Iitaka subadditivity, variation, and logarithmic additivity
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Proves Campana’s orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-\(\mathcal C\) manifolds with rational simple-normal-crossing boundaries. For projective fibrations \(f:U\to V\) of smooth complex quasi-projective varieties with connected fibers, general fiber F, and \(\bar\kappa(V)\ge0\), proves Popa’s inequality \(\bar\kappa(U)\ge\kappa(F)+\max\{\bar\kappa(V),\mathop{\mathrm{Var}}\nolimits (f)\}\), where variation measures the whole geometric generic fiber. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Nine other possible matches are listed below the card, among them Theorem 1.1(Logarithmic variation) in Logarithmic Kodaira dimension and whole-fiber variation and Corollary 1.2(Projective Iitaka–Viehweg inequality) in Logarithmic Kodaira dimension and whole-fiber variation.
Let X be a smooth compact connected complex manifold in Fujiki class \mathcal C, let \Delta be a rational SNC boundary, and let f:X\to Y be a surjective holomorphic map with connected fibers onto a normal compact irreducible complex space. For a very general smooth fiber F, put \Delta_F=\Delta|_F. Then
We prove Campana’s orbifold Iitaka subadditivity conjecture for rational simple normal crossing boundaries on compact manifolds in Fujiki class \(\mathcal C\), including coefficient one. Ordinary and logarithmic subadditivity follow.
We prove the logarithmic Iitaka–Viehweg inequality for projective surjective morphisms with connected fibers between smooth complex quasi-projective varieties whose base has nonnegative logarithmic Kodaira dimension. The variation measures the birational field of definition of the whole geometric generic fiber. This resolves Popa’s logarithmic variation conjecture positively.
We prove the reverse logarithmic Kodaira inequality for a surjective connected-fiber morphism \(f:(X,E)\to(Y,D)\) of smooth projective reduced simple-normal-crossing pairs, with \(\mathop{\mathrm{Supp}}\nolimits (f^*D)\subseteq\mathop{\mathrm{Supp}}\nolimits E\), such that X and every boundary stratum are smooth over \(Y\setminus\mathop{\mathrm{Supp}}\nolimits D\). Together with logarithmic subadditivity, the inequality gives additivity, including both negative-infinity cases. This resolves Popa’s logarithmic additivity conjecture positively in the projective reduced-SNC, stratum-smooth setting.
We prove the ordinary Iitaka subadditivity conjecture for surjective projective morphisms with connected fibers between smooth connected projective varieties over algebraically closed fields of characteristic zero. If F is the geometric generic fiber of \(f:X\to Z\), then \(\kappa(X)\geq\kappa(F)+\kappa(Z)\).
We prove the compact log-smooth Kähler case of b-semiampleness. Let \(f:Y\to X\) be a surjective holomorphic map with connected fibers between smooth compact connected Kähler manifolds, and let Δ be an effective rational divisor with simple normal crossing support and coefficients in \([0,1]\), with \(K_Y+\Delta\sim_{\mathbb Q}f^*L\) for \(L\in\mathop{\mathrm{Pic}}\nolimits (X)_{\mathbb Q}\). There is a smooth compact Kähler modification \(S\to X\) for which the threshold-moduli line satisfies \(M_{S_1}=\nu^*M_S\) in \(\mathop{\mathrm{Pic}}\nolimits (S_1)_{\mathbb Q}\) for every smooth compact Kähler modification \(\nu:S_1\to S\), and some positive multiple of MS is represented by a holomorphic line bundle generated by global sections. Horizontal components of coefficient one are allowed; neither projectivity nor a Campana orbifold Iitaka hypothesis is assumed.
One Lean statement in this family
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LogKodairaFiberNegative.lean, the release lists it as a supporting statement.
theoremveryGenerally_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 := bysorry
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three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express a supporting result (their term: “supporting-only”).
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)
“supporting-only”, in their words: The challenge statements state a lemma or auxiliary result, not the headline.
Their method: 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.
cert-machine finds that the Lean statements express a different result (their term: “DIFFERENT”).
The gap they name: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 −∞).
“DIFFERENT”, in their words: The challenge statements state a different theorem from the headline.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The paper studies logarithmic Kodaira additivity for connected-fiber morphisms of smooth projective reduced simple-normal-crossing pairs that are smooth on all boundary strata away from the base boundary. The linked formalization proves the negative-fiber branch: for a very general base point, if the logarithmic Kodaira dimension of the fiber is $-\infty$, then the total logarithmic Kodaira dimension equals the sum of the base and fiber dimensions and is $-\infty$. Every positive-degree logarithmic pluriform section on the total space then vanishes.
The finite-dimension and negative-base branches of the paper’s additivity theorem are outside this selected statement.
AI-drafted comparisons, with no recorded human review
Logarithmic Kodaira additivity in the negative-fiber branchLogKodairaFiberNegative
Unreviewed AI draft: states part of a paper’s theorem. States the negative-fibre branch of the logarithmic additivity theorem: if the fibre over a very general point has logarithmic Kodaira dimension −∞, so does the total space. The finite-dimension and negative-base branches are outside the statement.
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Logarithmic Kodaira additivity in the negative-fiber branchLogKodairaFiberNegative
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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Family 033The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements state a lemma or auxiliary result, not the headline.
Their method
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.
The challenge statements state a different theorem from the headline.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 264 s; peak memory 6.4 GiB).
Date
10 October 2026
Version
at the release’s recorded version
Their method
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.
Logarithmic Kodaira additivity in the negative-fiber branch
Role
the release lists it as a supporting statement
Paired with
Corollary 1.2 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.