VibeMathedMath problems solved with AI
Sign in

The codimension-one bound for singular sets of stationary integral varifolds

Let VV be a stationary integral mm-varifold in an open set U⊂Rm+nU\subset\mathbb R^{m+n} and Sing V\mathrm{Sing}\,V the support points near which VV is not an integer multiple of one smooth embedded minimal submanifold. Two mm-planes meeting along an (m−1)(m-1)-plane form a stationary varifold whose singular set is (m−1)(m-1)-dimensional, so no better bound is possible. Known cases included two-valued stationary Lipschitz graphs (Hirsch-Spolaor) and classes with controlled density under an epsilon-regularity assumption (Krummel-Minter-Wickramasekera). Brena, Decio and De Lellis (2025, Conjecture 1.1) conjecture the sharp bound, which is stronger than almost-everywhere regularity. Does every stationary integral mm-varifold satisfy dim⁡HSing V≤m−1\dim_{\mathcal H}\mathrm{Sing}\,V\le m-1, in every dimension and codimension?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Geometric measure theory: regularity of stationary varifolds
Posed by
Camillo Brena, Stefano Decio and Camillo De Lellis, Remarks and conjectures on stationary varifolds dedicated to Enrico Bombieri, Rend. Lincei Mat. Appl. 36 (2025), Conjecture 1.1
Year posed
2025
Years open
1y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every stationary integral mm-varifold VV in an open U⊂Rm+nU\subset\mathbb R^{m+n}, m,n≥1m,n\ge1, dim⁡HSing V≤m−1\dim_{\mathcal H}\mathrm{Sing}\,V\le m-1, with no stability, minimizing, orientability, density or codimension hypothesis; crossing planes show it is sharp. This implies Hm(Sing V)=0\mathcal H^m(\mathrm{Sing}\,V)=0. The proof combines the companion paper's signed-excess estimates with a fitting and frequency analysis around a smooth center and a measure-valued blow-up at higher-multiplicity planar tangents. It does not prove rectifiability or finer structure of the singular set, and does not cover Riemannian ambient metrics.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against Conjecture 1.1 of Brena-Decio-De Lellis as the manuscript quotes it; the proof was not refereed. The proof imports the signed-excess theorem and local estimates of the companion almost-everywhere regularity manuscript from the same release, itself unreviewed, so a gap there would carry over. Euclidean ambient space only. The release has no Lean formalization for this family, and the README cautions that unformalized results could have issues.

Sources

Changelog1 change

Discussion