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

What OpenAI claims

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. (Lean)

Papers (4)

1. Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions

   5 October 2026 · Preprint, not peer reviewed · PDF

   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.

2. The first lattice correction to Bloch's law

   5 October 2026 · Preprint, not peer reviewed · PDF

   We prove the first lattice correction to Bloch's law for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at every fixed spin S=1/2,1,3/2,…. The spontaneous-magnetization deficit agrees with the full ideal-magnon density up to o(β^-5/2). In addition to the leading Bloch term, this gives the correction 3ζ(5/2)(β S)^-5/2/(128π^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.

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

   5 October 2026 · Preprint, not peer reviewed · PDF

   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.

4. Spontaneous magnetization in the quantum Heisenberg ferromagnet

   24 September 2026 · Preprint, not peer reviewed · PDF

   For every dimension d ≥ 3 and every spin S ∈ {1/2,1,3/2,…}, 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.