Result 367, Partial differential equations

The critical dimension for the one-phase Bernoulli problem

Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most n−7n-7 in higher dimensions.

Lean formalization Classification or exact value

The bigger picture

Why it matters

An interface chosen by minimizing an energy can be smooth in low dimensions yet develop singular points in higher ones. The manuscript reports that, for the one-phase Bernoulli problem, the dividing line is dimension seven.

What changes?

The free boundary separates the region where a minimizing function is positive from where it is zero. A one-homogeneous function scales proportionally when distances are scaled; a global minimizer minimizes energy against changes in any bounded region. The manuscript claims that every nonzero such minimizer is flat, with a planar free boundary, in dimensions at most six, but a nonflat example exists in dimension seven. Consequently, interior free boundaries of local minimizers are smooth through dimension six.

What does that help mathematicians do?

For a local minimizer in dimension n at least seven, the reported bound limits the singular set to Hausdorff dimension at most n minus seven. This geometric notion of dimension also describes irregular sets. The bound is sharp, so it cannot be reduced in general. In dimension seven, singular points are locally finite, ruling out interior accumulation rather than merely asserting that the singular set has dimension zero.

Are there practical applications?

The immediate value is foundational: the result specifies where smooth-interface reasoning is justified and where singularities must be accommodated. For researchers studying this energy-minimization problem, it provides precise limits on the exceptional set where smoothness fails. The supplied material does not establish a numerical method or a practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The critical dimension for one-phase Bernoulli minimizers

September 24, 2026 50 pages

We prove that seven is the critical dimension for the one-phase Bernoulli problem: every nonzero one-homogeneous global minimizer in dimensions at most six is flat, while a nonflat one-homogeneous global minimizer exists in dimension seven. It follows that the interior free boundary of a local minimizer is smooth in dimensions at most six. In dimension n ≥ 7, its singular set has Hausdorff dimension at most n−7n-7, and this bound is sharp. In dimension seven, the singular set is locally finite.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026,
  author = {{OpenAI}},
  title = {{The critical dimension for one-phase Bernoulli minimizers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026.pdf}{OAI:The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/367.md.

The critical dimension for the one-phase Bernoulli problem

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The paper identifies seven as the critical dimension for nonflat one-homogeneous global minimizers of the one-phase Bernoulli problem. The linked formalization proves the existence side: in dimension seven there is a nonzero one-homogeneous global minimizer that is not a half-space solution.

The flatness classification in dimensions at most six and the resulting regularity and singular-set bounds are outside this selected statement.

Comparator links

Result Comparator statement
A nonflat one-homogeneous Bernoulli minimizer in dimension seven BernoulliNonflatInSeven.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.