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Math-2815 · mathematics · Analysis

Quadratic strong-operator paving for every MASA

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every MASA $A\subseteq M\subseteq B(H)$, self-adjoint $x\in M$ and $0<\varepsilon <1$, can one fix $r\leq 5\cdot 10^{8}\varepsilon ^-2$ before any strong neighborhood is chosen, so that for every finite $F\subset H$ and $\delta >0$ there are an r-partition $(p_i)$ of 1 in A, self-adjoint $a\in A$ with $||a||\leq ||x||$, and a projection $q\in M$ with $||(1-q)\xi ||<\delta $ for $\xi \in F$ and $||q(\sum p_ixp_i-a)q||\leq \varepsilon ||x||$, without an expectation or separability assumption?

Short English statement

For every MASA $A\subseteq M\subseteq B(H)$, self-adjoint $x\in M$ and $0<\varepsilon <1$, can one fix $r\leq 5\cdot 10^{8}\varepsilon ^-2$ before any strong neighborhood is chosen, so that for every finite $F\subset H$ and $\delta >0$ there are an r-partition $(p_i)$ of 1 in A, self-adjoint $a\in A$ with $||a||\leq ||x||$, and a projection $q\in M$ with $||(1-q)\xi ||<\delta $ for $\xi \in F$ and $||q(\sum p_ixp_i-a)q||\leq \varepsilon ||x||$, without an expectation or separability assumption?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2815-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2815

Manuscript 1 (family 300, 2026-09-25): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Quadratic strong-operator paving for every MASA

OpenAI's claim

Strong-operator paving over an arbitrary MASA is claimed with at most 500000000 $\varepsilon^{-2}$ projections for each self-adjoint element, uniformly over representations and without separability or expectation assumptions. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For $0<\varepsilon<1$, put $r_\varepsilon=\lceil400000000\varepsilon^{-2}\rceil+\lceil4/\varepsilon\rceil+1\le500000000\varepsilon^{-2}$. For every maximal abelian $A\subset M\subset B(H)$ and self-adjoint $x\in M$, OpenAI claims that for any finite $F\subset H$ and $\delta>0$ there are a unit partition $p_1,\ldots,p_{r_\varepsilon}\in A$, self-adjoint $a\in A$ with $\|a\|\le\|x\|$, and a projection $q\in M$ such that \[ \|(1-q)\xi\|<\delta\ (\xi\in F),\qquad \left\|q\left(\sum_ip_ixp_i-a\right)q\right\|\le\varepsilon\|x\|. \] The number is fixed before the inclusion and strong neighborhood, with no separability or conditional expectation premise.

Outline of the argument

Manuscript 1 (family 300, 2026-09-25): Rooted spectral costs from mixed determinant polynomials drive a measurable coloring on a Bernoulli extension, with cutoff bounds controlling nonsingular weights. One partition simultaneously handles the normalizer and its complement through free compression, and modular moment transfer returns it to actual MASA projections. Separate corner palettes then preserve the quadratic count while removing normal-expectation and separability assumptions.

Formal verification

Family 300 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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