The isoperimetric profile of the cubic three-torus

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Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are $4\pi/81$ and $1/\pi$, with exactly the adjacent two types minimizing at each transition. (OpenAI’s Lean note)

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Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

The paper states

Theorem 1.1

In The Isoperimetric Conjecture for the Cubic Flat Three-Torus, 24 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

For V\in(0,1), put v=\min(V,1-V). Among finite-perimeter subsets of \mathbb R^3/\mathbb Z^3 of volume V, the minimum perimeter is

I_{\rm unit}(V)= \min\left\{(36\pi)^{1/3}v^{2/3},\ 2\sqrt{\pi v},\ 2\right\}.

For 0<V\le1/2, every minimizing region is, up to an ambient isometry and a null set, one of the following:

\begin{array}{c|c} \text{volume range}&\text{region}\\ \hline 0<V\le 4\pi/81 & \text{a ball of radius }(3V/(4\pi))^{1/3}\\[2pt] 4\pi/81\le V\le1/\pi &\text{a solid circular tube of radius }\sqrt{V/\pi}\\ &\text{about a shortest closed geodesic}\\[2pt] 1/\pi\le V\le1/2 &\text{a slab of width }V\text{ between coordinate tori}. \end{array}

Both listed types occur at each common endpoint of adjacent ranges, and there are no further equality cases. For V>1/2, the minimizing regions are the complements of those at volume 1-V. At zero or full volume the conclusion is trivial.

Reported proof verification: One Lean statement in the release; one independent group reports that Comparator accepted a run for this file at the version shown here.
Papers and explanations: One current paper, readable here as a page beside its PDF.

Papers

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The Isoperimetric Conjecture for the Cubic Flat Three-Torus

24 September 2026, current edition.

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We prove the isoperimetric conjecture for the cubic flat three-torus and classify all minimizers, including the equality cases at the transition volumes $4\pi/81$ and $1/\pi$. The minimizing regions are balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements.

One Lean statement in this family

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`CubicTorus`

The paper states

Theorem 1.1

In The Isoperimetric Conjecture for the Cubic Flat Three-Torus, 24 September 2026

For V\in(0,1), put v=\min(V,1-V). Among finite-perimeter subsets of \mathbb R^3/\mathbb Z^3 of volume V, the minimum perimeter is

I_{\rm unit}(V)= \min\left\{(36\pi)^{1/3}v^{2/3},\ 2\sqrt{\pi v},\ 2\right\}.

For 0<V\le1/2, every minimizing region is, up to an ambient isometry and a null set, one of the following:

The words they share: 1/3, 2/3, 1/2, radius, real, shortest, unit, geodesic, slab, range, perimeter, types, finite, ambient, ball, complements, integer, null, cases, adjacent, tori, isometry, tube, width, listed, equality, sqrt, coordinate, min, closed, one, volume, region, pi, cubic, torus, flat, isoperimetric, circular.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Isoperimetric profile and all minimizers in the cubic flat three-torus

CubicTorus.lean, the release lists it as a supporting statement.

theorem unit_cubic_isoperimetric (V : ℝ) (hV : V ∈ Ioo 0 1) :    (∃ E : Set Torus, IsMinimizerAt V E) ∧    (∀ E : Set Torus, IsMinimizerAt V E →      perimeter E = ENNReal.ofReal (candidateProfile V)) ∧    (∀ E : Set Torus, IsMinimizerAt V E ↔ StandardRegionAt V E) := by  sorry

This 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.

One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program

cert-machine records it as “CERTIFIED”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

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.

AdmissibleField Model.lean, line 31
FinitePerimeter Model.lean, line 46
IsMinimizerAt Model.lean, line 50
CongruentAE Model.lean, line 73
SmallStandardRegion Model.lean, line 77
StandardRegionAt Model.lean, line 83

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

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 or imply the whole advertised result (their term: “full”).

Their note: unit_cubic_isoperimetric for every V ∈ (0,1): a minimizer exists; every minimizer has perimeter candidateProfile V = min((36π)^(1/3)v^(2/3), 2√(πv), 2) with v = min(V,1-V); and minimizers are exactly (up to isometry and null sets, with complements for V > 1/2) balls, tubes about a shortest geodesic, or slabs on the ranges V ≤ 4π/81, 4π/81 ≤ V ≤ 1/π, V ≥ 1/π, with both adjacent types allowed at the transition volumes.

Definitions the group asks readers to compare with the paper: perimeter E := ⨆ F admissible periodic C^∞ field with ‖F‖ ≤ 1, ofReal ∫_{cube ∩ E} div F (De Giorgi perimeter by hand on the cover) and torusVolume := map quotientMap (volume.restrict fundamentalCube): standard

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

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.

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cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization determines the isoperimetric profile and all minimizers in the unit cubic flat three-torus. At every volume $0<V<1$, a minimizer exists, and the minimizers, up to null-set changes, are exactly balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements in the appropriate volume ranges.
The classification includes all equality cases at the transition volumes $4\pi/81$ and $1/\pi$, and every minimizer has the stated candidate-profile perimeter.

AI-drafted comparisons, with no recorded human review

Isoperimetric profile and all minimizers in the cubic flat three-torus CubicTorus
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the isoperimetric profile of the unit cubic flat three-torus with all minimisers (balls, tubes, slabs and complements) and the equality cases.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

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Isoperimetric profile and all minimizers in the cubic flat three-torus `CubicTorus`

One report, 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

Other proofs and exact calculations

These reports examine claims in the papers using HOL Light, a different proof assistant and exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.

Switch volumes 4·PI/81 and 1/PI and the radii there, for any positive constant PI
HOL Light card 354-light
Reported by
Ben Knill, openai-math-hol-cards
Method
Proof of a selected step in HOL Light
Their verdict
“PASS” The report records a proof of a supporting fact.
What it covers
Cross-check of a Lean-covered result; on-screen value
Original report
Their report
The same with HOL’s real π: ties at 4π/81 (radii 1/3, 2/9) and at 1/π (area 2); the ball loses at 1/π
HOL Light card 354-pi
Reported by
Ben Knill, openai-math-hol-cards
Method
Proof of a selected step in HOL Light
Their verdict
“PASS” The report records a proof of a supporting fact.
What it covers
Cross-check; minimality is Lean’s
Original report
Their report
Proposition B.1 of The Isoperimetric Conjecture for the Cubic Flat Three-Torus, 24 September 2026
Reported by
cert-machine
Method
Exact computation
Their verdict
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: Proposition num:scalar-positive (B > 0 on [.64,1.02] u [1.62,2.25], B + S > 0 on [1.02,1.62]), decided independently, and the paper’s knot and gap tables; NOT the geometric reduction to these inequalities”
Original report
Their report

What these papers cite in the release, and what cites them

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Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

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11 references in The Isoperimetric Conjecture for the Cubic Flat Three-Torus

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
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