Long-Range Order in the Quantum Heisenberg Ferromagnet

Long-Range Order in the Quantum Heisenberg Ferromagnet

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Finite-range three-dimensional Heisenberg ferromagnets are claimed to obey the exact $T^{3/2}$ spontaneous-magnetization deficit law, with the dispersion determinant fixing the coefficient; For every positive quantum spin in dimensions $d\geq3$, low-temperature nearest-neighbor Heisenberg ferromagnets are claimed to admit zero-field KMS states with magnetization at least $S/4$; For the nearest-neighbor model, magnetization deficit is claimed equal to ideal-magnon density up to $o(\beta^{-5/2})$, including the stated $\zeta(5/2)$ lattice correction; At low fixed temperature, symmetric zero-field magnetization is claimed to approach a uniformly oriented vector of deterministic positive magnitude equal to the right pressure derivative. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Result 1.

Fix $S\in\{1/2,1,3/2,\ldots\}$. Let $J:\mathbb Z^3\to[0,\infty)$ have finite support, $J(0)=0$, $J(z)=J(-z)$, and support generating $\mathbb Z^3$. For the free-boundary Hamiltonian $H=-\tfrac12\sum_{x,y}J(y-x)\boldsymbol S_x\cdot\boldsymbol S_y$, let $m_{S,J}(\beta)$ be obtained by differentiating the limiting pressure from positive fields and evaluating at field zero. With \[ D_{S,J}=\frac S2\sum_zJ(z)zz^{\mathsf T}, \] the claimed low-temperature law is \[ \lim_{\beta\to\infty}\beta^{3/2}(S-m_{S,J}(\beta)) =\frac{\zeta(3/2)}{8\pi^{3/2}\sqrt{\det D_{S,J}}}. \] The thermodynamic limit precedes removal of the positive field.

Result 2.

For the coupling-one nearest-neighbor spin-$S$ quantum ferromagnet on $\mathbb Z^d$, with \[ H_\Lambda=-\sum_{\{x,y\}\subset\Lambda,\ |x-y|_1=1}\boldsymbol S_x\cdot\boldsymbol S_y, \] the source claims that every integer $d\ge3$ and $S\in\{1/2,1,3/2,\ldots\}$ admit a finite $\beta_0(d,S)>0$. At every finite $\beta\ge\beta_0$, the induced infinite-volume dynamics has a translation-invariant $\beta$-KMS state $\omega$ satisfying \[ \omega(S_0^z)\ge S/4. \] This chooses an ordered equilibrium state; it does not assign a nonzero one-site mean to the rotation-invariant state.

Result 3.

For the three-dimensional coupling-one nearest-neighbor spin-$S$ quantum ferromagnet, define spontaneous magnetization $m_S(\beta)$ by the thermodynamic pressure's right field derivative at zero. Put \[ \varepsilon_S(k)=2S\sum_{j=1}^3(1-\cos k_j),\qquad n_S^{\mathrm{sw}}(\beta)=\int_{[-\pi,\pi]^3} \frac{d^3k}{(2\pi)^3(e^{\beta\varepsilon_S(k)}-1)}. \] For each fixed $S\in\{1/2,1,3/2,\ldots\}$, the assertion is \[ \lim_{\beta\to\infty}(\beta S)^{5/2} [S-m_S(\beta)-n_S^{\mathrm{sw}}(\beta)]=0. \] Thus the full lattice spin-wave integral captures the magnetization deficit through the stated scale.

Result 4.

On the even-side torus $\Lambda_L=(\mathbb Z/L\mathbb Z)^3$, set $V=L^3$, $H_{S,L}=-\sum_{\{x,y\}\text{ nearest neighbors}}\boldsymbol S_x\cdot\boldsymbol S_y$, and $M_L^a=\sum_xS_x^a$. For each fixed $S\in\{1/2,1,3/2,\ldots\}$, the claim provides finite $\beta_0(S)$ so that any finite $\beta\ge\beta_0$ has positive spontaneous magnetization $m=p'_{S,\beta}(0+)$. With zero-field Gibbs expectation, \[ \lim_{L\to\infty,\ L\text{ even}} \left\langle\exp\!\left(V^{-1}\sum_{a=1}^3t_aM_L^a\right)\right\rangle_{\beta,L} =\frac1{4\pi}\int_{S^2}e^{m t\cdot n}\,d\sigma(n) =\frac{\sinh(m|t|)}{m|t|}\qquad(t\in\mathbb R^3). \] The value at $t=0$ is one. The same limit gives $V^{-2}\sum_{x,y}\langle\boldsymbol S_x\cdot\boldsymbol S_y\rangle\to m^2$.

Outline of the argument

Pinned exchange returns are compared with the one-magnon walk. Uniform sparse-pin and long-return remainder bounds isolate finite returns, whose small-momentum asymptotics produce the Bloch coefficient; volume, zero-field derivative, and temperature limits are taken in that order. Spin slots and exchange loops translate a simultaneous pin test into return-path estimates. Sparse obstacles are bypassed by local diffusion flows, and heat-flow leakage bounds close a pinning bootstrap, producing a magnetized thermodynamic equilibrium state.

Capacity comparison controls pinned return cycles on a period. Sparse-obstacle propagation and an ideal-potential profile identify comparison sources whose cancellation improves the error beyond the leading Bloch term, allowing the next dispersion coefficient to survive. Exchange-cycle pins and zero-free traces control torus comparisons at vanishing field. Removing the field across spatial scales and suppressing replica cross terms identifies the moment law; pressure tangents determine its radius without assigning joint measurements to noncommuting components.

Formal verification

For family 271, the selected theorem constructs low-temperature zero-field magnetized KMS states for all dimensions at least three and all positive spins. The Bloch expansion, lattice correction, and spherical moment law are separate unselected companions. The listed Comparator files are Heisenberg.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

• OpenAI, Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions, 2026-10-05; citation key OAI:Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, Spontaneous magnetization in the quantum Heisenberg ferromagnet, 2026-09-24; citation key OAI:Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, The first lattice correction to Bloch's law, 2026-10-05; citation key OAI:The-first-lattice-correction-to-Blochs-law-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet, 2026-10-05; citation key OAI:The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 271: Lean scope.
• Comparator Heisenberg.json.
• Comparator Heisenberg.lean.
• Family 271: Reasoning summary.
• OpenAI catalogue entry.

Long-Range Order in the Quantum Heisenberg Ferromagnet

Definitions and mathematical statement

At each site of ${\mathbb{Z}}^d$, spin-$S$ operators act on ${\mathbb{C}}^{2S+1}$, satisfy the angular-momentum commutation relations, and have $\mathbf S_x^2=S(S+1)I$. The finite-volume Hamiltonian is $H_\Lambda=-\sum_{\langle x,y\rangle\subset\Lambda}\mathbf S_x\cdot\mathbf S_y$. An infinite-volume equilibrium state is understood in the quantum Gibbs/KMS sense of the cited model. For each $d\ge3$ and positive half-integer $S$, the assertion is that some finite $\beta_0(d,S)$ satisfies \[ \forall\beta\ge\beta_0\text{ finite},\quad \exists\text{ translation-invariant zero-field equilibrium state }\omega_\beta, \quad\omega_\beta(\mathbf S_0)\ne0. \] The state is allowed to break rotational symmetry; its rotation-invariant mixture need not have nonzero magnetization. The coupling is fixed to one.

Formulation and concept references: [S1].

Short English statement

At sufficiently low positive temperature, must the quantum Heisenberg ferromagnet in every dimension at least three possess an equilibrium state with spontaneous magnetization?

Sources

• S1 Long range order for the quantum Heisenberg model, contributed27January1999. https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9901.HeisenbergFerr.html. Catalogue formulation source.
• E1 M. Correggi, A. Giuliani, and R. Seiringer, Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet, Communications in Mathematical Physics 339 $2015$, 279--307, Theorem 2.1, Section 3, and Proposition 5.2. https://arxiv.org/pdf/1312.7873v2; https://doi.org/10.1007/s00220-015-2402-0. Published result; arXiv version 2 is dated 10 August 2019. Consulted 24 September 2026.
• E2 E. R. Speer, Failure of reflection positivity in the quantum Heisenberg ferromagnet, Letters in Mathematical Physics 10 $1985$, 41--47, abstract, Introduction, and the finite-volume counterexamples. https://sites.math.rutgers.edu/~speer/homepage/pubs/LMP_10_41.pdf. Its specified regimes are not replaced here by a claim about every temperature or every possible reflection. Consulted 24 September 2026.
• E3 D. Ueltschi, Random loop representations for quantum spin systems, Journal of Mathematical Physics 54 $2013$, 083301; author manuscript arXiv:1301.0811, Theorem 3.2, the spin/loop correspondence of Section 3, and Section 7.1, especially equation $7.1$. https://arxiv.org/pdf/1301.0811. The spin-$1/2$ ferromagnetic representation is used, not the different models for which this paper proves long-range order. Consulted 24 September 2026.
• E4 B. Nachtergaele and R. Sims, Lieb--Robinson bounds in quantum many-body physics, in R. Sims and D. Ueltschi (eds.), Entropy and the Quantum, Contemporary Mathematics 529, American Mathematical Society $2010$, 141--176, Section 3.1, Theorem 3.1 and its proof (norm convergence of bounded-interaction dynamics on local observables). https://arxiv.org/pdf/1004.2086v1. Locality input for the thermodynamic-limit argument; the Gaussian-smearing passage of the KMS identity is given in the text. Consulted 24 September 2026.
• E5 D. Heson, S. Starr, and J. Thornton, Violation of ferromagnetic ordering of energy levels in spin rings for the singlet, arXiv:2307.12773v5, 7 January 2025, abstract and Section 2. https://arxiv.org/html/2307.12773v5. The manuscript distinguishes its largely numerical ordering claim from a rigorous singlet-sector result; neither is used as a proof of the lattice conjecture here. Consulted 24 September 2026.

Long-Range Order in the Quantum Heisenberg Ferromagnet

Definitions and mathematical statement

At each site of ${\mathbb{Z}}^d$, spin-$S$ operators act on ${\mathbb{C}}^{2S+1}$, satisfy the angular-momentum commutation relations, and have $\mathbf S_x^2=S(S+1)I$. The finite-volume Hamiltonian is $H_\Lambda=-\sum_{\langle x,y\rangle\subset\Lambda}\mathbf S_x\cdot\mathbf S_y$. An infinite-volume equilibrium state is understood in the quantum Gibbs/KMS sense of the cited model. For each $d\ge3$ and positive half-integer $S$, the assertion is that some finite $\beta_0(d,S)$ satisfies \[ \forall\beta\ge\beta_0\text{ finite},\quad \exists\text{ translation-invariant zero-field equilibrium state }\omega_\beta, \quad\omega_\beta(\mathbf S_0)\ne0. \] The state is allowed to break rotational symmetry; its rotation-invariant mixture need not have nonzero magnetization. The coupling is fixed to one.

Formulation and concept references: [S1].

Short English statement

At sufficiently low positive temperature, must the quantum Heisenberg ferromagnet in every dimension at least three possess an equilibrium state with spontaneous magnetization?

Research attempt

Current frontier and scope (24 September 2026).

The target is a symmetry-breaking equilibrium state at each sufficiently low positive temperature, not a magnetized finite-volume zero-field trace and not merely an ordered ground state. The original source [S1] distinguishes the quantum ferromagnet from the classical and quantum-antiferromagnetic problems. For the present isotropic model, Correggi--Giuliani--Seiringer prove the leading three-dimensional low-temperature free energy for every fixed spin. With the ground energy shifted to zero, their Theorem 2.1 gives \[ f^0(\beta,S)\sim -\frac{\zeta(5/2)}{8\pi^{3/2}}S^{-3/2}\beta^{-5/2}. \] This is a published free-energy theorem, not a theorem about spontaneous magnetization. [E1] Two method cautions matter. Speer proves failure of the standard reflection-positivity property in specified finite-volume spin-$1/2$ regimes, so that property cannot be imported wholesale from other spin models; this does not rule out every possible infrared argument. [E2] The January 2025 revision of Heson--Starr--Thornton studies energy-ordering reversals on even spin rings, with a largely numerical part and a rigorous singlet-sector result. It does not establish the all-dimensional lattice target. [E5] None of the sources inspected supplies the full assertion in this entry. The calculations below are independent deductions; any claim of their novelty would require a separate literature comparison.

Attack map.

The first route is to differentiate a spin-wave free-energy estimate in a field. It needs error control on the scale of the field itself, not only an asymptotic at zero field. The second route resolves the finite-volume trace into total-spin sectors, removes the rotational zero mode, and estimates the loss of total spin. It needs a volume-uniform bound on that loss. The third route, available directly for $S=1/2$, uses weighted interchange cycles and needs positive density of macroscopic cycles in that weighted model. These routes are related, but none of the missing estimates is a theorem supplied by a formal spin-wave substitution. The second route is pursued first. It produces a precise state-selection lemma and a conditional reduction. I then attempt a strong free-boson comparison that would close the reduction, and disprove that proposed comparison. A spatial twist supplies an obstruction even on the tori approximating the requested lattice, not just on an unrelated graph.

Attempt A: separate total-spin loss from global rotation.

Let $\Lambda_L=({\mathbb{Z}}/L{\mathbb{Z}})^d$, $L\geq3$, with its nearest-neighbor torus graph, and write $N=L^d$. Each unordered bond is counted once. Put \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-6f-70-65-72-61-74-6f-72-73} H^0_L=H_L+dNS^2I =\sum_{\langle x,y\rangle}(S^2I-\mathbf S_x\cdot\mathbf S_y), \quad \mathbf J=\sum_x\mathbf S_x,\quad M=J^3, \quad \mathcal J=\frac{\sqrt{I+4\mathbf J^2}-I}{2}. \end{equation} The largest eigenvalue of $\mathbf S_x\cdot\mathbf S_y$ is $S^2$, by addition of two spins, so $H^0_L\geq0$. The operator $\mathcal J$ has eigenvalue $j$ on a total-spin-$j$ representation, with $0\leq j\leq SN$. Define \[ D=SN I-\mathcal J. \] All three operators $H^0_L$, $\mathcal J$, and $M$ commute. The quantity $D$ measures loss of spin length, whereas $SN-M$ also counts a uniform rotation away from the chosen third axis. This difference cannot be ignored: rotational invariance gives \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-7a-65-72-6f-6d-6f-64-65} \langle M\rangle_{\beta,0,L}=0, \qquad \langle SN-M\rangle_{\beta,0,L}=SN \end{equation} for every finite volume and every $\beta$, including arbitrarily large $\beta$. A claim that the latter occupation is small in the zero-field trace is therefore false before any limiting argument is attempted. Lemma 202.1 (finite-volume field selection). For any finite $\mathrm{SU}(2)$-invariant Hamiltonian on $N$ spin-$S$ sites, and for $\beta,h>0$, define the thermal state using $H^0-hM$. Then \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-73-65-6c-65-63-74-69-6f-6e} \frac{\langle M\rangle_{\beta,h,L}}{N} \geq \frac{\langle\mathcal J\rangle_{\beta,0,L}}{N} -\frac{1}{N(e^{\beta h}-1)}. \end{equation} No reflection positivity or sign assumption on that Hamiltonian is needed for this finite-dimensional statement. Proof. Decompose the Hilbert space as $\bigoplus_j(V_j\otimes\mathcal M_j)$, where $V_j$ is the spin-$j$ irreducible representation. Schur's lemma gives $H^0=\bigoplus_j(I_{V_j}\otimes H_j)$. Set $w_j=\operatorname{Tr}_{\mathcal M_j}e^{-\beta H_j}\geq0$ and $x=\beta h$. The spin character is \[ \chi_j(x)=\sum_{m=-j}^j e^{xm} =e^{jx}\frac{1-e^{-(2j+1)x}}{1-e^{-x}}, \qquad \frac{\chi_j'(x)}{\chi_j(x)} =j-\frac1{e^x-1}+\frac{2j+1}{e^{(2j+1)x}-1} \geq j-\frac1{e^x-1}. \] The zero-field law of $j$ is proportional to $(2j+1)w_j$. The field reweights it by $a_j(x)=\chi_j(x)/(2j+1)$. For fixed $x>0$ this is nondecreasing in $j$ along the common integer or half-integer parity: passing from $j$ to $j+1$ adds two endpoints whose average is $\cosh((j+1)x)$, no smaller than the previous average. For any probability weights $p_j$, \[ \operatorname{Cov}_p(j,a_j) =\frac12\sum_{j,k}p_jp_k(j-k)(a_j-a_k)\geq0. \] Consequently the mean of $j$ cannot decrease under this reweighting. Averaging the character derivative gives $\langle M\rangle_{\beta,h,L}\geq \langle\mathcal J\rangle_{\beta,0,L}-(e^{\beta h}-1)^{-1}$. Division by $N$ proves the assertion.$\square$

The conditional reduction, including the equilibrium-state step.

For fixed $d,S,\beta$, consider the quantitative condition \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-48-57} \liminf_{L\to\infty} \frac{\langle\mathcal J\rangle_{\beta,0,L}}{N} \geq m_\beta>0. \tag{HW} \end{equation} Proposition 202.2. Condition (ra30006979:HW) implies the existence of a translation-invariant zero-field $\beta$-KMS state with $\omega_\beta(S_0^3)\geq m_\beta$. Proof. Choose $h_L=N^{-1/2}$. Equation (ra30006979:selection), together with $e^x-1\geq x$, gives \[ \langle S_0^3\rangle_{\beta,h_L,L} =\frac{\langle M\rangle_{\beta,h_L,L}}{N} \geq\frac{\langle\mathcal J\rangle_{\beta,0,L}}{N} -\frac1{\beta\sqrt N}. \] The equality uses torus translation invariance. By successive extraction on local matrix algebras, these states have a subsequence converging on every local observable. Positivity and consistency give a state on the quasi-local algebra. It is translation invariant and has the asserted spin expectation. Here is why the limit is an equilibrium state, rather than merely a positive functional. For bounded finite-range interactions the finite-volume dynamics converge in norm on local observables, uniformly on compact real-time intervals; this is the locality estimate proved in [E4], Section 3.1, Theorem 3.1. Periodic boundary bonds make no difference on any fixed local algebra in this limit: their distance from that algebra tends to infinity in the same locality estimate. Since $[H_L^0,M]=0$, adding $-h_LM$ composes the zero-field dynamics with an on-site spin rotation of angle $-h_Lt$. On an observable supported on $X$, its difference from the identity is bounded by $2S|X|\,|h_Lt|\|A\|$. Thus these field-dependent dynamics also converge to the zero-field dynamics, denoted $\alpha_t$. To justify passage of the KMS condition without asserting convergence at imaginary time for an unsmeared local observable, fix $a>0$ and set \[ B_{L,a}=\sqrt{a/\pi}\int_{{\mathbb{R}}} e^{-at^2}\alpha_t^{L,h_L}(B)\,\mathrm d t. \] These elements and their analytic translates converge in norm to the corresponding $B_a$ and $\alpha_z(B_a)$, because \[ \alpha_z^{L,h_L}(B_{L,a}) =\sqrt{a/\pi}\int_{{\mathbb{R}}} e^{-a(t-z)^2}\alpha_t^{L,h_L}(B)\,\mathrm d t \] has an integrable, volume-independent majorant for each fixed $z$. The finite-volume trace identity \[ \omega_L\left(A\alpha_{i\beta}^{L,h_L}(B_{L,a})\right) =\omega_L(B_{L,a}A) \] therefore passes to the limit. Gaussian-smoothed observables form the usual dense analytic test set for the KMS condition; allowing local $A,B$ and then their norm limits proves that the limiting state is $\beta$-KMS. This establishes the required equilibrium property at the fixed finite $\beta$.$\square$ In particular, it would suffice to prove, for each fixed $d\geq3$ and $S$, some $\rho_{d,S}(\beta)<S$ such that \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-64-65-66-65-63-74-67-6f-61-6c} \limsup_{L\to\infty}\frac{\langle D\rangle_{\beta,0,L}}{N} \leq\rho_{d,S}(\beta) \qquad\text{for every finite }\beta\geq\beta_0(d,S). \end{equation} Then (ra30006979:HW) holds with $m_\beta=S-\rho_{d,S}(\beta)$. This is a sufficient condition, not an unproved equivalence between a pointwise correlation limit and the original statement. Notice the order and the quantifiers: $\beta$ is fixed during the thermodynamic limit; a ground-state limit taken first does not suffice, and no monotonicity in $\beta$ has been assumed.

Attempt B: spin waves suggest a bound, but do not prove it.

For the quadratic spin-wave model with the coupling normalization of this entry, the nonzero momentum energies are \[ \epsilon_S(k)=2S\sum_{i=1}^d(1-\cos k_i),\qquad \rho_{\rm sw}(\beta)= \int_{[-\pi,\pi]^d}\frac{1}{e^{\beta\epsilon_S(k)}-1} \frac{\,\mathrm d k}{(2\pi)^d}. \] The elementary bound $\epsilon_S(k)\geq cS|k|^2$ gives, for $d>2$ and $\beta S\geq1$, \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-62-6f-73-65-64-65-6e-73-69-74-79} \rho_{\rm sw}(\beta) \leq C_d(\beta S)^{-d/2} \int_{{\mathbb{R}}^d}\frac{\,\mathrm d q}{e^{c|q|^2}-1} \leq C_d'(\beta S)^{-d/2}. \end{equation} The integral is finite precisely because near zero its radial integrand behaves like $r^{d-3}$. This calculation explains the favorable dimension threshold of this route. It estimates only the free Bose integral, not the interacting spin expectation. A strong enough proposed estimate would be \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-73-74-72-6f-6e-67-64-65-66-65-63-74} \limsup_{L\to\infty}\frac{\langle D\rangle_{\beta,0,L}}{N} \leq C_{d,S}(\beta S)^{-d/2} \quad(\beta S\geq1). \tag{SD} \end{equation} If (ra30006979:strongdefect) were proved, one could take \[ \beta_0(d,S)=S^{-1}\max\left\{1, (2C_{d,S}/S)^{2/d}\right\} \] and obtain a state with $\omega_\beta(S_0^3)\geq S/2$ for every finite $\beta\geq\beta_0$. Equation (ra30006979:strongdefect) is not proved here. The exact Holstein--Primakoff model has an occupation constraint and attractive interaction terms; the free-energy analysis in [E1], Section 3, does not discard them as an operator inequality giving (ra30006979:strongdefect). I tried a sharper comparison that would have made differentiation legitimate. For a finite torus, omit $k=0$ and let $\epsilon_1=\min_{k\ne0}\epsilon_S(k)$. The candidate was \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-66-61-6c-73-65-6d-67-66} \left\langle z^D\right\rangle_{\beta,0,L} \ \stackrel{?}{\leq}\ \prod_{k\ne0}\frac{1-e^{-\beta\epsilon_S(k)}} {1-ze^{-\beta\epsilon_S(k)}}, \qquad 1\leq z<e^{\beta\epsilon_1}. \end{equation} Both sides equal one at $z=1$. A valid bound of this form in a right-neighborhood of one would bound the first derivative, hence $\langle D\rangle$, by the finite-volume free-boson number. The latter per site tends to (ra30006979:bosedensity) for $d>2$: near zero the Riemann sums are controlled by the summable singularity $|k|^{-2}$. However, the full interval and finite-volume uniformity asserted in (ra30006979:falsemgf) are false, as shown next. Differentiating a false stronger statement cannot establish the desired weaker one.

Stress test I: a slow twist defeats energy coercivity.

Lemma 202.3 (low total spin at vanishing energy density). For each fixed $d,S$ and all sufficiently large $L$, there is a joint eigenvector of $H_L^0$ and $\mathcal J$ with \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-6c-6f-77-73-70-69-6e} j\leq N^{3/4},\qquad E\leq\frac{NS^2(1-\cos(2\pi/L))}{1-S/\sqrt N}. \end{equation} Proof. For every site set \[ \mathbf n_x=(\cos(2\pi x_1/L),\sin(2\pi x_1/L),0), \qquad \Psi_L=\bigotimes_x |S,\mathbf n_x\rangle, \] where $|S,\mathbf n\rangle$ is the spin coherent state with expectation $S\mathbf n$. These states exist by rotating a highest-weight vector. Their product expectations factor at different sites, and $\sum_x\mathbf n_x=0$. Only bonds in the first coordinate direction cost energy. Direct calculation therefore gives \begin{align} \langle H_L^0\rangle_{\Psi_L} &=NS^2(1-\cos(2\pi/L))\leq 2\pi^2S^2N/L^2, \label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-74-77-69-73-74-65-6e-65-72-67-79}\\ \langle\mathbf J^2\rangle_{\Psi_L} &=NS(S+1)+S^2\sum_{x\ne y}\mathbf n_x\cdot\mathbf n_y=NS. \label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-74-77-69-73-74-73-70-69-6e} \end{align} Since $\mathcal J^2\leq\mathbf J^2$, Cauchy--Schwarz yields $\langle\mathcal J\rangle_{\Psi_L}\leq\sqrt{NS}$ and hence $\langle D\rangle_{\Psi_L}\geq SN-\sqrt{NS}$. Already this disproves any operator inequality $H_L^0\geq cD$ with $c>0$ independent of $L$. To obtain an eigenvector rather than only a trial state, let $P=\mathbf1_{\{\mathcal J\leq N^{3/4}\}}$. Markov's inequality and (ra30006979:twistspin) imply \[ \|P\Psi_L\|^2\geq1-S/\sqrt N. \] The operator $P$ commutes with $H_L^0$, and $H_L^0\geq0$; therefore \[ \frac{\langle P\Psi_L,H_L^0P\Psi_L\rangle}{\|P\Psi_L\|^2} \leq\frac{\langle\Psi_L,H_L^0\Psi_L\rangle}{1-S/\sqrt N}. \] The minimum eigenvalue in the range of $P$ is at most this Rayleigh quotient. Simultaneous diagonalization inside that range proves (ra30006979:lowspin).$\square$ The lemma shows why a total energy-density bound cannot on its own force extensive total spin: states with $j/N\to0$ can have $E/N\to0$. It does not show that these states dominate a fixed-temperature trace. Their entropy and their relative thermal weights are exactly what an argument for (ra30006979:defectgoal) must still control. Nor does it exclude a bound with a factor $L^{-2}$, of the type used on boxes in [E1], Proposition 5.2; that vanishing factor is important.

Stress test II: the proposed fugacity comparison fails on the target tori.

On these tori the smallest nonzero quadratic spin-wave energy is \[ \epsilon_1=2S(1-\cos(2\pi/L)). \] For the joint eigenvector in (ra30006979:lowspin), let $D_*=SN-j$. For fixed $S$ and sufficiently large $N$, \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-68-61-6c-66-67-61-70} \frac{E}{\epsilon_1D_*} \leq\frac{1} {2(1-S/\sqrt N)(1-N^{-1/4}/S)}<\frac34. \end{equation} Now fix one such torus and put $z=e^{3\beta\epsilon_1/4}$. It lies in the purported domain of (ra30006979:falsemgf). Since $H_L^0\geq0$, its partition function is at most $(2S+1)^N$, while the single eigenvector contributes \[ \left\langle z^D\right\rangle_{\beta,0,L} \geq (2S+1)^{-N} \exp\{\beta(3\epsilon_1D_*/4-E)\} \longrightarrow\infty \quad\text{as }\beta\to\infty. \] In contrast, every factor on the right of (ra30006979:falsemgf) tends to one, since $\epsilon_S(k)-3\epsilon_1/4\geq\epsilon_1/4$ and there are only finitely many factors. This is an exact contradiction to that finite-volume comparison, for every fixed $d\geq3$ as well as for other dimensions. It is not a contradiction to (ra30006979:strongdefect): the test first fixes the volume and then lets $\beta$ grow, and its fugacity moves far away from one. A much weaker comparison of first derivatives after the correct thermodynamic limit is still conceivable. An exact four-spin check makes the mechanism transparent. A numerical scan of magnetization blocks on cycles of $4,6,8,10$ sites detected states with $E<\epsilon_1D$; the following calculation removes any reliance on that experiment. For a spin-$1/2$ four-cycle, set $\mathbf A=\mathbf S_1+\mathbf S_3$ and $\mathbf B=\mathbf S_2+\mathbf S_4$. Then \[ H^0=1-\mathbf A\cdot\mathbf B, \qquad E=1-\tfrac12\{j(j+1)-a(a+1)-b(b+1)\}. \] Taking opposite pairs to be singlets gives $a=b=j=0$, so $E=1$ and $D=2$, whereas the nonzero quadratic energies are $1,1,2$. The complete spin-addition calculation gives \[ Z=5+7e^{-\beta}+3e^{-2\beta}+e^{-3\beta},\qquad \langle z^D\rangle= \frac{5+(6z+z^2)e^{-\beta}+3ze^{-2\beta}+z^2e^{-3\beta}}{Z}. \] For $z=e^{3\beta/4}$ the same contradiction is explicit. This small cycle is a diagnostic for the comparison, not a counterexample to magnetization on ${\mathbb{Z}}^d$; the preceding twist argument is what carries the comparison failure to the relevant large tori.

Attempt C: a separate loop criterion, with the correct spin restriction.

For $S=1/2$, let $T_{xy}$ interchange the two tensor factors. Then \[ T_{xy}=2\mathbf S_x\cdot\mathbf S_y+\tfrac12I, \qquad H_L^0=\tfrac12\sum_{\langle x,y\rangle}(I-T_{xy}). \] Place independent Poisson transpositions of rate $1/2$ on each edge during $[0,\beta]$, and let $\pi$ be their chronological product. Expansion of the exponential gives $e^{-\beta H_L^0}=\mathbb E_{\rm Pois}T_\pi$. In a spin basis, $\operatorname{Tr}T_\pi=2^{c(\pi)}$, since a basis configuration fixed by $\pi$ is constant on each permutation cycle. Thus the probability law relevant to this quantum model is \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-6c-6f-6f-70-6c-61-77} \,\mathrm d\mathbb P_2(\pi)= \frac{2^{c(\pi)}}{\mathbb E_{\rm Pois}2^{c(\pi)}} \,\mathrm d\mathbb P_{\rm Pois}(\pi), \end{equation} not the unweighted interchange law. This is the Tóth-type representation discussed in [E3], Theorem 3.2 and Section 7.1; the rate $1/2$ here accounts for the coupling convention of this entry. Conditional on the cycle lengths $\ell_i$, independent signs $\sigma_i\in\{-1/2,1/2\}$ give $M=\sum_i\ell_i\sigma_i$ in the diagonal trace. Averaging the signs therefore proves the exact identity \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-6c-6f-6f-70-73-71-75-61-72-65} \langle M^2\rangle_{\beta,0,L} =\frac14\mathbb E_2\sum_i\ell_i^2. \end{equation} For example, if some $\varepsilon,\delta>0$ satisfy \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-39-37-39-3a-6c-6f-6f-70-67-6f-61-6c} \liminf_{L\to\infty}\mathbb E_2 \left[\frac1N\sum_{i:\ell_i\geq\varepsilon N}\ell_i\right]\geq\delta, \end{equation} then (ra30006979:loopsquare) gives $\liminf N^{-2}\langle M^2\rangle\geq\varepsilon\delta/4$. On a spin-$j$ sector, $M^2\leq j^2\leq(SN+1)j$, so \[ \frac{\langle\mathcal J\rangle}{N} \geq\frac{\langle M^2\rangle}{N(SN+1)}. \] For $S=1/2$, condition (ra30006979:HW) follows with $m_\beta=\varepsilon\delta/2$, and Proposition 202.2 supplies the required symmetry-breaking state at that $\beta$. Thus the bridge from the cycle criterion to the state formulation has been proved, not assumed as a pointwise-correlation equivalence. The attempted percolation shortcut fails. An added transposition can join two cycles or split one cycle; two successive transpositions on the same two vertices can undo one another. The event of a long cycle is not an increasing event of the Poisson bridges. Moreover the weight $2^{c(\pi)}$ changes in opposite directions under joining and splitting. A theorem about ordinary percolation or unweighted interchange therefore does not automatically prove (ra30006979:loopgoal). The missing weighted cycle estimate is not supplied here. The larger-spin target has an additional obstruction to this shortcut. For $S>1/2$, the bilinear interaction is not an affine function of the full-site transposition. For example, at $S=1$ the transposition has the same eigenvalue on pair-spin $0$ and $2$, while $\mathbf S_x\cdot\mathbf S_y$ has the different eigenvalues $-2$ and $1$ there. Simply changing $2^{c(\pi)}$ to $(2S+1)^{c(\pi)}$ would change the Hamiltonian. The all-$S$ state-selection reduction remains valid, but the displayed cycle criterion directly addresses only $S=1/2$. [E3], Sections 3.1 and 7.

Final stress test: why a free-energy asymptotic does not close the gap.

If $f_\beta(h)$ denotes free energy per site, magnetization is related to its one-sided slope at zero, not just its value there. Uniform small errors in values need not control this slope. The elementary concave, even functions \[ q_\delta(h)=-S\left(\sqrt{h^2+\delta^2}-\delta\right) \] satisfy $q_\delta(0)=q_\delta'(0)=0$ and approximate $-S|h|$ uniformly within $S\delta$. Nevertheless the limiting right derivative is $-S$, not zero. Thus even a very accurate value approximation may lose the relevant one-sided derivative. In particular, the zero-field asymptotic in [E1] cannot be differentiated into spontaneous magnetization without a separate error estimate at field scale. There is also a mesoscopic obstruction. Nearly aligned spins within every small box need not align their directions between boxes: the slow twist above has this property while its total spin is small. A boxwise spin-wave estimate must be supplemented by a bound on the relative orientations and their entropy. The selection lemma does not supply that bound. Reflection positivity, monotonicity of cycle lengths, or a bosonic partition-function domination is not assumed at the unresolved step.

Outcome.

Outcome: Conditional reduction. The finite-volume field-selection inequality and its zero-field KMS limit prove that (ra30006979:defectgoal), for every sufficiently large fixed finite $\beta$, would imply the exact state-existence target for each $d,S$. A macroscopic-cycle condition gives a second sufficient criterion for $S=1/2$. These are proved implications with explicit normalizations and limit orders; the required defect or cycle bound is still unproved. The counterexample search has a concrete, narrower outcome: coherent twists and an exact four-spin calculation disprove the proposed full fugacity comparison with zero-mode-removed free bosons. The twist also rules out volume-independent energy coercivity in total-spin loss. Neither construction is a counterexample to the original conjecture. What remains is a genuinely interacting, thermodynamic estimate of total-spin loss or weighted macroscopic cycles, not a change of the original all-dimensional, all-spin, positive-temperature statement.

Sources

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