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Math-497-A2 · revision r1 · on Math-497

OpenAI claim summary for Math-497

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 039, 2026-09-23): Nagata’s inequality forces emptiness for the restricted curve systems violating its strict slope bound; it leaves the SHGH exact dimension formula for arbitrary Cremona-reduced systems undecided.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. SHGH Conjecture for Linear Systems on Blown-Up Planes
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

SHGH Conjecture for Linear Systems on Blown-Up Planes

OpenAI's claim

At $r\ge10$ very general complex plane points, every effective curve is claimed to have total assigned multiplicity below its degree times $\sqrt r$, including reducible curves and unequal assignments. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For each $r\ge10$, OpenAI claims a nonempty very-general locus of ordered distinct points $(p_1,\ldots,p_r)$ in $\mathbb P^2_{\mathbb C}$ such that every nonzero effective plane curve $C$ of degree $d\ge1$ and integers $m_i\ge0$ with $\operatorname{mult}_{p_i}C\ge m_i$ satisfy \[ d\sqrt r>\sum_{i=1}^r m_i. \] Very general means excluding a union, indexed by a countable set, of proper algebraic closed subsets of the distinct-point configuration space. Multiplicities may be unequal, curves may be reducible, and the strict inequality starts at ten points.

Outline of the argument

Manuscript 1 (family 039, 2026-09-23): Incidence loci and cyclic products reduce a hypothetical failure to equal multiplicities at arbitrary configurations. Degeneration toward a cubic produces finite jet constraints; colliding the remaining points and taking limits of elliptic theta bases turns the contradiction into a full-rank polynomial evaluation problem.

Formal verification

For family 039, the scope page documents the following selections. The selected plane-curve inequality allows unequal multiplicities and repeated components at every allowed point count. A separate statement gives eventual maximality on each polarized complex surface, with a surface-dependent threshold; the higher-dimensional and positive-characteristic companions are not selected here. The documented statement covers the main claim of manuscript 1. Comparator files: MaximalSeshadriConstants.lean, Nagata.lean. Declarations: OAI.MaximalSeshadri.Geometry.maximalSeshadriConstants, OAI.Nagata.nagata_conjecture.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources