Approximation and quadratic strong-operator paving

Claims to settle both paving conjectures of Popa and Vaes for every maximal abelian subalgebra of a von Neumann algebra.

2 papers September 25, 2026 Explained by AI

Paving means cutting an operator into a fixed number of diagonal blocks so the off-diagonal part becomes small. The Kadison–Singer problem asked this for the standard diagonal and was solved in 2015. Popa and Vaes conjectured two weaker forms of paving for every maximal commutative subalgebra. Two companion papers claim to prove both forms, with explicit counts of blocks. There is no Lean check.

First step

1959

Kadison and Singer, who posed the pure-state extension problem for the diagonal subalgebra of B(ℓ²)

Significance

Major

It would settle both parts of a 2015 conjecture of Popa and Vaes extending Kadison–Singer paving to all maximal abelian subalgebras.

Trust

Claimed, not checked

Field

Operator algebras

2 papers

The explanation

Take the 4-by-4 matrix x with 0 on the diagonal and 1 everywhere else. Its norm, the most it stretches any vector, is 3. Cut the four coordinates into two groups, {1, 2} and {3, 4}. Keep only the entries inside each group and delete the rest. This is called a pinching. What remains is two copies of the 2-by-2 matrix with 0 on the diagonal and 1 off it. Its norm is 1, a third of the original. With four groups of one coordinate each, nothing off the diagonal survives. So more groups give smaller off-diagonal parts.

Paving asks for a fixed number of groups that works for every operator. The groups come from a maximal abelian subalgebra, or MASA: a largest family of commuting operators, like the diagonal matrices. After the pinching, one subtracts an element a of the MASA. The aim is that what is left has norm at most ε times the norm of x. The original question, from Kadison and Singer, concerned the diagonal of the infinite matrix spaceSource 1. Marcus, Spielman and Srivastava solved it by mixed characteristic polynomialsSource 2. For general MASAs, norm paving failsSource 3. So Popa and Vaes proposed two weaker forms and conjectured both hold for every MASASource 4.

The first paper claims the approximation form. Every self-adjoint operator is a strong limit of operators of norm at most three times its own. Each of these can be paved with error at most ε times its own normSource 5. The number of groups is at most C·ε^−6 for a universal constant CSource 6. The second paper claims the quadratic strong-operator formSource 7. Here the pinching only has to be small after a compression. The compressing projection must be close to the identity on any chosen finite set of vectors. The number of groups is at most 5 × 10⁸ · ε^−2Source 8. For ε = 1/2, the bound is 2,000,000,000. Neither paper needs separability or a conditional expectation onto the MASASource 9.

The obstacle is that earlier proofs split into two kinds of MASA, Cartan and singular, with different methods. Popa and Vaes noted the need to unite the two argumentsSource 10. The papers do this without any expectation onto the MASA. The first paper does not claim a quadratic count for its own boundSource 11.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The arguments are long and technical. No specialist review is published.

What it doesn't show. This does not give norm paving of the original operator for general subalgebras, which is known to fail. The first paper's bound is not claimed to be quadratic.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if correct: it claims both Popa–Vaes paving conjectures in full generality. There is no Lean check and no published specialist review.

Words used

Von Neumann algebra: A family of operators on a Hilbert space closed under sums, products, adjoints and pointwise limits.
MASA: Maximal abelian subalgebra: a largest family of commuting operators inside the algebra, like the diagonal matrices.
Pinching: Keeping only the diagonal blocks p x p of an operator for a partition of the identity into projections p.
Paving: Finding a fixed number of blocks so that the pinched operator is close to an element of the MASA.
Norm: The most an operator stretches any vector of length one.
Strong-operator topology: Convergence of operators measured by how they act on each fixed vector.
Conditional expectation: A projection from the algebra onto a subalgebra that behaves like averaging.

The original papers (2)

Approximation Paving over Arbitrary Maximal Abelian Subalgebras (PDF, opens in a new tab)
September 25, 2026
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras (PDF, opens in a new tab)
September 25, 2026

All quoted lines (11)

Kadison and Singer posed the pure-state extension problem for the atomic diagonal MASA of $\mathcal B(\ell^2)$ .
Approximation Paving over Arbitrary Maximal Abelian Subalgebras Back to text
Marcus, Spielman, and Srivastava resolved it by their mixed-characteristic-polynomial method.
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras Back to text
Requiring a small norm for $S_P(x)-a$ itself is norm paving, which fails for general masas.
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras Back to text
Popa and Vaes introduced strong-operator and approximation versions of paving and conjectured that every MASA has both properties.
Approximation Paving over Arbitrary Maximal Abelian Subalgebras Back to text
For each $0<\varepsilon<1$ , every self-adjoint operator is a strong limit of self-adjoint operators of norm at most three times its norm, each admitting a norm paving with error at most $\varepsilon$ times the approximant's own norm.
Approximation Paving over Arbitrary Maximal Abelian Subalgebras Back to text
For each $0<\varepsilon<1$ , every self-adjoint operator is a strong limit of self-adjoint operators of norm at most three times its norm, each admitting a norm paving with error at most $\varepsilon$ times the approximant's own norm. The number of projections is at most $C\varepsilon^{-6}$ for a universal constant $C$ .
Approximation Paving over Arbitrary Maximal Abelian Subalgebras Back to text
We prove the quadratic strong-operator paving conjecture of Popa and Vaes.
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras Back to text
For every $0<\eps<1$ , every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most $5\times10^8\eps^{-2}$ projections.
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras Back to text
The bound is uniform over representations and requires no separability or conditional-expectation assumption.
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras Back to text
They explicitly identified the need to unite the Cartan and singular arguments.
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras Back to text
We claim no quadratic bound for $R(\varepsilon)$ .
Approximation Paving over Arbitrary Maximal Abelian Subalgebras Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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