Spontaneous magnetization and thermal decay are undecidable at positive temperature

As the paper states it, in briefSpontaneous magnetization is undecidable at a fixed positive temperature for translation-invariant finite-range classical lattice models with strictly positive rational Boltzmann weights. A computable family on 2304 local states has maximal visible nine-colour Potts magnetization exactly zero or a fixed positive constant, attained in a translation-invariant Gibbs state. Its pressure in a uniform field is uniformly computable, but differentiability at zero field is undecidable with a fixed derivative-jump gap. Rational weights are asserted at zero field. We also prove undecidability of exponential decay of one specified thermal spin correlation, in every Gibbs state, on 512 local states. Both constructions couple critical visible spins to cellular-automaton replicas. Surjectivity cancels the collision corrections; nonsurjectivity gives strict subcritical bond domination and exponential decay of the specified correlation uniformly over states. The interaction range depends on the input, and the branches are fine-tuned at their critical or coexistence points. The results transfer to diagonal quantum interactions and survive a prescribed output-reset noise family, the same construction with reduced retention. Nonsurjective full-state uniqueness and exponential clustering of arbitrary local observables remain open.

• Partly answered Given a translation-invariant Hamiltonian, is it decidable whether the correlations in its thermal state decay exponentially with distance? posed by others

Verification

The construction and the magnetic comparison checked independently by a model from another developer; symbolic, exact and floating-point finite checks, and a further exact-arithmetic checker of the collision identities, the finite covariance comparison and the counterexample constants; an independent refutation-first check of the Potts extension, whose six repairs are in the paper.

Lean

None.

Peer review

Not yet.

Novelty

New, with named prior work. To our knowledge, the first fixed-positive-temperature undecidability result for spontaneous magnetization with strictly positive rational local weights and a fixed order-parameter gap, and the first for exponential decay of a specified thermal spin correlation with the state quantifier ranging over every Gibbs state; the paper calls these priority statements provisional. Prominent earlier many-body undecidability results concern low-energy properties: ground-state observables (Gu, Weedbrook, Perales and Nielsen), the spectral gap (Cubitt, Pérez-García and Wolf), phase diagrams (Bausch, Cubitt and Watson) and a ground-state phase transition at an uncomputable parameter (Purcell, Li and Cubitt). The reduction starts from Kari’s undecidability of two-dimensional cellular-automaton surjectivity; Marcovici and Taati, the closest neighbour, are credited for the noisy-automaton ingredient; the nine-colour input is the discontinuous transition of planar Potts models with more than four colours, due to Duminil-Copin, Gagnebin, Harel, Manolescu and Tassion. The zero-temperature limits studied by Chazottes and Hochman and by Gayral, Sablik and Taati are a different question.

As the paper states, its auxiliary arithmetic upper bound, fixed-template obstruction and L-model proposition were checked during the revision without a further independent-model audit, and the finite checks do not establish infinite-volume comparison, uniqueness, the KMS correspondence or undecidability; those rest on the written proofs.

Every check here is internal. How we check results.

Label checked 3 November 2026.

PDF

papers/thermal-undecidability-v1.0.0.pdf, version 1.0.0, 692,069 bytes, 36 pages.

SHA-256 9a7497baad502f54f88caa8bb3dbbea7a8535fc5711678d824259b927e5efb20

Of paper/main.pdf, 692,069 bytes, in the Zenodo archive; identical in the repository at that commit. The same file is hosted here: papers/thermal-undecidability-v1.0.0.pdf.

DOI

10.5281/zenodo.23924358; all versions 10.5281/zenodo.23924357

Code

github.com/Apsiape/thermal-undecidability

Commit

faf3f70d7289267e258bd68809e84ca76cbb8bda

The commit archived on Zenodo (Apsiape/thermal-undecidability, tag v1.0.0). The main branch has since recorded the DOI.

Licence

Paper and documentation CC BY 4.0; scripts MIT or Apache 2.0, at your option.

None so far.

All corrections, and what counts as one.

@misc{douglas2026thermalundecidability,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {Spontaneous magnetization and thermal decay are undecidable at positive temperature},
  year    = {2026},
  version = {1.0.0},
  doi     = {10.5281/zenodo.23924358},
  url     = {https://doi.org/10.5281/zenodo.23924358}
}

thermal-undecidability.bibAll papers as BibTeX