OpenAI · 300-02 · 2026-09-25 · Operator algebras
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras
preprints/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026
Family 300. The Popa–Vaes quadratic strong-operator paving conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with 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.
We prove the quadratic strong-operator paving conjecture of Popa and Vaes. For every , every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most projections. The bound is uniform over representations and requires no separability or conditional-expectation assumption.
Preprint source and Lean scope
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