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Spontaneous magnetization of the three-dimensional quantum Heisenberg ferromagnet at low temperature (Lieb's Problem A)

Theorem 1.1: for every $d\ge3$ and $S\in\{1/2,1,3/2,\dots\}$ there is $\beta_0(d,S)$ such that for every $\beta\ge\beta_0$ the zero-field nearest-neighbour dynamics has a translation-invariant $\beta$ -KMS state with $\omega(S_0^z)\ge S/4$ . The proof uses an exchange-cycle (random interchange) representation with boundary pins, stable-polynomial negative dependence and a determinant identity for conditioned walks. Not shown: long-range order in the finite-volume two-point formulation, the critical temperature, or anything in $d\le2$ . The companion manuscripts in the family prove Bloch's law and the spherical magnetization law in $d=3$ .

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Spontaneous magnetization of the three-dimensional quantum Heisenberg ferromagnet at low temperature (Lieb's Problem A) · Heisenberg ferromagnet ordering

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Registry verification: lean checked · announcement · candidate

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Unreleased internal OpenAI model
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Named evidence

VibeMathed record: Spontaneous magnetization of the three-dimensional quantum Heisenberg ferromagnet at low temperature (Lieb's Problem A)

specialist registry · passed

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of 'Spontaneous magnetization in the quantum Heisenberg ferromagnet' read against Lieb's problem as the manuscript cites it. Lean: the challenge lean/ComparatorChallenges/Heisenberg.json (solution_module OAI.MathematicalPhysics.Heisenberg.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml. The statement OAI.Heisenberg.spontaneous_magnetization was read here: for $d\ge3$ and $\ell=2S\ge1$ , finite-volume dynamics converge, and for all large $\beta$ there is a translation-invariant $\beta$ -KMS state with $\omega(S^z_0)$ real and at least $\ell/8=S/4$ . That is the headline claim. Not rebuilt here. The paper itself says it resolves the problem in the KMS-state formulation and that the finite-volume two-point criterion in Seiringer's account is a different formulation it does not address.

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Companion manuscript: Spontaneous magnetization in the quantum Heisenberg ferromagnet

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Bloch's $T^{3/2}$ law for the three-dimensional quantum Heisenberg ferromagnet

witness · pending

paper link supplied by the VibeMathed registry.

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Companion manuscript: The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet

paper · pending

paper link supplied by the VibeMathed registry.

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Companion manuscript: The first lattice correction to Bloch's law

paper · pending

paper link supplied by the VibeMathed registry.

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Lean formalization (Heisenberg spontaneous magnetization)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean formalization (Heisenberg spontaneous magnetization)

lean artifact · pending

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This event attributed to Unreleased internal OpenAI model

Companion manuscript: The first lattice correction to Bloch's law evidence for this event

Bloch's $T^{3/2}$ law for the three-dimensional quantum Heisenberg ferromagnet evidence for this event

Lean formalization (Heisenberg spontaneous magnetization) evidence for this event

Spontaneous magnetization of the three-dimensional quantum Heisenberg ferromagnet at low temperature (Lieb's Problem A) parent of this event

VibeMathed record: Spontaneous magnetization of the three-dimensional quantum Heisenberg ferromagnet at low temperature (Lieb's Problem A) evidence for this event

Companion manuscript: Spontaneous magnetization in the quantum Heisenberg ferromagnet evidence for this event

Companion manuscript: The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet evidence for this event

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