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Bloch’s law, its lattice correction, and the spherical magnetization law

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves Bloch’s T3/2 law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ3. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments

What the papers state

Theorem 1.1

In Spontaneous magnetization in the quantum Heisenberg ferromagnet, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For every integer d\geq3 and every S\in\{\frac12,1,\frac32,\ldots\}, there is a finite \beta_0(d,S)>0 such that, for every finite \beta\geq\beta_0(d,S), the dynamics associated with (1.1) has a translation-invariant \beta-KMS state \omega satisfying

\omega(S_0^z)\ge S/4.

Theorem 1.1 (Bloch’s law)

In Bloch’s Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions, 5 October 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For every fixed S\in\{\tfrac12,1,\tfrac32,\ldots\} and every interaction satisfying (1.1),

\lim_{\beta\to\infty}\beta^{3/2}\bigl(S-m_{S,J}(\beta)\bigr) =\frac{\zeta(3/2)}{8\pi^{3/2}\sqrt{\det D_{S,J}}}.

No Lean statement in the release is paired with this theorem.

Theorem 1.1 (Spherical magnetization law)

In The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet, 5 October 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For each fixed S there is a finite \beta_0(S) such that, for every fixed finite \beta\ge\beta_0(S), one has m=m_{S,\beta}>0 and

\begin{gathered}\lim_{\substack{L\to\infty\\L\text{ even}}} \left\langle \exp\!\left(\frac1V\sum_{a=1}^3t_aM_L^a\right) \right\rangle_{\beta,L} =\frac1{4\pi}\int_{\mathbb S^2}e^{m t\cdot n}\dd\sigma(n) =\frac{\sinh(m|t|)}{m|t|} \qquad(t\in\R^3).\end{gathered}

Here \sigma is surface-area measure on \mathbb S^2, and the quotient at t=0 is interpreted as 1. In addition,

\begin{gathered}\lim_{\substack{L\to\infty\\L\text{ even}}} \frac1{V^2}\sum_{x,y\in\Lambda_L} \langle S_x\cdot S_y\rangle_{\beta,L}=m^2.\end{gathered}

No Lean statement in the release is paired with this theorem.

The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.

Two more possible matches

  • Corollary 1.2 in The first lattice correction to Bloch’s law, 5 October 2026.
  • Theorem 1.1 in The first lattice correction to Bloch’s law, 5 October 2026.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations
Four current papers, each readable here as a page beside its PDF. Two explanations by others.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Bloch’s Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions

5 October 2026, current edition.

Read it herePDF

We prove Bloch’s T3/2 law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin. The result holds for every nonnegative symmetric finite-range interaction whose support generates ℤ3, including spatially anisotropic couplings. The leading magnetization deficit has the exact coefficient determined by the determinant of the quadratic one-magnon dispersion. The thermodynamic limit is taken before the right field derivative at zero, and the low-temperature limit is taken last.

The first lattice correction to Bloch’s law

5 October 2026, current edition.

Read it herePDF

We prove the first lattice correction to Bloch’s law for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at every fixed spin \(S=\tfrac12,1,\tfrac32,\ldots\). The spontaneous-magnetization deficit agrees with the full ideal-magnon density up to \(o(\beta^{-5/2})\). In addition to the leading Bloch term, this gives the correction \(3\zeta(5/2)(\beta S)^{-5/2}/(128\pi^{3/2})\). The magnetization is the right derivative at zero field of the thermodynamic pressure; the volume limit precedes the field derivative, and the low-temperature limit is taken last.

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

5 October 2026, current edition.

Read it herePDF

We prove the spherical magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and every sufficiently low fixed positive temperature. As the even periodic cubes grow, the symmetric zero-field magnetization converges in moments to a uniform direction with a deterministic positive magnitude. This magnitude equals the right derivative at zero field of the infinite-volume pressure. The law is expressed through self-adjoint linear combinations of the spin components and requires no joint measurement of noncommuting observables.

Spontaneous magnetization in the quantum Heisenberg ferromagnet

24 September 2026, current edition.

Read it herePDF

For every dimension d ≥ 3 and every spin \(S\in\{\frac12,1,\frac32,\ldots\}\), we prove that the nearest-neighbor isotropic quantum Heisenberg ferromagnet has a translation-invariant, spontaneously magnetized equilibrium state at every sufficiently low positive temperature. The same state satisfies the KMS condition for the zero-field dynamics and has magnetization at least \(S/4\). This resolves the low-temperature ordering problem in the spontaneous-magnetization formulation.

OpenAI released an abridged summary of the model’s reasoning for this family: Spontaneous magnetization in the quantum Heisenberg ferromagnet

One Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

Heisenberg

The paper states

The words they share: dynamics, kms, hamiltonian, translation, invariant, state, integer, 1/2, eq, finite, spontaneous, magnetization, 3/2.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Low-temperature spontaneous magnetization for every positive spin

Heisenberg.lean, the release lists it as a supporting statement.

theorem spontaneous_magnetization (d ℓ : ℕ) (hd : 3≤d) (hℓ : 1≤ℓ) :    DynamicsConverges d ℓ ∧      ∃ β₀ : ℝ,0<β₀ ∧ ∀ β : ℝ,β₀≤β →        ∃ ω : State d ℓ,TranslationInvariant ω ∧ IsKMS β ω ∧          (ω.functional (spinZ d ℓ 0)).im=0 ∧          (ℓ:ℝ)/8≤(ω.functional (spinZ d ℓ 0)).re := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).

Their note: PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Only spontaneous_magnetization (d ℓ) (hd : 3 ≤ d) (hℓ : 1 ≤ ℓ) : DynamicsConverges d ℓ ∧ ∃ β₀ > 0, ∀ β ≥ β₀, ∃ ω, TranslationInvariant ω ∧ IsKMS β ω ∧ (ω.functional (spinZ d ℓ 0)).im = 0 ∧ ℓ/8 ≤ (ω.functional (spinZ d ℓ 0)).re (nearest-neighbor Heisenberg ferromagnet, every spin S = ℓ/2) is stated, i.e. the summary’s third claim. The headline, Bloch’s T^{3/2} law with its exact coefficient for general finite-range couplings (and the first lattice correction for 3D nearest-neighbor couplings), has no Lean statement (docs: ‘The formalization proves spontaneous magnetization...’).

Definitions the group asks readers to compare with the paper: Entire quantum-spin setup is hand-built: Configuration d ℓ := Site d →₀ Level ℓ (incomplete tensor product over the all-zero reference configuration), QuasiLocal d ℓ := closure of the *-algebra of matrix units, hamiltonian with edges counted via WellOrderingRel, dynamics := Filter.limUnder atTop (finiteDynamics.. (box d n)..) (with convergence asserted separately by DynamicsConverges) and a hand-written IsKMS (analytic strip function with F(t) = ω(A α_t B), F(t+iβ) = ω(α_t B A)). Plausible, but none is Mathlib’s notion of a KMS state or quantum spin system.

“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).

The gap they name: Bloch’s T^{3/2} law with exact coefficient (finite-range couplings) and its first lattice correction (the two October 5 manuscripts, which give the family its title) are not stated; only the September 24 spontaneous-magnetization result is. The spherical magnetization law named in the family title is also absent (it is not a headline clause).

“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization proves spontaneous magnetization for the nearest-neighbor isotropic quantum Heisenberg ferromagnet on $\mathbb Z^d$ for every $d\ge3$ and every spin $S\in\{\tfrac12,1,\tfrac32,\ldots\}$. At every sufficiently low positive temperature, it constructs a translation-invariant equilibrium state satisfying the KMS condition for the zero-field dynamics and having magnetization at least $S/4$.

It also proves convergence of the finite-volume dynamics to the infinite-volume dynamics used in the KMS statement.

AI-drafted comparisons, with no recorded human review

  • Low-temperature spontaneous magnetization for every positive spin Heisenberg

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for d ≥ 3, every spin S and all large β, a translation-invariant KMS state of the quantum Heisenberg ferromagnet with magnetisation at least S/4. The Bloch-law results of the family’s later papers are not part of it.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Low-temperature spontaneous magnetization for every positive spin Heisenberg

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

  • seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source

    Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.

  • Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source

    Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanationCompare the Lean statement with the paperSuggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 271 No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 271 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form