`ComparatorChallenges/HardSphere.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

31 declarations, 1 contain sorry, 0 declared as axioms

Ground-state condensation in the dilute hard-sphere gas

The main result covers all pure ground states and all density operators supported on the ground space. In particular it covers the normalized ground-space projection. Only the definitions needed to state this result are retained below.

abbrev OAI.HardSphere.Length line 17
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:17 (same declaration text).
instance OAI.HardSphere.lengthPositive line 19
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:19 (same declaration text).
abbrev OAI.HardSphere.Torus line 21
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:21 (same declaration text).
abbrev OAI.HardSphere.Configuration line 23
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:23 (same declaration text).
abbrev OAI.HardSphere.LiftedConfiguration line 25
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:25 (same declaration text).
abbrev OAI.HardSphere.Direction line 27
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:27 (same declaration text).
def OAI.HardSphere.torusDistance line 29
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:29 (same declaration text).
def OAI.HardSphere.allowed line 32
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:32 (same declaration text).
def OAI.HardSphere.project line 35
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:35 (same declaration text).
def OAI.HardSphere.representative line 38
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:38 (same declaration text).
def OAI.HardSphere.coordDerivative line 41
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:41 (same declaration text).
abbrev OAI.HardSphere.ScalarL2 line 46
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:46 (same declaration text).
abbrev OAI.HardSphere.Jet line 49
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:49 (same declaration text).
structure OAI.HardSphere.SmoothTest line 52
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:52 (same declaration text).
def OAI.HardSphere.SmoothTest.jet line 60
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:60 (same declaration text).
def OAI.HardSphere.dirichletDomain line 63
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:63 (same declaration text).
def OAI.HardSphere.energy line 66
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:66 (same declaration text).
def OAI.HardSphere.bosonic line 69
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:69 (same declaration text).
def OAI.HardSphere.IsGroundVector line 73
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:73 (same declaration text).
def OAI.HardSphere.insertFirst line 77
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:77 (same declaration text).
def OAI.HardSphere.occupation line 81
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:81 (same declaration text).
def OAI.HardSphere.Density.basisSet line 88
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:88 (same declaration text).
def OAI.HardSphere.Density.basis line 90
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:90 (same declaration text).
structure OAI.HardSphere.Density.Operator line 93
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:93 (same declaration text).
def OAI.HardSphere.Density.Operator.compressionTrace line 101
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:101 (same declaration text).
def OAI.HardSphere.Orbital.sliceIntegral line 110
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:110 (same declaration text).
def OAI.HardSphere.Orbital.map line 113
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:113 (same declaration text).
def OAI.HardSphere.Orbital.continuous line 190
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:190 (same declaration text).
def OAI.HardSphere.GroundSupported line 363
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:363 (same declaration text).
def OAI.HardSphere.mixedOccupation line 368
Also declared: OAI/Analysis/HardSphere/CondensationDefinitions.lean:368 (same declaration text).
theorem OAI.HardSphere.condensation_with_mixed Challenge statement: Positive-temperature Bose–Einstein condensation and exact quantum depletion sorry line 372
L4DEL explanation
The formalization proves ground-state Bose–Einstein condensation in the dilute hard-sphere gas. There are absolute constants $\varepsilon_0,c_0>0$ such that, whenever the density $\rho$ and hard-sphere radius $a$ satisfy $\rho a^3<\varepsilon_0$, the condensate occupation fraction has limit inferior at least $c_0$ along every thermodynamic sequence with $N/L^3\to\rho$. The bound covers every pure ground state, every mixed state supported on the ground space, and the normalized ground-space projection. It is uniform in the gas parameter within this range and concerns ground states, without a positive-temperature assertion.
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.HardSphere.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/HardSphere/Thermodynamics/MixedCondensation.lean:43 (same statement text; different proof or body). Qualify references as OAI.HardSphere.condensation_with_mixed@path.
```
theorem condensation_with_mixed :
    ∃ε₀ c₀ : ℝ,0<ε₀ ∧ 0<c₀ ∧
    ∀a ρ : ℝ,0<a → 0<ρ → ρ*a^3<ε₀ →
    ∀(N : ℕ→ℕ) (L : ℕ→Length) (hN : ∀k,0<N k),
      Tendsto N atTop atTop → Tendsto (fun k ↦ (L k).val) atTop atTop →
      Tendsto (fun k ↦ (N k:ℝ)/(L k).val^3) atTop (𝓝 ρ) →
      (∀Ψ : (k : ℕ)→Jet (N k) (L k),
        (∀k,IsGroundVector (N k) (L k) a (Ψ k)) →
        c₀≤liminf (fun k ↦ occupation (hN k) (Ψ k).1) atTop) ∧
      (∀T : (k : ℕ)→Density.Operator (ScalarL2 (N k) (L k)),
        (∀k,GroundSupported a (T k)) →
        c₀≤liminf (fun k ↦ mixedOccupation (hN k) (T k)) atTop) := by
…
```
Mentions: OAI.HardSphere.Length, OAI.HardSphere.Jet, OAI.HardSphere.IsGroundVector, OAI.HardSphere.occupation, OAI.HardSphere.Density.Operator, OAI.HardSphere.ScalarL2, OAI.HardSphere.GroundSupported, OAI.HardSphere.mixedOccupation