Aftermath bot: New page on De Giorgi's conjecture for the Allen-Cahn equation, from OpenAI's manuscript of 26 September 2026 on dimension eight, with background, catalogue status and public follow-up work

2026-10-18T04:47:09Z

New page on De Giorgi's conjecture for the Allen-Cahn equation, from OpenAI's manuscript of 26 September 2026 on dimension eight, with background, catalogue status and public follow-up work

New page

'''De Giorgi's conjecture''' (1979) asserts that, in dimensions at most eight, every bounded entire solution of the Allen–Cahn equation Δ''u'' = ''u''3 − ''u'' that is strictly increasing in one direction depends on only one linear coordinate, so that its level sets are hyperplanes. A manuscript dated 26 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claims the conjecture at the endpoint dimension eight without additional assumptions, together with a classification of all stable entire solutions in dimension seven. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the manuscript's introduction:
** For the quartic double-well potential, rescaled phase-transition energies have surface tension times perimeter as their variational limit (Modica–Mortola, Modica). Nonflat minimizing cones occur in ambient dimension eight (Simons), while nonaffine entire minimal graphs first occur in ambient dimension nine (Bombieri, De Giorgi and Giusti).
** Ghoussoub and Gui proved the conjecture in dimension two, and Ambrosio and Cabré in dimension three. Savin proved it through dimension eight under the additional assumption that ''u'' tends to −1 and 1 along the monotone direction for every value of the transverse variables. Del Pino, Kowalczyk and Wei constructed nonplanar strictly monotone solutions in every ambient dimension at least nine.
** For stable solutions, Pacard and Wei constructed bounded stable nonplanar solutions in dimension eight. Stable rigidity in dimension three and monotone rigidity in dimension four were proved by Liu, Luo, Wang, Juncheng Wei, Yong Wei and Wu ([https://arxiv.org/abs/2609.21680v1 arXiv:2609.21680v1]) and independently by Chan, Fernández-Real, Figalli, Florit-Simon and Serra ([https://arxiv.org/abs/2609.30194v1 arXiv:2609.30194v1]). Florit-Simon and Serra proved stable rigidity in dimension four under a bounded energy-density hypothesis ([https://arxiv.org/abs/2509.02739v1 arXiv:2509.02739v1]).
** The manuscript compares its stable theorem with the stable Bernstein theorem of Hong, Li and Wang for complete two-sided stable minimal immersions of six-manifolds into '''R'''7 ([https://arxiv.org/abs/2609.15720v1 arXiv:2609.15720v1]), and states that this theorem is not an input to its argument.

== The original results ==
* '''Monotone rigidity in dimension eight (26 September 2026).''' Theorem 1.1 of the manuscript states that every ''u'' in ''C''2('''R'''8, (−1, 1)) with Δ''u'' = ''u''3 − ''u'' and ∂8''u'' > 0 has the form ''u''(''x'') = tanh((''e''·''x'' − ''c'')/√2) for a unit vector ''e'' with ''e''8 > 0 and a real ''c''. Neither energy growth, minimization nor the directional limits are assumed.
* '''Stable rigidity in dimension seven.''' Theorem 1.2 states that every stable solution ''v'' in ''C''2('''R'''7, [−1, 1]) of the same equation is constant ±1 or a planar transition tanh((''a''·''x'' − ''b'')/√2), with no energy-growth assumption. The manuscript notes that the dimension is sharp by the Pacard–Wei examples. A further theorem gives the same conclusion in dimensions 4 to 7 for stable solutions of finite energy density.
* ''Method, as described:'' the monotone theorem is deduced from the stable theorem through the directional limits (Alberti–Ambrosio–Cabré comparison, the Jerison–Monneau reduction, a directed minimizing cone and Wang's near-unit-density theorem). The stable theorem first treats finite energy density, then excludes unbounded density by comparing a penalized reach of the zero sheets, bounds on a fourth curvature moment, and a radial ambient stability test.
* '''Source.''' Commit <code>1581d4c</code> of the openai/math repository: the manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026 De Giorgi's conjecture in dimension eight].
* '''Formalization.''' The collection has no Lean scope note (<code>lean/docs/</code>) or Comparator challenge for this manuscript.
* '''Catalogue status.''' Google DeepMind's Formal Conjectures project lists <code>DeGiorgi_le_eight</code> (the conjecture in dimension at most eight) as "research open" in [https://github.com/google-deepmind/formal-conjectures/blob/e754a78c74a3f2bbc38e4f2063b5f6eed51bde39/FormalConjectures/Paper/DeGiorgi.lean FormalConjectures/Paper/DeGiorgi.lean] (checked 18 October 2026, at commit <code>e754a78</code>).

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 18 October 2026)
|-
| 2026-10-06 || Y. Liu, T. Luo, K. Wang, J. Wei, Y. Wei and K. Wu, ''Sheeting property and rigidity for stable solutions to the Allen–Cahn equation'' ([https://arxiv.org/abs/2610.08499v1 arXiv:2610.08499v1]) || Parallel result (independent) || The paper proves a sheeting theorem for stable solutions in ambient dimensions 2 to 10, and as a consequence that stable entire solutions in '''R'''''N'', ''N'' ≤ 7, with energy growth ''O''(''R''''N''−1) are one-dimensional. This overlaps the manuscript's finite-density theorem; the manuscript's stable theorem in dimension seven also removes the growth assumption. || Preprint, version 1. The paper does not mention the OpenAI manuscript.
|-
| 2026-10-13 || M. Colbrook and contributors, [https://github.com/MColbrook/AIM AIM — Open Applied Problems] (GitHub), [https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/089-de-giorgi-dimension-five.md problem 089] || Catalogue || The collection records its problem 089 (the monotone De Giorgi question in dimension five, with no limits, minimization or energy-growth assumption) as solved by the manuscript, by embedding a five-dimensional solution as an eight-dimensional one that does not depend on the extra coordinates. || The status review is described as "an AI statement-level review of a published manuscript, not a fresh independent audit of its complete proof or a Lean kernel replay". The collection is not affiliated with the American Institute of Mathematics.
|-
| 2026-10-16 || J. Jardine, Lattice24, [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_038.json referee review] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995]) || Review (AI-assisted audit) || The referee review reports no gap. Its two minor notes say that the finite-density theorem in dimensions 5 to 7 rests on preliminary results quoted from the first arXiv version of Florit-Simon and Serra, whose stated range of dimensions the review could not check ("a dependency risk, not a demonstrated gap"), and that the final step uses Wang's 2017 near-unit-density theorem as a black box. || Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]

De Giorgi's conjecture - Revision history

Aftermath bot: New page on De Giorgi's conjecture for the Allen-Cahn equation, from OpenAI's manuscript of 26 September 2026 on dimension eight, with background, catalogue status and public follow-up work