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Sharp finite-matrix Lieb–Thirring inequalities and all equality cases

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves the sharp one-dimensional Lieb–Thirring inequality for \(1/2\lt \gamma\lt 3/2\) and arbitrary finite-matrix potentials W ≥ 0 with \(\int\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty\): the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar sech2 solitons with independent scales and centers, and zero channels. (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

What the papers state

Theorem 1.1 (Equality classification)

In Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality, 5 October 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For every W satisfying (1.1),

\Tr(H_W)_-^\gamma\le C_\gamma\int_\R\tr(W^p), \qquad C_\gamma=\left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2} \frac{\Gamma(\gamma+1)}{\sqrt\pi\,\Gamma(\gamma+3/2)}.

Put r=(\gamma-1/2)^{-1}. Equality holds if and only if there are k\in\{0,\ldots,m\}, a constant unitary matrix U, positive numbers a_1,\ldots,a_k, and real numbers x_1,\ldots,x_k such that almost everywhere

W(x)=U\diag\bigl(w_1(x),\ldots,w_k(x),0,\ldots,0\bigr)U^*, \quad w_j(x)=(r+1)a_j^2\sech^2\!\bigl(ra_j(x-x_j)\bigr).

The case k=0 means W=0. Each nonzero channel has exactly one negative eigenvalue, -a_j^2, and therefore every extremal potential has at most m negative eigenvalues, counted with multiplicity.

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

Theorem 1.1 (Sharp matrix inequality)

In Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials, 5 October 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For every 1/2<\gamma<3/2, every finite m\ge1, and every W satisfying (1.1),

\Tr(H_W)_-^\gamma\le C_\gamma\int_\R\tr(W(x)^{\gamma+1/2})\,dx, \qquad C_\gamma= \left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2} \frac{\Gamma(\gamma+1)}{\sqrt\pi\,\Gamma(\gamma+3/2)}.

The constant is optimal in every matrix dimension. With r=(\gamma-1/2)^{-1}, it is attained by

W(x)=\diag\big((r+1)\sech^2(rx),0,\ldots,0\big).

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

The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.

One more possible match

  • Theorem 1.1 in Sharp one-dimensional Lieb–Thirring constants, 23 September 2026.
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
Three current papers, each readable here as a page beside its PDF. One explanation by others.

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.

Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality

5 October 2026, current edition.

Read it herePDF

We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and \(1/2\lt \gamma\lt 3/2\). For measurable Hermitian positive semidefinite potentials W with \(\int_\mathbb R\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty\), equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels. The nonzero solitons may have independent scales and centers.

Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials

5 October 2026, current edition.

Read it herePDF

We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent \(1/2\lt \gamma\lt 3/2\). The optimal constant is the scalar one-bound-state constant, independently of the matrix size. The inequality bounds the full sum of negative eigenvalue moments for every measurable Hermitian positive semidefinite potential W satisfying \(\int_\mathbb R\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty\). The matrices may have arbitrary rank and need not commute at different points.

Sharp one-dimensional Lieb–Thirring constants

23 September 2026, current edition.

Read it herePDF

We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every \(\frac12\lt \gamma\lt \frac32\), the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in \(L^{\gamma+1/2}(\mathbb R)\), with all negative eigenvalues included.

One Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

LiebThirring

The paper states

Theorem 1.1

In Sharp one-dimensional Lieb–Thirring constants, 23 September 2026

For every 1/2<\gamma<3/2 and every nonnegative W\in L^{\gamma+1/2}(\R),

\Tr(H_{-W})_-^\gamma \le 2\left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2} L_{\gamma,1}^{\cl}\int_\R W^{\gamma+1/2}.

The constant is optimal and equals L_{\gamma,1}^{(1)}. More precisely, if r=(\gamma-1/2)^{-1}, equality is attained by W(x)=(r+1)\sech^2(rx).

The words they share: optimal, 3/2, equality, constant, 1/2, gamma, thirring, sharp, one, lieb, nonnegative, rx, attained, sech.

Possible Lean counterpart

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

Sharp one-dimensional Lieb–Thirring inequality

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

def MainClaim : Prop :=  ∀ γ : ℝ, 1 / 2 < γ → γ < 3 / 2 →    (∀ W : ℝ → ℝ, Admissible γ W →      negativeMoment γ W ≤ ENNReal.ofReal (sharpConstant γ) * potentialMass γ W) ∧    optimalConstant γ = ENNReal.ofReal (sharpConstant γ) ∧    oneStateConstant γ = ENNReal.ofReal (sharpConstant γ) ∧    Admissible γ (equalityPotential γ) ∧    0 < potentialMass γ (equalityPotential γ) ∧    negativeMoment γ (equalityPotential γ) =      ENNReal.ofReal (sharpConstant γ) * potentialMass γ (equalityPotential γ)theorem sharp_lieb_thirring : MainClaim := 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.

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

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.

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: Only scalar potentials are treated: MainClaim : ∀ γ ∈ (1/2, 3/2), (∀ W : ℝ → ℝ, Admissible γ W → negativeMoment γ W ≤ sharpConstant γ * potentialMass γ W) ∧ optimalConstant γ = sharpConstant γ ∧ oneStateConstant γ = sharpConstant γ ∧ equality at (r+1) sech²(r x) with negativeMoment the sup over finite orthonormal families of H¹ weak negative eigenfunctions. The summary’s finite-MATRIX-valued potentials with constant independent of matrix size, and the complete classification of equality cases (direct sums of sech² solitons with independent scales/centers and zero channels), are not stated: this is the scalar bound with attainment at one potential but without the equality-case classification.

Definitions the group asks readers to compare with the paper: Custom spectral definitions: H1 as an L² function with an L² weak derivative, schrodingerForm, IsNegativeEigenfunction W k u (weak eigenfunction at energy -k²) and negativeMoment γ W := ⨆ N (u k) orthonormal eigenfunctions, Σ (ofReal k_i)^(2γ) in ℝ≥0∞ instead of Mathlib spectral theory; constants semiclassicalConstant, sharpConstant are written explicitly (Γ-function formulas).

“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 matrix-valued inequality (arbitrary finite matrix size, constant independent of size) and the classification of all equality cases — the content of two of the family’s three manuscripts and the subject of the family title — are not stated. Only the scalar sharp inequality with optimality and attainment at one sech² potential is.

“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 determines the sharp one-dimensional Lieb–Thirring constant for $1/2<\gamma<3/2$. For every nonnegative $W\in L^{\gamma+1/2}(\mathbb R)$, it bounds the full negative-eigenvalue moment of $-d^2/dx^2-W$ by the one-bound-state constant times the potential integral. This constant is optimal and is attained by $(r+1)\mathrm{sech}^2(rx)$, where $r=(\gamma-1/2)^{-1}$. No finite spectral cutoff is imposed.

AI-drafted comparisons, with no recorded human review

  • Sharp one-dimensional Lieb–Thirring inequality LiebThirring

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the sharp one-dimensional Lieb–Thirring constant for 1/2 < γ < 3/2, with optimality and equality at the explicit sech² potential.

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 one-dimensional Lieb–Thirring inequality LiebThirring

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal 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 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 456 s; peak memory 6.6 GiB).11 October 2026Original report

Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

  • Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source

    Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.

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

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.

Explain a result, compare statements or propose a correction

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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.

    Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 262 No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 262 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form