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 \(n-7\) in higher dimensions. (OpenAI’s Lean note)
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In The critical dimension for one-phase Bernoulli minimizers, 24 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Every nonzero one-homogeneous global minimizer of (1.1) in \R^d, 1\leq d\leq6, is flat. There is a nonflat one-homogeneous global minimizer in \R^7. Consequently,
In The critical dimension for one-phase Bernoulli minimizers, 24 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
Let n\ge2, let D\subset\R^n be open, and let u\in H^1_{\mathrm{loc}}(D) be a nonnegative local minimizer of the one-phase Bernoulli energy in D. Its interior free boundary is smooth outside \operatorname{Sing}(u). This set is empty when n\le6, is locally finite when n=7, and satisfies
For every n\ge7, a global minimizer attains this dimension bound.
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BernoulliNonflatInSevenThe paper states
In The critical dimension for one-phase Bernoulli minimizers, 24 September 2026
Every nonzero one-homogeneous global minimizer of (1.1) in \R^d, 1\leq d\leq6, is flat. There is a nonflat one-homogeneous global minimizer in \R^7. Consequently,
The words they share: flat, energy, nonzero, nonflat, homogeneous, minimizer, one, global, bernoulli.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
BernoulliNonflatInSeven.lean, the release lists it as a supporting statement.
theorem nonflat_in_seven : ∃ u : Space 7 → ℝ, GlobalMinimizer u ∧ Nonzero u ∧ OneHomogeneous u ∧ ¬ Flat u := by sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Test Model.lean, line 17WeakGradient Model.lean, line 21H1 Model.lean, line 28H10 Model.lean, line 35H1Loc Model.lean, line 43LocalMinimizer Model.lean, line 55GlobalMinimizer Model.lean, line 66OneHomogeneous Model.lean, line 69Nonzero Model.lean, line 72Flat Model.lean, line 76Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
Their note: nonflat_in_seven: there is u : ℝ⁷ → ℝ that is a nonnegative global minimizer of ∫ |∇u|² + 1_{u>0} (ballwise comparison with H¹ competitors agreeing on the boundary, all balls), nonzero, 1-homogeneous and not a half-space solution max(x·e,0): the existence half of ‘seven is the first dimension’. The other half (flatness of such minimizers in dimensions ≤ 6) and the consequences (smooth free boundaries through dimension six, singular set dimension ≤ n-7) are not stated (docs: ‘outside this selected statement’).
Definitions the group asks readers to compare with the paper: LocalMinimizer/GlobalMinimizer written with hand-built H¹, H¹₀ (L²-approximation by compactly supported smooth functions) and the exact energy ∫ ‖G‖² + if 0 < u x then 1 else 0 with an explicit weak-gradient representative G; flatness, homogeneity and nonvanishing are a.e. statements
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name: Flatness of one-homogeneous global minimizers in dimensions ≤ 6 (the ‘first’ in the headline) and the regularity and singular-set consequences.
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
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.
A nonflat one-homogeneous Bernoulli minimizer in dimension seven BernoulliNonflatInSeven
Unreviewed AI draft: states part of a paper’s theorem. States the existence half of the main theorem: a nonflat one-homogeneous global minimiser in dimension seven. Flatness in dimensions at most six and the singular-set corollary are outside it.
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BernoulliNonflatInSevenTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
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Proposition 4.1 of The critical dimension for one-phase Bernoulli minimizers, 24 September 2026
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Why this passage is shown
Theorem 1.1
In The critical dimension for one-phase Bernoulli minimizers, 24 September 2026
Why this passage is shown
Corollary 1.2 (Sharp singular-set bound)
In The critical dimension for one-phase Bernoulli minimizers, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, BernoulliNonflatInSeven
Proof-verification report
MathVet, BernoulliNonflatInSeven
Report using another proof or calculation method
Proposition 4.1 of The critical dimension for one-phase Bernoulli minimizers, 24 September 2026
Exact calculations for finite claims, cert-machine
9814e4cLean statement
BernoulliNonflatInSeven