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Kalton's Problem 2: does a bi-Lipschitz copy of $c_0$ force a linear copy of $c_0$ ?

Theorem 1.1: there is a separable real Banach space $Z$ with no closed subspace isomorphic to $c_0$ and an onto bi-Lipschitz map $F:Z\oplus_\infty c_0\to Z$ . Restricting $F$ to the $c_0$ summand embeds $c_0$ bi-Lipschitzly in $Z$ , so Problem 2 has a negative answer. With Aharoni's theorem, $Z$ is bi-Lipschitz universal for separable metric spaces, and $Z$ , $Z\oplus_\infty c_0$ form a second separable Lipschitz-isomorphic, linearly non-isomorphic pair. Real scalars only; it does not address analogous questions for other classical spaces.

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 Problem 2: does a bi-Lipschitz copy of $c_0$ force a linear copy of $c_0$? · Kalton's c0 embedding problem

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Kalton's Problem 2: does a bi-Lipschitz copy of $c_0$ force a linear copy of $c_0$ ?

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No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against Kalton's Problem 2 as the manuscript quotes it; Kalton's survey itself was not opened here. Lean-checked on the Comparator challenge C0Absorption (OAI.C0Absorption.main_result, OAI/Analysis/C0Absorption/Main.lean), listed in the release's formalization catalogue. Its statement, read here, asserts a complete separable real normed space $Z$ with no bounded-below linear map from $c_0$ , a surjective bi-Lipschitz map $Z\times c_0\to Z$ (sup-norm product), a bi-Lipschitz map $c_0\to Z$ , bi-Lipschitz universality for separable metric spaces, and no continuous linear equivalence $Z\times c_0\simeq Z$ : the headline claim in full. Permitted axioms: propext, Quot.sound, Classical.choice. The statement was not independently audited and the development was not rebuilt here.