The critical dimension of the one-phase Bernoulli free boundary problem: is it seven?

For nonnegative $v\in H^1(B)$ the one-phase Bernoulli (Alt-Caffarelli) energy is $J(v;B)=\int_B(|\nabla v|^2+\mathbf 1_{\{v>0\}})\,dx$ . By Weiss's monotonicity formula and dimension reduction, the regularity of minimizing free boundaries is governed by the least dimension $d_*$ admitting a nonzero, nonflat, one-homogeneous global minimizer: free boundaries are smooth below $d_*$ and singular sets have dimension at most $n-d_*$ . Caffarelli, Jerison and Kenig settled dimension three, Jerison and Savin (2015) proved flatness through dimension four, so $d_*\ge5$ , and De Silva and Jerison (2009) constructed a nonflat minimizing cone in dimension seven, so $d_*\le7$ , by analogy with the Simons cone for minimal surfaces. Is $d_*=7$ , that is, is every one-homogeneous global minimizer in dimensions five and six flat?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Free boundary problems; Alt-Caffarelli functional
Posed by: Open after De Silva and Jerison (2009) and Jerison and Savin (2015); the manuscript names no single poser
Year posed: —
Years open: —
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 40 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: every nonzero one-homogeneous global minimizer of the one-phase Bernoulli energy in $\mathbb R^d$ , $1\le d\le6$ , is flat, $u=(x\cdot e)_+$ , and a nonflat one-homogeneous global minimizer exists in $\mathbb R^7$ , so $d_*=7$ . Corollary 1.2: the interior free boundary of a local minimizer in $\mathbb R^n$ is smooth for $n\le6$ , its singular set is locally finite for $n=7$ and has Hausdorff dimension at most $n-7$ for $n\ge7$ , sharp by products of the De Silva-Jerison cone. Method: a Hessian-shape test function and vector field on the spherical link, with an exactly verified polynomial inequality. Not shown: classification of stable but non-minimizing cones, two-phase or vector-valued versions, or rigidity of non-homogeneous entire stable solutions.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'The critical dimension for one-phase Bernoulli minimizers' (September 24, 2026). Its key inequality is checked by an exact finite certificate, with Python reproduction scripts shipped in the release.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question: every nonzero one-homogeneous global minimizer in dimensions $1\le d\le6$ is flat and a nonflat one exists in dimension seven, so $d_*=7$ . The new content is flatness in dimensions five and six; existence in dimension seven is De Silva and Jerison's 2009 cone. Lean: lean/ComparatorChallenges/BernoulliNonflatInSeven.json exists with solution module OAI.Analysis.BernoulliCone.Main present, not in the formalization catalogue; the statement read here asserts only the existence of a nonzero, one-homogeneous, nonflat global minimizer in $\mathbb R^7$ , the already known half, and its scope note puts flatness in dimensions at most six outside it. So this entry is unreviewed. The flatness proof rests on a universal tensor inequality verified by an exact finite certificate; the release's reproduction scripts were not run here and, by their own README, do not certify the correspondence between formulas and code.

The critical dimension of the one-phase Bernoulli free boundary problem: is it seven?

What was actually shown

What the AI did

Verification

Sources

Discussion