AI Proved What?

Episode 271, October 2026

A law for magnets, proved at last?

Four AI preprints claim spontaneous magnetization and Bloch’s law for the quantum Heisenberg ferromagnet

Covers result 271 in OpenAI’s catalog: all 4 papers. See its connections to other results and the work it cites.

Play Episode 271: A law for magnets, proved at last? A block of gold arrows pointing up above a gently falling teal curve.

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Chapters

  1. 0:00The question
  2. 1:18The claims
  3. 2:50How the proofs go
  4. 3:32Where it stands

The claim, in plain words

The simplest quantum model of a magnet is Heisenberg’s ferromagnet: spins on a cubic lattice, each coupled to its neighbours so that they prefer to point the same way. The equations treat every direction alike, so the magnet has to choose one spontaneously. In one and two dimensions this cannot happen at any positive temperature; in three, it should. In 1976 Dyson, Lieb and Simon announced a proof, but it contained an error. Their method works for the antiferromagnet with larger spins, where neighbours point opposite ways, but fails for the ferromagnet. The problem stayed on open-problem lists, Simon’s in 1984 and Lieb’s in 1999, and in 2025 Seiringer called it “the famous open problem”. There is also a textbook formula: in 1930 Bloch predicted that at low temperature spin waves reduce the magnetization below its maximum by an amount growing like T3/2.

Result 271 is four OpenAI preprints, 176 pages, dated September 24 and October 5, 2026. The first claims spontaneous magnetization: for nearest-neighbour Heisenberg ferromagnets in every dimension from three up and every spin, at low enough temperature, there is an equilibrium state magnetized to at least a quarter of the maximum. It uses its own formulation of the problem; a companion proves the averaged two-point version in three dimensions. The second claims Bloch’s law with its exact coefficient in three dimensions, for every spin and any finite-range ferromagnetic couplings that connect the whole lattice, taking the infinite volume first, then zero field, then low temperature. A third derives the next term for nearest neighbours, Dyson’s lattice correction at T5/2; in Dyson’s units its coefficient is exactly Dyson’s. A fourth shows that in a large box with no field, the magnetization has a fixed length and a uniformly random direction.

What it doesn’t claim

Dyson’s interaction term or anything beyond the first correction, an explicit temperature range, the Curie temperature, or Bloch’s law outside three dimensions. The order of limits matters: on a 3×3×3 cube at temperature 0.1, free spin waves lose almost nothing; in infinite volume they lose about 1% of the maximum (our calculation).

Where it stands

Posted as preprints
OpenAI, four papers, 176 pages, dated September 24 and October 5, 2026
Peer reviewed
Not yet
Formalized in Lean
In part: the magnetization theorem, not Bloch’s law (docs)

Last checked October 19, 2026. The preprints have not been revised. The magnetization theorem is formalized in Lean, about 25,000 lines with no unproved steps, which we did not build. The three papers on Bloch’s law, the lattice correction and the spherical law are not formalized, and OSVerify has not reviewed any of them. We found no comment from specialists.

How the proofs go

  1. Loops around imaginary time. The proofs split each spin into slots and write the thermal state as loops winding around a circle of imaginary time. Every loop touching the edge of a box is pinned up, which costs only surface energy. The question becomes whether a loop deep inside reaches the edge, and that is a random walk among sparse obstacles, which escapes in three dimensions.
  2. Where Bloch’s coefficient comes from. A return of the walk after n windings contributes in proportion to n−3/2, and summing over windings gives ζ(3/2), the number at the heart of Bloch’s coefficient. The first winding alone gives 38% of it.

The numbers in the video

  • Ours: a random walker on the cubic lattice returns home only 34% of the time; the spin-wave sum converges in three dimensions but grows without bound in two.
  • Ours: Bloch’s coefficient ζ(3/2)/(8π3/2) = 0.0586 and the lattice correction 3ζ(5/2)/(128π3/2) = 0.0056; in Dyson’s units the correction is exactly Dyson’s (3π/4)ζ(5/2).
  • Ours: the free spin-wave loss per site at temperature 0.1 for spin 1/2, from a 3×3×3 cube up to infinite volume.

Read the source

The 4 papers in this result

OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.

Transcript

Read the full script

0:00The question

Cool iron below its Curie point and its atomic spins line up in domains: a magnet. The simplest quantum model is Heisenberg’s ferromagnet: spins on a cubic lattice, each coupled to its neighbours so that they prefer to point the same way. The equations treat every direction alike, so the magnet has to choose one spontaneously. In one and two dimensions this cannot happen at any positive temperature; in three, it should. Proving it has been famously hard. In 1976 Dyson, Lieb and Simon announced a proof, but it contained an error. Their method works for the antiferromagnet with larger spins, where neighbours point opposite ways, but fails for the ferromagnet. The problem stayed on open-problem lists: Simon’s in 1984, Lieb’s in 1999, and in 2025 Seiringer called it the famous open problem. There is also a textbook formula. In 1930 Bloch predicted that at low temperature, spin waves reduce the magnetization below its maximum by an amount growing like temperature to the three halves. Our computer shows why three dimensions matter: a random walker on the cubic lattice returns home only 34% of the time, and the spin-wave sum converges in three dimensions but grows without bound in two.

1:18The claims

Result 271 is four OpenAI preprints, 176 pages, dated September 24 and October 5, 2026. The first claims spontaneous magnetization: for nearest-neighbour Heisenberg ferromagnets in every dimension from three up and every spin, at low enough temperature there is an equilibrium state magnetized to at least a quarter of the maximum. It uses its own formulation of the problem; a companion proves the averaged two-point version in three dimensions. The second claims Bloch’s law with its exact coefficient in three dimensions, for every spin and any finite-range aligning couplings that connect the whole lattice, taking the infinite volume first, then zero field, then low temperature. A third derives the next term for nearest neighbours, Dyson’s lattice correction at the five halves power; our computer confirms that, in Dyson’s units, its coefficient is exactly Dyson’s. A fourth shows that in a large box with no field, the magnetization has a fixed length and a uniformly random direction. Not claimed: Dyson’s interaction term or anything beyond the first correction, an explicit temperature range, the Curie temperature, or Bloch’s law outside three dimensions. The order of limits matters: on a 3×3×3 cube at temperature 0.1, our free spin waves lose almost nothing; in infinite volume they lose about 1% of the maximum.

2:50How the proofs go

The proofs split each spin into slots and write the thermal state as loops winding around a circle of imaginary time. Every loop touching the edge of a box is pinned up, which costs only surface energy. The question becomes whether a loop deep inside reaches the edge, and that is a random walk among sparse obstacles, which escapes in three dimensions. For Bloch’s law, a return of the walk after n windings contributes in proportion to n^(−3/2), and summing over windings gives ζ(3/2), the number at the heart of Bloch’s coefficient. Our computer shows the sum filling up: the first winding alone gives 38%.

3:32Where it stands

The preprints are unrefereed and unchanged since release. The magnetization theorem is formalized in Lean, about 25,000 lines with no unproved steps, which we did not build. The three papers on Bloch’s law, the lattice correction and the spherical law are not formalized, and OSVerify has not reviewed any of them. We found no comment from specialists. If they hold, a 50-year-old open problem is solved in the papers’ own terms, and a 96-year-old textbook law is finally a theorem for this model.

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.