Failure of integer-degree harmonic dimension comparison

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Disproves Yau’s proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ³ has at least $(k+2)^2$ independent harmonic functions of growth at most k, exceeding the Euclidean count $(k+1)^2$. The metric may depend on k. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments

The paper states

Theorem 1.1

In A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison, 26 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.

The other possible matches are Theorem 1.1 in A counterexample to integer-degree harmonic dimension comparison () and Corollary 1.2 in the same paper ().

For every v\in(4/9,1) and c\in(1,9v/4), there is an integer k_0(v,c) such that for every integer k\geq k_0(v,c) there is a complete smooth metric g_k on \mathbb R^3, Euclidean near the origin, with \operatorname{Ric}_{g_k}\geq0, \operatorname{AVR}(g_k)=v, and

h_k(\mathbb R^3,g_k)\geq c(k+1)^2>(k+1)^2 =h_k(\mathbb R^3,g_{\mathrm E}).

The Ricci curvature is positive outside a compact set. The metric has uncountably many pairwise nonisometric pointed tangent cones at infinity. There are points tending to infinity and planes containing the radial direction at those points whose sectional curvature is negative. The metric is chosen after k and may depend on it.

No Lean statement in the release is paired with this theorem.

Reported proof verification: One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations: Two current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

A counterexample to integer-degree harmonic dimension comparison

25 September 2026, current edition.

Read it here PDF

For some even n ≥ 8 and integer k ≥ 2, we construct a complete smooth metric on ℝⁿ with nonnegative Ricci curvature whose space of real harmonic functions of pointwise polynomial growth at most k has dimension larger than the Euclidean harmonic-polynomial dimension. The metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. This answers the integer-degree form of Yau’s dimension comparison question in the negative.

A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison

26 September 2026, current edition.

Read it here PDF

For every $4/9\lt v\lt 1$ and $1\lt c\lt 9v/4$, all sufficiently large integers k admit a complete smooth metric on ℝ³ with nonnegative Ricci curvature, asymptotic volume ratio v, and at least $c(k+1)^2$ linearly independent real harmonic functions of pointwise growth at most k. This answers Yau’s integer-degree dimension comparison question negatively in dimension three, with a fixed-factor excess over the Euclidean count. For each $1\lt c\lt 9/4$, these metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. The metric may depend on k.

One Lean statement in this family

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`HarmonicGrowth`

The paper states

Theorem 1.1

In A counterexample to integer-degree harmonic dimension comparison, 25 September 2026

There are an even integer n\geq8, an integer k\geq2, and a complete smooth metric g on \mathbb R^n such that \operatorname{Ric}_g\geq0 and

h_k(\mathbb R^n,g)>h_k(\mathbb R^n,g_{\mathrm E}).

The metric is Euclidean near the origin, and its asymptotic volume ratio satisfies

0<\lim_{R\to\infty} \frac{\operatorname{vol}_g B_g(0,R)}{\omega_nR^n}<1,

where \omega_n is the volume of the Euclidean unit ball.

The words they share: even, unit, omega, ball, ratio, volume, euclidean, origin, complete, near, smooth, integer, real, metric, dimension, harmonic, asymptotic.

Possible Lean counterpart

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

Counterexample to harmonic dimension comparison

HarmonicGrowth.lean, the release lists it as a main result.

def MainClaim : Prop :=  ∃ (n k : ℕ) (g : SmoothMetric n), Even n ∧ 8 ≤ n ∧ 2 ≤ k ∧    Complete g ∧ RicciNonnegative g ∧ EuclideanNearOrigin g ∧ PositiveSubunitAVR g ∧    ∃ u : Fin (euclideanDimension n k + 1) → (Space n → ℝ),      LinearIndependent ℝ u ∧ ∀ i, HarmonicGrowth g k (u i)⋮axiom mainStatement : MainClaim⋮theorem main : MainClaim := mainStatement

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

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 a weaker statement than the advertised result (their term: “weaker-statement”).

Their note: MainClaim is ∃ n (even, ≥ 8), k ≥ 2 and one complete smooth metric on ℝⁿ with Ric ≥ 0, Euclidean near 0, asymptotic volume ratio in (0,1), and euclideanDimension n k + 1 independent harmonic functions of growth ≤ k: a single higher-dimensional counterexample, as the docs say. The summary’s claim (complete metrics on ℝ³ with ≥ (k+2)² harmonic functions of growth ≤ k for every sufficiently large integer k) is not implied. The challenge file also contains axiom mainStatement : MainClaim and theorem main := mainStatement instead of sorry, a non-standard stub shape; no axiom exists in the solution cone.

Definitions the group asks readers to compare with the paper: axiom mainStatement : MainClaim in the challenge file (the only challenge here with an axiom and no sorry); Hand-written Riemannian geometry on ℝⁿ: ricci, laplaceBeltrami, distance := sInf of C¹ path lengths, Complete, PositiveSubunitAVR (ball volume/(ω_n R^n) → a ∈ (0,1)), HarmonicGrowth (|u| ≤ C(1+d(0,x))^k) and euclideanDimension n k := C(n+k-1,k) + C(n+k-2,k-1) (dimension of Euclidean harmonic polynomials of degree ≤ k)

“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker 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.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).

The gap they name: The three-dimensional statement — on R³, for every sufficiently large k, ≥ (k+2)² independent harmonic functions of growth ≤ k — is not stated. The Lean states a single (n, k) with n even ≥ 8 and k ≥ 2, with euclideanDimension(n,k) + 1 functions.

“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.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalized result disproves the proposed Euclidean upper comparison for dimensions of harmonic functions with integer polynomial growth. It constructs one complete smooth metric with nonnegative Ricci curvature on an even-dimensional Euclidean space, Euclidean near the origin and with asymptotic volume ratio strictly between zero and one, admitting more independent harmonic functions of the prescribed growth degree than the Euclidean count. The construction uses dimension $16$ and degree $50000$. Separate tangent-cone and cone-spectrum conclusions are not included.

AI-drafted comparisons, with no recorded human review

Counterexample to harmonic dimension comparison HarmonicGrowth
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem with an explicit witness (dimension 16, degree 50000): a complete metric with nonnegative Ricci curvature on R^n with more harmonic functions of degree-k growth than Euclidean space. The challenge file states the claim through a placeholder axiom; Comparator admits only the three standard axioms in the library proof.

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.

Counterexample to harmonic dimension comparison `HarmonicGrowth`

Three 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
Neal McBurnett, openai-math-lean-tooling	“accepted”	Re-run through Comparator with Lean’s kernel: accepted (the check took 1096 s; peak memory 7.0 GiB).	11 October 2026	Original report

Other proofs and exact calculations

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.

A counterexample to integer-degree harmonic dimension comparison, 25 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: the supplementary exact spectral selection at n = 16, k = 50000 (finite-selection.tex) — every parameter inequality, the eigenvalue data, the multiplicities, every square-root bound and the three printed integers; not the analytic construction of the metric nor the lemmas the eigenvalues and multiplicities come from”
Original report
Their report

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

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison cites A counterexample to integer-degree harmonic dimension comparison.

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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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 counterexample and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Author not stated. Source

grwtsk, [AUDIT.00] Audit the formal trust surface of the OpenAI math release (issue 2), 9 October 2026. Source
The challenge HarmonicGrowth declares an axiom mainStatement in place of a sorry; it remains a challenge-only file, and an independent run later accepted its solution.

Questions awaiting review

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Family 361 The 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

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Family 361 MathVet: its request is printed in full under the outside readers. Open the form

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