Nagata’s conjecture and maximal Seshadri constants
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Proves Nagata’s strict inequality \(\sum_i m_i\lt d\sqrt r\) for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal multipoint Seshadri constants \((L^n/r)^{1/n}\) for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every integer r\ge10, there is a countable union E_r of proper Zariski-closed subsets of U_r, with nonempty complement, such that the following holds for every tuple (p_1,\ldots,p_r)\notin E_r. If C is a nonzero effective plane curve of degree d\ge1 and m_1,\ldots,m_r\in\Z_{\ge0} satisfy \mult_{p_i}C\ge m_i, then
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Let X/k be a smooth integral projective variety over an algebraically closed field of characteristic p>0, with \dim X=n\ge3, and let L be ample. There is an integer r_0=r_0(X,L) such that
\varepsilon_r(X,L)=\left(\frac{L^n}{r}\right)^{1/n}
\qquad\text{for every integer }r\ge r_0.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
Two more possible matches
Theorem 1.1in Maximal Multipoint Seshadri Constants in Higher Dimensions.
Theorem 1.1in Maximal Seshadri constants on arbitrary polarized surfaces, 23 September 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than \(d\sqrt r\). The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata’s conjecture in its strict, nonhomogeneous form.
We prove that, for every smooth integral complex projective surface S and every ample line bundle L, the multipoint Seshadri constant at r very general points equals \(\sqrt{L^2/r}\) for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.
Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3. We prove that there is a threshold \(r_0=r_0(X,L)\) such that for every integer \(r\ge r_0\), the ordinary multipoint Seshadri constant at r very general points equals the volume bound \((L^n/r)^{1/n}\). This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.
Let L be an ample line bundle on a smooth integral projective variety X over an algebraically closed field of positive characteristic, with \(\dim X=n\ge3\). We prove that there is a threshold \(r_0=r_0(X,L)\) such that for every integer \(r\ge r_0\), the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound \((L^n/r)^{1/n}\). The same conclusion holds in dimension two. This establishes the positive-characteristic form of the qualitative Nagata–Biran–Szemberg assertion at geometric generic tuples.
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For every smooth integral complex projective surface S and every ample line bundle L on S, there is an integer r_0=r_0(S,L) with the following property. For each integer r\ge r_0, there is a countable union Z_r of proper Zariski-closed subsets of U_r, with nonempty complement, such that for every \mathbf p\in U_r\setminus Z_r the class
\pi^*L-\sqrt{\frac{L^2}{r}}\sum_{i=1}^r E_i
is nef on the blow-up at \mathbf p. Consequently,
\varepsilon(S,L;\mathbf p)=\sqrt{\frac{L^2}{r}}.
The words they share: ample, bundle, class, closed, complement, complex, countable, integral, line, nonempty, projective, proper, smooth, surface, union, zariski.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
MaximalSeshadriConstants.lean, the release lists it as a main result.
theoremmaximalSeshadriConstants (S : Surface) (L : LineBundle S.scheme) (hL : LineBundle.IsAmple S.scheme L) : EventualMaximalSeshadri S L := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
For every integer r\ge10, there is a countable union E_r of proper Zariski-closed subsets of U_r, with nonempty complement, such that the following holds for every tuple (p_1,\ldots,p_r)\notin E_r. If C is a nonzero effective plane curve of degree d\ge1 and m_1,\ldots,m_r\in\Z_{\ge0} satisfy \mult_{p_i}C\ge m_i, then
d\sqrt r>\sum_{i=1}^r m_i.
The words they share: effective, plane, curve, zariski, nonzero, nonempty, countable, degree, sqrt, closed, integer, conjecture, nagatas, complement, union, proper.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Other mathematical statements in OpenAI’s Lean library
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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 part of the advertised result (their term: “partial”).
Their note:Nagata (r >= 10, very general, arbitrary multiplicities) fully stated; maximal multipoint Seshadri constants only for smooth projective surfaces over C, not (L^n/r)^{1/n} in dimension n >= 2 and not over algebraically closed fields of positive characteristic.
Definitions the group asks readers to compare with the paper:Seshadri side is custom scheme theory: selfIntersection S L := χ(L^2) - 2χ(L) + χ(O) (= L^2 by Riemann-Roch, checked), curveDegree := χ(L|_C) - χ(O_C), cohomology := Abelian.Ext (structureSheaf X) M n with cohomologyDimension := Module.finrank ℂ (junk 0 if infinite-dimensional), IsBlowup by universal property, curveMultiplicity via idealOrder of the stalk kernel; Nagata side defines the curve equation by Classical.choose of a sorry’d existence theorem (curveEquation) that is itself a checked theorem_name
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:maximal multipoint Seshadri constants in dimension n >= 3 (complex, very general points) and in positive characteristic (all n >= 2, geometric generic tuple) are claimed by the headline but not stated; Lean states only smooth complex projective surfaces
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
Nagata’s conjecture asserts that a nonzero effective plane curve of degree $d$, with multiplicities at least $m_i$ at $r\ge10$ very general points, satisfies $\sum_i m_i<d\sqrt r$.
The formalization establishes this inequality simultaneously for all curves and multiplicity vectors outside one countable union of proper Zariski-closed exceptional sets with nonempty complement. Reducible curves, repeated components, and unequal multiplicities are included, in both effective-curve and homogeneous-polynomial formulations. The formalization also includes the passage from homogeneous effective cycles to equations.
The maximality question asks whether the multipoint Seshadri constant reaches the upper bound $\sqrt{L^2/r}$ at very general points once $r$ is sufficiently large. The formalized result establishes this for every smooth integral complex projective surface $S$ and ample line bundle $L$.
For every $r$ above a threshold depending on $(S,L)$, it gives a countable union of proper closed exceptional sets with nonempty complement. Outside it, the constant is $\sqrt{L^2/r}$ and the boundary class defining this Seshadri constant on the point blowup is nef. The threshold is not explicit or uniform over all surfaces.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the Seshadri paper: for every smooth projective surface and ample line bundle, the multipoint Seshadri constant at very general points is maximal once r exceeds a threshold depending on the pair.
Nagata’s conjectureNagata
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: Nagata’s inequality for all curves and multiplicity vectors at r ≥ 10 points outside one countable union of proper closed subsets.
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Identify the theorem the catalogue describes
Family 039No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded.Open the form
Address a question raised in an independent assessment
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Why this passage is shown
Theorem 1.1
In Maximal Seshadri constants on arbitrary polarized surfaces, 23 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
closed, complex, projective, smooth
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Independent assessment
MathVet
Their term
“partial”
What their term means
Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 less than the headline; the missing part comes from a paper the Lean note does not link.
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 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.