Approximation and quadratic strong-operator paving

What OpenAI claims

Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with O(ε^-2) blocks. The norm bound holds after compression by a projection arbitrarily close to the identity in the strong topology, resolving the Popa–Vaes quadratic paving conjecture.

Papers (2)

1. Approximation Paving over Arbitrary Maximal Abelian Subalgebras

   25 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove the approximation-paving conjecture of Popa and Vaes for every maximal abelian subalgebra of a complex von Neumann algebra. For each 0 < ε < 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 ε times the approximant's own norm. The number of projections is at most Cε^-6 for a universal constant C. No separability or conditional-expectation hypothesis is required.

2. Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras

   25 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove the quadratic strong-operator paving conjecture of Popa and Vaes. For every 0 < ε < 1, every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most 5×10⁸ε ^-2 projections. The bound is uniform over representations and requires no separability or conditional-expectation assumption.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.