`ComparatorChallenges/Nagata.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

48 declarations, 3 contain sorry, 0 declared as axioms

Nagata's strict inequality for very general complex plane configurations

The exceptional family is chosen before every curve and multiplicity vector. Closedness means common multihomogeneous zero loci on the product of projective planes, restricted to the open set of ordered distinct configurations.

def OAI.Nagata.AffineMultiplicity.pointIdeal line 42
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/AffineMultiplicity.lean:22 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.pointIdeal@path.
def OAI.Nagata.AffineMultiplicity.orderAtLeast line 45
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/AffineMultiplicity.lean:27 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.orderAtLeast@path.
def OAI.Nagata.AffineMultiplicity.translateHom line 49
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/TaylorMultiplicity.lean:60 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.translateHom@path.
theorem OAI.Nagata.AffineMultiplicity.translateHom_C line 53
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/TaylorMultiplicity.lean:63 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.translateHom_C@path.
theorem OAI.Nagata.AffineMultiplicity.translateHom_X line 57
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/TaylorMultiplicity.lean:67 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.translateHom_X@path.
theorem OAI.Nagata.AffineMultiplicity.translateHom_neg_comp line 61
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/TaylorMultiplicity.lean:71 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.translateHom_neg_comp@path.
theorem OAI.Nagata.AffineMultiplicity.translateHom_comp_neg line 67
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/TaylorMultiplicity.lean:77 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.translateHom_comp_neg@path.
def OAI.Nagata.AffineMultiplicity.translateEquiv line 71
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/TaylorMultiplicity.lean:81 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.translateEquiv@path.
theorem OAI.Nagata.AffineMultiplicity.translateEquiv_apply line 77
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/TaylorMultiplicity.lean:87 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.translateEquiv_apply@path.
theorem OAI.Nagata.AffineMultiplicity.translate_ne_zero line 80
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/TaylorMultiplicity.lean:173 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.translate_ne_zero@path.
theorem OAI.Nagata.AffineMultiplicity.translated_support_nonempty line 87
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/TaylorMultiplicity.lean:180 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.translated_support_nonempty@path.
def OAI.Nagata.AffineMultiplicity.ordinaryMultiplicity The least total degree of a nonzero Taylor coefficient line 92
Source documentation
The least total degree of a nonzero Taylor coefficient.
L4DEL explanation
The least total degree of a nonzero Taylor coefficient.
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)
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/TaylorMultiplicity.lean:189 (different declaration text). Qualify references as OAI.Nagata.AffineMultiplicity.ordinaryMultiplicity@path.
```
def ordinaryMultiplicity (point : σ → K) (polynomial : MvPolynomial σ K)
    (nonzero : polynomial ≠ 0) : ℕ :=
…
```
abbrev OAI.Nagata.ProjectiveGeometry.PlanePoint line 101
Also declared: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:36 (same declaration text).
abbrev OAI.Nagata.ProjectiveGeometry.OrderedDistinctPoints line 103
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:39 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.OrderedDistinctPoints@path.
structure OAI.Nagata.ProjectiveGeometry.MultihomogeneousEquation line 106
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:113 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.MultihomogeneousEquation@path.
def OAI.Nagata.ProjectiveGeometry.equationVanishes line 112
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:122 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.equationVanishes@path.
def OAI.Nagata.ProjectiveGeometry.IsProductZariskiClosed line 118
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:130 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.IsProductZariskiClosed@path.
def OAI.Nagata.ProjectiveGeometry.IsConfigurationZariskiClosed line 122
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:135 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.IsConfigurationZariskiClosed@path.
abbrev OAI.Nagata.ProjectiveGeometry.ChartVariables line 127
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:217 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.ChartVariables@path.
def OAI.Nagata.ProjectiveGeometry.dehomogenize line 129
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:220 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.dehomogenize@path.
def OAI.Nagata.ProjectiveGeometry.chartCoordinates line 135
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:227 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.chartCoordinates@path.
def OAI.Nagata.ProjectiveGeometry.multiplicityAtLeast Local multiplicity lower bounds are membership in powers of the point ideal line 139
Source documentation
Local multiplicity lower bounds are membership in powers of the point ideal.
L4DEL explanation
Local multiplicity lower bounds are membership in powers of the point ideal.
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)
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:248 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.multiplicityAtLeast@path.
```
def multiplicityAtLeast (polynomial : MvPolynomial (Fin 3) ℂ)
    (point : PlanePoint) (multiplicity : ℕ) : Prop :=
…
```
Mentions: OAI.Nagata.ProjectiveGeometry.PlanePoint
theorem OAI.Nagata.ProjectiveGeometry.chart_exists line 145
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:254 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.chart_exists@path.
abbrev OAI.Nagata.Workers.W30.TernaryPolynomial line 157
Also declared: OAI/AlgebraicGeometry/PlaneCurves/Factorization.lean:27 (same declaration text).
def OAI.Nagata.Workers.W30.equationMultiplicity line 161
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Factorization.lean:49 (different declaration text). Qualify references as OAI.Nagata.Workers.W30.equationMultiplicity@path.
def OAI.Nagata.Workers.W30.associatePrincipalIdeal line 165
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Factorization.lean:77 (different declaration text). Qualify references as OAI.Nagata.Workers.W30.associatePrincipalIdeal@path.
def OAI.Nagata.Workers.W30.equationIdealCycle line 169
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Factorization.lean:89 (different declaration text). Qualify references as OAI.Nagata.Workers.W30.equationIdealCycle@path.
theorem OAI.Nagata.Workers.W30.exists_positive_degree_equation_of_homogeneous_cycle Challenge statement: Nagata's conjecture sorry line 174
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 2 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.AlgebraicGeometry.PlaneCurves.Nagata as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/IdealCycles.lean:242 (different declaration text). Qualify references as OAI.Nagata.Workers.W30.exists_positive_degree_equation_of_homogeneous_cycle@path.
```
theorem exists_positive_degree_equation_of_homogeneous_cycle
    (cycle : Ideal TernaryPolynomial →₀ ℕ) (nonzero : cycle ≠ 0)
    (homogeneous : ∀ ideal ∈ cycle.support,
      ideal.IsPrime ∧ ideal ≠ ⊥ ∧ ideal.IsPrincipal ∧
        ideal.IsHomogeneous (MvPolynomial.homogeneousSubmodule (Fin 3) ℂ)) :
    ∃ polynomial : TernaryPolynomial,
      polynomial ≠ 0 ∧ polynomial.IsHomogeneous polynomial.totalDegree ∧
        0 < polynomial.totalDegree ∧ equationIdealCycle polynomial = cycle := by
…
```
Mentions: OAI.Nagata.Workers.W30.TernaryPolynomial
abbrev OAI.Nagata.W16.PlanePolynomial line 188
Also declared: OAI/AlgebraicGeometry/PlaneCurves/Forms.lean:26 (same declaration text).
structure OAI.Nagata.W16.NonzeroHomogeneousForm line 190
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Forms.lean:68 (different declaration text). Qualify references as OAI.Nagata.W16.NonzeroHomogeneousForm@path.
theorem OAI.Nagata.W16.homogeneous_eval_scale line 195
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Forms.lean:45 (different declaration text). Qualify references as OAI.Nagata.W16.homogeneous_eval_scale@path.

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
theorem OAI.Nagata.W16.eval_dehomogenize line 211
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Forms.lean:265 (different declaration text). Qualify references as OAI.Nagata.W16.eval_dehomogenize@path.
theorem OAI.Nagata.W16.eval_dehomogenize_normalized line 224
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Forms.lean:278 (different declaration text). Qualify references as OAI.Nagata.W16.eval_dehomogenize_normalized@path.

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
theorem OAI.Nagata.W16.eval_eq_zero_of_dehomogenize_eq_zero line 244
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Forms.lean:296 (different declaration text). Qualify references as OAI.Nagata.W16.eval_eq_zero_of_dehomogenize_eq_zero@path.

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
theorem OAI.Nagata.W16.eq_zero_of_dehomogenize_eq_zero line 255
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Forms.lean:307 (different declaration text). Qualify references as OAI.Nagata.W16.eq_zero_of_dehomogenize_eq_zero@path.

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
theorem OAI.Nagata.W16.dehomogenize_ne_zero line 268
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Forms.lean:318 (different declaration text). Qualify references as OAI.Nagata.W16.dehomogenize_ne_zero@path.

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
def OAI.Nagata.Workers.W10.EffectivePlaneCurve Nonzero finite cycles of homogeneous prime plane ideals, with natural weights line 283
Source documentation
Nonzero finite cycles of homogeneous prime plane ideals, with natural weights.
L4DEL explanation
Nonzero finite cycles of homogeneous prime plane ideals, with natural weights.
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)
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/EffectiveCurves.lean:67 (same statement text; different proof or body). Qualify references as OAI.Nagata.Workers.W10.EffectivePlaneCurve@path.
```
def EffectivePlaneCurve := {cycle : Ideal TernaryPolynomial →₀ ℕ //
…
```
def OAI.Nagata.Workers.W10.curveEquation An equation realizing the entire cycle, including repeated components line 289
Source documentation
An equation realizing the entire cycle, including repeated components.
L4DEL explanation
An equation realizing the entire cycle, including repeated components.
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)
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/EffectiveCurves.lean:72 (different declaration text). Qualify references as OAI.Nagata.Workers.W10.curveEquation@path.
```
def curveEquation (curve : EffectivePlaneCurve) : TernaryPolynomial :=
…
```
theorem OAI.Nagata.Workers.W10.curveEquation_spec line 293
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/EffectiveCurves.lean:75 (different declaration text). Qualify references as OAI.Nagata.Workers.W10.curveEquation_spec@path.
theorem OAI.Nagata.Workers.W10.curveEquation_nonzero line 300
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/EffectiveCurves.lean:80 (different declaration text). Qualify references as OAI.Nagata.Workers.W10.curveEquation_nonzero@path.
def OAI.Nagata.Workers.W10.curveDegree line 303
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/EffectiveCurves.lean:83 (different declaration text). Qualify references as OAI.Nagata.Workers.W10.curveDegree@path.
theorem OAI.Nagata.Workers.W10.curveEquation_homogeneous line 305
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/EffectiveCurves.lean:88 (different declaration text). Qualify references as OAI.Nagata.Workers.W10.curveEquation_homogeneous@path.
def OAI.Nagata.Workers.W10.curveForm line 309
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/EffectiveCurves.lean:95 (different declaration text). Qualify references as OAI.Nagata.Workers.W10.curveForm@path.
def OAI.Nagata.ProjectiveGeometry.formOrdinaryMultiplicity line 316
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/ProjectiveMultiplicity.lean:202 (different declaration text). Qualify references as OAI.Nagata.ProjectiveGeometry.formOrdinaryMultiplicity@path.
def OAI.Nagata.W15.curveOrdinaryMultiplicity line 329
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/CycleMultiplicity.lean:397 (different declaration text). Qualify references as OAI.Nagata.W15.curveOrdinaryMultiplicity@path.
def OAI.Nagata.FullNagata line 338
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Configuration.lean:312 (same statement text; different proof or body). Qualify references as OAI.Nagata.FullNagata@path.

theorem OAI.Nagata.nagata_conjecture Challenge statement: Nagata's conjecture sorry line 352

L4DEL explanation

This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.

The linked formalization note covers 2 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
The repository's comparator configuration selects OAI.AlgebraicGeometry.PlaneCurves.Nagata as the released solution module checked against this declaration.

Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)

Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Nagata.lean:148 (different declaration text). Qualify references as OAI.Nagata.nagata_conjecture@path.

theorem nagata_conjecture :
    ∀ count : ℕ, 10 ≤ count →
      ∃ exceptional : ℕ → Set (OrderedDistinctPoints count),
        (∀ index, IsConfigurationZariskiClosed (exceptional index) ∧
          exceptional index ≠ Set.univ) ∧
        (∃ points : OrderedDistinctPoints count, ∀ index, points ∉ exceptional index) ∧
        ∀ points : OrderedDistinctPoints count, (∀ index, points ∉ exceptional index) →
          ∀ curve : Workers.W10.EffectivePlaneCurve, ∀ multiplicities : Fin count → ℕ,
            (∀ index, multiplicities index ≤ W15.curveOrdinaryMultiplicity curve (points.val index)) →
            (∑ index, (multiplicities index : ℝ)) <
              (Workers.W10.curveDegree curve : ℝ) * Real.sqrt (count : ℝ) := by
…

theorem OAI.Nagata.nagata_conjecture_coordinates Challenge statement: Nagata's conjecture sorry line 365
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 2 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.AlgebraicGeometry.PlaneCurves.Nagata as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/AlgebraicGeometry/PlaneCurves/Nagata.lean:162 (same statement text; different proof or body). Qualify references as OAI.Nagata.nagata_conjecture_coordinates@path.
```
theorem nagata_conjecture_coordinates : FullNagata := by
…
```