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

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.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 26, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Kalton's question: does the Lipschitz-free space of every uniformly discrete metric space have the bounded approximation property? · Kalton's BAP question for uniformly discrete free spaces

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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

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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 $\ell_1$ .