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Lean statement

Logarithmic Kodaira additivity in the negative-fiber branch

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.ReverseLogKodaira.SmoothProjectiveVariety.veryGenerally_fiber_negative_branch

Lean statement file: LogKodairaFiberNegative.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 033 link it.

See the candidate theoremWho ran the proof-verification programReport a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem veryGenerally_fiber_negative_branch    {X Y : SmoothProjectiveVariety} {E : X.ReducedSNCBoundary} {D : Y.ReducedSNCBoundary}    (f : StratumSmoothFibration X Y E D) :    VeryGenerally Y fun y => y.point ∈ D.complement ∧      ∀ F : FiberModel f.toBoundaryFibration y, F.boundary.kodaira = ⊥ →        E.kodaira = D.kodaira + F.boundary.kodaira ∧        ∀ (m : ℕ), 0 < m → ∀ s : X.RationalPluriform m, s ∈ E.sections m → s = 0 := by sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

  • abbreviation OAI.ReverseLogKodaira.complexBase line 16

    abbrev complexBase : Scheme := Spec (CommRingCat.of ℂ)

  • abbreviation OAI.ReverseLogKodaira.projectiveGrading line 18

    abbrev projectiveGrading (N : ℕ) := MvPolynomial.homogeneousSubmodule (Fin (N + 1)) ℂ

  • abbreviation OAI.ReverseLogKodaira.complexProjectiveSpace line 20

    abbrev complexProjectiveSpace (N : ℕ) : Scheme := Proj (projectiveGrading N)

  • definition OAI.ReverseLogKodaira.projectiveConstants line 22

    def projectiveConstants (N : ℕ) : ℂ →+* projectiveGrading N 0 where  toFun r := ⟨MvPolynomial.C r, MvPolynomial.isHomogeneous_C (Fin (N + 1)) r⟩  map_zero' := Subtype.ext (map_zero MvPolynomial.C)  map_one' := Subtype.ext (map_one MvPolynomial.C)  map_add' x y := Subtype.ext (map_add MvPolynomial.C x y)  map_mul' x y := Subtype.ext (map_mul MvPolynomial.C x y)

  • definition OAI.ReverseLogKodaira.complexProjectiveStructure line 29

    def complexProjectiveStructure (N : ℕ) : complexProjectiveSpace N ⟶ complexBase :=  Proj.toSpecZero (projectiveGrading N) ≫ Spec.map (CommRingCat.ofHom (projectiveConstants N))

  • structure OAI.ReverseLogKodaira.SmoothProjectiveVariety line 33

    structure SmoothProjectiveVariety where  scheme : Scheme  structural : scheme ⟶ complexBase  dimension : ℕ  integral : IsIntegral scheme  smooth : SmoothOfRelativeDimension dimension structural  projective : ∃ (N : ℕ) (i : scheme ⟶ complexProjectiveSpace N),    IsClosedImmersion i ∧ i ≫ complexProjectiveStructure N = structural

  • abbreviation OAI.ReverseLogKodaira.SmoothProjectiveVariety.RationalCanonical line 82

    abbrev RationalCanonical (X : SmoothProjectiveVariety) :=  ⋀[X.scheme.functionField]^X.dimension (KaehlerDifferential ℂ X.scheme.functionField)

  • abbreviation OAI.ReverseLogKodaira.SmoothProjectiveVariety.RationalPluriform line 85

    abbrev RationalPluriform (X : SmoothProjectiveVariety) (m : ℕ) :=  ⨂[X.scheme.functionField]^m X.RationalCanonical

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.regularPluriformLattice line 91

    def regularPluriformLattice (X : SmoothProjectiveVariety) (U : X.scheme.affineOpens)    [Nonempty U.1] (m : ℕ) : Submodule Γ(X.scheme, U.1) (X.RationalPluriform m) :=  Submodule.span Γ(X.scheme, U.1) <| Set.range fun a : Fin m → Fin X.dimension → Γ(X.scheme, U.1) =>    PiTensorProduct.tprod X.scheme.functionField fun j =>      exteriorPower.ιMulti X.scheme.functionField X.dimension fun i =>        KaehlerDifferential.D ℂ X.scheme.functionField (algebraMap _ _ (a j i))

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.IsCartierIdeal line 100

    def IsCartierIdeal (X : SmoothProjectiveVariety) (I : X.scheme.IdealSheafData) : Prop :=  ∀ x : X.scheme, ∃ (U : X.scheme.affineOpens), x ∈ U.1 ∧    ∃ t : Γ(X.scheme, U.1), t ≠ 0 ∧ I.ideal U = Ideal.span {t}

  • structure OAI.ReverseLogKodaira.SmoothProjectiveVariety.ReducedSNCBoundary line 104

    structure ReducedSNCBoundary (X : SmoothProjectiveVariety) where  count : ℕ  components : Fin count → X.scheme.IdealSheafData  cartier : ∀ i, X.IsCartierIdeal (components i)  transverse : ∀ S : Finset (Fin count), S.Nonempty →    IsEmpty ((S.sup components).subscheme) ∨      (S.card ≤ X.dimension ∧ SmoothOfRelativeDimension (X.dimension - S.card)        ((S.sup components).subschemeι ≫ X.structural))  union_cartier : X.IsCartierIdeal (∏ i, components i)

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.ReducedSNCBoundary.ideal line 116

    def ideal {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) : X.scheme.IdealSheafData :=  ∏ i, E.components i

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.ReducedSNCBoundary.support line 120

    def support {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) : Set X.scheme :=  E.ideal.support

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.ReducedSNCBoundary.IsLogPluriform line 123

    def IsLogPluriform {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary)    (m : ℕ) (s : X.RationalPluriform m) : Prop :=  ∀ (U : X.scheme.affineOpens) (_ : Nonempty U.1) (t : Γ(X.scheme, U.1)),    t ≠ 0 → E.ideal.ideal U = Ideal.span {t} →      (algebraMap Γ(X.scheme, U.1) X.scheme.functionField t)^m • s ∈        X.regularPluriformLattice U m

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.ReducedSNCBoundary.sections line 153

    def sections {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) (m : ℕ) :    Submodule ℂ (X.RationalPluriform m) where  carrier := E.IsLogPluriform m  zero_mem' := E.isLogPluriform_zero m  add_mem' := E.isLogPluriform_add m  smul_mem' := E.isLogPluriform_smul m

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.ReducedSNCBoundary.HasImageDimensionAtLeast line 162

    def HasImageDimensionAtLeast {X : SmoothProjectiveVariety}    (E : X.ReducedSNCBoundary) (k : ℕ) : Prop :=  ∃ (m : ℕ), 0 < m ∧ ∃ s : X.RationalPluriform m,    s ∈ E.sections m ∧ s ≠ 0 ∧ ∃ r : Fin k → X.scheme.functionField,      AlgebraicIndependent ℂ r ∧ ∀ i, r i • s ∈ E.sections m

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.ReducedSNCBoundary.kodaira line 170

    def kodaira {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) : WithBot ℕ∞ :=  ⨆ (k : ℕ) (_ : E.HasImageDimensionAtLeast k), ((k : ℕ∞) : WithBot ℕ∞)

  • structure OAI.ReverseLogKodaira.SmoothProjectiveVariety.ComplexPoint line 176

    structure ComplexPoint (X : SmoothProjectiveVariety) where  hom : complexBase ⟶ X.scheme  over_base : hom ≫ X.structural = 𝟙 complexBase

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.complexBasePoint line 181

    def complexBasePoint : complexBase := ⟨⊥, Ideal.isPrime_bot⟩

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.ComplexPoint.point line 184

    def ComplexPoint.point {X : SmoothProjectiveVariety} (y : X.ComplexPoint) : X.scheme :=  y.hom complexBasePoint

  • definition OAI.ReverseLogKodaira.SmoothProjectiveVariety.ReducedSNCBoundary.complement line 188

    def ReducedSNCBoundary.complement {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) :    X.scheme.Opens := E.ideal.support.compl

  • structure OAI.ReverseLogKodaira.SmoothProjectiveVariety.BoundaryFibration line 193

    structure BoundaryFibration (X Y : SmoothProjectiveVariety)    (E : X.ReducedSNCBoundary) (D : Y.ReducedSNCBoundary) where  hom : X.scheme ⟶ Y.scheme  over_base : hom ≫ Y.structural = X.structural  surjective : Surjective hom  connected : ∀ y : Y.ComplexPoint, ConnectedSpace ↥(CategoryTheory.Limits.pullback hom y.hom)  support : hom ⁻¹' D.support ⊆ E.support

  • structure OAI.ReverseLogKodaira.SmoothProjectiveVariety.StratumSmoothFibration line 203

    structure StratumSmoothFibration (X Y : SmoothProjectiveVariety)    (E : X.ReducedSNCBoundary) (D : Y.ReducedSNCBoundary)    extends BoundaryFibration X Y E D where  smooth : Smooth (hom.resLE D.complement (hom ⁻¹ᵁ D.complement) le_rfl)  strata : ∀ (S : Finset (Fin E.count)), S.Nonempty →    Smooth (((S.sup E.components).subschemeι ≫ hom).resLE D.complement      (((S.sup E.components).subschemeι ≫ hom) ⁻¹ᵁ D.complement) le_rfl)

  • structure OAI.ReverseLogKodaira.SmoothProjectiveVariety.FiberModel line 213

    structure FiberModel {X Y : SmoothProjectiveVariety} {E : X.ReducedSNCBoundary}    {D : Y.ReducedSNCBoundary} (f : BoundaryFibration X Y E D) (y : Y.ComplexPoint) where  variety : SmoothProjectiveVariety  inclusion : variety.scheme ⟶ X.scheme  square : IsPullback inclusion variety.structural f.hom y.hom  boundary : variety.ReducedSNCBoundary  boundary_pullback : boundary.ideal = E.ideal.comap inclusion

1 more definitions are reached the same way; the full file lists them.

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: additivity, logarithmic, pairs and theorem.

Statements in The reverse logarithmic Kodaira inequality and additivity come first because the release lists only that paper for this Lean statement.

Corollary 1.2 of The reverse logarithmic Kodaira inequality and additivity

Corollary (Logarithmic additivity)

For the pairs in Theorem thm:upper-bound,

\kappa(X,K_X+E)=\kappa(Y,K_Y+D)+\kappa(F,K_F+E_F).
Section 1, Introduction Read it in the paper TeX source, line 51
Show the TeX
For the pairs in Theorem~\ref{thm:upper-bound},
\begin{equation}
 \kappa(X,K_X+E)=\kappa(Y,K_Y+D)+\kappa(F,K_F+E_F).
\end{equation}

Other possible paper matches, closest wording first

  • Proposition 3.1 of The reverse logarithmic Kodaira inequality and additivity. Words of the Lean statement, its names split into words, that also appear in the theorem: range, regular, homogeneous, grade, tensor, lattice and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: outside, logarithmic and finite.
  • Proposition 4.1 of The reverse logarithmic Kodaira inequality and additivity. Words of the Lean statement, its names split into words, that also appear in the theorem: scalar, polynomial, union, homogeneous, group, empty and 24 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
  • Theorem 7.8 of The reverse logarithmic Kodaira inequality and additivity. Words of the Lean statement, its names split into words, that also appear in the theorem: log, reduced, snc, model, surjective, complex and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: outside and logarithmic.
  • Theorem 8.1 of Logarithmic Kodaira dimension and whole-fiber variation. Words of the Lean statement, its names split into words, that also appear in the theorem: relative, dimension, log, zero, fibration, map and 14 more. Found in the paper’s title, which its main theorem states: kodaira. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
  • Proposition 6.3 of The reverse logarithmic Kodaira inequality and additivity. Words of the Lean statement, its names split into words, that also appear in the theorem: support, relative, pullbacks, prime, log, reduced and six more.
  • Proposition 2.7 of Orbifold and logarithmic Iitaka subadditivity. Words of the Lean statement, its names split into words, that also appear in the theorem: least, grading, supported, tensoring, lattice, images and 18 more.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

  • Accepted Their term: “CERTIFIED”.

    Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.

    Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

  • Accepted Their term: “comparator-sandboxed-pass”.

    Re-run through Comparator in a sandboxed virtual machine: accepted.

    Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

  • Accepted Their term: “accepted”.

    Re-run through Comparator with Lean’s kernel: accepted (the check took 264 s; peak memory 6.4 GiB).

    Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 033’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

  • MathVet

    Finds a supporting or a different result stated Their term: “supporting-only”.

    The challenge statements state a lemma or auxiliary result, not the headline. Their record

    Their note: Only the very-general negative-fibre (kappa(F) = -infinity) branch of log additivity in the projective reduced-SNC stratum-smooth setting; no Campana orbifold Iitaka statement, no Popa/variation inequality, no kappa >= 0 branch.

    Definitions the group asks readers to compare with the paper: Entire framework is custom scheme theory: SmoothProjectiveVariety, ReducedSNCBoundary, kodaira := ⨆ (k : ℕ) (_ : E.HasImageDimensionAtLeast k), (k : WithBot ℕ∞) (Iitaka dimension via algebraically independent ratios of log pluriforms), VeryGenerally; the conclusion quantifies ∀ F : FiberModel ..., so it is vacuous if no FiberModel can exist (non-vacuity is not visible from the statement)

  • Carlos Toledo’s cert-machine

    Finds a supporting or a different result stated Their term: “DIFFERENT”.

    The challenge statements state a different theorem from the headline. Their record

    Their note: Neither headline sentence is stated. The family's only challenge states the negative-fibre branch of logarithmic Kodaira ADDITIVITY (for a very general complex base point y outside the base boundary, if the fibre pair has log Kodaira dimension ⊥ = −∞ then κ̄(X,E) = κ̄(Y,D) + κ̄(F) = −∞ and every positive-degree log pluriform on X vanishes), for stratum-smooth surjective connected-fibre morphisms of smooth projective reduced-SNC pairs — a result of the third manuscript ('The reverse logarithmic Kodaira inequality and additivity'), which the headline does not mention (only the family title's word 'logarithmic additivity' does). It neither implies nor is implied by the headline: subadditivity is a lower bound on κ̄(U) and says nothing when κ(F) = −∞, while the Lean statement is an upper bound (equality to −∞).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The finite-dimension and negative-base branches of the paper's additivity theorem are outside this selected statement.”

An AI agent drafted an interpretation of that note, classifying this statement as “states part of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

245 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 53 to its own declarations, 95 to Mathlib’s documentation. 33 declared names also link to the same declaration in OpenAI’s library. 5 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. 4 of them could name more than one declaration. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAI noncomputable sectionopen CategoryTheory AlgebraicGeometryopen scoped TensorProductnamespace ReverseLogKodairaattribute [local instance] MvPolynomial.gradedAlgebraabbrev complexBase : Scheme := Spec (CommRingCat.of ℂ)abbrev projectiveGrading (N : ℕ) := MvPolynomial.homogeneousSubmodule (Fin (N + 1)) ℂabbrev complexProjectiveSpace (N : ℕ) : Scheme := Proj (projectiveGrading N)def projectiveConstants (N : ℕ) : ℂ →+* projectiveGrading N 0 where  toFun r := ⟨MvPolynomial.C r, MvPolynomial.isHomogeneous_C (Fin (N + 1)) r⟩  map_zero' := Subtype.ext (map_zero MvPolynomial.C)  map_one' := Subtype.ext (map_one MvPolynomial.C)  map_add' x y := Subtype.ext (map_add MvPolynomial.C x y)  map_mul' x y := Subtype.ext (map_mul MvPolynomial.C x y)def complexProjectiveStructure (N : ℕ) : complexProjectiveSpace N ⟶ complexBase :=  Proj.toSpecZero (projectiveGrading N) ≫ Spec.map (CommRingCat.ofHom (projectiveConstants N)) structure SmoothProjectiveVariety where  scheme : Scheme  structural : scheme ⟶ complexBase  dimension : ℕ  integral : IsIntegral scheme  smooth : SmoothOfRelativeDimension dimension structural  projective : ∃ (N : ℕ) (i : scheme ⟶ complexProjectiveSpace N),    IsClosedImmersion i ∧ i ≫ complexProjectiveStructure N = structuralattribute [instance] SmoothProjectiveVariety.integral SmoothProjectiveVariety.smoothnamespace SmoothProjectiveVarietyinstance (X : SmoothProjectiveVariety) : Smooth X.structural :=  SmoothOfRelativeDimension.smooth X.dimension X.structural def globalConstants (X : SmoothProjectiveVariety) : ℂ →+* Γ(X.scheme, ⊤) :=  X.structural.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of ℂ)).inv.hom def constants (X : SmoothProjectiveVariety) (U : X.scheme.Opens) : ℂ →+* Γ(X.scheme, U) :=  (X.scheme.presheaf.map (homOfLE (le_top (a := U))).op).hom.comp X.globalConstantsinstance (X : SmoothProjectiveVariety) (U : X.scheme.Opens) : Algebra ℂ Γ(X.scheme, U) :=  (X.constants U).toAlgebrainstance (X : SmoothProjectiveVariety) : Nonempty (⊤ : X.scheme.Opens) :=  ⟨⟨genericPoint X.scheme, by trivial⟩⟩ def functionFieldConstants (X : SmoothProjectiveVariety) : ℂ →+* X.scheme.functionField :=  (X.scheme.germToFunctionField ⊤).hom.comp X.globalConstantsinstance (X : SmoothProjectiveVariety) : Algebra ℂ X.scheme.functionField :=  X.functionFieldConstants.toAlgebralemma germ_constants (X : SmoothProjectiveVariety) (U : X.scheme.Opens) [Nonempty U] (c : ℂ) :    algebraMap Γ(X.scheme, U) X.scheme.functionField (algebraMap ℂ Γ(X.scheme, U) c) =      algebraMap ℂ X.scheme.functionField c := by  change X.scheme.presheaf.germ U (genericPoint X.scheme) _    (X.scheme.presheaf.map (homOfLE (le_top (a := U))).op (X.globalConstants c)) =      X.scheme.presheaf.germ ⊤ (genericPoint X.scheme) _ (X.globalConstants c)  exact X.scheme.presheaf.germ_res_apply (homOfLE le_top) (genericPoint X.scheme) _ _instance (X : SmoothProjectiveVariety) (U : X.scheme.Opens) [Nonempty U] :    IsScalarTower ℂ Γ(X.scheme, U) X.scheme.functionField :=  IsScalarTower.of_algebraMap_eq fun c => (X.germ_constants U c).symmabbrev RationalCanonical (X : SmoothProjectiveVariety) :=  ⋀[X.scheme.functionField]^X.dimension (KaehlerDifferential ℂ X.scheme.functionField)abbrev RationalPluriform (X : SmoothProjectiveVariety) (m : ℕ) :=  ⨂[X.scheme.functionField]^m X.RationalCanonicalinstance (X : SmoothProjectiveVariety) (m : ℕ) : AddCommGroup (X.RationalPluriform m) :=  Module.addCommMonoidToAddCommGroup X.scheme.functionFielddef regularPluriformLattice (X : SmoothProjectiveVariety) (U : X.scheme.affineOpens)    [Nonempty U.1] (m : ℕ) : Submodule Γ(X.scheme, U.1) (X.RationalPluriform m) :=  Submodule.span Γ(X.scheme, U.1) <| Set.range fun a : Fin m → Fin X.dimension → Γ(X.scheme, U.1) =>    PiTensorProduct.tprod X.scheme.functionField fun j =>      exteriorPower.ιMulti X.scheme.functionField X.dimension fun i =>        KaehlerDifferential.D ℂ X.scheme.functionField (algebraMap _ _ (a j i)) def IsCartierIdeal (X : SmoothProjectiveVariety) (I : X.scheme.IdealSheafData) : Prop :=  ∀ x : X.scheme, ∃ (U : X.scheme.affineOpens), x ∈ U.1 ∧    ∃ t : Γ(X.scheme, U.1), t ≠ 0 ∧ I.ideal U = Ideal.span {t}structure ReducedSNCBoundary (X : SmoothProjectiveVariety) where  count : ℕ  components : Fin count → X.scheme.IdealSheafData  cartier : ∀ i, X.IsCartierIdeal (components i)  transverse : ∀ S : Finset (Fin count), S.Nonempty →    IsEmpty ((S.sup components).subscheme) ∨      (S.card ≤ X.dimension ∧ SmoothOfRelativeDimension (X.dimension - S.card)        ((S.sup components).subschemeι ≫ X.structural))  union_cartier : X.IsCartierIdeal (∏ i, components i)namespace ReducedSNCBoundarydef ideal {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) : X.scheme.IdealSheafData :=  ∏ i, E.components i def support {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) : Set X.scheme :=  E.ideal.supportdef IsLogPluriform {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary)    (m : ℕ) (s : X.RationalPluriform m) : Prop :=  ∀ (U : X.scheme.affineOpens) (_ : Nonempty U.1) (t : Γ(X.scheme, U.1)),    t ≠ 0 → E.ideal.ideal U = Ideal.span {t} →      (algebraMap Γ(X.scheme, U.1) X.scheme.functionField t)^m • s ∈        X.regularPluriformLattice U mlemma isLogPluriform_zero {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) (m : ℕ) :    E.IsLogPluriform m 0 := by  intro U hU t ht hI  simp only [smul_zero, Submodule.zero_mem]lemma isLogPluriform_add {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) (m : ℕ)    {s r : X.RationalPluriform m} (hs : E.IsLogPluriform m s) (hr : E.IsLogPluriform m r) :    E.IsLogPluriform m (s + r) := by  intro U hU t ht hI  simpa only [smul_add] using (X.regularPluriformLattice U m).add_mem    (hs U hU t ht hI) (hr U hU t ht hI)lemma isLogPluriform_smul {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) (m : ℕ)    (c : ℂ) {s : X.RationalPluriform m} (hs : E.IsLogPluriform m s) :    E.IsLogPluriform m (c • s) := by  intro U hU t ht hI  have h := (X.regularPluriformLattice U m).smul_mem    (algebraMap ℂ Γ(X.scheme, U.1) c) (hs U hU t ht hI)  rw [IsScalarTower.algebraMap_smul Γ(X.scheme, U.1)] at h  rwa [smul_comm c] at h def sections {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) (m : ℕ) :    Submodule ℂ (X.RationalPluriform m) where  carrier := E.IsLogPluriform m  zero_mem' := E.isLogPluriform_zero m  add_mem' := E.isLogPluriform_add m  smul_mem' := E.isLogPluriform_smul m def HasImageDimensionAtLeast {X : SmoothProjectiveVariety}    (E : X.ReducedSNCBoundary) (k : ℕ) : Prop :=  ∃ (m : ℕ), 0 < m ∧ ∃ s : X.RationalPluriform m,    s ∈ E.sections m ∧ s ≠ 0 ∧ ∃ r : Fin k → X.scheme.functionField,      AlgebraicIndependent ℂ r ∧ ∀ i, r i • s ∈ E.sections m def kodaira {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) : WithBot ℕ∞ :=  ⨆ (k : ℕ) (_ : E.HasImageDimensionAtLeast k), ((k : ℕ∞) : WithBot ℕ∞)end ReducedSNCBoundary structure ComplexPoint (X : SmoothProjectiveVariety) where  hom : complexBase ⟶ X.scheme  over_base : hom ≫ X.structural = 𝟙 complexBase def complexBasePoint : complexBase := ⟨⊥, Ideal.isPrime_bot⟩ def ComplexPoint.point {X : SmoothProjectiveVariety} (y : X.ComplexPoint) : X.scheme :=  y.hom complexBasePoint def ReducedSNCBoundary.complement {X : SmoothProjectiveVariety} (E : X.ReducedSNCBoundary) :    X.scheme.Opens := E.ideal.support.compl structure BoundaryFibration (X Y : SmoothProjectiveVariety)    (E : X.ReducedSNCBoundary) (D : Y.ReducedSNCBoundary) where  hom : X.scheme ⟶ Y.scheme  over_base : hom ≫ Y.structural = X.structural  surjective : Surjective hom  connected : ∀ y : Y.ComplexPoint, ConnectedSpace ↥(CategoryTheory.Limits.pullback hom y.hom)  support : hom ⁻¹' D.support ⊆ E.support structure StratumSmoothFibration (X Y : SmoothProjectiveVariety)    (E : X.ReducedSNCBoundary) (D : Y.ReducedSNCBoundary)    extends BoundaryFibration X Y E D where  smooth : Smooth (hom.resLE D.complement (hom ⁻¹ᵁ D.complement) le_rfl)  strata : ∀ (S : Finset (Fin E.count)), S.Nonempty →    Smooth (((S.sup E.components).subschemeι ≫ hom).resLE D.complement      (((S.sup E.components).subschemeι ≫ hom) ⁻¹ᵁ D.complement) le_rfl) structure FiberModel {X Y : SmoothProjectiveVariety} {E : X.ReducedSNCBoundary}    {D : Y.ReducedSNCBoundary} (f : BoundaryFibration X Y E D) (y : Y.ComplexPoint) where  variety : SmoothProjectiveVariety  inclusion : variety.scheme ⟶ X.scheme  square : IsPullback inclusion variety.structural f.hom y.hom  boundary : variety.ReducedSNCBoundary  boundary_pullback : boundary.ideal = E.ideal.comap inclusion def VeryGenerally (Y : SmoothProjectiveVariety) (P : Y.ComplexPoint → Prop) : Prop :=  ∃ Z : ℕ → TopologicalSpace.Closeds Y.scheme,    (∀ n, Z n ≠ ⊤) ∧ ∀ y : Y.ComplexPoint, (∀ n, y.point ∉ Z n) → P yend SmoothProjectiveVarietyend ReverseLogKodairanamespace ReverseLogKodaira.SmoothProjectiveVarietytheorem veryGenerally_fiber_negative_branch    {X Y : SmoothProjectiveVariety} {E : X.ReducedSNCBoundary} {D : Y.ReducedSNCBoundary}    (f : StratumSmoothFibration X Y E D) :    VeryGenerally Y fun y => y.point ∈ D.complement ∧      ∀ F : FiberModel f.toBoundaryFibration y, F.boundary.kodaira = ⊥ →        E.kodaira = D.kodaira + F.boundary.kodaira ∧        ∀ (m : ℕ), 0 < m → ∀ s : X.RationalPluriform m, s ∈ E.sections m → s = 0 := by sorryend ReverseLogKodaira.SmoothProjectiveVarietyendend OAI
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The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

The reverse logarithmic Kodaira inequality and additivity

Theorem (Reverse logarithmic Kodaira inequality)

Under these hypotheses,

\kappa(X,L_X)\leq\kappa(Y,L_Y)+\kappa(F,L_F).

If either term on the right is -\infty, then H^0(X,mL_X)=0 for every m>0.

Section 1, Introduction Read it in the paper TeX source, line 30
Show the TeX
Under these hypotheses,
\begin{equation}
 \kappa(X,L_X)\leq\kappa(Y,L_Y)+\kappa(F,L_F).
\end{equation}
If either term on the right is $-\infty$, then
$H^0(X,mL_X)=0$ for every $m>0$.
Corollary (Logarithmic additivity)

For the pairs in Theorem thm:upper-bound,

\kappa(X,K_X+E)=\kappa(Y,K_Y+D)+\kappa(F,K_F+E_F).
Section 1, Introduction Read it in the paper TeX source, line 51
Show the TeX
For the pairs in Theorem~\ref{thm:upper-bound},
\begin{equation}
 \kappa(X,K_X+E)=\kappa(Y,K_Y+D)+\kappa(F,K_F+E_F).
\end{equation}
Proposition

Let f,E,D satisfy the hypotheses of Theorem thm:upper-bound, and let 0\ne s\in H^0(X,mL_X) for some m>0. Then \kappa(Y,L_Y)\geq0. If \kappa(Y,L_Y)=0 and s vanishes identically on X_y for some y\in V, there are an integer a>0 and a nonzero \tau\in H^0(Y,aL_Y) with \tau(y)=0.

Section 1, Introduction Read it in the paper TeX source, line 78
Show the TeX
Let $f,E,D$ satisfy the hypotheses of Theorem~\ref{thm:upper-bound}, and
let $0\ne s\in H^0(X,mL_X)$ for some $m>0$. Then
$\kappa(Y,L_Y)\geq0$. If $\kappa(Y,L_Y)=0$ and $s$ vanishes identically
on $X_y$ for some $y\in V$, there are an integer $a>0$ and a nonzero
$\tau\in H^0(Y,aL_Y)$ with $\tau(y)=0$.
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 033 link it.
Solution module
OAI.AlgebraicGeometry.LogKodaira.FiberNegative
Allowed axioms
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
34 local vocabulary declarations
Lines of Lean, without comments or blank lines
149
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

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Plectis Atlas

Logarithmic Kodaira additivity in the negative-fiber branch

Family 033: Iitaka subadditivity, variation, and logarithmic additivity

Candidate pairing

Corollary (Logarithmic additivity)

For the pairs in Theorem thm:upper-bound,

\kappa(X,K_X+E)=\kappa(Y,K_Y+D)+\kappa(F,K_F+E_F).

The reverse logarithmic Kodaira inequality and additivity

TeX label cor:additivity, build/sections/introduction.tex, line 51

The Lean statement says

theorem veryGenerally_fiber_negative_branch    {X Y : SmoothProjectiveVariety} {E : X.ReducedSNCBoundary} {D : Y.ReducedSNCBoundary}    (f : StratumSmoothFibration X Y E D) :    VeryGenerally Y fun y => y.point ∈ D.complement ∧      ∀ F : FiberModel f.toBoundaryFibration y, F.boundary.kodaira = ⊥ →        E.kodaira = D.kodaira + F.boundary.kodaira ∧        ∀ (m : ℕ), 0 < m → ∀ s : X.RationalPluriform m, s ∈ E.sections m → s = 0 := by sorry

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Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.three independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

34 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/LogKodairaFiberNegative.html

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