Model and result

Fix \(S\in\{\tfrac12,1,\tfrac32,\ldots\}\). Let \(S^1,S^2,S^3\) be the irreducible spin-\(S\) matrices, normalized by \[[S^a,S^b]=i\sum_c\epsilon_{abc}S^c, \qquad \sum_{a=1}^3(S^a)^2=S(S+1)I.\] For even \(L\ge4\), write \(\Lambda_L=(\mathbb Z/L\mathbb Z)^3\) and \(V=L^3\). On \(\bigotimes_{x\in\Lambda_L}\mathbb C^{2S+1}\), consider \[ H_{S,L}=-\sum_{\{x,y\}\text{ nearest neighbors}} \sum_{a=1}^3 S_x^aS_y^a, \qquad M_L^a=\sum_{x\in\Lambda_L}S_x^a, \tag{1}\] where each unordered bond is counted once. We use the Gibbs expectation \[\langle A\rangle_{\beta,L} =\frac{\mathop{\mathrm{Tr}}(Ae^{-\beta H_{S,L}})}{\mathop{\mathrm{Tr}}e^{-\beta H_{S,L}}}.\] The pressure and its right derivative at zero field are \[\begin{align*} p_{S,\beta}(h) &=\lim_{\substack{L\to\infty\\L\text{ even}}} \frac1{\beta V}\log\mathop{\mathrm{Tr}} e^{-\beta(H_{S,L}-hM_L^3)},\tag{2}\\ m_{S,\beta}&=p_{S,\beta}'(0+). \tag{3}\end{align*}\] The pressure limit exists for this bounded finite-range interaction, and convexity gives the right derivative. The volume limit in (2) precedes removal of the field.

The statement uses self-adjoint linear combinations of the spin components. It requires no joint finite-volume measurement of these noncommuting operators. Both the spin and the positive temperature remain fixed throughout the volume limit.

Context and antecedents

The dimensional distinction in this problem is fundamental. Mermin and Wagner (Mermin and Wagner 1966) excluded spontaneous ferro- or antiferromagnetic order at positive temperature for finite-range isotropic Heisenberg models in dimensions one and two. In higher dimensions, infrared bounds proved continuous-symmetry breaking for classical models (Fröhlich et al. 1976) and ordering for quantum antiferromagnets (Dyson et al. 1978). These quantum antiferromagnetic results do not supply the isotropic ferromagnetic ordering input used here. Another line of work establishes the low-temperature free-energy asymptotic predicted by spin-wave theory (Correggi et al. 2015). That asymptotic concerns a different limit from the field derivative or the zero-field magnetization distribution at a fixed positive temperature.

The spherical distribution itself has clear antecedents. Ueltschi’s description of three-dimensional loop models (Ueltschi 2017, sec. 8.1) derives the transform \(\sinh(hm)/(hm)\) for the spin-\(\tfrac12\) ferromagnet conditional on a conjectured Poisson–Dirichlet law for macroscopic loops. Björnberg, Fröhlich, and Ueltschi (Björnberg et al. 2020, Theorem 2.1) prove the spherical transform for the complete-graph quantum Heisenberg model at arbitrary fixed spin. Our setting has nearest-neighbor interactions on the three-dimensional lattice. We prove its magnetization law without assuming a distribution for macroscopic loop lengths.

The existence of spontaneous magnetization and the identification of its zero-field distribution are distinct questions. The companion manuscript (OpenAI 2026b) constructs magnetized equilibrium states for the isotropic nearest-neighbor quantum Heisenberg ferromagnet in dimensions at least three, for every fixed spin at sufficiently low positive temperatures. The companion (OpenAI 2026a) establishes the low-temperature asymptotic of the pressure-defined magnetization for nonnegative symmetric finite-range interactions whose support generates \(\mathbb Z^3\). We use finite exchange and geometric estimates from these works. The present argument adds spatial control on periodic tori and a pressure argument excluding translation-ergodic equilibrium states with different magnetization magnitudes.

The exchange representation builds on permutation and random-walk representations of quantum spins (Powers 1976; Conlon and Solovej 1991), Tóth’s random-stirring representation (Tóth 1993), and its higher-spin formulation through symmetric slots (Nachtergaele 1994). The finite pin estimates used below draw on the stability and negative dependence theory of Borcea, Brändén, and Liggett (Borcea et al. 2009). Our zero-free matrix products use the contraction principle associated with Asano’s and Ruelle’s developments of Lee–Yang theory (Asano 1970; Ruelle 1971, 1973); the contraction needed here is proved explicitly. Mean entropy, equilibrium states, and pressure tangents are treated within the variational framework of Lanford and Robinson (Lanford and Robinson 1968a, 1968b). We recall the required facts and prove the additional replica and harmonic-selection statements.

The two parts of the proof

Here is the obstruction that the proof must overcome. Let \(J_L\) denote the total-spin label in the rotation-invariant Gibbs state. Conditional on \(J_L\), any component of \(M_L\) is uniform on \(-J_L,-J_L+1,\ldots,J_L\). A fixed positive field and the definition of \(m\) already imply \[ \mathbb P(J_L/V>m+\epsilon)\longrightarrow0 \qquad(\epsilon>0). \tag{6}\] Thus it remains to prove \[ \liminf_{L\to\infty}\frac{\mathbb E[J_L(J_L+1)]}{V^2}\ge m^2. \tag{7}\] Together these bounds force \(J_L/V\to m\) in probability and give (4). Section 8 supplies this last representation-theoretic argument in detail.

Two facts produce the lower bound. First, the squared magnetization averaged on large local boxes is controlled by the squared magnetization averaged on the whole torus, with an error vanishing as the local scale grows. This prevents local magnetization from canceling through spatial variation on larger scales. Second, every translation-ergodic equilibrium state has magnetization vector of length \(m\). Here an equilibrium state maximizes mean entropy minus \(\beta\) times the physical energy density among translation-invariant states; an ergodic state is an extreme point of that invariant state space. Their variational and averaging properties are recalled in Section 6. The second fact identifies the local magnetization length to which the first fact applies.

Both facts use the same weak exponential estimate. To describe it, let \(\mathcal P\) be a rectangular partition whose cell sides are comparable to a large scale \(R\le L\). For a deterministic site function \(f\) with \(|f_x|\le1\) and \(\sum_{x\in C}f_x=0\) in each cell, put \(D_f=\sum_xf_xS_x^3\). We prove, for \(b_R\asymp R^{-5/2}\) and a fixed \(\kappa>0\), that \[ \log\langle e^{\theta_R D_f}\rangle_{\beta,L} \le C_\beta b_R V, \qquad \theta_R=\kappa b_R\log R. \tag{8}\] In particular the left side divided by \(\theta_R V\) tends to zero uniformly in these tests. In nested rectangular partitions, apply this bound to signs of differences between a cell average and its parent average. A union bound and a telescoping identity then control all spatial scales. Applied to a product of two local zero-sum spin tests in different replicas, it forces the expectation of that product to vanish in every equilibrium state of the replicated system: finite-volume restrictions of such a state have relative entropy per site tending to zero with respect to the product Gibbs state.

Obtaining (8) is the spatial part of the proof. Sections 2–[sec:spatial-comparison] use exchange cycles and a field of strength \(b_R\) in a perpendicular direction to bound \(R^{-5/2}V^{-1}\langle D_f^2\rangle_{b_R}\) by a negative power of \(R\). The field gives a killed one-particle kernel an invertible resolvent. A sparse-pin estimate controls the holes left after exposing unpinned cycles; a comparison on this random set then uses the zero sum of \(f\) in each cell. Section 5 turns the resulting variance bound into (8) through a zero-free complex strip.

The remaining argument identifies the magnitude. Negative squares of large-box magnetization averages would distinguish states of different lengths, but the corresponding perturbed Gibbs operators do not directly fit the exchange zero-freeness argument. Section 7 realizes these perturbations as first-order averages of random products of real matrices acting on the \(2S\) spin-\(\tfrac12\) slots representing each site. Let \(Z(z)\) be the normalized trace of such a product with complex field \(z\). For every realization, \(Z(z)\) is nonzero on \(\mathop{\mathrm{Re}}z>0\). Shared Gaussian variables couple its moments to replicated pressures; the zero-sum cross terms just described disappear from their pressure derivatives. Moment bounds then select harmonic limits of normalized \(\log|Z(z)|\). Analyticity of their restrictions to the real axis excludes coexistence of different ergodic magnetization lengths. More concretely, an ergodic state of length \(a\), oriented along the auxiliary field, contributes the affine function \(xa-a^2\) to a limiting pressure tangent. Large positive \(x\) selects the branch \(xm-m^2\); analytic continuation of that branch cannot dominate a competing branch of smaller length at every positive \(x\). The spatial estimate, the replica argument, and the passage from random zero-free functions to harmonic pressure tangents are the main additional constructions of this paper.

Conventions

We suppress \(S\) in expectations and usually suppress \(\beta,L\) when no confusion is possible. Write \(X,Y,Z\) for the site spin components \(1,2,3\), in the standard representation where \(X,Z\) are real and \(Y\) is imaginary. Set \(H_0=-\beta H_{S,L}\). Constants \(c,C\) may depend on the fixed spin and on fixed geometric choices; \(C_\beta\) may also depend on the fixed inverse temperature. We first choose a sufficiently large finite \(\beta\), and only then choose lower thresholds for \(L\) and spatial scales. All torus volume limits are through even \(L\).

The finite exchange inputs are developed in Section 2, and the deterministic lattice estimates imported from the companions are stated with their hypotheses where they are used. Positivity of \(m\) follows directly from our torus sparse-pin estimate in Corollary 11; no sharp low-temperature asymptotic is needed. We increase \(\beta_0(S)\) as needed below.

Symmetric slots and permutation layers

Put \(\ell=2S\) and let \(\mathcal I=\Lambda_L\times\{1,\ldots,\ell\}\) be the set of \(N=\ell V\) slots. On \((\mathbb C^2)^{\otimes\mathcal I}\) write \(s_i^a\) for the Pauli matrices divided by two, and let \[P_* =\prod_{x\in\Lambda_L}\frac1{\ell!} \sum_{\pi\in\mathfrak S_\ell}\mathcal U_{\pi,x}\] be the projection onto the tensor product of the symmetric site subspaces. Here \(\mathcal U_{\pi,x}\) permutes the slots at \(x\). On this subspace \(S_x^a=\sum_{i:\,x_i=x}s_i^a\) is the spin-\(S\) representation. In particular, if compression is needed for an individual slot operator, then \[ P_*s_i^aP_* = \ell^{-1}S_{x_i}^aP_*. \tag{9}\]

For each site bond use all \(\ell^2\) slot edges above it. On these edges put independent Poisson exchanges of rate \(1/2\) during \([0,\beta)\), and at the seam from \(\beta\) to \(0\) insert an independent uniform slot permutation at every site. Let \(\pi\) be the resulting permutation of \(\mathcal I\) and \(\mathbb P_{\rm b}\) its underlying picture law. If \(\mathcal E\) is the set of slot edges, then \[ e^{H_0}P_*=e^{\beta|\mathcal E|/4} \mathbb E_{\rm b}\mathcal U_\pi, \qquad H_0=-\beta H_{S,L}. \tag{10}\] On the left \(H_0\) is lifted to the slot space before compression. Indeed, \(\boldsymbol s_i\cdot\boldsymbol s_j=\frac12\mathcal U_{(ij)}-\frac14I\), and the mean chronological Poisson product has generator \(\frac12\sum_{e\in\mathcal E}(\mathcal U_e-I)\). The all-slot sum above each bond commutes with \(P_*\), while the seam average inserts \(P_*\). This proves (10). This is Tóth’s exchange representation (Tóth 1993), with the higher-spin construction of Nachtergaele (Nachtergaele 1994, sec. 2.2); see also (OpenAI 2026a, Proposition 2.1).

If \(A\) is a polynomial in the physical site spins, let \(\widehat A\) be its slot lift obtained by replacing \(S_x^a\) by \(\sum_{i:\,x_i=x}s_i^a\). The physical trace is then \[ \mathop{\mathrm{Tr}}_{\rm phys}(Ae^{H_0}) =\mathop{\mathrm{Tr}}_{\rm slot}(\widehat A e^{H_0}P_*). \tag{11}\] Following lines through repetitions of one sampled picture gives its cycles. In this slot trace with diagonal spin insertions a cycle has a constant up or down color. Without insertions its contribution is two. We take slot-time points immediately after an instantaneous operation, including the seam. In finite volume there are almost surely finitely many marks, with distinct times avoiding any prescribed finite set of cuts.

We call a permutation law on a finite set admissible if it is a distributional limit of finite products of independent Bernoulli transpositions and deterministic permutations. Products here are ordered; no commutation is assumed. This class includes finite-time Poisson exchanges at arbitrary nonnegative rates, uniform permutations on specified subsets, and independent compositions of these operations. For the Poisson assertion, divide time into small intervals and apply each edge’s Bernoulli indicator of at least one mark in a fixed order. The approximation agrees with the picture unless an interval contains at least two marks, whose probability tends to zero. For a uniform permutation, run positive-rate exchanges on every pair of the given subset and let time tend to infinity. This irreducible random walk on the finite permutation group converges to its uniform invariant law.

We use the following finite stability facts. A nonzero polynomial is stable if it does not vanish when all variables have positive imaginary parts. For every admissible law of a permutation \(\pi\) of a finite set \(I\), \[ \mathbb E\prod_{i\in I}(u_i+v_{\pi(i)}) \quad\hbox{is stable}. \tag{12}\] Identification of variables preserves stability; differentiation, real specialization, and bounded-degree coefficientwise limits preserve it up to the zero polynomial. Finally, if a random subset \(A\subseteq I\) has a stable generating polynomial \(G_A(\boldsymbol z)=\mathbb E\prod_{j\in A}z_j\), then for every \(J\subseteq I\), \[ \Pr(J\subseteq A)\le\prod_{j\in J}\Pr(j\in A). \tag{13}\] These are the partial-symmetrization and negative-dependence facts in (Borcea et al. 2009, Theorems 4.20 and 4.9); the exact finite formulations and their elementary proofs appear in (OpenAI 2026b, Lemmas 3.2–3.6). In particular, the facts apply on a disjoint union of time layers, without any spatial assumptions.

Pins and exposure

The finite constructions in this subsection allow independent Poisson exchanges on an arbitrary finite slot graph, together with finitely many prescribed deterministic permutations, Bernoulli transpositions, and uniform permutations on prescribed subsets at fixed times. All random operations are independent under the base law. Their order at coincident prescribed times is fixed, and a slot-time point at such a time lies after the entire ordered bundle of operations. Thus extra stationary lines and extra instantaneous exchanges are included.

A pin is a specified slot-time point required to have up color. For a finite deterministic pin set \(E\), let \(n_E\) be the number of cycles avoiding \(E\), let \(\mathcal B_E\) be their union, and define \[ \mathcal Z_E=\mathbb E_{\rm b}2^{n_E},\qquad \mathbb P_E(d\omega) =\mathcal Z_E^{-1}2^{n_E(\omega)}\mathbb P_{\rm b}(d\omega). \tag{14}\] Cycles meeting pins are up; the others are colored independently and fairly. For a nonempty single-time set \(T\) disjoint from \(E\), put \(P=E\cup T\) and let \(\sigma\) take each pin in \(P\) to the next pin on its directed cycle, at strictly positive elapsed time. Set \[ r(E,T)=\frac1{|T|}\sum_{i\in T} \mathbb P_P\{\sigma(i)\in T\}. \tag{15}\] The same sampled picture is repeated in defining \(\sigma\).

To describe exposure, label lines at time zero and reveal all labels on cycles missing \(P\), their trajectories through a period, and all their endpoint images. The trajectories record their positions before and after each elementary operation, including intermediate stages of an ordered bundle at one time. Denote these data by \(\mathbf a\). The slots not occupied by exposed labels are called holes. For fixed exposure, define a fresh law \(\mathbb Q_{\mathbf a}\) on the holes as follows. Exchanges whose two endpoints are holes have their original independent Poisson clocks. At a transposition exchanging an exposed label with a hole, that hole moves into the vacated slot. These forced transports are deterministic bijections between successive hole layers. A partly revealed uniform permutation is completed by a uniform bijection between the remaining source and target slots; a prescribed deterministic permutation restricts to a deterministic bijection between those hole layers. An elementary Bernoulli transposition touching an exposed label has its outcome revealed; one touching only holes retains its independent Bernoulli law. Let \(G_P\) denote the event that every cycle of this fresh picture meets \(P\).

The conditional next-pin first moment

The conditioning in Lemma 2(ii) couples entire cycles. The next result explains exactly why its next-pin first moment can nevertheless be computed by a walk with fresh transitions. Matrices in this subsection use the column-forward convention: \(\Pi e_i=e_{\pi(i)}\).

Apply the lemma at a fixed exposure by cutting at all pin times and at the finitely many prescribed changes in hole sets. Identify the hole layers, which have equal cardinality. The unconditioned interval maps are independent admissible bijections; their direct sum followed by the cyclic shift of the layers is an admissible permutation of the layered set. Its mean matrix gives the hole walk: jumps at the original edge rates (\(1/2\) for physical edges), prescribed deterministic restrictions and forced transports, fresh residual uniform choices at each prescribed uniform operation (including the seam), and fresh hole-only Bernoulli outcomes. Repeat this environment periodically, making independent transition choices on every passage, including later passages through the same physical time interval. If \(\tau_T^+\) is the first positive visit to \(T\) and \(\tau_E\) the first visit to \(E\), then \[ |T|r(E,T)=\mathbb E_P\sum_{i\in T} \Pr^{\rm walk}_{\mathbf a,i}(\tau_T^+<\tau_E). \tag{24}\] This follows by Lemma 3 under \(\mathbb Q_{\mathbf a}(\,\cdot\mid G_P)\) and then Lemma 2(ii). The identity concerns a conditional first moment. The original repeated permutation paths retain their cycle dependence; it is the determinant identity that replaces their next-pin mean by fresh Markov transitions.

Disk contractions and trace zero-freeness

We next use (12) for complex matrix insertions. In any common qubit basis the polynomial of a permutation layer, evaluated between input kets \((1,x_i)^T\) and algebraic output covectors \((1,y_i)\), without complex conjugation, is \[ \mathbb E\prod_i(1+x_i y_{\pi(i)}). \tag{25}\] It has no zeros when all \(|x_i|,|y_i|<1\). To see this, substitute \[u_i=i\frac{1+x_i}{1-x_i},\qquad v_j=i\frac{1-y_j}{1+y_j}\] in (12). Both families are in the upper half-plane, and \(u_i+v_{\pi(i)}=2i(1+x_i y_{\pi(i)})/ [(1-x_i)(1+y_{\pi(i)})]\); the product of denominators is independent of \(\pi\).

Choose now the \(s^2\) eigenbasis with vectors \((1,\pm i)^T/\sqrt2\) in the \(s^3\) coordinates. The following three slot maps will cover both the field comparison and the later pressure argument.

The next section uses a small positive field to supply pins. The finite identities proved here reduce uniform bounds on their avoidance to estimates for the corresponding hole walk.

Soft pins and the ghost representation

Rotate coordinates so that the field selects the up color. If \(q\in(0,1)\), \(w=1-q=e^{-b}\), its single-slot factor, after removing a scalar, is \[ \mathop{\mathrm{diag}}(1,w)=wI+qP_{\mathrm{up}}. \tag{28}\] Thus we may independently select a pin on each real slot at the cut with probability \(q\), and then weight the exchange picture by \(2^{c}\), where \(c\) counts the cycles missing all pins. Pinned cycles have the up color; the other cycles receive independent fair colors. Expanding (28) proves that the resulting spin law is the Gibbs law with exponent \(H_0+b\sum_xY_x\), after rotating the up axis back to \(Y\).

For the induction we need a fixed pin set. Add one stationary ghost slot for each real slot, pin each ghost at time \(\beta/2\), and, just after the seam at time zero, independently transpose each real slot with its ghost with probability \(q\). The time-zero layer is taken after these switches. A selected switch inserts the corresponding ghost line into the real cycle; that cycle then contains a ghost pin. An unselected ghost remains a separate pinned cycle and contributes weight one. Consequently every switch configuration has exactly the weight of the corresponding soft-pin configuration. This remains true with any finite set of deterministic pins on real slot points. Figure 1 illustrates the two cases.

Let \(E\) denote all the ghost pins together with any finite set of deterministic real-slot pins. We write \(\mathbb P_E\) for the normalized picture law with density proportional to \(2^{c_E}\), and \(\mathcal B_E\) for the slot-time points on cycles missing \(E\). These cycles contain only real slots. At coincident prescribed operations, a slot point means the point just after the entire ordered bundle, as in Section 2. Distances between sites are wrapped supremum distances, denoted \(d_L\); balls are denoted by \(B_h(z)\).

Fix \(\alpha=1/100\) and a sufficiently large constant \(C_s\ge6\). A finite collection of real slot points is sparse if its site coordinates, counted with multiplicity, satisfy \[ \#\{i:x_i\in B_h(z)\}\le C_s(1+h)^\alpha \qquad(z\in\Lambda_L,\ h\ge0). \tag{29}\] The same definition applies to lists of centers. In particular, different slots at the same site count separately. The freedom to choose \(C_s\) large will permit the finite collections of nearby test points used in Section [sec:spatial-comparison].

The proof is a finite downward induction on deterministic pins. Its larger-pin hypothesis gives regular hole graphs; diffusion and a flow-energy estimate then make next-pin returns rare enough to remove the test pins by Lemma 2.

If \(T\) is empty the assertion is immediate; if it meets \(E\), the event is impossible. We henceforth consider nonempty \(T\cap E=\varnothing\), put \(P=E\cup T\), and expose the cycles missing \(P\) under \(\mathbb P_P\). All other slot points are called holes. Lemmas 2 and 3 apply to this enlarged picture. Here are the details concerning ghosts.

For any proposed set of exposed time-zero real labels, reveal its trajectories, its seam images, and the switch outcomes encountered by those trajectories. Closure into cycles avoiding \(P\) requires every such switch to be absent. Separate Poisson clocks according to whether they touch the currently exposed labels, as in the exposure disintegration of Lemma 2. Conditional on the revealed histories, the unused clocks are fresh on pairs of holes; the remaining seam images are uniform completions within each site; and switches on the other real–ghost pairs are still independent Bernoulli switches. When an exposed label jumps through a hole, that hole is transported in the opposite direction. A mark between two exposed labels does not move a hole. Requiring these to be all the missing cycles conditions the fresh picture on every remaining cycle meeting \(P\). The exposed cycles contribute only their fixed factor \(2^{c_P}\); therefore this description gives the conditional law, including its all-hit conditioning. That conditioning has positive probability at almost every exposure.

Every ghost is a hole and is pinned. Missing real cycles never use a switch, so the number of real holes is constant in time. The seam does not change their number at any site. In particular the factorial-mark estimate in Lemma 2 remains valid: inserting a real-edge transposition changes the number of missing cycles by at most one, even in the presence of switches.

By Lemma 3, conditional next-pin probabilities are first-hit probabilities for the unconditioned fresh walk, with fresh transitions on each repeated traversal. Delete the other real-pin stops and kill that walk on entering a ghost. This only increases the probability that its next pin belongs to \(T\): after entering a ghost, the walk encounters its pin before its next opportunity to return to a real slot. The one-turn real-hole kernel at the time of \(T\) is \[ U=wK,\qquad U_+=U,\quad U_-=U^*,\quad Q=\tfrac12(U+U^*), \qquad E_y=\langle y,(I-Q)y\rangle\ge q\lVert y\rVert_2^2. \tag{31}\] Here and throughout this section matrices are column-forward. The matrix \(K\) is doubly stochastic: its factors are symmetric hole-walk heat kernels, forced transport bijections, and uniform remaining seam completions. Exactly one switch layer is crossed per turn, and its real-to-real mean is \(wI\), which proves \(U=wK\). All factors, their adjoints, and spatially killed versions contract counting \(1\)-, \(2\)-, and supremum norms. For example, a nonnegative matrix whose row and column sums are at most one satisfies \(\sum_i|(Av)_i|^2\le\sum_{i,j}A_{ij}|v_j|^2\le\lVert v\rVert_2^2\). This also proves the last inequality in (31).

The deterministic geometric input

We state precisely the infinite-lattice results that will be transferred to the torus. An available-slot graph on \(\mathbb Z^3\) has between zero and \(\ell\) slots at each site, all \(\ell\) slots outside a finite set, and all pairs of available slots joined across every nearest-neighbor site bond. A site with no available slot is blocked. Join blocked sites at supremum distance at most eight and write \(\mathcal C(w)\) for the component of a blocked site \(w\). Fix \(\nu=1/300\). The constants \(M_*\) and \(A_0\) below depend only on this lattice geometry and \(\ell\).

A graph is regular at \(z\) with radius \(A\ge A_0\) if

1. every ball \(B_{M_*}(v)\), with \(v\in B_{11A}(z)\), contains fewer than six blocked sites;

2. each blocked \(w\notin B_{10A}(z)\) satisfies \(\max_{v\in\mathcal C(w)}|v-w|_\infty \le(1+|w-z|_\infty)^\nu\).

The anchor itself need not have an available slot. One may take \(M_*\ge\max\{48,\rho(3,44)+2\}\), where the local-connectivity radius \(\rho(3,44)\) has the following property: two unblocked sites at distance at most 44 can be joined inside \(B_{\rho(3,44)}\) about the first site if that ball contains fewer than six blocked sites. The path has length at most \((2\rho(3,44)+1)^3\). This is (OpenAI 2026b, Lemma 5.5 and Equation (5.13)); regularity is (OpenAI 2026a, Definition 3.2), equivalently (OpenAI 2026b, Definition 5.3). Take \(A_0\ge M_*+47\) large enough for the following two conclusions.

These are (OpenAI 2026a, Proposition 3.3 and Lemma 3.4); the underlying geometric constructions are (OpenAI 2026b, Propositions 5.6 and 6.2). Their hypotheses are deterministic and impose no relation between \(A\) and a time scale. Only these stated conclusions of the infinite-lattice results will be used here.

At a given exposure and physical time, make such a graph for each torus anchor \(z\): lift the real holes in \(B_{\lfloor L/4\rfloor}(z)\) to \(\mathbb Z^3\), and declare every slot outside this box available. We say the slice is regular at \(z\) when this graph is regular. We shall use \[ A=\lceil C_{\mathrm{loc}}\beta\rceil, \qquad 11A+M_*<\lfloor L/4\rfloor, \tag{34}\] where \(C_{\mathrm{loc}}\) is fixed sufficiently large below. Thus the Nash inequality for functions supported in \(B_A(z)\) uses exactly the torus edges, with zero values beyond the support.

From grid pins to regular slices

Choose a finite circular grid \(\mathcal G\) containing zero, with successive gaps at most \(p^{12}\). For a set \(P\) of ghost pins and real pins on this grid, let \(\mathsf I(P)\) assert (30) under \(\mathbb P_P\) for every sparse test at one grid time. In this subsection we assume \(\mathsf I(P)\). This is the hypothesis for a larger pin set in the eventual downward induction; no estimate for a smaller pin set is assumed.

At time \(t\), write \(\mathcal O_t\) for the sites without a real hole. At a grid time, simultaneous blocking of prescribed sparse sites costs at most \(p^n\): choose one fixed slot at each site and apply \(\mathsf I(P)\). If any chosen slot is pinned, the event is impossible. The same conclusion can therefore be used without checking disjointness separately.

Cutting off the flows

Localized diffusion controls a direct one-turn return. A return after many turns will instead be controlled by an energy inequality. The following finite-torus form of the sparse-source estimate isolates the only error introduced by replacing a pinned boundary with ghost killing.

Return energy and completion of the induction

We now prove Proposition 6. Fix a grid and argue by downward induction over its finite set of real slot points. When every such point is pinned, \(\mathsf I(E)\) is immediate. Assume the assertion for every strict superset of \(E\), and choose a nonempty sparse grid test \(T\) disjoint from \(E\). Then \(P=E\cup T\) satisfies \(\mathsf I(P)\), so all the preceding blocking and diffusion estimates hold under \(\mathbb P_P\).

At an exposure under that law put \(m_T=|T|\) and \(\zeta=\mathbf 1_T\). Let \(F=1\) on \(T\), and elsewhere let \(F\) be the probability that the ghost-killed walk with kernel \(U\) ever hits \(T\) at a later cut. Explicitly, start with \(F_0=\zeta\) and iterate \(F_{n+1}=1\) on \(T\), \(F_{n+1}=U^*F_n\) off \(T\). These vectors increase to \(F\), with \(0\le F\le1\) and \(U^*F=F\) off \(T\). Set \(g=F-\zeta\), so \(0\le g\le1\) and \(g=0\) on \(T\).

Let \(r(E,T)\) be the mean, over \(\mathbb P_P\) and the uniform starting point in \(T\), of the indicator that its next pin is in \(T\). The pin inequality of Lemma 2 is \[ \mathbb P_E(T\subseteq\mathcal B_E)\le(2r(E,T))^{m_T}. \tag{41}\] Conditionally on the exposure, deletion of the other real-pin stops bounds the sum of these next-pin probabilities by \(\sum_{i\in T}(U^*F)_i\). Harmonicity off \(T\) gives \[\begin{align*} \sum_{i\in T}(U^*F)_i &=m_T-\langle F,(I-Q)F\rangle\\ &=\langle\zeta,Q\zeta\rangle+2\langle Q\zeta,g\rangle-E_g. \tag{42}\end{align*}\] Indeed the quadratic forms of \(U,U^*,Q\) agree on real vectors, and \(\langle\zeta ,g\rangle=0\). In particular no reversibility of \(U\) has been assumed.

The first term is controlled by one-turn diffusion. Start a unit mass at each \(i\in T\) and localize at its site, in either direction. Lemma 9 bounds discarded mass in expectation by \(Ce^{-c\beta}\). On a good exposure there are at most \(C_s(1+A)^\alpha\) target points of \(T\) in its localized region, each receiving at most \(C\beta^{-3/2}\). A nongood exposure has probability at most \(100\Delta_A\) and still carries total mass at most one. Thus \[ \mathbb E_P\langle\zeta,Q\zeta\rangle \le Cm_T\bigl(e^{-c\beta}+\Delta_A+A^\alpha\beta^{-3/2}\bigr). \tag{43}\]

To absorb the linear term in (42), split the traversal in direction \(d\in\{+,-\}\) at elapsed time \(u\in I=[\beta/3,2\beta/3]\): \[U_d=B_d(u)C_d(u),\qquad Y_d(u)=B_d(u)^*g.\] The factors use matching hole layers at physical time \(t_d(u)=\theta+u\) or \(\theta-u\) modulo \(\beta\), where \(\theta\) is the cut of \(T\). Each instantaneous factor is assigned to exactly one side; the finitely many exceptional split times do not affect the integrals. As \(u\) decreases, \(Y_d(u)\) accumulates adjoint factors from the endpoint. Its continuous norm loss is twice its Dirichlet energy, and instantaneous factors contract. Consequently \[ \int_I\mathcal D_{t_d(u)}(Y_d(u))\,\mathrm du \le\tfrac12\bigl(\lVert g\rVert_2^2-\lVert U_d^*g\rVert_2^2\bigr) \le E_g. \tag{44}\] The second inequality follows from the exact identity \[E_g-\tfrac12(\lVert g\rVert_2^2-\lVert U_d^*g\rVert_2^2) =\tfrac12\lVert g-U_d^*g\rVert_2^2,\] because \(\langle g,U_d^*g\rangle=\langle g,Qg\rangle\). Also \(\lVert Y_d(u)\rVert_\infty\le1\) and \(\lVert Y_d(u)\rVert_2\le\lVert g\rVert_2\).

For each \(i\in T\), propagate its unit mass by the starting portion \(C_d(u)\), with localization at \(z_i=x_i\), and retain it only when the slice is regular at \(z_i\). Call the resulting sum \(M_d(u)\), and put \[J_d=|I|^{-1}\int_I\langle M_d(u),Y_d(u)\rangle\,\mathrm du.\] The original pairing \(\langle C_d(u)\zeta,Y_d(u)\rangle=\langle U_d\zeta,g\rangle\) is constant in \(u\). Supremum contraction, localization, and the deterministic-time bound (38) show that \[ \mathbb E_P\left|\langle U_d\zeta,g\rangle-J_d\right| \le Cm_T(e^{-c\beta}+\Delta_A). \tag{45}\] This uses Fubini and only regularity at the split, without requiring a good exposure over the entire turn.

Apply Lemma 10 to the retained masses. Cauchy–Schwarz in \(u\), (44), and \(E_g\ge q\lVert g\rVert_2^2\) give the pointwise bound \[ |J_d|\le C\left((m_TA^\alpha/\beta)^{1/2} +m_TL^{1/2-k}/\sqrt q\right)\sqrt{E_g}. \tag{46}\] Since \(2\langle Q\zeta,g\rangle=\langle U_+\zeta,g\rangle+\langle U_-\zeta,g\rangle\), we complete the square in \(\sqrt{E_g}\) to obtain \[J_++J_--E_g \le C\left(m_TA^\alpha/\beta+m_T^2L^{1-2k}/q\right).\] Combining this with (43) and (45), and dividing by \(m_T\), yields \[ r(E,T)\le C\left(e^{-c\beta}+\Delta_A +A^\alpha\beta^{-3/2}+A^\alpha/\beta +m_TL^{1-2k}/q\right). \tag{47}\]

Every constant so far is independent of the grid, its number of points, and the induction pin set. Sparseness gives \(m_T\le C L^\alpha\); because \(q\ge L^{-5/2}\), the last term of (47) is at most \[C L^{\alpha+7/2-2k}=C L^{-47/100}.\] The remaining terms, with \(A=\lceil C_{\mathrm{loc}}\beta\rceil\), are \(o(p)\) as \(\beta\to\infty\): their polynomial orders are \(\beta^{-12/5}\), \(\beta^{-149/100}\), and \(\beta^{-99/100}\), besides stretched-exponential terms. Choose \(\beta_0\) sufficiently large first. For each fixed \(\beta\ge\beta_0\), choose \(L_*(\beta)\) sufficiently large for (34) and for the last term to make \(2r(E,T)\le p\). Equation (41) then proves \(\mathsf I(E)\), completing the finite induction.

For arbitrary finite extra pins and a test at any deterministic time, choose a grid containing their times and zero, and refine it to mesh at most \(p^{12}\). Uniformity in the grid gives the proposition without a limiting procedure. In particular the blocking, component-tail, and regularity conclusions above now hold under every allowed pin law. All time integrals are measurable: the finite-volume histories have finitely many jumps, their kernels are products of measurable heat and instantaneous matrices, and the hitting function is an increasing limit of finite-horizon probabilities. No choice of the auxiliary flows has to be measurable, since only their deterministic pairing inequality is used.

The susceptibility as a hole resolvent

Condition on \(\mathbf a\). The fresh real-hole motion consists of rate-\(1/2\) jumps across the currently available slot edges, the forced hole transports at exposed jumps, and a uniform completion of the seam permutation within each site. Its one-turn kernel, without killing or conditioning on its cycles, is denoted by \(K\). In this section we use the row-forward convention: \(K_{ij}\) is the probability of going from \(i\) to \(j\). Thus \(K\) is doubly stochastic. Set \[ D=I-wK,\qquad D^{-1}=\sum_{r\ge0}w^rK^r. \tag{49}\] This inverse exists and is entrywise nonnegative. Empty hole sets use empty matrices, determinant one, and sums zero.

Define the bounded real vector \[ u=qD^{-1}f_{\mathcal H},\qquad e_i=\sum_{j\in\mathcal H}K_{ij}(u_i-u_j)^2. \tag{55}\] Since \(qD^{-1}\) is stochastic, \(|u_i|\le1\). Double stochasticity and \(Du=qf_{\mathcal H}\) imply \[ \frac12\sum_i e_i =\langle u,(I-K)u\rangle =\frac q w\bigl(\langle u,f_{\mathcal H}\rangle-\lVert u\rVert_2^2\bigr) \le CqN. \tag{56}\] No reversibility of \(K\) is used here.

We will approximate \(u\) by a constant on the holes in each cell. The missing slots mean that \(f_{\mathcal H}\) no longer has exact cellwise cancellation. The following estimate supplies its replacement.

A graph recording communication between holes

The small traversal energy in (56) must now control spatial variation. A hole can be carried by the exposed trajectories even when the fresh walk makes no jumps. We retain that motion explicitly in a graph on the cut holes.

Ignore fresh walk jumps and follow only forced transports from the cut to the seam. Choose one allowed residual seam matching within each site. The resulting lines, called skeleton lines, give a permutation \(\phi\) of \(\mathcal H\). Define an undirected graph \(\Gamma\) on \(\mathcal H\) with the following edges:

1. an endpoint edge from \(i\) to every possible final cut hole under a residual seam matching, including \(\phi(i)\);

2. a contact edge between \(i,j\) whenever their skeleton lines occupy adjacent sites, with an available slot edge between them, throughout some interval of positive length.

Self-edges may be omitted. All choices of representatives and matching can be made by fixed orderings of the finite sets.

Every edge of \(\Gamma\) also has a spatial realization. An endpoint edge follows one skeleton line and then the seam, which does not change its site. A contact edge follows its first skeleton line to a contact time, crosses one site edge, and follows the second line back to its cut. Such a realization consists of at most two pieces of actual marks in chronological or reverse chronological order, and one extra site edge. This observation will prevent an edge leaving a small obstacle from making a long excursion or winding around the torus.

Coarse obstacles and local communication

Choose \(d=\lceil C_0\beta\rceil\), with the fixed constant \(C_0\) large enough below. Partition each coordinate circle into integer intervals of lengths in \([d,2d]\). Their product boxes are indexed by a three-dimensional coordinate torus. Fine distances refer to site coordinates and coarse distances to these box indices; both use the supremum norm and the periodic convention. A site is blocked if it has no cut hole.

We use the following deterministic input from (OpenAI 2026b, Lemma 5.5), also recalled in (OpenAI 2026a, sec. 3). For each fixed \(a\) there is \(\rho(3,a)\ge a+1\) such that two unblocked sites at distance at most \(a\) can be joined by a nearest-neighbor unblocked path in the ball of radius \(\rho(3,a)\) about the first, of length at most \((2\rho(3,a)+1)^3\), provided that ball contains fewer than six blocked sites. We use \(a=44\) and fix \(M\ge\max\{48,\rho(3,44)+2\}\). The result concerns arbitrary blocked sets; we apply it in local lifts to the lattice. Increase \(C_0\), if necessary, so that \(d\) dominates these fixed geometric constants.

Let \(0<\delta<\min\{1,\beta/2\}\); its value in terms of \(R\) will be chosen only after the comparison is proved. Put \(j_* =\lfloor d/20\rfloor\). Declare a coarse box bad when at least one of the following occurs:

1. some \(M\)-ball centered within fine distance \(10d\) of the box contains at least six blocked sites;

2. starting within that \(10d\)-neighborhood, a site path of \(j_*\) inter-site steps is supplied by distinct actual marks in chronological or reverse chronological order during one period;

3. among marks on edges incident to the \(15d\)-neighborhood, there is a mark within \(\delta\) of \(0\) or \(\beta\), or two distinct marks at times less than \(\delta\) apart.

The last event will be called a short-time flag. Slot compatibility is deliberately not required in (ii); this makes the condition constrain every skeleton trajectory. Join bad boxes at coarse distance at most eight. All probabilities below use the original joint picture law. The bad events may depend on the full picture, even though \(K\) and \(u\) depend only on its exposure.

The first assertion converts ordinary paths through good boxes into energy bounds. The second is the essential control on a hole trapped among bad boxes: any escape through a good boundary must take place locally. We next build those boundaries on a finite portion of the covering lattice, including when the cell under consideration is the whole torus.

Filled hulls and the comparison function

We recall precisely the deterministic hull construction that we import from (OpenAI 2026b, Lemma 6.1 and Section 6.1, Equations (6.2)–(6.3)). For any finite blocked set \(\mathcal Z\subset\mathbb Z^3\), join its points at supremum distance at most eight. For each component \(C\), thicken it to \(A_C=\{z:\mathop{\mathrm{dist}}(z,C)\le4\}\) and fill the finite nearest-neighbor components of \(\mathbb Z^3\setminus A_C\). The resulting hull \(H_C\) is contained in the coordinate bounding box of \(A_C\). The hulls are disjoint or nested, so the maximal ones are disjoint. Their exterior vertex boundaries consist of unblocked vertices and are connected by unblocked nearest-neighbor paths in the unit enlargement of that bounding box. At an edge from \(H_C\) to its complement the exterior vertex has distance five from \(C\) and at least four from every other blocked component.

Choose an exterior-boundary representative \(r_H\) for each maximal hull and define \(\rho(z)=r_H\) on \(H\), fixing other vertices. Apply this map once. Each lattice edge has a route between its two images, using only unblocked vertices and lying in the original edge together with the enlarged boxes of its incident hulls. For completeness, if its endpoints belong to distinct maximal hulls \(H,H'\), the route is \[r_H\longrightarrow v\longrightarrow u\longrightarrow r_{H'} \qquad(u\in H,\ v\in H').\] The first and last portions run along the respective exterior boundaries, and the middle portion reverses the original edge. A representative can lie in another hull, so \(\rho\) is not assumed idempotent. This is why the once-applied convention matters. These inputs concern arbitrary finite blocked sets and require no regular-anchor hypothesis.

Let \(n\ge1\) be an integer parameter and assume \(10^6dn\le R\le L\). For a cell \(B\), lift its coordinate intervals to \(\mathbb Z^3\), and take a coarse rectangular window containing all boxes meeting that lift, padded by \(1000n\) coarse units. Copy the torus box data periodically into the window and declare precisely its bad boxes blocked; outside the window there are no blocked boxes. Call \(B\) usable if every component of this finite blocked set has diameter at most \(2n\). Lemma 16 gives \[ \mathbb P(B\text{ is unusable}) \le C_\beta R^3e^{-cn^{\lambda_1}}. \tag{62}\] To verify the periodic issue in this bound, a component of diameter greater than \(2n\) has, from one of its vertices, a path to distance \(n\). Truncate at its first reach. The path stays within \(n+8\) coarse units of its root; the assumed bound \(10^6dn\le L\) ensures that its projection preserves these distances and gives a torus bad component reaching \(n\). The window has at most \(C_\beta R^3\) vertices, including its periodic copies, which proves the union bound. The hull construction therefore takes place on a finite blocked subset of the covering lattice.

For usable \(B\), apply the preceding lattice hull construction. Each enlarged bounding box has diameter at most \(2n+10\); simple routes in the union of at most two such boxes have length at most \(Cn^3\). All hulls and routes used near \(B\) are strictly inside the padded window. There the finite-set designation “unblocked” means that the corresponding torus coarse box is good. Choose representative cut holes periodically for all good boxes and define on fine sites of the lifted cell \[ U_B(x)=u_{i(\rho(z_x))}, \tag{63}\] where \(z_x\) is the coarse box containing \(x\). Thus \(|U_B|\le1\). This is a function on all sites, even when some sites have no hole.

Figure 2 illustrates the two local comparisons used in this argument: energy is charged at cut starts, and an exit from a hull reaches a nearby surviving boundary route.

Energy, exceptional sets, and the final estimate

The construction has supplied a bounded comparison function on each usable cell. Its lattice gradients are controlled by the total traversal energy, and a hole fails to match that function only near a short-time flag or in a small closed class. We now sum these three contributions and choose the auxiliary scales.

Curvature of a logarithmic Laplace transform

The analytic step uses the following strip lemma. Its positivity is essential: small curvature at the origin will then control curvature on a real interval whose length grows logarithmically with the spatial scale.

A zero-field exponential bound

Comparison through nested partitions

For a rectangular cell \(C\) and an axis \(a\), write \[\overline S_C^a=\frac1{|C|}\sum_{x\in C}S_x^a.\] All averages in a single fixed axis commute. The next proposition compares their ordinary squares; it makes no joint-measurement assertion about the three spin components.

The variational principle and its tangents

For a density matrix \(\rho\), let \(\mathsf S(\rho)=-\mathop{\mathrm{Tr}}(\rho\log\rho)\). If \(Q_n=\{0,\ldots,n-1\}^3\) and \(\omega\in\mathcal S_{\rm TI}\), its mean entropy and unperturbed interaction expectation per site are \[\mathfrak s(\omega)=\lim_{n\to\infty}\frac{\mathsf S(\omega_{Q_n})}{|Q_n|}, \qquad e_0(\omega)=\beta\sum_{r=1}^3 \omega(\mathbf S_0\cdot\mathbf S_{e_r}).\] For a self-adjoint local observable \(a\), let \(A_L=\sum_{x\in\Lambda_L}\tau_x(a)\), interpreted periodically for \(L\) larger than its support, and set \[P_a(t)=\lim_{L\to\infty}\frac1V \log\mathop{\mathrm{Tr}}e^{H_0+tA_L},\qquad t\in\mathbb R.\] The limit uses even \(L\), as elsewhere. Finite range and boundedness make free and periodic boundary conditions equivalent for this limit: the difference of their exponents has norm \(O(L^2)\).

The following standard variational facts are recalled with proofs to specify the state space on which the later tangent is taken; see (Lanford and Robinson 1968a, Theorems 2–3) and (Lanford and Robinson 1968b, Theorem 3).

In the field direction \(Y\), the exponent-normalized pressure is \(\beta p_{S,\beta}(b/\beta)\). Thus \[ m=\max_{\omega\in\mathcal E_\beta}\omega(Y_0). \tag{74}\] Rotation invariance gives the same maximum in every unit direction. Consequently the vector \(\mathbf m(\omega)=(\omega(X_0),\omega(Y_0),\omega(Z_0))\) has length at most \(m\) for every \(\omega\in\mathcal E_\beta\). The face maximizing \(\omega(Y_0)\) has an extreme point, so an ergodic equilibrium state attains \(m\). Section 7 will prove that every ergodic equilibrium state has this same magnetization magnitude.

The following mean-ergodic consequence of extremality is standard for asymptotically commuting lattice translations; see (Kastler and Robinson 1966, Theorem 3).

Products of zero-sum observables in distinct replicas

Take \(j\) copies of the spin system, with superscripts \([r]\) denoting the replica. The unperturbed exponent is \(\sum_{r=1}^jH_0^{[r]}\), and its pressure is \(jP_0\). Every translation-invariant equilibrium state for this interaction has each marginal in \(\mathcal E_\beta\). Indeed, entropy subadditivity between replicas bounds its variational value by the sum of the marginal variational values, each at most \(P_0\); equality of the total forces equality for each marginal. Conversely, products of states in \(\mathcal E_\beta\) are replicated equilibrium states, because their entropies and interaction expectations add.

For real coefficients \(v_i\) on finitely many qubit slots, define \[D_v^a=\sum_i v_i s_i^a, \qquad B_v^a=P_*D_v^aP_*\] as a physical local operator. On a site’s symmetric subspace, \(P_*s_{(x,k)}^aP_*=S_x^a/\ell\). Thus \[ B_v^a=\sum_x c_x S_x^a, \qquad c_x=\frac1\ell\sum_{k=1}^{\ell}v_{(x,k)}, \qquad \sum_xc_x=0\quad\hbox{if }\sum_iv_i=0. \tag{75}\] The next lemma concerns products of these compressed operators in different replicas. It does not identify \(P_*(D_v^a)^2P_*\) with \((B_v^a)^2\).

Thus, for a finite-range replica perturbation that is a sum of single-replica terms and the cross terms in Lemma 25, its pressure tangent depends only on the single-replica terms. The construction in the next section uses this fact to study negative squares of block magnetizations.

A Laplacian construction for the negative squares

Recall that there are \(\ell=2S\) slots per site and that \(P_*\) projects onto the physical spin space. For a real vector \(v=(v_i)\) on finitely many slots, write \[D_v^a=\sum_i v_i s_i^a.\] Our generators will be \(iD_v^2\) and \(D_v^1,D_v^3\), with \(\sum_i v_i=0\). All three are real matrices in the \(s^3\) basis, and their exponentials with real coefficients are products of single-slot matrices in \(SL_2(\mathbb R)\). Squaring \(iD_v^2\) contributes a negative square in the \(Y\)-direction. The other two generators contribute positive squares; the reduction of the static coupling will cancel their unwanted nearest-neighbor coefficients.

All quadratic coefficients below use ordered slot pairs: \[(D_v^a)^2=\sum_{i,j}v_iv_j s_i^a s_j^a.\] For a cube \(C\), let \(L_C^{\rm full}\) be the graph Laplacian of the complete graph on its \(\ell|C|\) slots. Let \(L_C^{\rm bond}\) be the Laplacian with an edge between every pair of slots whose sites are nearest neighbors inside \(C\). Thus an off-diagonal Laplacian entry is \(-1\) on an edge. These matrices are extended by zero when used on a larger slot set.

The bond graph of \(C_B\) is connected. Consequently one can choose \(c_B>0\) so that, for every \(C\in\mathcal C_B\), \[ Q_C=c_B L_C^{\rm bond}-B^{-6}L_C^{\rm full}\ge0. \tag{81}\] Indeed, both matrices annihilate the constant vector, and on its orthogonal complement the bond Laplacian is positive definite while \(L_C^{\rm full}=\ell B^3I\). Decompose \(Q_C\) spectrally as \(\sum_v vv^{\mathsf T}\). Every vector in this sum is real and has zero sum; use the generators \(iD_v^2\) for this list.

To cancel the remaining static reduction in the \(X,Z\) directions, put \(n_{B,W}=(W-B+1)^3\) and, for \(C\in\mathcal C_W\), consider \[ R_C=D_BW^{-6}L_C^{\rm full} -\frac{c_B}{n_{B,W}} \sum_{\substack{C'\in\mathcal C_B\\C'\subseteq C}} L_{C'}^{\rm bond}. \tag{82}\] The constant \(D_B\) can be chosen so that \(R_C\ge0\) for every \(W\ge2B\). Here is a uniform bound. Each slot belongs to at most \(B^3\) small cubes, and its degree in each bond graph is at most \(6\ell\). The sum of the bond Laplacians therefore has norm at most \(12\ell B^3\). Since \(n_{B,W}\ge W^3/8\), the second term in (82) has norm at most \(96c_B\ell B^3/W^3\). On the zero-sum subspace, the first term is \(D_B\ell/W^3\) times the identity. Thus, for example, \[ D_B=96c_BB^3+1 \tag{83}\] suffices. Decompose each \(R_C\) as \(\sum_v vv^{\mathsf T}\) with real zero-sum vectors, and use both generators \(D_v^1\) and \(D_v^3\). Choose the lists on one cube of each size and translate them. The resulting families are translation invariant.

Let \(T_{B,L}\) be the physical compression of the static reduction \[ -c_B\sum_{C\in\mathcal C_B} \sum_{\substack{i,j\text{ slots in }C\\ x_i,x_j\text{ nearest neighbors}}} \sum_{a=1}^3s_i^a s_j^a, \tag{84}\] where the inner sum is ordered. Each unordered site bond belongs to \((B-1)B^2\) translates of \(C_B\), so \[H_{0,t}:=H_0+tT_{B,L} =\bigl(\beta-2tc_B(B-1)B^2\bigr) \sum_{\{x,y\}\text{ bond}}\mathbf S_x\cdot\mathbf S_y.\] For fixed \(B\) this remains ferromagnetic for all sufficiently small \(t>0\), uniformly in \(L>2W\).

Denote all the generators just constructed by \(G_\alpha\). Their quadratic contribution is compressed after squaring: \[ T_{B,L}+\sum_\alpha P_*G_\alpha^2P_* =A_{B,W,L}+C_{B,W}VI \tag{85}\] on physical space, for a real scalar \(C_{B,W}\) independent of \(L>2W\). To check the identity, first consider distinct slots \(i,j\) in the cube \(C\) under consideration. If \(E_C(i,j)\) is the indicator of an edge of the bond graph, the coefficients from a small cube in the \(Y\)-direction are \[\underbrace{-c_BE_C(i,j)}_{\text{static reduction}} +\underbrace{\bigl(c_BE_C(i,j)-B^{-6}\bigr)}_{-(Q_C)_{ij}} =-B^{-6}.\] For either the \(X\)- or the \(Z\)-direction, the contribution from a large cube is \[(R_C)_{ij}=-D_BW^{-6} +\frac{c_B}{n_{B,W}} \sum_{\substack{C'\in\mathcal C_B\\C'\subseteq C}}E_{C'}(i,j).\] Every small cube lies in exactly \(n_{B,W}\) large cubes. After summing over translates, the second term cancels the static reduction, leaving the coefficient \(-D_BW^{-6}\) on each ordered pair in each large cube. These are precisely the off-diagonal coefficients of \(A_{B,W,L}\). The remaining terms have \(i=j\) and are scalar because \((s_i^a)^2=I/4\); translation invariance gives the scalar in (85). In particular, the identity does not replace \(P_*G_\alpha^2P_*\) by \((P_*G_\alpha P_*)^2\), which would in general give different single-replica terms.

Nonvanishing random traces and replicated moments

Fix \(B,W\) and a sufficiently small \(t>0\). For an even \(L>2W\) and an integer \(k\ge1\), take independent real centered Gaussian variables \(\lambda_{r,\alpha}\), \(1\le r\le k\), of variance \(2t/k\). Fix an order of the generators and define a slot-space layer \[U_r(z)=\left(\prod_\alpha e^{\lambda_{r,\alpha}G_\alpha}\right) e^{(tz/k)\sum_i s_i^2}.\] Only the complete layer is compressed. With all products taken in their fixed order, define the normalized random entire function \[ \mathcal Z_{L,k,t}(z)= \frac{e^{-tC_{B,W}V}}{\mathop{\mathrm{Tr}}e^{H_0}} \mathop{\mathrm{Tr}}_{\rm phys}\prod_{r=1}^k \left[e^{H_{0,t}/k}\bigl(P_*U_r(z)P_*\bigr)\right]. \tag{86}\] Gaussian exponential moments ensure that all finite products of these functions are integrable, locally uniformly in their arguments.

Every realization of (86) is nonzero on \(\mathbb H\). Although the pulse supports overlap, their fixed ordered product is still a tensor product of single-slot matrices: factors on different slots commute and can be regrouped without changing the order on any one slot. All pulse matrices on a slot lie in \(SL_2(\mathbb R)\) and hence act as disk automorphisms in the \(s^2\) basis, by Lemma 5. On every slot the remaining factor \(e^{tzs^2/k}\) sends the closed disk strictly inside the disk when \(\Re z>0\). Their composition therefore has the strict mapping property of Lemma 4. To identify the permutation layers, write \(E\) for the full-slot exponential of the static interaction with time step \(1/k\). It commutes with \(P_*\), and cyclicity gives \[\mathop{\mathrm{Tr}}_{\rm phys}\prod_{r=1}^k [e^{H_{0,t}/k}(P_*U_r(z)P_*)] =\mathop{\mathrm{Tr}}_{\rm slots}\prod_{r=1}^k[(EP_*)U_r(z)].\] Each \(EP_*\) has nonnegative exchange rates and a symmetric seam, so it is a permutation layer of the class allowed in that lemma. The identity uses \(P_*^2=P_*\) and does not require \(U_r\) to preserve physical space. This proves samplewise nonvanishing, and in particular \[ z\longmapsto\frac{1}{tV}\log|\mathcal Z_{L,k,t}(z)| \quad\text{is harmonic on }\mathbb H. \tag{87}\]

Complex conjugation has a specific sign here. The static matrices, the projections and all pulses are real in the \(s^3\) basis, whereas \(s^2\) is purely imaginary. Therefore, for the same Gaussian sample, \[ \overline{\mathcal Z_{L,k,t}(z)} =\mathcal Z_{L,k,t}(-\overline z). \tag{88}\] The argument on the right may lie outside \(\mathbb H\); the function is entire, so this causes no difficulty. In particular its second moment uses the pair \((z,-\overline z)\), not \((z,\overline z)\).

We next compute products of traces using replicas, always sharing the Gaussian variables between replicas. For fixed \(j\) and arguments \(z_1,\ldots,z_j\in\mathbb C\), the finite-dimensional product formula gives \[ \begin{split} \lim_{k\to\infty}\mathbb E\prod_{r=1}^j\mathcal Z_{L,k,t}(z_r) &=\mathcal R^{(j)}_{L,t}(z_1,\ldots,z_j)\\ &:=\frac{\mathop{\mathrm{Tr}}\exp\!\left\{ \sum_{r=1}^j\bigl(H_0^{[r]}+tA_{B,W,L}^{[r]} +tz_rM_L^{2,[r]}\bigr) +tA_{{\rm cross},L}^{(j)}\right\}} {(\mathop{\mathrm{Tr}}e^{H_0})^j}. \end{split} \tag{89}\] The limit is locally uniform in the arguments. To verify both the limit and its cross terms, let \(P^{(j)}\) be the product of the physical projections in all replicas. For one shared Gaussian variable, \[\mathbb E\exp\!\left(\lambda\sum_{r=1}^jG_\alpha^{[r]}\right) =\exp\!\left(\frac tk \left(\sum_{r=1}^jG_\alpha^{[r]}\right)^2\right).\] Independence allows these expectations to be taken in the prescribed pulse order. After the outer compression, the first-order term is the following operator on physical replica space: \[ P^{(j)}\left(\sum_{r=1}^jG_\alpha^{[r]}\right)^2P^{(j)} =\sum_{r=1}^j(P_*G_\alpha^2P_*)^{[r]} +2\sum_{r<s}(P_*G_\alpha P_*)^{[r]} (P_*G_\alpha P_*)^{[s]}. \tag{90}\] For an imaginary-direction generator \(G_\alpha=iD_v^2\), the cross product on the right is \[-(P_*D_v^2P_*)^{[r]}(P_*D_v^2P_*)^{[s]}.\] For the other two directions it has the positive sign. Thus \[A_{{\rm cross},L}^{(j)} =2\sum_\alpha\sum_{r<s}(P_*G_\alpha P_*)^{[r]} (P_*G_\alpha P_*)^{[s]}\] is self-adjoint, finite range and translation invariant. Every one of its local summands is a real multiple of a cross term in Lemma 25, since all the vectors \(v\) have zero sum.

Using (85) and distributing the scalar correction in (86) among the \(k\) steps, the expected replicated step is \(I+k^{-1}\mathcal H+O(k^{-2})\), where \(\mathcal H\) is the exponent in (89). Here \(B,W,j,t,L\) are fixed and the error is locally uniform in \((z_1,\ldots,z_j)\). Independence between time steps makes the expected product the \(k\)th power of this deterministic step. The matrix product limit (Trotter 1959) proves (89).

For real \(x_1,\ldots,x_j\), the numerator defining \(\mathcal R^{(j)}_{L,t}\) is positive. The thermodynamic limit followed by the right pressure tangent gives \[ \lim_{t\downarrow0}\lim_{L\to\infty} \frac{1}{tV}\log\mathcal R^{(j)}_{L,t}(x_1,\ldots,x_j) =\sum_{r=1}^j s_{B,W}(x_r). \tag{91}\] Indeed, by (73), the left side is the maximum of the perturbation density over equilibrium states of the unperturbed replicated interaction. Lemma 25 makes the cross density vanish in every such state. Each marginal belongs to \(\mathcal E_\beta\), giving the upper bound by the sum on the right. The product of maximizing single-replica states is an equilibrium state and attains that sum. This proves equality.

We also need an upper bound at complex arguments. For self-adjoint matrices \(A,B\), \[ |\mathop{\mathrm{Tr}}e^{A+iB}|\le\mathop{\mathrm{Tr}}e^A. \tag{92}\] For completeness, apply the product formula to \((e^{A/n}e^{iB/n})^n\) and then Schatten Hölder: its trace norm is at most \(\|e^{A/n}\|_n^n=\mathop{\mathrm{Tr}}e^A\), since the intervening factors are unitary. Taking the limit proves the inequality. Applied to (89), it bounds its absolute value by \(\mathcal R^{(j)}_{L,t}(\Re z_1,\ldots,\Re z_j)\).

The limits in (91) are uniform on compact sets of real arguments in their stated order. Indeed the normalized log traces are \(S\)-Lipschitz in each real argument, uniformly in \(L,t\), by the norm bound \(\|M_L^2\|\le SV\); finite nets promote pointwise convergence to uniform convergence. The same bound holds for their limits. Combining this observation with (88), (89) and (92), we obtain the following useful finite parameter statement. Given a compact \(K\subset\mathbb H\), finitely many positive real points \(x_1,\ldots,x_q\), and \(\epsilon>0\), one can choose \(t>0\) sufficiently small, then an even \(L\) sufficiently large, and finally \(k\) sufficiently large, so that, with \(T=tV\) and \(\mathcal Z=\mathcal Z_{L,k,t}\), \[\begin{align*} \mathbb E|\mathcal Z(z)|^2 &\le \exp\{T(2s_{B,W}(\Re z)+\epsilon)\},\qquad z\in K, \tag{93}\\ \left|T^{-1}\log\mathbb EU-2\sum_{i=1}^q s_{B,W}(x_i)\right| &\le\epsilon, \tag{94}\\ \left|T^{-1}\log\mathbb EU^2-4\sum_{i=1}^q s_{B,W}(x_i)\right| &\le\epsilon, \qquad U=\prod_{i=1}^q|\mathcal Z(x_i)|^2. \tag{95}\end{align*}\] The parameter \(T\) can be required to exceed any prescribed number, by enlarging \(L\) after \(t\) has been fixed. For the last two estimates, use the real argument lists \((x_1,-x_1,\ldots,x_q,-x_q)\) and twice that list, together with the evenness of \(s_{B,W}\). These products are nonnegative random variables at finite \(k\). Their limits at fixed \(t,L\) are strictly positive, so the locally uniform product limit also gives the logarithmic estimates. For the first estimate it gives the required upper bound after the finite-volume parameters have been fixed, even when its exponential right side is small.

This order of choices is essential: the matrix product limit is taken at fixed volume; the pressure limit then precedes differentiation at \(t=0\). The finite estimates are implemented in the reverse order, choosing \(t\), then \(L\), then \(k\). No zero-free assertion about an averaged partition function has been used.

Selecting harmonic logarithms

We isolate the analytic step to explain how moment information retains the samplewise nonvanishing just proved.

All ergodic equilibrium states have the same radius

The negative square in (78) now separates two hypothetical magnetization radii: a state with smaller radius would dominate at small positive \(x\), while the largest radius must dominate at large \(x\). Harmonicity prevents this change.