Kalton's question: does the Lipschitz-free space of every uniformly discrete metric space have the bounded approximation property?

For a pointed metric space $(M,d,o)$ , the Lipschitz-free space $\mathcal F(M)$ is the closed span of the evaluations $\delta_x$ in $\mathrm{Lip}_0(M)^*$ . Kalton showed that if $M$ is uniformly discrete ( $d(x,y)\ge\theta>0$ for $x\ne y$ ) then $\mathcal F(M)$ has the approximation property, and Dalet proved the metric approximation property for countable proper spaces. Kalton asked, and Godefroy-Ozawa (Question 1) and Godefroy's survey (Problem 6.2) restated: if $M$ is uniformly discrete, must $\mathcal F(M)$ have the bounded approximation property, that is, can the finite-rank approximants of the identity be chosen with a uniform norm bound?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Banach space theory; Lipschitz-free spaces
Posed by: Nigel Kalton (Collectanea Mathematica 2004, remarks after Proposition 4.4); restated by Godefroy and Ozawa (2014, Question 1) and Godefroy (2015 survey, Problem 6.2)
Year posed: 2004
Years open: 22y
Solved: 2026-09-26
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 24 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: there is a countable, unbounded, nonproper pointed metric space with $d(x,y)\ge1$ for distinct points whose real Lipschitz-free space has AP but fails $\lambda$ -BAP for every finite $\lambda$ . It is built as a wedge of bounded blocks (Theorem 1.2) in which a finite anchor set defeats every finite-rank operator of norm at most $p$ . Consequences stated in the paper: for each $p$ , real $\ell_1$ has an equivalent norm with $2B_p$ -BAP failing $p$ -BAP (for $p=1$ failing MAP), and $\mathcal F(M)$ is a separable Banach space that is not approximable in Kalton's nonlinear sense (Kalton 2012, Problem 1). Real scalars only; the proper countable case is excluded by Dalet's theorem.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorems 1.1 and 1.2 were read against Kalton's question. lean/docs/330.md points to ComparatorChallenges/DiscreteLipschitzFree.json (theorem OAI.DiscreteFree.source_endpoints, solution module OAI.Analysis.LipschitzFree.Main); the solution file exists at the pinned commit and the challenge is not in formalization.yaml. The statement was read here: the block theorem together with a main statement asserting a countable pointed metric space with all distinct points at distance at least 1, unbounded and not proper, whose real free space (built from scratch in the file) has AP and fails $\Lambda$ -BAP for every $\Lambda\ge1$ . That is the headline claim. Not rebuilt here. A separate challenge, RealL1Renorming, covers the quantitative renorming of $\ell_1$ .

Kalton's question: does the Lipschitz-free space of every uniformly discrete metric space have the bounded approximation property?

What was actually shown

What the AI did

Verification

Sources

Discussion