`ComparatorChallenges/LiebThirring.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 120 curated AI-written annotations and 1273 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 9 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. Those 9 annotations were revised on 2026-10-14 to bind all of them to exact page-one quotations from 6 pinned OpenAI manuscripts, the matching pinned formalization notes, and named historical primary sources; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. The 3 manuscript-grounded matrix-multiplication annotations were composed from pinned OpenAI manuscripts and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Six additional target-specific annotations for Catalan's constant, the irrationality exponent of pi, dyadic Erdos similarity, Unique Games, quintic Lienard systems, and Yau nodal growth were composed from pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for Seymour's second-neighborhood conjecture, Cannon's conjecture, Koebe circle-domain uniformization, and circle-domain rigidity were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for simple Lebesgue spectrum on the three-torus, Rokhlin multiple mixing, the nine-dimensional Borsuk counterexample, and Tingley's sphere-isometry problem were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the Abhyankar-Sathaye counterexample, the circulant Hadamard conjecture, the Eilenberg-Ganea counterexample, and short Egyptian fractions were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and quotations from Sathaye, Turyn with Storer, Eilenberg with Ganea, and Erdos in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the complete Crouzeix inequality, unrestricted Euclidean five-coloring, the general Mahler conjecture, and Thompson F nonamenability were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, the matching formalization notes, and linked historical sources by Crouzeix, de Grey, Mahler, Cannon, Floyd, and Parry; no human review is claimed. Four additional target-specific annotations for algorithmic strong thin trees, Barnette's Hamiltonian-cycle conjecture, the explicit Artin CAT(0) counterexample, and generalized amplitude-damping capacity were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, matching formalization notes, and linked historical sources by Anari with Goddyn, Arts with Hopman and Timmermans, Charney, and Khatri with Sharma and Wilde; no human review is claimed. Fourteen solution-level annotations for Seymour's second-neighborhood theorem were added on 2026-10-14 from the pinned 15-page manuscript, covering the exact-neighborhood model, minimal-counterexample reduction, subset deficit, transitive blowup, pruning, extremal contradiction, and proof-bearing final theorem. The subset and blowup annotations also bind exact quotations from Tyler Seacrest (2019) and Brantner, Brockman, Kay, and Snively (2009); no human review is claimed. Thirty-seven declaration-level annotations were added on 2026-10-14 for the rest of ComparatorChallenges/AlgorithmicThinTrees.lean, covering its multigraph semantics, stack-machine cost model, explicit and binary-multiplicity encodings, output certificate, and complete algorithmic proposition. They are bound to the pinned OpenAI manuscript and linked earlier thin-tree sources by Luis Goddyn, Nima Anari, and Shayan Oveis Gharan; the annotation explicitly preserves that the concluding comparator theorem still has body by sorry, and no human review is claimed. Thirty-one supporting declarations in ComparatorChallenges/UniqueGamesTheorem.lean were upgraded or added on 2026-10-14, completing curated coverage of all thirty-two declarations in that file. The annotations cover the finite game model, Boolean formula and parser, binary-vector translations, explicit output serialization, finite-alphabet machine contract, and gap theorem; they distinguish manuscript-backed mathematics from Lean-specific canonical parsing and unary game output encoding, bind exact OpenAI manuscript quotations, and credit Subhash Khot through his 2010 survey. The comparator theorem remains explicitly labeled as having body by sorry; no human review is claimed. The source-derived generator was hardened on 2026-10-14 so a formalization note covering multiple comparator results no longer copies its combined Scope prose onto each declaration; those targets now use declaration-specific source comments when available and otherwise remain explicitly bounded to the exact Lean statement. The remaining source-derived annotations were generated mechanically from pinned formalization notes and source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the source prose.

Credit: OpenAI's pinned repository says an unreleased internal model produced the manuscripts and supporting proof artifacts. The collection builds on the mathematics community's accumulated problems, results, language, and formal libraries; each manuscript's BibTeX and bibliography are the authority for individual scholarly credit.

Project sources: Pinned openai/math documentation, Pinned formalization catalogue

16 declarations, 1 contain sorry, 0 declared as axioms

The form domain is H¹(R;C), represented by an L² function together with its L² weak derivative. Null-set changes do not affect any definition below. negativeMoment is the extended nonnegative sum over negative eigenvalues, counting multiplicities: equivalently the supremum over all finite orthonormal families of weak eigenfunctions of the form. This does not assume in advance finiteness of the moment or a finite number of eigenvalues.

structure OAI.SharpLiebThirring.H1 line 23
Also declared: OAI/Analysis/LiebThirring/Model.lean:23 (same declaration text).
def OAI.SharpLiebThirring.l2Pairing line 31
Also declared: OAI/Analysis/LiebThirring/Model.lean:31 (same declaration text).
def OAI.SharpLiebThirring.schrodingerForm The polarized form of -d²/dx² - W, on exactly H¹ line 34
Source documentation
The polarized form of -d²/dx² - W, on exactly H¹.
L4DEL explanation
The polarized form of -d²/dx² - W, on exactly H¹.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Analysis/LiebThirring/Model.lean:34 (same declaration text).
```
def schrodingerForm (W : ℝ → ℝ) (u v : H1) : ℂ :=
…
```
Mentions: OAI.SharpLiebThirring.H1
def OAI.SharpLiebThirring.IsNegativeEigenfunction Weak eigenfunction at the negative energy -k² line 39
Source documentation
Weak eigenfunction at the negative energy -k².
L4DEL explanation
Weak eigenfunction at the negative energy -k².
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Analysis/LiebThirring/Model.lean:39 (same declaration text).
```
def IsNegativeEigenfunction (W : ℝ → ℝ) (k : ℝ) (u : H1) : Prop :=
…
```
Mentions: OAI.SharpLiebThirring.H1
def OAI.SharpLiebThirring.IsOrthonormalFamily line 42
Also declared: OAI/Analysis/LiebThirring/Model.lean:42 (same declaration text).
def OAI.SharpLiebThirring.negativeMoment An unlimited finite-family-size supremum, in ℝ≥0∞, includes the entire sum line 46
Source documentation
An unlimited finite-family-size supremum, in ℝ≥0∞, includes the entire sum.
L4DEL explanation
An unlimited finite-family-size supremum, in ℝ≥0∞, includes the entire sum.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Analysis/LiebThirring/Model.lean:46 (same declaration text).
```
def negativeMoment (γ : ℝ) (W : ℝ → ℝ) : ℝ≥0∞ :=
…
```
def OAI.SharpLiebThirring.oneStateMoment The moment retaining at most one negative eigenvalue line 52
Source documentation
The moment retaining at most one negative eigenvalue.
L4DEL explanation
The moment retaining at most one negative eigenvalue.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Analysis/LiebThirring/Model.lean:52 (same declaration text).
```
def oneStateMoment (γ : ℝ) (W : ℝ → ℝ) : ℝ≥0∞ :=
…
```
def OAI.SharpLiebThirring.Admissible line 56
Also declared: OAI/Analysis/LiebThirring/Model.lean:56 (same declaration text).
def OAI.SharpLiebThirring.potentialMass line 59
Also declared: OAI/Analysis/LiebThirring/Model.lean:59 (same declaration text).
def OAI.SharpLiebThirring.semiclassicalConstant line 62
Also declared: OAI/Analysis/LiebThirring/Model.lean:62 (same declaration text).
def OAI.SharpLiebThirring.sharpConstant line 65
Also declared: OAI/Analysis/LiebThirring/Model.lean:65 (same declaration text).
def OAI.SharpLiebThirring.optimalConstant line 68
Also declared: OAI/Analysis/LiebThirring/Model.lean:68 (same declaration text).
def OAI.SharpLiebThirring.oneStateConstant line 72
Also declared: OAI/Analysis/LiebThirring/Model.lean:72 (same declaration text).
def OAI.SharpLiebThirring.equalityPotential line 76
Also declared: OAI/Analysis/LiebThirring/Model.lean:76 (same declaration text).
def OAI.SharpLiebThirring.MainClaim The full spectral bound, optimal constants, and attainment at the explicit potential line 81
Source documentation
The full spectral bound, optimal constants, and attainment at the explicit potential.
L4DEL explanation
The full spectral bound, optimal constants, and attainment at the explicit potential.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Analysis/LiebThirring/Model.lean:81 (same declaration text).
```
def MainClaim : Prop :=
…
```
theorem OAI.SharpLiebThirring.sharp_lieb_thirring Challenge statement: Sharp finite-matrix Lieb–Thirring inequalities and all equality cases sorry line 94
Source documentation
Full sharp Lieb–Thirring bound, both optimal constants, and equality at the explicit potential; no finite spectral cutoff is imposed.
L4DEL explanation
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.
1. This is an exact comparator target linked by the pinned repository's formalization note for this result.
2. The repository's comparator configuration selects OAI.Analysis.LiebThirring.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/LiebThirring/Main.lean:38 (same statement text; different proof or body). Qualify references as OAI.SharpLiebThirring.sharp_lieb_thirring@path.
```
theorem sharp_lieb_thirring : MainClaim := by
…
```
Mentions: OAI.SharpLiebThirring.MainClaim