Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy
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Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for \(0\lt s\lt 2\) and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy. (OpenAI’s Lean note)
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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.
Six other possible matches are listed below the card, among them Theorem 1.1(Universal energy minimality) in Universal optimality of the triangular lattice and Corollary 1.3(Spherical logarithmic energy) in Triangular minimality for planar Coulomb renormalized energy.
For every smooth nonnegative completely monotone function g on (0,\infty) and every locally finite planar set \mathcal C of centered disk density one,
We prove that the density-one triangular lattice minimizes the lower energy per particle for every nonnegative completely monotone function of squared distance, among all locally finite planar configurations of centered disk density one. The comparison includes infinite energies. The proof constructs sharp Gaussian Fourier minorants using an atomic interpolation certificate.
We prove universal energy minimality for the triangular lattice in the plane. Among locally finite configurations of centered density one, it minimizes the lower limit of centered-ball energy averages for every nonnegative completely monotone function of squared distance, including when the energy is infinite. We also prove triangular minimality for planar logarithmic and Riesz renormalized energies, with \(0\lt s\lt 2\) in the Riesz case, and the corresponding jellium minima. The proof uses sharp Gaussian Fourier bounds, positive mixtures, and heat-kernel comparison.
We resolve the planar Cohn–Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density \(\pi/(2\sqrt3)\). We construct a radial Schwartz certificate and prove its global sign conditions using rigorous interval arithmetic and analytic estimates. The same certificate recovers the classical uniqueness of the triangular packing among periodic equality cases.
We prove the Sandier–Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background. Combined with Bétermin and Sandier’s asymptotic formula, this also proves the Brauchart–Hardin–Saff conjecture for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere. The proof uses a direct Voronoi-cell comparison with rigorous interval arithmetic.
Four Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
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AtomicGaussian.lean, the release lists it as a supporting statement.
theoremsharp_gaussian_minorants_atomic (α : ℝ) (hα : 0 < α) : ∃ f : SchwartzMap Plane ℝ, Radial f ∧ (∀ x, f x ≤ gaussian α x) ∧ (∀ w, (realFourier f w).im = 0 ∧ 0 ≤ (realFourier f w).re) ∧ (∀ a ∈ A, a ≠ 0 → f a = gaussian α a) ∧ (∀ w ∈ dualA, w ≠ 0 → realFourier f w = 0) ∧ (1 ≤ α → Construction.HasAtomicConstruction α f) := bysorry
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.
PlanarPacking.lean, the release lists it as a supporting statement.
theoremsharp_fourier_certificate : ∃ f : SchwartzMap SharpPlanar.Plane ℝ, SharpPlanar.SharpCertificate f := bysorry
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.
Atlas has no report of an independent group using Comparator to verify OpenAI’s proof of this Lean statement at the version shown here. Who ran the proof-verification program
TriangularEnergy.lean, the release lists it as a supporting statement.
theoremuniversal_energy_minimum (g : ℝ → ℝ) (C : Set Plane) (hg : AdmissiblePotential g) (hC : LocallyFinite C) (hd : DensityOne C) : latticeEnergy g ≤ energy g C ∧ latticeEnergy g = energy g A := bysorry
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.
TriangularGaussian.lean, the release lists it as a supporting statement.
theoremsharp_gaussian_minorants_with_construction (α : ℝ) (hα : 0 < α) : ∃ f : SchwartzMap Plane ℝ, Radial f ∧ (∀ x, f x ≤ gaussian α x) ∧ (∀ w, (realFourier f w).im = 0 ∧ 0 ≤ (realFourier f w).re) ∧ (∀ a ∈ A, a ≠ 0 → f a = gaussian α a) ∧ (∀ w ∈ dualA, w ≠ 0 → realFourier f w = 0) ∧ (1 ≤ α → AtomicConstruction α f) := bysorry
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.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither 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:Universal minimality of the triangular lattice for lower-limit energy per particle is stated; the renormalized Riesz (0<s<2) and Coulomb minimality, the Sandier-Serfaty and Brauchart-Hardin-Saff (spherical log energy) claims are not.
Definitions the group asks readers to compare with the paper:Custom energy/diskEnergy (liminf of per-particle disk energy in ℝ≥0∞, with diskPoints := if h : finite then h.toFinset else ∅), DensityOne, latticeEnergy := ∑' over nonzero lattice points; AtomicGaussian/TriangularGaussian carry a huge bespoke ‘atomic construction’ (explicit rational constants such as h = 17/50, H = 27/50, residue list, 20x23 jet matrices) inside the conclusion HasAtomicConstruction α f, unreadable as a statement
“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 — UNLINKED”).
The gap they name:Triangular minimality for unit-background renormalized Riesz energies (0<s<2) and for the planar Coulomb renormalized energy, hence the Sandier–Serfaty and two-dimensional Brauchart–Hardin–Saff conclusions. These come from two of the family’s four manuscripts (‘Universal optimality of the triangular lattice’ and ‘Triangular minimality for planar Coulomb renormalized energy’), neither of which the scope note links. PlanarPacking (Cohn–Elkies sharpness) is an extra result the headline does not mention.
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The formalization proves that the density-one triangular lattice minimizes lower energy per particle among all locally finite planar configurations of centered-disk density one, for every nonnegative completely monotone function of squared distance. Infinite energies are allowed in the comparison.
It also constructs sharp radial Schwartz minorants for every Gaussian potential: each minorant lies below the Gaussian, has nonnegative real Fourier transform, agrees with the Gaussian at nonzero triangular-lattice points, and vanishes on nonzero dual-lattice points after Fourier transformation. For the stated parameter range, the formalization includes the explicit atomic interpolation construction.
The planar Cohn–Elkies sharpness conjecture asks whether the two-point Fourier method attains the optimal circle-packing density. The formalization constructs a radial Schwartz function $f$ on $\mathbb R^2$ with $\widehat f(0)=1$, $f(0)=2/\sqrt3$, $\widehat f$ real and nonnegative everywhere, and $f(x)\le0$ whenever $\|x\|\ge1$. These are the sharp Fourier certificate conditions yielding density $\pi/(2\sqrt3)$. The separate uniqueness statement for periodic equality cases is outside this selected theorem.
AI-drafted comparisons, with no recorded human review
Sharp Gaussian minorants from an atomic certificateAtomicGaussian
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s sharp Gaussian minorant theorem for every α > 0, including the atomic interpolation construction for α ≥ 1.
Sharp Fourier certificate for planar circle packingPlanarPacking
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the packing paper: a radial Schwartz function meeting the sharp Cohn–Elkies conditions for density π/(2√3). The uniqueness statement for periodic equality cases is outside it.
Universal energy minimality of the triangular latticeTriangularEnergy
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s universal energy theorem: the density-one triangular lattice minimises lower energy per particle for every nonnegative completely monotone potential among configurations of centred-disk density one.
Gaussian Fourier minorants with their constructionTriangularGaussian
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s sharp Gaussian minorant theorem for every α > 0 with the construction for α ≥ 1; it differs from AtomicGaussian only in how the construction is packaged.
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.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Sharp Gaussian minorants from an atomic certificateAtomicGaussian
Two 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.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Lemma 4.1 of An atomic certificate for triangular-lattice universal optimality
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: every inequality of Lemma lem:finite-certificate (the two 20x20 interval blocks and their inverses, the parameter polytope, the jet / curvature / Taylor envelopes, all 659 Bernstein row minima) and the printed rational consequences; not the analytic interpolation, sign propagation or energy-transfer arguments, hence not the universal-optimality theorem”
Proposition 3.3 of Universal optimality of the triangular lattice
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 prop:finite-certificate through its three stronger tables (27 power, 30 column, 37310 Bernstein and 10 tail comparisons) on enclosures of the exact quadrature arrays, the same tables on an exact execution of the floor-rounded procedure of Appendix C, the exact node constants, and the scalar budgets; not the analytic interpolation, sign-propagation, energy-transfer or mixture arguments, hence not the theorem”
Lemmas 3.2 and 4.1 of A sharp Fourier certificate for planar circle packing
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: every finite certificate the proof cites -- Lemma interpolation:finite (the two inverse certificates via W_+-, the exterior block, the 204 residuals, the table norms) and Lemma signs:bernstein (2436 Bernstein coefficients), recomputed with an independent quadrature and error bound, the exact node data and envelope constants, and every scalar substitution of certificates.tex:487-572; NOT the theorem, whose infinite inversion, sign argument and normalisation are analytic”
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component -- the whole finite certificate of Section sec:computation (Proposition comp:finite-soundness), by an independent route: the 38 lattice coefficients (two ways) and |c_1| < 1; lambda, mu, K, B(l), f_0(l), f(l), -pi B(l) - lambda/2 (closed-form/Fejer route) each inside its saved pair; the exterior signs comp:tail-signs (K < 0 included); the finite-core premise; the global derivative bounds L_1..L_4; the outer primitive assertion (2 g_d(x) > K off int D_0) and the negative-part assertion on the paper’s 2186 x 3612 grid; the Hessian assertions on the 601 x 401 cells of D_0; the printed arithmetic of the truncation sections. Not the headline: analytic, not finite: the reduction (sec:reduction), the geometry and dual identity (sec:geometry), Lemma cal:stationarity (used by the Hessian step), the truncation estimates of sec:tails (their printed arithmetic is checked in part 7; the sweeps add their 10^-11 query error, which is conditional on the finite-core premise decided here), Corollary cor:bhs. The paper’s own radial table (midpoint propagation) and its K-integral G_ss mesh are not reproduced: the table is built here by another route (closed forms on [b, R_0], Taylor integration on [R_0, l]) and K by Fejer quadrature.”
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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Explain a result, compare statements or propose a correction
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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 “Career-defining publication”. The page says that nothing on it is verified.
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the planar packing result to Green’s problem 42, recording a dilation by two.
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Run Comparator on OpenAI’s proof
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Identify the theorem the catalogue describes
Family 090The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Lemma 4.1 of An atomic certificate for triangular-lattice universal optimality
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: every inequality of Lemma lem:finite-certificate (the two 20x20 interval blocks and their inverses, the parameter polytope, the jet / curvature / Taylor envelopes, all 659 Bernstein row minima) and the printed rational consequences; not the analytic interpolation, sign propagation or energy-transfer arguments, hence not the universal-optimality theorem”
Proposition 3.3 of Universal optimality of the triangular lattice
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: Proposition prop:finite-certificate through its three stronger tables (27 power, 30 column, 37310 Bernstein and 10 tail comparisons) on enclosures of the exact quadrature arrays, the same tables on an exact execution of the floor-rounded procedure of Appendix C, the exact node constants, and the scalar budgets; not the analytic interpolation, sign-propagation, energy-transfer or mixture arguments, hence not the theorem”
Lemmas 3.2 and 4.1 of A sharp Fourier certificate for planar circle packing
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: every finite certificate the proof cites -- Lemma interpolation:finite (the two inverse certificates via W_+-, the exterior block, the 204 residuals, the table norms) and Lemma signs:bernstein (2436 Bernstein coefficients), recomputed with an independent quadrature and error bound, the exact node data and envelope constants, and every scalar substitution of certificates.tex:487-572; NOT the theorem, whose infinite inversion, sign argument and normalisation are analytic”
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component -- the whole finite certificate of Section sec:computation (Proposition comp:finite-soundness), by an independent route: the 38 lattice coefficients (two ways) and |c_1| < 1; lambda, mu, K, B(l), f_0(l), f(l), -pi B(l) - lambda/2 (closed-form/Fejer route) each inside its saved pair; the exterior signs comp:tail-signs (K < 0 included); the finite-core premise; the global derivative bounds L_1..L_4; the outer primitive assertion (2 g_d(x) > K off int D_0) and the negative-part assertion on the paper’s 2186 x 3612 grid; the Hessian assertions on the 601 x 401 cells of D_0; the printed arithmetic of the truncation sections. Not the headline: analytic, not finite: the reduction (sec:reduction), the geometry and dual identity (sec:geometry), Lemma cal:stationarity (used by the Hessian step), the truncation estimates of sec:tails (their printed arithmetic is checked in part 7; the sweeps add their 10^-11 query error, which is conditional on the finite-core premise decided here), Corollary cor:bhs. The paper’s own radial table (midpoint propagation) and its K-integral G_ss mesh are not reproduced: the table is built here by another route (closed forms on [b, R_0], Taylor integration on [R_0, l]) and K by Fejer quadrature.”