`ComparatorChallenges/KoebeCircleDomains.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

15 declarations, 2 contain sorry, 0 declared as axioms

Circle-domain uniformization and rigidity under conformal removability.

abbrev OAI.CircleDomainRigidity.Sphere line 13
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:13 (same declaration text).
abbrev OAI.CircleDomainRigidity.Mobius line 14
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:14 (same declaration text).
def OAI.CircleDomainRigidity.sphereInv line 16
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:16 (same declaration text).
def OAI.CircleDomainRigidity.finiteCoord line 19
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:19 (same declaration text).
def OAI.CircleDomainRigidity.chart line 21
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:21 (same declaration text).
def OAI.CircleDomainRigidity.chartSymm line 24
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:24 (same declaration text).
def OAI.CircleDomainRigidity.IsConformalAt line 27
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:27 (same declaration text).
def OAI.CircleDomainRigidity.IsConformalOn line 31
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:31 (same declaration text).
def OAI.CircleDomainRigidity.IsConformalEquivalence line 34
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:34 (same declaration text).
def OAI.CircleDomainRigidity.IsRoundClosedDisk line 39
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:39 (same declaration text).
def OAI.CircleDomainRigidity.IsCircleDomain line 44
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:44 (same declaration text).
def OAI.CircleDomainRigidity.OrientationPreserving line 49
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:49 (same declaration text).
def OAI.CircleDomainRigidity.IsConformallyRemovable line 52
Also declared: OAI/Analysis/CircleDomains/LengthArea.lean:52 (same declaration text).
theorem OAI.Problem047.koebe_circle_domain Challenge statement: Koebe's unrestricted circle-domain uniformization sorry line 62
L4DEL explanation
every domain in the Riemann sphere is conformally equivalent to a circle domain
— OpenAI Koebe circle-domain manuscript (abstract)
Any domain Q in C, whose boundary has at most countably many components, is conformally homeomorphic to a circle domain
— Zheng-Xu He and Oded Schramm (1993), Theorem 0.1
Every nonempty connected open subset of the Riemann sphere is conformally equivalent to a circle domain, with no countability restriction on its complementary components.
1. The input U is any open connected set in the sphere. The output is a set V and a sphere map f such that V is a circle domain and f is a conformal equivalence from U to V.
2. IsCircleDomain V says that V is open and connected and that each connected component of its complement is either a singleton or the Möbius image of a closed round unit disk.
3. He and Schramm proved the countably connected case in 1993. The pinned manuscript states the unrestricted result; this distinction matters because the Lean theorem contains no countability assumption. Its comparator body is sorry, paired with OAI.Analysis.CircleDomains.Main.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI Koebe circle-domain manuscript (abstract), Pinned OpenAI formalization note 071, Zheng-Xu He and Oded Schramm (1993), Theorem 0.1
Distinct definitions of this name: OAI/Analysis/CircleDomains/Main.lean:10 (same statement text; different proof or body). Qualify references as OAI.Problem047.koebe_circle_domain@path.

circle domain: a domain in the Riemann sphere whose complementary components are round closed disks or single points
```
theorem koebe_circle_domain :
    ∀ U : Set Sphere, IsOpen U → IsConnected U →
      ∃ (V : Set Sphere) (f : Sphere → Sphere),
        IsCircleDomain V ∧ IsConformalEquivalence f U V := by
…
```
theorem OAI.Problem047.removability_implies_rigidity Challenge statement: Conformal removability implies circle-domain rigidity sorry line 68
L4DEL explanation
a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components
— OpenAI circle-domain rigidity manuscript (abstract)
two rigidity conjectures of He and Schramm are in fact equivalent, at least for a large family of circle domains
— Malik Younsi (2015), arXiv v1 abstract
If the boundary of a circle domain is conformally removable, then every conformal equivalence from it to another circle domain is the restriction of a Möbius transformation, regardless of how many complementary components there are.
1. The theorem separately assumes that U and V are circle domains, that frontier U is conformally removable, and that f is a conformal equivalence from U to V.
2. IsConformallyRemovable quantifies over orientation-preserving sphere homeomorphisms conformal off the compact set. The conclusion supplies one Möbius transformation M agreeing with f at every point of U.
3. This is only the removability-to-rigidity direction associated with the He–Schramm program; it is not an equivalence and it does not claim the converse. The body is sorry, paired with OAI.Analysis.CircleDomains.Main.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI circle-domain rigidity manuscript (abstract), Pinned OpenAI formalization note 071, Malik Younsi (2015), arXiv v1 abstract
Distinct definitions of this name: OAI/Analysis/CircleDomains/Main.lean:16 (same statement text; different proof or body). Qualify references as OAI.Problem047.removability_implies_rigidity@path.

conformally removable: a compact set outside which every orientation-preserving conformal sphere homeomorphism must already be Möbius everywhere
conformally rigid circle domain: a circle domain whose conformal equivalences to other circle domains are restrictions of Möbius transformations
```
theorem removability_implies_rigidity :
    ∀ (U V : Set Sphere) (f : Sphere → Sphere),
      IsCircleDomain U → IsCircleDomain V →
      IsConformallyRemovable (frontier U) → IsConformalEquivalence f U V →
      ∃ M : Mobius, ∀ p ∈ U, f p = M • p := by
…
```