The Popa-Vaes paving conjecture over arbitrary maximal abelian subalgebras: strong-operator, approximation and quadratic paving

Let $A\subseteq M$ be a maximal abelian $*$ -subalgebra (MASA) of a von Neumann algebra. For a partition $P=(p_1,\dots,p_r)$ of $1$ in $A$ put $S_P(x)=\sum p_ixp_i$ . After Marcus-Spielman-Srivastava solved Kadison-Singer paving for the diagonal MASA of $B(\ell^2)$ , Popa and Vaes showed norm paving fails for general MASAs and introduced weaker notions: $x$ is so-pavable if $\|q(S_P(x)-a)q\|\le\varepsilon\|x\|$ for some $a\in A$ and a projection $q$ arbitrarily close to $1$ strongly, with $r$ depending only on $\varepsilon$ ; approximation paving instead norm-paves a strongly close bounded perturbation of $x$ . Popa and Vaes proved these for type I algebras, amenable Cartan inclusions and profinite actions with $O(\varepsilon^{-4})$ bounds, and quadratic bounds for singular MASAs in ultraproducts. Conjecture 2.8: is every MASA in every von Neumann algebra so-pavable and approximation-pavable, and can so-paving always be achieved with $O(\varepsilon^{-2})$ projections?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Operator algebras: paving and maximal abelian subalgebras
Posed by: S. Popa and S. Vaes, Paving over arbitrary MASAs in von Neumann algebras, Anal. PDE 8 (2015), Conjecture 2.8
Year posed: 2015
Years open: 11y
Solved: 2026-09-25
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 20 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Quadratic paper, Theorem 1.1: for $0<\varepsilon<1$ and $r_\varepsilon\le5\times10^8\varepsilon^{-2}$ , every self-adjoint $x$ in any von Neumann algebra $M$ is $(\varepsilon,r_\varepsilon)$ so-pavable over every MASA $A\subseteq M$ , with no separability or conditional expectation assumed; corollary: quadratic norm paving in Ocneanu ultrapowers of expected countably decomposable MASAs. Approximation paper, Theorem 1.1: every self-adjoint $x$ is a strong limit of self-adjoint $y$ with $\|y\|\le3\|x\|$ , each norm-pavable to error $\varepsilon\|y\|$ with at most $C\varepsilon^{-6}$ projections. Not shown: norm paving of the original operator (false in general), optimal numerical constants, or a quadratic rate for approximation paving.

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. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The two manuscripts of the family prove the two parts of the conjecture with independent arguments.

Verification

No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts against Popa-Vaes Conjecture 2.8. The quadratic paper proves 2.8(2): every self-adjoint element is so-pavable over every MASA with at most $5\times10^8\varepsilon^{-2}$ projections, which also gives the so-paving half of 2.8(1). The approximation paper proves the approximation-paving half of 2.8(1) with $O(\varepsilon^{-6})$ projections. The proofs were not refereed; they rely on the Ravichandran-Srivastava mixed determinantal multipaving bound and on measured-equivalence-relation and amalgamated-free-product constructions. The quadratic exponent is shown optimal by a Popa-Vaes example. No Lean formalization is supplied for this family.

The Popa-Vaes paving conjecture over arbitrary maximal abelian subalgebras: strong-operator, approximation and quadratic paving

What was actually shown

What the AI did

Verification

Sources

Discussion