`ComparatorChallenges/RadialDensityInterval.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

14 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.RadialTransition.Position line 12
Also declared: ComparatorChallenges/RadialTransition.lean:12 (same declaration text), OAI/Probability/RadialTransition/Model.lean:12 (same declaration text).
abbrev OAI.RadialTransition.Potential line 13
Also declared: ComparatorChallenges/RadialTransition.lean:13 (same declaration text), OAI/Probability/RadialTransition/Model.lean:13 (same declaration text).
def OAI.RadialTransition.cube line 15
Also declared: ComparatorChallenges/RadialTransition.lean:15 (same declaration text), OAI/Probability/RadialTransition/Model.lean:15 (same declaration text).
def OAI.RadialTransition.configurationCube line 18
Also declared: ComparatorChallenges/RadialTransition.lean:18 (same declaration text), OAI/Probability/RadialTransition/Model.lean:18 (same declaration text).
def OAI.RadialTransition.energy line 21
Also declared: ComparatorChallenges/RadialTransition.lean:21 (same declaration text), OAI/Probability/RadialTransition/Model.lean:21 (same declaration text).
def OAI.RadialTransition.partition line 24
Also declared: ComparatorChallenges/RadialTransition.lean:24 (same declaration text), OAI/Probability/RadialTransition/Model.lean:24 (same declaration text).
def OAI.RadialTransition.packingNumber line 28
Also declared: ComparatorChallenges/RadialTransition.lean:28 (same declaration text), OAI/Probability/RadialTransition/Model.lean:28 (same declaration text).
def OAI.RadialTransition.IsPackingDensity line 33
Also declared: ComparatorChallenges/RadialTransition.lean:33 (same declaration text), OAI/Probability/RadialTransition/Model.lean:33 (same declaration text).
def OAI.RadialTransition.IsBoundedPotential line 36
Also declared: ComparatorChallenges/RadialTransition.lean:36 (same declaration text), OAI/Probability/RadialTransition/Model.lean:36 (same declaration text).
def OAI.RadialTransition.StableWith line 39
Also declared: ComparatorChallenges/RadialTransition.lean:39 (same declaration text), OAI/Probability/RadialTransition/Model.lean:39 (same declaration text).
def OAI.RadialTransition.AlgebraicDecayWith line 42
Also declared: ComparatorChallenges/RadialTransition.lean:42 (same declaration text), OAI/Probability/RadialTransition/Model.lean:42 (same declaration text).
def OAI.RadialTransition.IsCanonicalFreeEnergy line 45
Also declared: ComparatorChallenges/RadialTransition.lean:45 (same declaration text), OAI/Probability/RadialTransition/Model.lean:45 (same declaration text).
def OAI.RadialTransition.StrictTemperatureCorner line 51
Also declared: ComparatorChallenges/RadialTransition.lean:51 (same declaration text), OAI/Probability/RadialTransition/Model.lean:51 (same declaration text).
theorem OAI.RadialTransition.densityInterval Challenge statement: A common radial phase transition over a density interval sorry line 57
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 3 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Probability.RadialTransition.DensityInterval as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Probability/RadialTransition/DensityInterval.lean:33 (same statement text; different proof or body). Qualify references as OAI.RadialTransition.densityInterval@path.
```
theorem densityInterval :
    ∃ (p : ℝ) (φ : Potential) (B C η βc : ℝ) (f : ℝ → ℝ → ℝ),
      0 < p ∧ IsPackingDensity p ∧
      IsBoundedPotential φ ∧ Continuous φ ∧
      0 < B ∧ 0 < C ∧ StableWith φ B ∧ AlgebraicDecayWith φ C ∧
      0 < η ∧ Ioo (5*p/3 - η) (5*p/3 + η) ⊆ Ioi 0 ∧
      βc ∈ Icc (7/8:ℝ) (9/8) ∧
      ∀ ρ ∈ Ioo (5*p/3 - η) (5*p/3 + η),
        IsCanonicalFreeEnergy φ ρ (fun β => f β ρ) ∧
        StrictTemperatureCorner (fun β => f β ρ) βc := by
…
```
Mentions: OAI.RadialTransition.Potential, OAI.RadialTransition.IsPackingDensity, OAI.RadialTransition.IsBoundedPotential, OAI.RadialTransition.StableWith, OAI.RadialTransition.AlgebraicDecayWith, OAI.RadialTransition.IsCanonicalFreeEnergy, OAI.RadialTransition.StrictTemperatureCorner