[\begin{split} &\sum_{i,j}u_i(e)\bar u_j(e) X_b(\bar u_i)X_b(u_j)\\ &\qquad= X_b\left(\sum_i u_i(e)\bar u_i\right) X_b\left(\sum_j\bar u_j(e)u_j\right) =X_b(t_e)X_b(\bar t_e). \end{split}\] The other rows follow by the same finite summation; their explicit expressions distinguish the conjugated and unconjugated contractions.

For each of the first four rows, the pointwise coefficient bound and Cauchy–Schwarz, followed by (25), give an absolute expectation at edge \(e\) of at most \(\beta^2\rho_e^2\). The coefficient may depend on the spins; its pointwise absolute value is bounded before applying Cauchy–Schwarz. These four rows contribute at most \(4\beta^2\|T\|_{\mathrm{HS}}^2\). The next two rows contribute at most \(2\beta^2\|T\|_{\mathrm{HS}}^2\), since both contractions have absolute value \(|T_{el}|^2\).

For each of the four \(H_3X\) rows, (25) bounds the total absolute expectation by \[\beta^{3/2}\sum_e T_{ee}\rho_e \le\beta^{3/2} \left(\sum_e T_{ee}^2\right)^{1/2} \left(\sum_e\rho_e^2\right)^{1/2} \le\beta^{3/2}\|T\|_{\mathrm{HS}}^2.\] Here \(T_{ee}=\sum_i|u_i(e)|^2\ge0\), and the squared diagonal entries form part of the Hilbert–Schmidt sum. The last row is bounded by \(\beta\sum_e T_{ee}^2\le\beta\|T\|_{\mathrm{HS}}^2\).

Finally, the sum on the left of (21) is real: complex conjugation interchanges \(i,j\) and permutes the arguments of the symmetric fourth derivative. Keeping the nonnegative square (23), and subtracting the bounds for the remaining terms, proves (21) with exactly \(K_{\mathrm{rot}}(\beta)\) from (12).

Combining (20) and (21) in (18) proves (11). Only the upper bound and gradient invariance of \(R\) were used; no positivity property of \(R\) is required. ◻

Proof. For a real profile \(\phi\) and one mode on shell \(r_p\), direct substitution on a positive coordinate-\(j\) edge gives \[ S(f_p,\overline{f_p})_e =-i r_p^{-3}\phi(x/N)\phi((x+e_j)/N) \sin\left(\frac{2\pi p_j}{16N}\right). \tag{36}\] If the group is indexed by \(m\ne j\), its frequency set is invariant under \(p_j\mapsto-p_j\), so its sum vanishes. For \(m=j\), the negative half-plane gives the negative of the positive-half-plane sine sum. Interchanging coordinates identifies the positive sums for \(j=1,2\). This proves the identity in (33).

On shell \(r\), retain only the \(2r+1\) frequencies with \(p_1=r\) and \(|p_2|\leq r\). Since \[\sin\left(\frac{\pi r}{8N}\right)\geq\frac r{4N} \qquad(1\leq r\leq N),\] their contribution to \(\lambda_N\) is at least \((2r+1)/(4r^2)\geq1/(2r)\). This gives its asserted lower bound.

For an edge in direction \(m\), let \(z=(x+y)/(2N)\). Centered Taylor expansion, with uniform bounds on the fixed smooth functions, gives \[h_e=N^{-1}\partial_m\Theta(z)+O(N^{-3}),\qquad \phi(x/N)\phi(y/N)=\phi(z)^2+O(N^{-2}).\] Using (30) yields \(|h_e-\widetilde h_e|\leq C N^{-3}\). There are \(O(N^2)\) annulus edges, and \(|h_e|+|\widetilde h_e|\leq C/N\). The three estimates in (34) follow, also after restriction to \(E\).

For a mode on shell \(r\), the product rule for a discrete difference and \(|e^{iv}-1|\leq|v|\) give \[ \|f_i\|_\infty\leq Cr^{-3/2},\qquad \|df_i\|_2\leq Cr^{-3/2}(1+r)\leq C' r^{-1/2}. \tag{37}\] There are \(O(r)\) modes on shell \(r\), including all four groups. For shells \(s\leq r\), choose the bound in the minimum that uses the supremum norm of the shell-\(r\) mode. The cost of a pair is at most \(Cr^{-3}s^{-1}\); all \(O(rs)\) pairs therefore cost at most \(Cr^{-2}\). Summing both shell orderings gives \[ \sum_{i,j}\min\left\{ \|f_i\|_\infty^2\|df_j\|_2^2, \|f_j\|_\infty^2\|df_i\|_2^2 \right\} \leq C\sum_{r=1}^N\sum_{s=1}^r r^{-2} \leq C'(1+\log N). \tag{38}\]

It remains to bound \(T\). Applying the triangle inequality mode by mode would give only \(\|T\|_{\mathrm{HS}}\leq\sum_i\|df_i\|_2^2=O(N)\); we use Fourier orthogonality for the sharper bound. Work first in a single profile group and let \(U\) be the matrix whose \(p\)-th column is \(r_p^{1/2}df_p\). Embed the annulus in the discrete torus \(Q_N=(\mathbb Z/(16N)\mathbb Z)^2\). Its representatives may be chosen in \([-8N,8N-1]^2\); all profiles vanish near the boundary of this square, so they also define torus functions. Let \(F\) have columns \[F_{x,p}=\exp\left(\frac{2\pi i\,p\cdot x}{16N}\right), \qquad x\in Q_N,\] using the frequencies of this group. They are distinct modulo \(16N\), and orthogonality of these unnormalized waves gives \[ F^*F=(16N)^2I,\qquad \|F\|_{\mathrm{op}}=16N. \tag{39}\] For direction \(j\), let \(P_j\) restrict functions on \(Q_N\) to the origins of the coordinate-\(j\) edges in \(E\). Let \(M_v\) denote multiplication by \(v\) on \(Q_N\), and set \[\Delta_j\phi(x)=\phi((x+e_j)/N)-\phi(x/N), \qquad \phi^{(j)}(x)=\phi((x+e_j)/N).\] The directional block of \(U\) has the exact factorization \[\begin{align*} U_j={}&P_jM_{\Delta_j\phi}F \operatorname{diag}(r_p^{-1})\\ &+P_jM_{\phi^{(j)}}F \operatorname{diag}\left( \frac{\exp(2\pi i p_j/(16N))-1}{r_p}\right). \tag{40}\end{align*}\] Here \(\|P_j\|_{\mathrm{op}}\leq1\), \(\|\Delta_j\phi\|_\infty\leq C/N\), and the last diagonal factor has operator norm at most \(C/N\), since \(|p_j|\leq r_p\). Equation (39) now gives \(\|U_j\|_{\mathrm{op}}\leq C\) and hence \(\|U\|_{\mathrm{op}}\leq C\) after combining the two directions.

The contribution of this group is \[T_{\mathrm{group}}=U\operatorname{diag}(r_p^{-1})U^*.\] The finite-matrix inequality \(\|ADB\|_{\mathrm{HS}}\leq \|A\|_{\mathrm{op}}\|D\|_{\mathrm{HS}}\|B\|_{\mathrm{op}}\) therefore implies \[\|T_{\mathrm{group}}\|_{\mathrm{HS}}^2 \leq C\sum_p r_p^{-2} \leq C'\sum_{r=1}^N r\,r^{-2} \leq C''(1+\log N).\] The triangle inequality over the four groups gives the same order for \(\|T\|_{\mathrm{HS}}^2\). Together with (38), this proves (35). ◻

Proof of Proposition 5. Apply Proposition 4 in the law with twist \(t\), using the family (31). The left side is \[R_t(-i\lambda_N\widetilde h,i\lambda_N\widetilde h) =\lambda_N^2R_t(\widetilde h,\widetilde h),\] because \(R_t\) is bilinear and \(\widetilde h\) is real. Lemma 6 gives, uniformly in \(t\), \[ R_t(\widetilde h,\widetilde h) \leq\frac{C_\beta(1+\log N)}{(\log N)^2} \leq\frac{C'_\beta}{\log N}. \tag{41}\]

We must transfer this upper bound to \(h\) without treating \(R_t\) as a positive form. If \(R_t(h,h)\leq0\), the desired positive upper bound already holds. Otherwise, the defining second-derivative identity and the pointwise bound on \(H_{2,t}\) give \[ \langle X_t(hc)^2\rangle_t <-\langle H_{2,t}(hc,hc)\rangle_t \leq\beta\|h\|_2^2\leq C\beta. \tag{42}\] Put \(d=h-\widetilde h\). The current is a real random variable on real edge forms, and \[|X_t(dc)|\leq\beta\|d\|_1\leq\frac{C\beta}{N}.\] Consequently ordinary Cauchy–Schwarz for random variables, followed by (42), yields \[\begin{align*} \left|\langle X_t(hc)^2-X_t(\widetilde h c)^2\rangle_t\right| &\leq 2\langle X_t(hc)^2\rangle_t^{1/2} \|X_t(dc)\|_\infty+\|X_t(dc)\|_\infty^2\\ &\leq\frac{C_\beta}{N}. \tag{43}\end{align*}\] The local quadratic term satisfies the stronger pointwise bound \[\begin{align*} |H_{2,t}(hc,hc)-H_{2,t}(\widetilde h c,\widetilde h c)| &\leq\beta\sum_{e\in E}|d_e|\,|h_e+\widetilde h_e|\\ &\leq\frac{C\beta}{N^2}. \tag{44}\end{align*}\] Equations (41)–[eq:annulus-local-error] and the formula for \(R_t\) imply in both cases that \[ -\frac{Z''(t)}{Z(t)}=R_t(h,h) \leq\frac{C_\beta}{\log N}+\frac{C_\beta}{N} \leq\delta_N,\qquad \delta_N=\frac{C'_\beta}{\log N}. \tag{45}\] All constants are independent of the boundary twist, the retained vertices and edges, and the admissible strengths.

Finally, \(Z\) is positive, smooth, and \(2\pi\)-periodic. Set \(M=\max_t Z(t)\) and choose \(t_0\) with \(Z(t_0)=M\). Then \(Z'(t_0)=0\), and (45) implies \(Z''(t)\geq-\delta_N Z(t)\geq-\delta_N M\). For any \(t\), choose an orientation \(\epsilon\in\{-1,1\}\) and \(0\leq\ell\leq\pi\) with \(t=t_0+\epsilon\ell\) modulo \(2\pi\). Twice integrating the second-derivative bound for \(v\mapsto Z(t_0+\epsilon v)\) gives \[Z(t)\geq M-\frac{\delta_N M\ell^2}{2} \geq M\left(1-\frac{\pi^2\delta_N}{2}\right).\] Taking the minimum and enlarging \(C_\beta\) proves (26). Since \(Z(\pi)/Z(0)\geq\min Z/\max Z\), it also proves (27). ◻

Proof. For one coordinate, association follows by taking independent copies \(X,X'\) and writing \[2\operatorname{Cov}(F(X),H(X)) =\mathbb E\big[(F(X)-F(X'))(H(X)-H(X'))\big]\geq0.\] Proceed by induction on the number of coordinates. Write a point as \((u,t)\), with \(t\) the final coordinate. Each conditional density \(w(u,t)\) is log-supermodular in \(u\), and its law is associated by induction. For \(t'\geq t\), log-supermodularity implies that \(w(u,t')/w(u,t)\) is increasing in \(u\): apply the defining inequality to \((u,t')\) and \((v,t)\) when \(u\leq v\). Tilting the conditional law at \(t\) by this increasing likelihood ratio, and applying its association, shows that the conditional law at \(t'\) stochastically dominates it. The ratios are bounded in the continuous case by positivity and compactness.

If \(F(u,t)\) and \(H(u,t)\) are increasing, their conditional expectations are consequently increasing functions of \(t\). The decomposition \[\operatorname{Cov}(F,H) =\mathbb E\big[\operatorname{Cov}(F,H\mid t)\big] +\operatorname{Cov}\big(\mathbb E[F\mid t],\mathbb E[H\mid t]\big)\] has nonnegative terms by induction and the one-coordinate case. This proves both assertions. ◻

Proof. The mean inequality follows by expanding the Gibbs exponential. For Ising spins, every reference moment of a sign monomial is either zero or one. For planar spins, each cosine of an angle difference has nonnegative Fourier coefficients, so every product of such cosines has a nonnegative constant Fourier coefficient.

For covariances, take two independent copies of the Gibbs system. If \(T_e\) and \(T_f\) are two edge energies and primes denote the second copy, then \[ 2\operatorname{Cov}(T_e,T_f) =\mathbb E\big[(T_e-T'_e)(T_f-T'_f)\big]. \tag{52}\] In the planar reference integral, make the substitution \(\theta=\varphi+\psi\), \(\theta'=\varphi-\psi\) modulo \(2\pi\). Independent uniform \(\varphi,\psi\) produce independent uniform \(\theta,\theta'\): this is a surjective homomorphism of compact tori, and therefore preserves Haar probability. For an oriented edge \(g\), write \(\varphi_g=\varphi_x-\varphi_y\), and similarly for \(\psi_g\). The doubled density numerator becomes \[\exp\left(2\sum_{g\in E}L_g\cos\varphi_g\cos\psi_g\right), \qquad T_e-T'_e=-2\sin\varphi_e\sin\psi_e.\] Consequently the unnormalized numerator on the right of (52) is \[4\sum_{a\in\mathbb N_0^E} \left(\prod_{g\in E}\frac{(2L_g)^{a_g}}{a_g!}\right) \left( \int\sin\varphi_e\sin\varphi_f \prod_{g\in E}(\cos\varphi_g)^{a_g}\,\mathrm d\varphi \right)^2\geq0,\] where the integral uses product Haar probability. Each integral is real; the possible multiplicity of the torus map introduces no factor because the measures are probabilities.

For Ising reference signs \(d,d'\), set \(u=(d+d')/2\) and \(v=(d-d')/2\) at each site. Their sitewise values are \((1,0),(-1,0),(0,1),(0,-1)\), with equal probabilities. Every mixed monomial in \(u,v\) has nonnegative expectation: a monomial involving positive powers of both vanishes, an odd pure moment vanishes, and an even pure moment is nonnegative. The doubled density is \[\exp\left(2\sum_{g=\{x,y\}}K_g(u_xu_y+v_xv_y)\right),\] and an edge-energy difference is \(2(u_xv_y+v_xu_y)\). Expanding the exponential and the product of two differences gives only nonnegative coefficients multiplying nonnegative reference moments. All expansions above are absolutely integrable on the finite products, proving the covariance assertion. ◻

Proof. For either partition function, with couplings \(J_e(r)\) and edge energies \(T_e\), differentiation in distinct site variables \(r_x,r_y\) gives \[\begin{align*} \partial_x\partial_y\log Z(J(r)) &=\sum_e\langle T_e\rangle\,\partial_x\partial_y J_e\\ &\quad+\sum_{e,f}\operatorname{Cov}(T_e,T_f) (\partial_xJ_e)(\partial_yJ_f). \end{align*}\] For \(J=K\), the first derivatives are nonnegative. For \(J=L\), they are nonpositive, so the products in the second sum are again nonnegative. The distinct-site mixed derivatives of individual couplings vanish unless \(e=\{x,y\}\), in which case \[\partial_x\partial_y K_{xy}=b_{xy}, \qquad \partial_x\partial_y L_{xy} =b_{xy}\frac{r_xr_y}{q_xq_y}\geq0\] in the open amplitude cube. Lemma 8 therefore shows that every distinct-site mixed derivative of \(\log\rho\) is nonnegative there. An increment of \(\log\rho\) in one coordinate is thus nondecreasing in every other coordinate. For two points \(u,v\), compare the successive increments from \(u\wedge v\) to \(u\) with the same coordinate increments from \(v\) to \(u\vee v\). Their sum gives the log-supermodular inequality in the open cube, and continuity extends it to the closed cube. The possible singularity of derivatives at \(r_x=1\) is immaterial. The density itself is positive and continuous, so Lemma 7 applies. ◻

Proof. Conditional on \(r\), summing the signs in (50) gives a bond weight proportional to \[ 2^{k(\eta)}\prod_{e\in E}p_e^{\eta_e}(1-p_e)^{1-\eta_e}, \tag{53}\] where \(k(\eta)\) counts all open components, including singletons. The function \(k\) is supermodular: the change on opening an edge is \(-1\) if its endpoints were disconnected and \(0\) otherwise, and this change increases as other bonds are opened. Hence the weight is log-supermodular and its law is associated by Lemma 7. Increasing any nondegenerate parameter \(p_e\) tilts the weight by an increasing likelihood ratio. Association therefore implies stochastic increase of the conditional bond law in each \(p_e\), and consequently in \(r\). A zero parameter forces its edge closed; restriction to the remaining edges, or continuity, handles these degenerate cases.

For bounded increasing functions \(F(r,\eta)\) and \(H(r,\eta)\), conditional association makes \(\mathbb E[\operatorname{Cov}(F,H\mid r)]\) nonnegative. Their conditional means are increasing in \(r\), by stochastic monotonicity of the bond law and their direct monotonicity in \(r\). Their covariance is nonnegative by Lemma 9. The total covariance identity proves the claim. ◻

Proof. For \(0<\varepsilon<1\), condition the free system on \(r_x\geq1-\varepsilon\) and \(\tau_x=1\) for every \(x\in B\). Conditional on \(r,\eta\), the probability of the sign requirement is \(2^{-k_B(\eta)}\), where \(k_B\) is the number of open components meeting \(B\). Thus the induced law of \(r,\eta\) is tilted by \[T_\varepsilon(r,\eta) =\mathbf1_{\{r_x\geq1-\varepsilon\ \text{for all }x\in B\}} 2^{-k_B(\eta)}.\] Opening an edge cannot increase \(k_B\); this also holds when it merges a component meeting \(B\) with one not meeting \(B\). Consequently \(T_\varepsilon\) is increasing. Lemma 10 gives \[\frac{\mathbb E[\mathbf1_A T_\varepsilon]}{\mathbb E[T_\varepsilon]} \geq\mathbb P(A).\]

As \(\varepsilon\downarrow0\), the conditioned reference measures on the positive caps converge to point masses at \(s_*\). The finite-volume Gibbs weight is bounded, positive, and continuous. The spin-to-bond kernel is also continuous, since its edge-opening probability is \[1-\exp\big(-2b_{xy}\max\{s_x^1s_y^1,0\}\big).\] In particular there is no discontinuity at a zero first coordinate. Integrating any event of the finitely many bonds and passing to the limit proves the asserted comparison for exact pinning. ◻

Proof. First work in one restricted annulus graph. Pin both of its layers to \(s_*\), and let \(Z(0)\) be its partition function. Let \(Z(\pi)\) be the partition function with the inner layer changed to \(-s_*\) and the outer layer left at \(s_*\). Then \[ \mathbb P_{\mathrm{pinned}}(\text{crossing}) =1-\frac{Z(\pi)}{Z(0)}. \tag{54}\] Indeed, in the expansion (50), every open component missing the pinned layers has two sign choices. A component meeting a pinned layer has one sign choice, provided all prescribed signs on it agree. Changing the inner signs therefore removes exactly the bond configurations joining the two layers and changes the weight of none of the remaining configurations. All other factors are unchanged: the boundary amplitudes are \(r=1,q=0\), and the reference normalization for the unpinned signs is the same. This argument holds at every interior amplitude configuration and hence after integration. If a layer has no vertices, both sides of (54) are zero.

By Proposition 5, the right side of (54) is bounded uniformly by a nonnegative sequence tending to zero. Enlarge it slightly to obtain a strictly positive sequence \(p_N\) with the same limit.

For several annuli, pin all their present boundary-layer sites to \(s_*\) simultaneously. The intersection of their crossing events is increasing, so Proposition 11 bounds its free probability by its pinned probability. In the pinned system, each unpinned annulus region is separated from every vertex outside that annulus by the pinned layers: a nearest-neighbor path cannot jump across either layer. The unpinned annulus spins are therefore independent with precisely the required fixed boundary values. For each resulting annulus marginal, apply Equation (54) to the partition functions of that restricted annulus graph. Boundary-only interactions contribute constants, and the independent sampling of edge bonds preserves independence of the crossing events. Their probabilities multiply, proving the bound.

This separation uses only edges present in \(G\). Missing vertices or edges, including those beyond the finite volume, cannot create an unseparated nearest-neighbor interaction. Pinning the layer sites that are present and applying Proposition 5 to the restricted subgraph proves the assertion also for truncated boxes. ◻

Proof. Translate the lattice so that \(x=0\). Choose a finite integer \(N\geq2\) so large that, for the sequence in Proposition 12, \[ 0<p:=p_N<1, \qquad \alpha:=4p^{1/25}<1. \tag{55}\] Partition \(\mathbb Z^2\) into boxes \(z+\{0,\ldots,N-1\}^2\), with coarse vertices \(z\in N\mathbb Z^2\). Suppose \(0\) and \(y\) are joined by an open path and \(d:=\|y\|_\infty>4N\). Recording the visited boxes and erasing loops produces a simple nearest-neighbor coarse path from the origin’s box to the box containing \(y\). If its length is \(k\), then \[ k\geq d/N-1. \tag{56}\]

Each vertex \(z\) of this simple coarse path corresponds to a box visited by the original open path, at some vertex strictly within sup distance \(N\) of \(z\). At least one of the original endpoints is at distance greater than \(2N\) from \(z\): otherwise their mutual distance would be at most \(4N\). A segment of the original path from the visited vertex to that endpoint thus contains a crossing of \(D_N(z)\): take its first hit on the outer layer and its last preceding hit on the inner layer. The intervening segment stays in the annulus. It does not matter whether that segment survived coarse loop erasure; its edges are still open.

Among the \(k+1\) distinct coarse vertices, one of the 25 residue classes modulo \(5N\) in the two coordinates contains at least \((k+1)/25\) vertices. Their closed outer boxes of sup radius \(2N\) are disjoint, because distinct centers in the class differ by at least \(5N\) in some coordinate. Choose one maximizing class deterministically for each candidate coarse path. Proposition 12 bounds the probability that this path has all its required crossings by \(p^{(k+1)/25}\). There are at most \(4^k\) coarse paths of length \(k\) from the initial box. The union bound and (56) therefore give \[\begin{align*} \mathbb P(0\leftrightarrow y) &\leq p^{1/25}\sum_{k\geq\lceil d/N-1\rceil}\alpha^k\\ &\leq \frac{p^{1/25}}{\alpha(1-\alpha)} \exp\left(-\frac{-\log\alpha}{N}\,d\right). \end{align*}\] Since \(d\geq\|y\|_2/\sqrt2\), take \[m_0=\frac{-\log\alpha}{\sqrt2\,N}>0.\] Increasing the finite prefactor to cover \(d\leq4N\) proves the result. Every estimate was uniform in the finite subgraph and its strengths. ◻

Proof. Represent a uniform sphere point by \((U,V)/\sqrt{|U|^2+|V|^2}\), where \(U\in\mathbb R^3\) and \(V\in\mathbb R^{n-3}\) have independent standard Gaussian coordinates. The direction \(U/|U|\) is uniform on \(S^2\) and is independent of \((|U|,V)\). Since \(v=V/\sqrt{|U|^2+|V|^2}\) depends only on these latter variables, the three-component direction is independent of \(v\) under the reference law. Applying this representation independently at all sites proves that, conditional on all \(v_x\), the reference directions remain independent and uniform on \(S^2\).

The conditioned interaction is \[\sum_{\{x,y\}\in E}b_{xy}v_x\cdot v_y +\sum_{\{x,y\}\in E}b_{xy}\rho_x\rho_y s_x\cdot s_y.\] The first sum is constant under the conditional law and cancels from its normalization. The second gives the asserted couplings. The event \(\rho_x=0\) has reference probability zero and still has Gibbs probability zero because the finite Gibbs density is positive and bounded. Alternatively, choosing any independent reference direction at such a site would decouple it and leave all spin products unchanged. ◻

Proof of Theorem 1 and Corollary 2. For \(n=3\), Equation (51) and Proposition 13 give \[0\leq\langle s_x^1s_y^1\rangle_{G,b}^{(3)} \leq A_0e^{-m_0\|x-y\|_2}.\] For \(n>3\), apply Lemma 14. Conditional on all \(v_x\), the first-coordinate correlation is \(\rho_x\rho_y\) times the corresponding three-component first-coordinate correlation. The latter is nonnegative and satisfies the same bound, uniformly in the conditional strength array. Since \(0\leq\rho_x\rho_y\leq1\), averaging gives, for every \(n\ge3\), \[0\leq\langle\sigma_x^1\sigma_y^1\rangle_{G,b}^{(n)} \leq A_0e^{-m_0\|x-y\|_2}.\] The original free spin law is invariant under coordinate permutations, so \[\langle\sigma_x\cdot\sigma_y\rangle_{G,b}^{(n)} =n\langle\sigma_x^1\sigma_y^1\rangle_{G,b}^{(n)}\geq0.\] Taking \(A=nA_0\) and \(m=m_0\) proves Theorem 1.

The constants are independent of the volume, so the same estimate passes to the limsup in (3). If a subsequence converges locally weakly, the expectation of the bounded continuous local function \(\sigma\mapsto\sigma_0\cdot\sigma_x\) converges along it. Its limit therefore satisfies the same bound. This proves Corollary 2 without a uniqueness assertion. ◻