`ComparatorChallenges/BoltzmannNonuniqueness.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

39 declarations, 1 contain sorry, 0 declared as axioms

Nonuniqueness for the periodic hard-sphere Boltzmann equation

Theorem 1.1, with the full entropy-class admissibility of Definition 2.1 and strong initial L¹ continuity with integrable collision gain and loss.

abbrev OAI.BoltzmannNonuniqueness.Velocity line 19
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:24 (same declaration text).
abbrev OAI.BoltzmannNonuniqueness.Position line 20
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:25 (same declaration text).
abbrev OAI.BoltzmannNonuniqueness.Phase line 21
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:26 (same declaration text).
abbrev OAI.BoltzmannNonuniqueness.Density line 22
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:27 (same declaration text).
abbrev OAI.BoltzmannNonuniqueness.Sphere line 23
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:28 (same declaration text).
def OAI.BoltzmannNonuniqueness.sphereArea Ordinary surface area, rather than normalized probability on the sphere line 26
Source documentation
Ordinary surface area, rather than normalized probability on the sphere.
L4DEL explanation
Ordinary surface area, rather than normalized probability on the sphere.
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/MathematicalPhysics/Boltzmann/Model.lean:31 (same declaration text).
```
def sphereArea : Measure Sphere := (volume : Measure Velocity).toSphere
```
Mentions: OAI.BoltzmannNonuniqueness.Sphere, OAI.BoltzmannNonuniqueness.Velocity
def OAI.BoltzmannNonuniqueness.normalComponent line 28
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:33 (same declaration text).
def OAI.BoltzmannNonuniqueness.scatterLeft line 30
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:35 (same declaration text).
def OAI.BoltzmannNonuniqueness.scatterRight line 32
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:37 (same declaration text).
def OAI.BoltzmannNonuniqueness.collisionKernel line 34
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:39 (same declaration text).
def OAI.BoltzmannNonuniqueness.gain line 36
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:41 (same declaration text).
def OAI.BoltzmannNonuniqueness.loss line 42
Also declared: OAI/MathematicalPhysics/Boltzmann/Model.lean:47 (same declaration text).
def OAI.BoltzmannNonuniqueness.collision line 47
Distinct definitions of this name: OAI/MathematicalPhysics/Boltzmann/Model.lean:52 (same statement text; different proof or body). Qualify references as OAI.BoltzmannNonuniqueness.collision@path.
abbrev OAI.BoltzmannNonuniqueness.Evolution line 52
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:23 (same declaration text).
abbrev OAI.BoltzmannNonuniqueness.SpaceTimePhase line 53
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:24 (same declaration text).
abbrev OAI.BoltzmannNonuniqueness.LiftedSpaceTimePhase line 54
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:25 (same declaration text).
def OAI.BoltzmannNonuniqueness.torusProjection line 56
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:27 (same declaration text).
def OAI.BoltzmannNonuniqueness.torusRepresentative A representative in the fundamental half-open unit cube, used only to evaluate periodic derivatives line 60
Source documentation
A representative in the fundamental half-open unit cube, used only to evaluate periodic derivatives. Periodicity makes those derivatives independent of this choice.
L4DEL explanation
A representative in the fundamental half-open unit cube, used only to evaluate periodic derivatives. Periodicity makes those derivatives independent of this choice.
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/MathematicalPhysics/Boltzmann/Admissibility.lean:31 (same declaration text).
```
def torusRepresentative (x : Position) : Velocity :=
…
```
Mentions: OAI.BoltzmannNonuniqueness.Position, OAI.BoltzmannNonuniqueness.Velocity
def OAI.BoltzmannNonuniqueness.maxwellian line 63
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:39 (same declaration text).
def OAI.BoltzmannNonuniqueness.entropyMoment line 66
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:46 (same declaration text).
def OAI.BoltzmannNonuniqueness.FiniteEntropyDatum The convention 0 |log 0| = 0 is automatic in this real-valued moment line 70
Source documentation
The convention 0 |log 0| = 0 is automatic in this real-valued moment.
L4DEL explanation
The convention 0 |log 0| = 0 is automatic in this real-valued moment.
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/MathematicalPhysics/Boltzmann/Admissibility.lean:50 (same declaration text).
```
def FiniteEntropyDatum (f : Density) : Prop :=
…
```
Mentions: OAI.BoltzmannNonuniqueness.Density
def OAI.BoltzmannNonuniqueness.BoundedVelocitySupport line 73
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:53 (same declaration text).
def OAI.BoltzmannNonuniqueness.energy line 76
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:56 (same declaration text).
def OAI.BoltzmannNonuniqueness.momentum line 78
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:58 (same declaration text).
def OAI.BoltzmannNonuniqueness.relativeEntropy line 80
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:60 (same declaration text).
def OAI.BoltzmannNonuniqueness.extendedEntropyProduction Lower semicontinuous extension of (A-B) log(A/B) on the nonnegative quadrant line 85
Source documentation
Lower semicontinuous extension of (A-B) log(A/B) on the nonnegative quadrant. The off-quadrant value is immaterial for nonnegative densities.
L4DEL explanation
Lower semicontinuous extension of (A-B) log(A/B) on the nonnegative quadrant. The off-quadrant value is immaterial for nonnegative densities.
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/MathematicalPhysics/Boltzmann/Admissibility.lean:65 (same declaration text).
```
def extendedEntropyProduction (A B : ℝ) : ℝ≥0∞ :=
…
```
def OAI.BoltzmannNonuniqueness.entropyDissipation line 90
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:70 (same declaration text).
structure OAI.BoltzmannNonuniqueness.Renormalization The renormalizations of Definition 2.1, with the right derivative at zero line 101
Source documentation
The renormalizations of Definition 2.1, with the right derivative at zero.
L4DEL explanation
The renormalizations of Definition 2.1, with the right derivative at zero.
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/MathematicalPhysics/Boltzmann/Admissibility.lean:81 (same declaration text).
```
structure Renormalization where
…
```
structure OAI.BoltzmannNonuniqueness.KineticTest A smooth test, periodic in the spatial variable and compact in time and velocity line 109
Source documentation
A smooth test, periodic in the spatial variable and compact in time and velocity.
L4DEL explanation
A smooth test, periodic in the spatial variable and compact in time and velocity.
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/MathematicalPhysics/Boltzmann/Admissibility.lean:89 (same declaration text).
```
structure KineticTest where
…
```
def OAI.BoltzmannNonuniqueness.KineticTest.transportDerivative line 116
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:96 (same declaration text).
structure OAI.BoltzmannNonuniqueness.MassTest Smooth local mass tests have no velocity cutoff, since their velocity dependence is introduced only by the transport derivative line 122
Source documentation
Smooth local mass tests have no velocity cutoff, since their velocity dependence is introduced only by the transport derivative.
L4DEL explanation
Smooth local mass tests have no velocity cutoff, since their velocity dependence is introduced only by the transport derivative.
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/MathematicalPhysics/Boltzmann/Admissibility.lean:102 (same declaration text).
```
structure MassTest where
…
```
def OAI.BoltzmannNonuniqueness.MassTest.transportDerivative line 127
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:107 (same declaration text).
def OAI.BoltzmannNonuniqueness.relativeDensity line 131
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:111 (same declaration text).
def OAI.BoltzmannNonuniqueness.renormalizedIntegrand line 133
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:113 (same declaration text).
def OAI.BoltzmannNonuniqueness.nonnegativeTimes line 140
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:120 (same declaration text).
def OAI.BoltzmannNonuniqueness.boundedTimes line 141
Also declared: OAI/MathematicalPhysics/Boltzmann/Admissibility.lean:121 (same declaration text).
structure OAI.BoltzmannNonuniqueness.AdmissibleGlobal Full global admissibility, including both separately renormalized collision terms, local mass, total momentum, and the every-time energy and entropy inequalities line 145
Source documentation
Full global admissibility, including both separately renormalized collision terms, local mass, total momentum, and the every-time energy and entropy inequalities.
L4DEL explanation
Full global admissibility, including both separately renormalized collision terms, local mass, total momentum, and the every-time energy and entropy inequalities.
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/MathematicalPhysics/Boltzmann/Admissibility.lean:125 (same declaration text).
```
structure AdmissibleGlobal (f₀ : Density) (F : Evolution) : Prop where
…
```
Mentions: OAI.BoltzmannNonuniqueness.Density, OAI.BoltzmannNonuniqueness.Evolution
structure OAI.BoltzmannNonuniqueness.StrongEarly Additional regularity asserted on the common initial interval line 194
Source documentation
Additional regularity asserted on the common initial interval. The eLpNorm statement is strong L¹ continuity and cannot be made vacuous by a nonintegrable Bochner integral.
L4DEL explanation
Additional regularity asserted on the common initial interval. The eLpNorm statement is strong L¹ continuity and cannot be made vacuous by a nonintegrable Bochner integral.
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/MathematicalPhysics/Boltzmann/Admissibility.lean:174 (same declaration text).
```
structure StrongEarly (T : ℝ) (F : Evolution) : Prop where
…
```
Mentions: OAI.BoltzmannNonuniqueness.Evolution
theorem OAI.BoltzmannNonuniqueness.nonuniqueness Challenge statement: Nonuniqueness with local conservation for the hard-sphere Boltzmann equation sorry line 204
L4DEL explanation
The formalization proves nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions on $\mathbb T^3\times\mathbb R^3$ with the same nonnegative initial density. The data have bounded velocity support and finite mass, energy, and absolute entropy. Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities. On a common initial interval they are strongly continuous in $L^1$, with integrable collision gains and losses. These are the conditions of the selected periodic result; later local-conservation refinements are outside it.
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.MathematicalPhysics.Boltzmann.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/MathematicalPhysics/Boltzmann/Main.lean:13 (same statement text; different proof or body). Qualify references as OAI.BoltzmannNonuniqueness.nonuniqueness@path.
```
theorem nonuniqueness :
    ∃ f₀ : Density, BoundedVelocitySupport f₀ ∧
      ∃ (F G : Evolution) (T : ℝ), AdmissibleGlobal f₀ F ∧ AdmissibleGlobal f₀ G ∧
        StrongEarly T F ∧ StrongEarly T G ∧
        ∃ t ∈ Set.Icc 0 T, ¬(F t =ᵐ[MeasureTheory.volume] G t) :=
…
```
Mentions: OAI.BoltzmannNonuniqueness.Density, OAI.BoltzmannNonuniqueness.BoundedVelocitySupport, OAI.BoltzmannNonuniqueness.Evolution, OAI.BoltzmannNonuniqueness.AdmissibleGlobal, OAI.BoltzmannNonuniqueness.StrongEarly